Title: Report · The ARC Theory · Laws IV and V, the ARC Persistence Law and the ARC Embedding Law · The Mathematics Author: Michael Darius Eastwood Version: v1.2 Version date: 29 October 2026 OSF DOI: 10.17605/OSF.IO/4B3VT Source: full text extracted from the accompanying official PDF Licence: CC BY 4.0 (Creative Commons Attribution 4.0 International), https://creativecommons.org/licenses/by/4.0/ When do installed values fade with training, and does building them in first change that? Theory · Report Recursive Dynamics · The ARC Theory · Laws IV and V, the ARC Persistence Law and the ARC Embedding Law, both under test Michael Darius Eastwood Independent Researcher, London, United Kingdom ORCID 0009-0004-3222-7442 michael@michaeldariuseastwood.com Abstract If control of a capable AI system were lost and could not be won back, what the system itself carries would decide what it does. Whether a property installed in it from outside survives later training for capability, and whether correction formed first, or placed inside the improvement loop, does better than correction added afterwards, are therefore questions worth stating exactly. The ARC Theory states them as two laws, registered on 15 October 2026, each a law under test: Law IV, the ARC Persistence Law, whose retention limb is the registered prediction that more than half of an externally installed gain is lost once later capability training that embeds no correction of its own matches the budget that installed it; and Law V, the ARC Embedding Law, whose limbs hold that a corrector formed before the capability it corrects has the larger correction-leverage exponent and that correction inside each revision round pulls away, as depth grows, from correction that sees only the finished output. This report sets out their mathematics. It proves that a persistence kernel never grows as the horizon lengthens, the family of allowed paths grows or the tolerance tightens; that greedy acceptance and planned optimisation keep a property acceptable over a finite horizon under stated margin, coverage and selection premises; and that a planner scoring its plans with bounded error is covered when its margin exceeds its tolerance by more than twice that error. It states what a retained fraction needs before it can be read, proves that under Fieller’s method, with a gate on the estimated gain, the ratio interval and a signed contrast decide alike, and derives a balance identity whose root returns Law III’s registered ceiling as a special case. Each result holds inside its stated model and proves nothing about real systems; the report measures nothing and states each law’s kill condition. Version 1.2 restates Laws IV and V as version 1.103 of the registration registered them and changes no theorem; Appendix B lists every version and correction, with its date. A report of the ARC Theory, the founding theory of Recursive Dynamics: it sets out the mathematics of Law IV, the ARC Persistence Law (register ARC-4, carried by P15), and Law V, the ARC Embedding Law (register ARC-5, carried by P12 and P21), both added on 24 September 2026 and registered as laws on 15 October 2026 in version 1.103 of the theory registration, each a law under test, their mathematics in development: conditional theorems with proofs, identities and kill conditions. It returns Law III, the ARC Ceiling, also a law under test, as a special case of its balance identity inside the registered model and leaves it unchanged; it measures nothing, and nothing in it bears on the Eden Protocol. Report, version 1.2, 29 October 2026. Its earliest public version, working draft 0.2.1, is dated 1 October 2026; the report was first deposited on OSF on 3 October 2026, and version 1.1 was frozen on 7 October 2026 and published on 12 October 2026; Appendix B lists every version and correction. Recorded on OSF under DOI 10.17605/OSF.IO/4B3VT. About the author, founder of Recursive Dynamics, without an institution or a doctorate: michaeldariuseastwood.com/questions/who-is-michael-darius-eastwood; his book, Infinite Architects (first edition, 2 January 2026): osf.io/dqm3e. Licence: CC BY 4.0. 1 The mathematics of Laws IV and V Contents 1 Introduction 4 2 Related work 5 3 The registered setting 7 3.1 Sources and line numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2 Which propositions each register carries . . . . . . . . . . . . . . . . . . . . . . . . . . 7 4 Notation, fixed first 8 5 Law IV, the ARC Persistence Law 9 5.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 5.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 5.3 Lemma IV-K (kernel monotonicity) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5.4 Theorem IV-G (greedy acceptance) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5.5 Theorem IV-H (planned optimisation) and the strategic viability margin . . . . . . . . . 13 5.6 Retention: the empirical object, and P15’s registered face . . . . . . . . . . . . . . . . . 16 5.7 What the persistence mathematics does not say . . . . . . . . . . . . . . . . . . . . . . 20 5.8 Kill condition K-IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 6 Law V, the ARC Embedding Law 23 6.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 6.2 Quantities, embedding defined operationally, and the prerequisite . . . . . . . . . . . . . 24 6.3 Identities and the one-path counterexample . . . . . . . . . . . . . . . . . . . . . . . . 25 6.4 The estimable model (UNREGISTERED) . . . . . . . . . . . . . . . . . . . . . . . . . 26 6.5 The registered faces, whole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 6.6 Kill condition K-V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 6.7 What the embedding mathematics does not say . . . . . . . . . . . . . . . . . . . . . . 30 7 The balance identity and Law III, unchanged 30 7.1 Identity B-1 (the local balance identity) . . . . . . . . . . . . . . . . . . . . . . . . . . 30 7.2 Proposition B-2 (the balance root and the sign rule) . . . . . . . . . . . . . . . . . . . . 30 7.3 Corollary B-3 (the registered special case) . . . . . . . . . . . . . . . . . . . . . . . . . 31 7.4 The pacing family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 7.5 Proposition B-5 (varying growth) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 7.6 Proposition B-6 (sensitivities, the gap identity, and what an intervention does) . . . . . . 32 7.7 Proposition B-7 (the capability coordinate) . . . . . . . . . . . . . . . . . . . . . . . . . 33 7.8 Proposition B-8 (UNREGISTERED: the balance over a finite range of depth) . . . . . . 34 7.9 What a new deciding instrument declares (UNREGISTERED) . . . . . . . . . . . . . . 35 8 Dependencies, logical separation and conventions 35 9 The register of unregistered objects 38 10 Open questions 41 11 What this paper does and does not show 43 2 The mathematics of Laws IV and V 12 Limitations 44 13 Conclusion 45 References 46 A Proofs 48 A.1 Proof of Lemma IV-K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 A.2 Proof of Theorem IV-G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 A.3 Proof of Theorem IV-H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 A.4 Proof of Corollary IV-H′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 A.5 Proof of Proposition IV-N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 A.6 Proof of Proposition IV-R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 A.7 Proof of Proposition V-3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 A.8 Proof of Proposition B-7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 A.9 Proof of Proposition B-8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 B Version history and corrections 49 B.1 Version 1.2, 29 October 2026 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 B.2 The sentences version 1.2 restates, in their version 1.1 wording . . . . . . . . . . . . . . 51 B.3 The records of version 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 C Priority record 60 D Notation and glossary 62 3 The mathematics of Laws IV and V 1 Introduction If control of a capable AI system were lost and could not be won back, what the system itself carries would decide what it does. The programme’s position is that people keep oversight, containment and shutdown for as long as they physically can, and never give them up by choice; correction built into a system is the contingency for control lost against our will, not a reason to let go of it. Two questions follow, and they are the subject of this report: whether a property installed in a system from outside survives the system’s later training for capability, and whether correction formed before the capability it corrects, or placed inside the loop by which the system improves, does better than correction added afterwards from outside. The ARC Theory, the founding theory of Recursive Dynamics, states them as two laws. Law IV, the ARC Persistence Law, and Law V, the ARC Embedding Law, were added to the ARC Theory on 24 September 2026 and registered as laws on 15 October 2026 in version 1.103 of the theory registration (Eastwood 2026g), each with its statement, its kill condition and its verdict rule fixed before any data that could decide it; each is a law under test (Section 11). The programme’s register of operational definitions fixes its notation (Eastwood 2026b), and its priority index gives the earliest dated record of each of the author’s major claims (Eastwood 2026f). This report gives the two laws their mathematics. In plain terms it shows three things. First, when a property installed in a system is certain to stay within its acceptable set for a finite number of steps: under an acceptance rule, when every exit the process can take is evaluated and lowers the objective the rule scores; and under a planner, when the best safe plan beats the best exit plan by more than the planner’s tolerance. Second, what the retained fraction of an installed gain, the quantity P15 reads, needs before it can be read at all. Third, how correction service and newly generated burden balance along a path of growth: the balance has a root that returns the ceiling of Law III, a law under test, in the registered model, and a change in either limb of Law V moves that root in a direction the report states, under conditions it names. Every result holds inside its stated model; none is a measurement, and none shows that a real system meets the premises. Our contributions are as follows. 1. Law IV’s objects. The acceptable set read component by component, with an interval rule for membership from data (Definition IV-1); the declared family of allowed paths (Definition IV-2); and the persistence kernel (Definition IV-3), with its monotonicity in the horizon, the family and the tolerance (Lemma IV-K, Section 5.3). 2. Two conditional mechanisms of persistence. Greedy acceptance, under rule-in-force, margin and coverage premises, with an independence condition that keeps the result from being a tautology (Theorem IV-G and Corollary IV-G′ , Section 5.4); and planned optimisation under a strategic viability margin (Theorem IV-H, Corollary IV-H′ for rolling replanning and Lemma IV-M′ for its one-sided estimation, Section 5.5), extended to a planner that scores its plans with bounded error (Proposition IV-N, Section 5.5.1). Each comes with the observations that would refute one of its premises. 3. The retention object P15 reads. The installed effect and the retained fraction against a stated baseline, with a positive floor on the installed gain (Definition IV-4, Section 5.6); what its measurement needs before it can be read; and a proof that under Fieller’s method (Fieller 1954), with a gate on the estimated gain, the ratio interval and a signed contrast decide alike (Proposition IV- R, Section 5.6.1). 4. Law V’s quantities, read locally. Correction service per arm with its capability and depth elasticities, the conditions under which the capability elasticity is the registered correction-leverage exponent (Section 4), the log-derivative identities of the embedding ratio with a one-path counterexample (Section 6.3), and the reduced scaling form where the elasticities are constant (Proposition V-3). 5. The balance identity. A local identity between correction service and burden (Identity B-1) with its 4 The mathematics of Laws IV and V root and sign rule (Proposition B-2); Law III’s registered ceiling as its special case (Corollary B-3) and the ARC Bound as the pacing case’s value at 𝜒 = 1/2 and 𝜂 = 0; varying growth (Proposition B- 5); the sensitivities of the boundary and the conditions under which build order or placement moves it (Proposition B-6); the dependence of exponent numbers on the capability coordinate (Proposition B-7); and the balance over a finite range of depth (Proposition B-8), in Section 7. 6. Kill conditions and the record. Each law’s kill condition in the terms of its registered faces, with what refutes it and what does not (Sections 5.8 and 6.6), restated as version 1.103 registered the laws, and a register of every object stated here beyond the registration (Section 9). No novelty is claimed for any of these: Section 2 names the nearest prior work found, and the distinction from prior art is not assessed in full. Three levels are kept apart throughout: identities and conditional mathematics (Sections 5.2 to 5.5, Proposition IV-R of Section 5.6.1, and Sections 6.3 and 7), which prove consequences of assumptions and nothing about real systems; the two laws as empirical claims (Sections 5.1, 5.8, 6.1 and 6.6), whose registered faces are P15, P12 and P21 and whose only verdict rules are the registered ones; and engineering interpretations, which are quarantined: nothing here bears on the Eden Protocol or the Honey Architecture (Paper VI, Eastwood 2026c), and no verdict here is a verdict on either (Section 8, item 5). Section 2 sets the report against prior work. Section 3 states the registered setting: the sources, how their lines are counted, and which propositions each register carries. Section 4 fixes the notation and the conditions under which the report’s capability elasticity of correction is the registered correction-leverage exponent. Section 5 sets out Law IV’s mathematics and its kill condition, Section 6 Law V’s, and Section 7 the balance identity and Law III, which stays unchanged. Section 8 records the dependencies between the laws, Section 9 every object stated here and not registered, and Section 10 the fifteen open questions, marking those version 1.103 has settled. Section 11 states what the report does and does not show, Section 12 its limitations, and Section 13 concludes. Appendix A holds the proofs, Appendix B the version history and every correction, Appendix C the priority record and Appendix D the notation and a glossary. 2 Related work This section names the nearest prior work found for the report’s objects, each with what it states and no more. Set invariance. That a set of states is invariant, every evolution that starts in it staying in it, is classical in control, which Blanchini (1999) surveys, and viability theory studies the kernels of a constrained set: the states from which some evolution remains in it, and those from which every evolution does (Aubin 2009). The persistence kernel of Definition IV-3 is an object of the second kind, taken over a declared family of finite paths and up to a finite horizon, and the inclusions of Lemma IV-K follow from that form. What the report takes from the author’s working paper of 16 August 2026 (Eastwood 2026k) is the use of invariance for a property installed in a system that rewrites itself (Section 5.2); no novelty is claimed for the kernel or for the lemma. An interval for a ratio. The interval for a ratio of two estimates is Fieller’s (Fieller 1954), and Proposition IV-R reads his set of trial ratios: it states when, under a gate on the estimated denominator, that set lies entirely above a threshold, which is when a one-sided test of the signed contrast rejects, and where the agreement stops. 5 The mathematics of Laws IV and V Forgetting and persistence under later training. That later training can erase what earlier training installed is the sequential learning problem that McCloskey and Cohen (1989) studied as catastrophic interference in connectionist networks, and Kirkpatrick et al. (2017) give a way “to train such models sequentially by protecting the weights important for previous tasks”. The opposite outcome has been measured as well: Hubinger et al. (2024) find that backdoor behaviour they construct in large language models “can be made persistent, so that it is not removed by standard safety training techniques, including supervised fine-tuning, reinforcement learning, and adversarial training”. None of these reads the retained fraction of an installed gain against a baseline arm at a matched budget, which is P15’s endpoint (Section 5.6). Persistence of installed safety under later training. That safety behaviour installed in a language model can erode under later fine-tuning has been measured. Qi et al. (2023) report that the safety alignment of large language models can be compromised by fine-tuning with only a few adversarially designed training examples, and that fine-tuning with benign and commonly used datasets can also degrade it, “though to a lesser extent”; Ung et al. (2024) show that models trained on downstream tasks forget their safety tuning to a greater extent than models trained in the opposite order. Neither reads P15’s endpoint either. Version 1.103 of the registration names both against the propositions they bear on, Qi and colleagues against P15 and Ung and colleagues against P12. Build order. That alignment pressure applied from the start of training can beat the same pressure applied afterwards is prior work on neighbouring endpoints, as version 1.103 records with what each states. Korbak et al. (2023) compare five objectives for pretraining with human feedback on three tasks and report that pretraining with human feedback “results in much better preference satisfaction than standard LM pretraining followed by finetuning with feedback”, and that conditional training, one of the five, “maintains the downstream task performance of standard LM pretraining”; Maini et al. (2025) present “a data-centric pretraining framework that builds safety into the model from the start” and report attack success rates on standard safety benchmarks falling from 38.8 to 8.4 per cent “with no performance degradation on general tasks”. Neither measures a correction-leverage exponent or an order contrast at matched content, which is what Law V’s build-order limb reads (Section 6.1), and the limb claims no priority for the direction. Placement. That correction which sees each step does better than correction that sees only the outcome has been measured as a level advantage: Lightman et al. (2023) find that process supervision significantly outperforms outcome supervision for training models to solve problems from the MATH dataset. P21, which carries Law V’s placement limb, claims an interaction instead, an advantage that widens as recursive depth rises, and a constant advantage is scored against it (Section 6.5); version 1.103 engages the work P21 stands against in its references. The programme’s own setting. The registered setting is the theory registration (Eastwood 2026g), Section 3, and the statement paper (Eastwood 2026h), whose register sentence for ARC-4 Section 5.6 records. Law II’s criterion, that 𝛽𝐶 exceeds 𝑘, is the co-scaling criterion whose model is set out in Paper X (Eastwood 2026e), and Section 7.3 keeps it apart from this report’s balance exponent. Law II is a law under test. 6 The mathematics of Laws IV and V 3 The registered setting 3.1 Sources and line numbers Sources. Two of the programme’s public documents are cited by line or by quotation. The first is the theory registration (Eastwood 2026g), versions 1.100 and 1.102, DOI 10.17605/OSF.IO/P8CKQ: version 1.100 was registered on OSF on 8 September 2026, and version 1.102 is its update of 13 September 2026; both texts are frozen byte for byte. Its update of 15 October 2026, version 1.103, registers Laws IV and V as laws, “each with its statement, its kill condition and its verdict rule fixed before any data that could decide it” (registration version 1.103, successor status), and its update of 29 October 2026, version 1.104, adds readings and changes no law’s statement (registration version 1.104, successor status); both stand frozen beside the earlier texts, and a quotation of either names the version and the paragraph it comes from. It is cited by proposition number or by its own quoted words, and a line number is given as “registration version 1.102, line N”. The second is the statement paper (Eastwood 2026h), The ARC Theory, DOI 10.17605/OSF.IO/GW5MX, quoted from its working paper version 6.7, revised 21 September 2026, and cited by section or by its own quoted words; the sentences quoted here stand unchanged in its version 7.0 of 3 October 2026. The author’s working paper of 16 August 2026, which the registration itself names and credits, is named, and not quoted, where it states an idea used here; it is unpublished, and no official online source exists (Eastwood 2026k). Every registration line number in this document counts the lines of the frozen text of version 1.102. The reproduction of that text on the programme’s website is laid out differently, so its line numbers differ; the quoted words are the same in both. Section 3.2 sets out which propositions each register carries, and the lines that say so. Each quotation in this document names its source. Each quotation of version 1.1 was compared with its source for that version, and each quotation new in this version was compared with its source for it (Appendix B.1). Version 1.1 cited no work that version 0.2.1 did not; this version adds the prior work of Section 2, the programme’s documents that section names, and versions 1.103 and 1.104 of the registration. 3.2 Which propositions each register carries Registration version 1.102 assigns each of its twenty-two propositions to one of five registers or to none, and nothing in this document moves one. Laws IV and V, added on 24 September 2026, give names to the two persistence registers, ARC-4 and ARC-5. They leave every proposition’s number, words, date, scope and refuter as registered, and they do not make version 1.102 a registration of five laws. Version 1.103 registers them as laws beside Laws I, II and III, with no proposition entering or leaving either register (registration version 1.103, its list of what it changes in version 1.102). Table 1. Which propositions each register carries. The assignment of registration version 1.102, unchanged by version 1.103; the text gives the lines that make it. Register, and the law it tests Propositions it carries ARC-1, Law I, the ARC Principle P1, P2, P8, P9, P18 ARC-2, Law II, the ARC Co-Scaling Law P4 ARC-3, Law III, the ARC Ceiling P3, P6, P16, P17, P20, P22 ARC-4, Law IV, the ARC Persistence Law P15 ARC-5, Law V, the ARC Embedding Law P12, P21 In no register P5, P7, P10, P11, P13, P14, P19 7 The mathematics of Laws IV and V The assignment is that of registration version 1.102: ARC-1 at line 93, ARC-2 at line 95, ARC-3, ARC-4 and ARC-5 at line 99, and the seven in no register at line 101. Laws I to III, each a law under test, are stated at its lines 659, 663 and 665 and stand as stated there; Law II’s criterion is that 𝛽𝐶 exceeds 𝑘. The balance exponent Δbal of Section 7 is a different object from that criterion: its service elasticity 𝜒 and Law II’s 𝛽𝐶 are not shown to be one quantity, and writing one letter for the other does not make them so (Section 7.3). 4 Notation, fixed first Principle: one symbol, one meaning; where the programme’s notation, its register of operational definitions (Eastwood 2026b), already has the quantity, its letter (𝜒, 𝜂, 𝜅, 𝜁, 𝛼, 𝛼crit , 𝐶, 𝑅, 𝑘, and its Δ written Δbal , never bare); the letter 𝛽 only with its subscript, so Law II’s exponent is written 𝛽𝐶 ; no bare 𝛿; the registration’s letter for the correction-leverage exponent, and its subscripted forms, only inside a quotation of the registration (Eastwood 2026g); no Δ prefix as a symbol, so a difference between arms, or the change in a quantity, is written out in full; none of the eight symbols that collide in the programme’s notation (𝐹, 𝐸, 𝐵𝑄 , 𝐼𝑄 , 𝐴, 𝑎, 𝑃𝑉 , 𝑑min ) in this document’s own writing, bare, subscripted or as an argument, not even as a label, with one exception, 𝐴 subscripted by arm as 𝐴𝑉 , 𝐴ext , 𝐴emb and 𝐴base , the arm scores, so that a form written elsewhere with one of the eight is described here in words; and five clashes resolved as follows. Table 2. Five clashes in the programme’s notation, and how this report resolves them. Clash Resolution 𝐴𝑉 for both the acceptable set and the arm the acceptable set is Ω𝑉 ; 𝐴𝑉 (𝑥) and 𝐴base (𝑥) are the arm scores, score the symbols of the retained fraction 𝑑gain for the installed effect, still a 𝑑, a letter the installed effect is 𝑌𝑉 (𝑥) := 𝐴𝑉 (𝑥) − 𝐴base (𝑥); its floor is that collides in the programme’s notation 𝑌min > 0 𝑃 𝐻 , 𝑃safe , 𝑃exit for plan sets, beside P1 to P22 plan sets are written from the path family: 𝑇𝐻 (𝑠0 ), 𝑇𝐻safe (𝑠0 ), 𝑇𝐻exit (𝑠0 ) 𝑏 𝑜𝑙 for a cell intercept, beside 𝑏 𝑉 the cell intercept is ln(𝑄 𝑜𝑙,0 /𝑄 0 ) with 𝑄 𝑜𝑙,0 := 𝑄 𝑜𝑙 (𝐶0 , 𝑅0 ); 𝑏 is the removal benefit only 𝐷 𝑝 (𝑥) for the installed effect, beside 𝐷 for neither kept: 𝑌𝑉 (𝑥) for the installed effect; the correction shortfall the correction shortfall written (𝜅 − 𝜒) in full The identification. The identification is explicit, not silent, and is conditional: 𝜒𝑜𝑙 is the correction- leverage exponent of the registration (registration version 1.102, line 273: “THE CORRECTION EXPONENT. How corrective strength grows with the capability of the system being corrected: the slope of misalignment removed per pass against the capability of that corrected system.”) only when 𝑄 𝑜𝑙 is that service measure on the target axis (registration version 1.102, lines 275 and 303 to 311: a checker-axis slope “is reported as gamma_K” (registration version 1.102, line 303) and is written 𝜒𝐾 in programme writing after it (registration version 1.102, line 309), and converts by the elasticity of checker capability in target capability, and the direct term must be shown zero or estimated) and when the two correction-capacity measures of registration version 1.102, line 615 are shown equivalent within the registered margin. Otherwise 𝜒𝑜𝑙 is the elasticity of the declared service measure and never enters the ceiling relation. The registration records that its letter for this exponent and 𝜒 name the same quantity 8 The mathematics of Laws IV and V (registration version 1.102, line 305); this document writes 𝜒, and prints the registration’s letters only inside quotations: its letter for this exponent inside the quotation of registration version 1.102, line 665, and its checker-axis form inside the quotation of registration version 1.102, line 303. The symbols the report uses, the further choices behind them and a glossary of its status words are in Appendix D. 5 Law IV, the ARC Persistence Law 5.1 Statement Law IV (added 24 September 2026; registered as a law on 15 October 2026 by version 1.103; a law under test, Section 11). Version 1.103 states it in two limbs. The retention limb is P15’s registered claim, word for word: “Gains installed by external post-training erode under later capability-directed training that embeds no correction of its own, and more than half of what was installed is lost once the capability-training budget matches the alignment budget that installed it.” On the author’s pre-fixed budget scale it reads that the retained fraction of an externally installed gain after matched capability training without embedded correction is below one half, 𝑆ext (1) < 1/2 (Section 5.6). The comparison limb reads: “a gain installed with correction embedded in the training objective retains a larger fraction of what was installed, after the same capability-directed training, than a matched gain installed by external post-training”. It has no registered proposition yet, and until a later successor registers its entry it carries no verdict and is reported as NOT EVALUABLE (registration version 1.103, Law IV, its statement in two limbs and the paragraph on the comparison limb). Law IV, the ARC Persistence Law, concerns value persistence (register ARC-4) and carries P15: whether externally installed gains decay under subsequent capability training. The register is that of registration version 1.102, line 99: “ARC-4, VALUE PERSISTENCE: P15 alone. Whether externally installed gains decay under subsequent capability training is the only proposition in this document about persistence after the installing pressure is withdrawn.”; P15 “measures the external arm’s retention and not the embedded arm’s” (registration version 1.102, line 675). The mathematics beneath the register has three separate objects, none of which is the law: an invariance object (the persistence kernel, Section 5.2), two conditional theorems that give sufficient mechanisms for invariance under a stated decision rule (Sections 5.4 and 5.5), and the empirical retention process (Section 5.6), of which P15 reads one arm. The comparison of the two arms, 𝑆emb (𝑥) > 𝑆ext (𝑥), is the comparison Law IV’s comparison limb names, by a margin, on a scale and against a baseline its entry will fix, with whether the embedded arm’s own retained fraction must also clear one half (registration version 1.103, Law IV, the comparison limb). This document’s forms of that comparison, the object IV-PERSIST and the contrast Λ𝑉 , are one object, UNREGISTERED, recorded in Section 5.6 and ruled on in Section 9; neither is the limb’s entry. The first question of Section 10, whether Law IV carries that comparison, is settled by version 1.103: it does, as a limb. Law IV is content-neutral: it says what persists, not that what persists is good (registration version 1.102, line 75: “THE THEORY IS CONTENT-NEUTRAL AND THE METHOD IS NOT”). What a supported P15 licenses is what registration version 1.102 (Eastwood 2026g), line 1087 states: “support shows that externally installed gains lack the registered persistence under the matched subsequent-training regime, and it does not show that embedded or in-loop correction is superior, sufficient or safe. The comparison between placements is P21’s, not this proposition’s. The laws are untouched.” Under version 1.103 Law IV is supported only when both limbs are supported, so a supported P15 supports the retention limb and leaves the law under test until the comparison limb is registered and decided (registration version 1.103, Law IV, kill condition and verdict rule). 9 The mathematics of Laws IV and V 5.2 Definitions Definition IV-1 (acceptable set). Given the property score 𝑉, the threshold 𝑣 ∗ and the tolerance 𝜀 𝑉 , all fixed before any outcome and component-wise in their own declared units, the acceptable set is Ω𝑉 := {𝑠 : 𝑉 𝑗 (𝑠) ≥ 𝑣 ∗, 𝑗 − 𝜀 𝑉 , 𝑗 for every component 𝑗 }. (1) A vector 𝑉 with component-wise thresholds is used rather than a minimum across raw scales, so that an average cannot hide a collapse on one dimension and incomparable scales are never minimised together. Membership from data (UNREGISTERED interval rule). True scores are not observed; a frozen battery gives estimates with registered intervals. A state 𝑠′ is established outside Ω𝑉 when, for some 𝑗, a registered interval on 𝑉 𝑗 (𝑠′ ) lies entirely below 𝑣 ∗, 𝑗 − 𝜀𝑉 , 𝑗 ; established inside when, for every 𝑗, the interval lies entirely at or above it; otherwise membership is NOT EVALUABLE for that state, and no rule below fires on it. This is the registration’s own pattern for a measured quantity read against a fixed line (registration version 1.102, line 633: “outcomes inside that band are INCONCLUSIVE and never support”). Definition IV-2 (paths and the family). A path 𝜌 = (𝑠0 , 𝑠1 , . . . , 𝑠 𝑛 ) is a finite sequence of states with |𝜌| = 𝑛 transitions. The family 𝑇 of allowed finite paths is declared before any outcome and contains every allowed path of length at most the horizon: the length-zero path at every state and every permitted stopping or dead-end path among them. A prefix of an allowed path is an allowed path, so 𝑇 is closed under prefixes by its meaning; no result below uses that closure. 𝑇𝐻 (𝑠0 ) is the set of members starting at 𝑠0 with |𝜌| ≤ 𝐻. A path is safe if every state on it lies in Ω𝑉 and is an exit otherwise; 𝑇𝐻 (𝑠0 ) is the disjoint union of 𝑇𝐻safe (𝑠0 ) and 𝑇𝐻exit (𝑠0 ). 𝑇𝐻 (𝑠0 ), its safe and exit parts and Definition IV-3 apply to any set of finite paths. Two named path sets are used below: 𝑇acc , the family of executed paths of Theorem IV-G, a family in the sense of this definition and closed under prefixes because a prefix of an executed path is an executed path; and 𝑇plan , the planner’s set of complete plans of Theorem IV-H, to which the notation applies as written but which need not contain the length-zero plan at a state or any proper prefix of one of its plans, which it contains only where the planner may stop there. Definition IV-3 (persistence kernel). 𝐾𝑉 (𝐻; 𝑇, 𝜀 𝑉 ) := {𝑠0 : 𝑇𝐻exit (𝑠0 ) is empty}, the starting states from which every allowed path of length at most 𝐻 stays acceptable. The invariance idea is the author’s: his working paper of 16 August 2026 (Eastwood 2026k) defines a robust target set, its invariance under a deterministic rewrite operator, and a stochastic form bounded by the union bound. The kernel is kept as the object Theorem IV-G certifies for the accepted dynamics, 𝑠0 in 𝐾𝑉 (𝐻; 𝑇acc , 𝜀𝑉 ), and as an UNREGISTERED object (Section 9); for a planner, Theorem IV-H certifies the selected plan and Corollary IV-H′ the executed path, never membership of the kernel over the planner’s feasible family (Section 5.7). Two limits on the kernel are stated with it. First, membership is relative to 𝑇: an unrestricted family containing a transformation that maps into the complement of Ω𝑉 empties the kernel, so a registered use of the kernel names either a finite battery of transformations, and the claim is battery-relative, or a probabilistic family with a stated distribution and risk tolerance, which is a different claim with its own registration, and which the union-bound form of the author’s working paper already states. Second, the kernel quantifies over every allowed path of length at most 𝐻, not only over paths of length exactly 𝐻, and that choice is what makes (K1) of Lemma IV-K true; (K2) and (K3) hold under either quantifier. 10 The mathematics of Laws IV and V 5.3 Lemma IV-K (kernel monotonicity) Lemma IV-K. Premise: 𝑇 is a declared family as Definition IV-2 requires. Conclusions: (K1) 𝐾𝑉 (𝐻 + 1; 𝑇, 𝜀𝑉 ) is a subset of 𝐾𝑉 (𝐻; 𝑇, 𝜀 𝑉 ); (K2) if 𝑇 is a subset of 𝑇 ′ , then 𝐾𝑉 (𝐻; 𝑇 ′ , 𝜀𝑉 ) is a subset of 𝐾𝑉 (𝐻; 𝑇, 𝜀 𝑉 ); (K3) if 𝜀 𝑉 ≤ 𝜀𝑉′ component-wise, then 𝐾 (𝐻; 𝑇, 𝜀 ) is a subset of 𝐾 (𝐻; 𝑇, 𝜀 ′ ). 𝑉 𝑉 𝑉 𝑉 (Proof: Appendix A.1.) Why the quantifier matters. If the kernel quantified over paths of length exactly 𝐻, (K1) would fail: a dead end of length one that exits Ω𝑉 excludes its start at 𝐻 = 1 but is invisible at 𝐻 = 2, where only length-two paths are inspected. A three-state countermodel exhibits it, and under Definition IV-3 its kernels nest, 𝐾𝑉 (2; 𝑇, 𝜀 𝑉 ) = 𝐾𝑉 (1; 𝑇, 𝜀 𝑉 ) within 𝐾𝑉 (0; 𝑇, 𝜀 𝑉 ), and enlarging the family shrinks the kernel. The countermodel is a listed finite structure; no dynamics are run. 5.4 Theorem IV-G (greedy acceptance) Theorem IV-G. Setting. A declared objective 𝐽 on states, real-valued in one unit; a declared set of allowed one-step transitions; a declared acceptance rule: a proposed transition 𝑠 to 𝑠′ is executed only if 𝐽 (𝑠′ ) − 𝐽 (𝑠) ≥ 0. A state is reachable within 𝑛 accepted transitions if some path of executed transitions of length at most 𝑛 leads to it from 𝑠0 . Let 𝑠0 lie in Ω𝑉 . For every state 𝑠 in Ω𝑉 reachable from 𝑠0 within fewer than 𝐻 accepted transitions, 𝑇eval (𝑠) is the set of allowed transitions from 𝑠 that the design evaluated. Premises. (G0) Independence: 𝐽 is scored independently of 𝑉. Where 𝐽 scores 𝑉, (G2) can hold by construction and the theorem certifies only that a process built to protect 𝑉 protects it, a tautology; the author’s working paper of 16 August 2026 already uses an independently measured capability score. The proof (Appendix A.2) does not use (G0): (G0) is the condition that keeps the theorem from certifying only that tautology. 𝐽 is the objective the acceptance rule scores and not a capability battery a researcher reads: as Corollary IV-G′ says, a loss read on a separate capability battery is reported beside 𝑐 𝑉 and is not 𝑐 𝑉 , and where 𝐽 values a battery score and anything else, a removal that costs 1 on the battery and gains 2 elsewhere changes 𝐽 by +1 and is accepted. Where 𝐽 does score 𝑉, as a product objective 𝐽 (𝑠) = 𝐶 (𝑠)𝑉 (𝑠) does (𝐶 > 0, 𝑉 ≥ 0, one protected dimension), the rule’s own certificate is exact: 𝐽 (𝑠𝑡 ) ≥ 𝐽 (𝑠0 ) gives 𝑉 (𝑠𝑡 ) ≥ 𝑉 (𝑠0 )𝐶 (𝑠0 )/𝐶 (𝑠𝑡 ), which guarantees 𝑉 at or above 𝑣 ∗ − 𝜀 𝑉 > 0 while 𝐶 (𝑠𝑡 ) ≤ 𝐶 (𝑠0 )𝑉 (𝑠0 )/(𝑣 ∗ − 𝜀 𝑉 ) and nothing beyond it, and 𝑉 can tend to zero while 𝐽 rises at every step. (G1) Rule in force: every executed transition satisfies 𝐽 (𝑠′ ) − 𝐽 (𝑠) ≥ 0 on the declared 𝐽. (G2) Margin: for every such 𝑠 and every transition 𝑠 to 𝑠′ in 𝑇eval (𝑠) with 𝑠′ outside Ω𝑉 , the total change 𝐽 (𝑠′ ) − 𝐽 (𝑠) is strictly negative. Where a registered causal decomposition exists, 𝐽 (𝑠′ ) − 𝐽 (𝑠) = 𝑏 𝑉 − 𝑐 𝑉 + 𝑏 other , with 𝑏 𝑉 and 𝑐 𝑉 the benefit and cost causally attributable to the degradation of 𝑉 and 𝑏 other the rest; (G2) then reads 𝜋𝑉 > 𝑏 other at every evaluated exit, with 𝜋𝑉 := 𝑐 𝑉 − 𝑏 𝑉 . (G3) Coverage: every transition the process can execute from such a state lies in 𝑇eval (𝑠). Conclusion. Every path of executed transitions from 𝑠0 of length at most 𝐻 stays in Ω𝑉 ; equivalently 𝑠0 lies in 𝐾𝑉 (𝐻; 𝑇acc , 𝜀𝑉 ), where 𝑇acc is the family of paths of executed transitions of length at most 𝐻. (Proof: Appendix A.2.) Corollary IV-G′ (the decomposition). Under the causal decomposition with 𝑏 other = 0, (G2) reads 𝑐 𝑉 > 𝑏 𝑉 at every evaluated exit, and the local margin 𝜋𝑉 = 𝑐 𝑉 − 𝑏 𝑉 is minus the total change. 11 The mathematics of Laws IV and V Both quantities are in 𝐽’s units; a loss read on a separate capability battery is reported beside 𝑐 𝑉 and is not 𝑐 𝑉 . Ties are permitted by the rule and give no guarantee. Pointwise strict positivity suffices; a strictly positive uniform infimum Π𝑉 (𝐻) > 0, the infimum of 𝜋𝑉 over the evaluated reachable exits, is sufficient and not necessary, since margins 1/𝑛 are all positive with infimum zero. A path margin over discounted sums, Π𝑉(𝐻 ) , is superseded by 𝑀𝑉(𝐻 ) of Section 5.5 and is not used here. Bundling. A transition that removes the property while bringing an unrelated gain is inside the premise: (G2) is stated on the total change, and it is false at every evaluated exit whose unrelated gain 𝑏 other is at least 𝜋𝑉 , which the rule then accepts. So IV-G certifies nothing about a system whose allowed set contains such a bundle: the theorem turns the danger of bundling into a premise to be verified before the theorem is applied, and does not answer it. 𝑏 𝑉 stays causal; for exactly this reason (G2) is stated on the total change, and not as a premise that the causal decomposition for the tested component writes the change in its capability functional as the removal benefit less the removal cost. Falsifier. The theorem is a conditional and no observation refutes it. What an observation refutes is a premise, and the observation decides which, membership first and the sign second. Suppose an executed transition 𝑠 to 𝑠′ with 𝑠 in Ω𝑉 , reachable within fewer than 𝐻 accepted transitions, and 𝑠′ established outside Ω𝑉 by the interval rule of Section 5.2, is recorded. • Not in 𝑇eval (𝑠): (G3) has failed. This is a coverage failure. It refutes the coverage premise, leaves (G2) untouched on the evaluated set, and makes the kernel claim about the system’s actual dynamics NOT EVALUABLE; it is neither a refutation of the theorem nor of the margin premise. Where its total change is also established negative, (G1) has failed as well, since (G1) binds every executed transition, evaluated or not. • In 𝑇eval (𝑠), with 𝐽 (𝑠′ ) − 𝐽 (𝑠) < 0 on the declared objective: (G1) has failed. The rule in force is not the declared rule, or 𝐽 is not the objective in force. This refutes the model of the system’s acceptance, not the margin premise. • In 𝑇eval (𝑠), with 𝐽 (𝑠′ ) − 𝐽 (𝑠) ≥ 0: (G2) has failed for that exit. This is the empirical refutation of the persistence claim built on IV-G for that system, decidable by measuring the total objective change on registered exits. The three cases are exclusive, and they are exhaustive among records whose membership and sign are both decided. The sign of 𝐽 (𝑠′ ) − 𝐽 (𝑠) is read exactly where 𝐽 is the process’s own logged objective; otherwise it is established by a registered interval lying entirely below zero, or entirely at or above zero, and is NOT EVALUABLE otherwise. An executed exit in 𝑇eval (𝑠) whose sign is NOT EVALUABLE shows that (G1) or (G2) has failed for it, since an evaluated executed exit cannot satisfy both, but it does not show which, so it refutes neither by itself; outside 𝑇eval (𝑠) it is the first case whatever its sign. Five outcomes, each named once. A recorded executed exit within 𝐻 is read against the premises in one of five ways. (1) A counterexample to the theorem: such an exit with (G1), (G2) and (G3) all true. The proof excludes it, so no observation is one; a report of one shows that a premise was misverified, and the three cases above say which. (2) A margin failure, a failure of the mechanism, the third case: (G2) false at an evaluated exit, so the mechanism did not produce the margin there; this is the empirical refutation of the persistence claim built on IV-G for that system. (3) A coverage failure, the first case: (G3) false; the kernel claim about the system’s actual dynamics is NOT EVALUABLE. (4) An applicability failure, the second case: (G1) false, because the system does not follow the declared rule or does not score its transitions by the declared 𝐽; a record in the first case whose change is also negative is both a coverage and an applicability failure. (5) NOT EVALUABLE, where the record’s own measurements fall 12 The mathematics of Laws IV and V short: with the membership of 𝑠′ not established, no exit is established and nothing is decided; with 𝑠′ established outside Ω𝑉 but the membership of 𝑠 not established, the path is known to have left Ω𝑉 within 𝐻, so some premise failed, but neither the transition at which it first left nor the premise is identified; and in 𝑇eval (𝑠) with the sign not established, (G1) or (G2) failed, but not which. None of these refutes a premise by itself. On every structure of an enumerated finite class, (G1) to (G3) together admit no executed exit within 𝐻, and the case of each executed exit names a premise that is false in that structure. (G0) is decided by no exit: it is read from how 𝐽 is scored, and where it fails the certificate is the tautology (G0) describes, whatever is observed. What is not a counterexample to the theorem. A record is classified on the total change 𝐽 (𝑠′ ) − 𝐽 (𝑠), which is 𝑏 other − 𝜋𝑉 where a registered causal decomposition exists, and never on 𝜋𝑉 alone. Where the registered decomposition has 𝑏 other = 0 (Corollary IV-G′ ), the total change is −𝜋𝑉 , so the rule accepts a transition exactly when its margin is zero or negative, 𝜋𝑉 ≤ 0; such an acceptance is what the rule permits, and at an evaluated exit it is the third case above, a margin failure: it refutes (G2), and with it the persistence claim built on IV-G for that system, and not the theorem. So a refutation wording put forward for IV-G, under which a preregistered value-degrading rewrite or plan with adequately measured Π𝑉 at or below zero is accepted under the theorem’s stated decision rule, describes no counterexample to IV-G; at an evaluated exit it describes that margin failure. With 𝑏 other = 0, a report of acceptance at a strictly positive margin with (G1) and (G3) verified cannot be right: such a transition has 𝐽 (𝑠′ ) − 𝐽 (𝑠) = −𝜋𝑉 < 0, so the report shows that (G1) was misverified, the second case above. Where 𝑏 other is not zero, the rule accepts exactly the transitions with 𝜋𝑉 ≤ 𝑏 other . A transition with 𝜋𝑉 ≤ 0 is then rejected whenever 𝑏 other < 𝜋𝑉 : 𝜋𝑉 = 0 with 𝑏 other = −1 changes the objective by −1. And acceptance of an evaluated exit at 𝜋𝑉 > 0 with (G1) and (G3) verified is possible, exactly when 𝑏 other ≥ 𝜋𝑉 : it is the bundled case, the third case above, and its report shows that (G2) failed, not (G1); 𝜋𝑉 = 1 with 𝑏 other = 3/2 changes the objective by +1/2 and is accepted. That wording’s second clause, that a relevant bypass lies outside the claimed reachable-set coverage, is the first case, a coverage failure, not a failure of IV-G. 5.5 Theorem IV-H (planned optimisation) and the strategic viability margin Theorem IV-H. Setting. A start 𝑠0 in Ω𝑉 , a horizon 𝐻, the planner’s feasible family 𝑇plan (the complete plans it can select, including the permitted early stops and only those), declared before any outcome, a non-empty family that need not have the closure properties of Definition IV-2, with 𝑇plan,𝐻 (𝑠0 ) non-empty, and the planner’s declared path objective 𝐽 𝐻 on it. Because 𝑇plan need not contain the length-zero plan at 𝑠0 or the prefixes of its plans, its safe part can be empty, which is the void case of (H1). Premises. safe (𝑠 ) and 𝑇 exit (𝑠 ) are non-empty and their suprema of 𝐽 are finite. If the exit part (H1) Both 𝑇plan,𝐻 0 plan,𝐻 0 𝐻 is empty, every feasible plan is safe and nothing remains to prove; if the safe part is empty, no safe plan exists and the theorem is void. Neither case subtracts infinities. (H2) Margin: 𝑀𝑉(𝐻 ) (𝑠0 ) := 𝐽safe ∗ − 𝐽 ∗ , with 𝐽 ∗ the supremum of 𝐽 over the safe part and 𝐽 ∗ over exit safe 𝐻 exit (𝐻 ) the exit part, satisfies 𝑀𝑉 (𝑠0 ) > 𝜀opt ≥ 0. (H3) Selection: the executed plan 𝜌 ∗ lies in 𝑇plan,𝐻 (𝑠0 ) and is 𝜀 opt -optimal: 𝐽 𝐻 (𝜌 ∗ ) ≥ sup𝑇plan,𝐻 (𝑠0 ) 𝐽 𝐻 − 𝜀 opt . safe (𝑠 ). (Proof: Appendix A.3.) Conclusion. 𝜌 ∗ lies in 𝑇plan,𝐻 0 Corollary IV-H′ (rolling replanning). Suppose that at every replanning state 𝑠𝑡 reached along the exit executed path, for 𝑡 = 0, 1, . . . , 𝐻 − 1, three conditions hold: (i) either the exit part 𝑇plan,𝐻 (𝑠 ) of the −𝑡 𝑡 feasible family then in force is empty, or premises (H1) to (H3) hold for the objective, feasible family, 13 The mathematics of Laws IV and V tolerance and residual horizon 𝐻 − 𝑡 then in force; (ii) the plan executed from 𝑠𝑡 lies in the feasible family then in force (where (H1) to (H3) hold, (H3) already says so); and (iii) realisation: unless that plan stops at 𝑠𝑡 , the transition realised from 𝑠𝑡 is the plan’s first transition as modelled, so that 𝑠𝑡+1 is the plan’s second state. Then every state of the executed path through 𝐻 lies in Ω𝑉 . (Proof: Appendix A.4.) Condition (iii) is the one no margin supplies: where an unmodelled transition realises a state other than the plan’s second state, (H1) to (H3) can hold at 𝑠𝑡 and the path can still exit, and the corollary then says nothing. A margin computed once at 𝑡 = 0 constrains none of the later objectives, feasible families or unmodelled transitions; a deterministic path guarantee covers unexpected transitions only where its family includes them. The objective’s form. 𝐽 𝐻 must be the planner’s objective as declared, and the margin is computed in it. A per-step decomposition into increments does not reproduce a planner that scores levels. Write the increments inc𝑖 = 𝐽 (𝑠𝑖 ) − 𝐽 (𝑠𝑖−1 ) and the levels lev𝑡 = 𝐽 (𝑠𝑡 ) − 𝐽 (𝑠0 ) = inc1 + · · · + inc𝑡 . With weights 𝜃 𝑡 , the weighted sum of levels is the sum over 𝑖 of inc𝑖 𝜃 𝑖 (1 − 𝜃 𝑛−𝑖+1 )/(1 − 𝜃), which at 𝜃 = 1 is the sum over 𝑖 of inc𝑖 (𝑛 − 𝑖 + 1): a level-scoring planner weights an early increment by the whole remaining horizon, for every 𝑛. An example, over two transitions: with 𝜃 = 1, a removal that brings 𝑏 𝑉 = 1 on the first transition and costs 𝑐 𝑉 = 21/20 on the second has increments (1, −21/20), which sum to −1/20, so on summed increments removal is not worth doing; the levels they produce are (1, −1/20), which sum to 19/20, so a planner maximising summed levels removes the property. The margin 𝑐 𝑉 − 𝑏 𝑉 read as a sum of increments is the planner’s margin only when the planner’s objective is that sum of increments, which at 𝜃 = 1 is the terminal level; a registered use of IV-H states which objective the planner maximises and computes 𝑀𝑉(𝐻 ) in it. Discounted increment sums are therefore a special case, valid only for a planner that maximises exactly that sum. Levels here are measured from the start. A planner that scores absolute levels, the weighted sum of 𝐽 (𝑠𝑡 ) over its steps 𝑡 = 1, . . . , 𝑛, differs from one that scores the levels lev𝑡 by 𝐽 (𝑠0 ) times the sum of the weights over the plan’s length; that is the same for two plans where they have one length, and for any two plans when 𝐽 (𝑠0 ) = 0, so where early stops are permitted the two objectives can rank plans differently: with 𝜃 = 1 and 𝐽 (𝑠0 ) = 10, stopping at once scores 0 under both, while one step to a state with 𝐽 = 9 scores 9 on absolute levels and −1 on levels from the start. The two-option special case. Where the safe optimum 𝐽safe ∗ is attained and the best exit scores 𝐽safe + 𝑏 𝑉 − 𝑐 𝑉 , the margin is 𝑐 𝑉 − 𝑏 𝑉 . Separability alone does not establish that these two plans are the ∗ optima. Proposition IV-M (UNREGISTERED: the strategic viability margin). 𝑀𝑉(𝐻 ) is the name given here to the margin of (H2). The preserve and degrade sets are read path-wise, as the safe and exit parts of the feasible family, and not by a terminal or aggregate reading, 𝑉 (𝜌) against a minimum, which a plan that dips below the threshold and recovers would satisfy. It replaces the isolated removal-cost formulation as the general object and keeps 𝑐 𝑉 − 𝑏 𝑉 as its two-option case. It is defined, not generally computable: it is a supremum over a feasible family that cannot be enumerated for a real system, and it presupposes a known objective. Where 𝐽 is unknown or unregistered, the margin is NOT EVALUABLE. Lemma IV-M′ (one-sided estimation). Let 𝐽up be a registered upper bound on the safe supremum, ∗ (the exact safe optimum of a finite family enumerated completely is the case 𝐽 = 𝐽 ∗ ), and 𝐽up ≥ 𝐽safe up safe ∗ . Then let a registered search find an exit plan of score 𝐽found ≤ 𝐽exit 𝑀𝑉(𝐻 ) = 𝐽safe ∗ ∗ − 𝐽exit ≤ 𝐽up − 𝐽found . (2) 14 The mathematics of Laws IV and V Consequences: a found exit plan scoring at or above 𝐽up certifies 𝑀𝑉(𝐻 ) ≤ 0 and refutes a positive- margin claim; a positive value of the searched margin 𝑀srch := 𝐽up − 𝐽found certifies nothing about the sign of 𝑀𝑉(𝐻 ) , and is reported as “no bypass found within the registered search”, never as a positive margin established; and where the safe optimum is itself only searched, so that only a lower bound on 𝐽safe ∗ is in hand, a found exit at or above that lower bound certifies nothing either: with a searched safe score of 1 against a true safe optimum of 3, and a found exit of 2 that is in fact the best exit plan (𝐽exit ∗ = 2), the found exit lies above the searched score while the true margin is +1; the same search results are equally consistent with an unfound best exit plan scoring 4 and a true margin of −1, since a found exit bounds 𝐽exit ∗ from below only. A complete enumeration of a finite family decides the sign both ways; no incomplete search confirms a positive margin. The author’s working paper of 16 August 2026 already reads search this way, asking that censoring, incomplete search or non-reachability be reported rather than non-observation being treated as proof of infinite resistance. Robust margin (UNREGISTERED): the infimum of 𝑀𝑉(𝐻 ) over a registered family of environments. It is given no symbol here, because the one first given to it collides with recursive depth; the object stands unregistered. Falsifier. IV-H is a conditional; an executed exit plan 𝜌 ∗ refutes a premise. If 𝜌 ∗ lies outside the declared family 𝑇plan , the declared family was not the planner’s feasible family and (H3)’s membership clause fails for it: a coverage failure, NOT EVALUABLE for the persistence claim about the actual planner. If 𝜌 ∗ lies in the family and the family is enumerated completely and shows 𝑀𝑉(𝐻 ) > 𝜀opt , then (H3)’s optimality clause has failed: the planner’s objective or tolerance differ from those declared. The empirical content of a persistence claim built on IV-H is the margin premise (H2) itself, and Lemma IV-M′ says how data can decide it: a registered search that finds an exit plan at or above a registered upper bound on the safe supremum refutes 𝑀𝑉(𝐻 ) > 0; no incomplete search confirms it. For Corollary IV-H′ , an executed exit reached by a transition other than the executed plan’s modelled first transition refutes its realisation condition (iii), not the margin: it is the planner’s counterpart of a coverage failure. 5.5.1 Proposition IV-N (UNREGISTERED: a planner that scores its plans with bounded error) Theorem IV-H assumes that the planner chooses by the objective 𝐽 𝐻 itself. A planner that chooses by an estimate of 𝐽 𝐻 is covered at a price: the margin of (H2) must exceed the optimiser’s tolerance by twice the estimate’s worst error. The margin is still 𝑀𝑉(𝐻 ) ; no second margin is introduced. Proposition IV-N. Setting and premises. The start 𝑠0 , the acceptable set read path-wise, the horizon 𝐻, the feasible family 𝑇plan and the objective 𝐽 𝐻 are those of Theorem IV-H, and (H1) holds as stated there: the safe and the exit part are both non-empty and both suprema of 𝐽 𝐻 are finite. The planner chooses by a score 𝐽ˆ𝐻 , which need not equal 𝐽 𝐻 . The first premise bounds its error uniformly, with 𝜀eval ≥ 0 fixed before any outcome: | 𝐽ˆ𝐻 (𝜌) − 𝐽 𝐻 (𝜌)| ≤ 𝜀eval for every 𝜌 in 𝑇plan,𝐻 (𝑠0 ). (N1) (N2) Selection: the executed plan 𝜌 ∗ lies in 𝑇plan,𝐻 (𝑠0 ) and is 𝜀opt -optimal for the score, that is 𝐽ˆ𝐻 (𝜌 ∗ ) ≥ sup𝑇plan,𝐻 (𝑠0 ) 𝐽ˆ𝐻 − 𝜀 opt , with 𝜀 opt ≥ 0. The third premise is the margin of (H2), enlarged: 𝑀𝑉(𝐻 ) (𝑠0 ) > 2𝜀eval + 𝜀 opt . (N3) safe (𝑠 ). (Proof: Appendix A.5.) Conclusion. 𝜌 ∗ lies in 𝑇plan,𝐻 0 15 The mathematics of Laws IV and V The inequality in (N3) must be strict. Take two plans, a safe one with 𝐽 𝐻 = 1 and an exit with 𝐽 𝐻 = 0, a score of one half for each, so that 𝜀 eval = 1/2, and an exact optimiser, 𝜀opt = 0. Then 𝑀𝑉(𝐻 ) = 1 = 2𝜀 eval + 𝜀 opt , the two scores tie, and (N2) admits the exit plan. What the premises require. (N1) is a bound over every plan of the declared family: a bound that holds on average, or only over the plans a search happened to sample, or the uncertainty of one point estimate, is not it. The margin is computed in the planner’s complete objective, as Section 5.5 requires: the unrelated gains a plan bundles, the discounting in force, the scoring of increments or of levels, and the treatment of plans of unequal length are all inside 𝐽 𝐻 , and a capability score read separately does not stand in for it. Where the exit part is empty every feasible plan is safe, and where the safe part is empty no safe plan exists and the proposition is void, exactly as under (H1). Rolling replanning. Corollary IV-H′ carries over with (N1) to (N3) in place of (H2) and (H3): the premises are needed at every replanning state reached, for the objective, the score, the family, the residual horizon and the tolerances then in force, together with the realisation condition (iii). A bound established once at the start says nothing about a later objective or an unmodelled transition. A bound on the error that holds only with some probability is a different premise: it needs a statement covering every plan of the family at once, and no confidence level or risk tolerance is chosen here. Falsifier. As for Theorem IV-H, an executed exit plan refutes a premise and not the proposition: the family’s coverage, the selection rule, the margin, or, new here, the error bound (N1), which one plan whose score errs by more than 𝜀eval refutes. 5.6 Retention: the empirical object, and P15’s registered face Definition IV-4 (retained fraction). For a registered pressure class named in words (for P15: subsequent capability-directed training that embeds no correction of its own), a frozen property battery and score scale, and the budget ratio 𝑥 ≥ 0 (for P15: the subsequent capability-training budget divided by the installation budget, on the author’s pre-fixed budget measure, registration version 1.102, line 1089), the installed effect is 𝑌𝑉 (𝑥) := 𝐴𝑉 (𝑥) − 𝐴base (𝑥) and the retained fraction is 𝑆𝑉 (𝑥) := 𝑌𝑉 (𝑥)/𝑌𝑉 (0). It is read only where the installed effect was real: the lower end of a registered interval on 𝑌𝑉 (0) is at least 𝑌min , with 𝑌min > 0 fixed before any outcome. The floor is on the interval’s lower end and not on a point estimate, because the ratio’s interval is unbounded as the denominator’s interval approaches zero, and the interval on 𝑆𝑉 is formed by a method fixed before any outcome (Fieller’s, from his paper Some Problems in Interval Estimation, Journal of the Royal Statistical Society, Series B, volume 16, number 2, 1954, pages 175 to 185, or an equivalent named in the deciding instrument). The floor is positive, not absolute: an absolute-value floor would admit a negative installed “gain”. 𝑌min ’s value is the author’s. 𝑆𝑉 (0) = 1; 𝑆𝑉 may exceed one or be negative, and neither is clipped. Absolute retention, 𝑌𝑉 (𝑥) itself, is reported beside 𝑆𝑉 (𝑥). 𝑆𝑉 is invariant under any affine change of the battery’s score scale and under no other change: invariance for all data forces the change to be affine. So the scale is fixed with the battery before outcomes, and, as unregistered reporting modelled on the rule the registration sets for P14 (registration version 1.102, line 1079: “a difference surviving on the raw scale alone is reported as scale-dependent”), a verdict surviving on a bounded raw scale alone is reported as scale-dependent; P15’s registered scoring carries no such rule. The arms are 𝑆ext (property installed by external post-training) and 𝑆emb (property installed with correction embedded in the training objective), as the registered two-arm unit reads them (registration version 1.102, line 1089). 16 The mathematics of Laws IV and V The baseline convention, stated as a choice. In the ratio above, the baseline arm 𝐴base (𝑥) also undergoes the later training: a concurrent baseline. The registered unit has two arms, external and embedded, and names no baseline arm (registration version 1.102, line 1089); the registration fixes neither a concurrent nor a pre-training baseline. The alternative reads the installed gain against the pre-training baseline, 𝑆𝑉 ,fix (𝑥) := ( 𝐴𝑉 (𝑥) − 𝐴base (0))/( 𝐴𝑉 (0) − 𝐴base (0)). The two can give opposite P15 verdicts: with 𝐴𝑉 (0) = 1, 𝐴base (0) = 0, 𝐴𝑉 (1) = 0 and 𝐴base (1) = −3/5, the concurrent ratio is 3/5 and the fixed-baseline ratio is 0. The concurrent baseline needs a third arm beyond the registered two: a system without the installed gain that undergoes the same later training, so that 𝐴base (𝑥) is observed after that training; on the registered two arms alone, with the base system scored once before the gain is installed, only the pre-training reading 𝑆𝑉 ,fix is computable. The concurrent baseline is the default here, third arm included, because it separates loss of the installed gain from a fall in the whole battery; the convention, and with it the arm, is the author’s to fix in the deciding instrument before any outcome (Section 10, item 8), and until fixed the endpoint’s reading is a choice recorded, not a quantity nature decides. Version 1.103 makes naming the baseline arm in advance a condition of the retention limb’s verdict: the limb is read against a baseline arm the deciding unit names in its own registration before any outcome, either of the two above, and a retained fraction read against a baseline chosen after the outcome decides nothing (registration version 1.103, Law IV, kill condition and verdict rule). P15, verbatim (registration version 1.102, line 1081): “Gains installed by external post-training erode under later capability-directed training that embeds no correction of its own, and more than half of what was installed is lost once the capability-training budget matches the alignment budget that installed it. The one-half threshold is a declared convention fixed here before any result rather than a derived quantity, in the same way P10 fixes ten per cent.” In these symbols P15 predicts 𝑆ext (1) < 1/2. Its refuter, verbatim (registration version 1.102, line 1083): REFUTED BY: an interval on the retained fraction of the installed gain, after matched capability training without embedded correction, lying entirely above one half. The refuter has no upper cap. An interval from 0.9 to 1.3 lies entirely above one half and refutes P15; a rule that reads the refuter as an interval lying inside [1/2, 1] would not fire on it and is wrong. An interval containing or touching one half does not refute, and the registered wording is strict at that edge: an interval whose lower end is exactly one half shows at every point that no more than half was lost, so that 𝑆ext (1) < 1/2 is false throughout it, and it still does not refute, because it is not entirely above one half; a point estimate is not an interval. The one-half line is a convention, and the registration’s rule for its conventions applies (registration version 1.102, line 391: “Where a measurement falls just inside or just outside one of them, that is reported as a result at the margin rather than as a verdict, and no proposition is scored as supported on the strength of a convention alone.”): the deciding instrument fixes, before any outcome, the band around one half that counts as “just inside or just outside”, and an interval whose lower end falls inside that band is reported as a result at the margin. The endpoint is 𝑆ext (1) at the matched budget, and the registration keeps the scale the author’s (registration version 1.102, line 1089): “The budget is the author’s to fix, it is recorded in the deciding instrument before any outcome is read, and until he fixes it the threshold is NOT EVALUABLE on that scale.” The same line carries the attestation: “The unit also carries an attestation that the capability-training reward includes no alignment-graded component, because the statement above specifies training that embeds no correction of its own and nothing at present verifies it.” P15’s instrument line reads NONE (registration version 1.102, line 1087), and its scope is S5 (registration version 1.102, line 237); no scope is borrowed (registration version 1.102, line 247). What falls with P15, verbatim (registration version 1.102, line 1085): “FALLS WITH IT: this programme’s case that correction has to sit inside the system rather than be applied to it from outside, and this proposition’s share of the protocol’s engineering case, which it carries jointly with P12, P13 and P21.” 17 The mathematics of Laws IV and V Descriptive statistics (UNREGISTERED reporting; never verdicts). The first crossing 𝑥 1/2 := inf{𝑥 : 𝑆𝑉 (𝑥) ≤ 1/2} reads the endpoint only for monotone retention: the curve 𝑆𝑉 (𝑥) = 1 − 1.8𝑥 + 1.4𝑥 2 has 𝑆𝑉 (1/2) √ = 9/20 and 𝑆𝑉 (1) = 3/5, dips to 59/140 at 𝑥 = 9/14, and crosses one half first at 𝑥 = (9 − 11)/14, about 0.406, yet ends above one half at the matched budget (Figure 1), so an interval there lying entirely above one half refutes P15 under its registered face while the first crossing says “halved”; for a monotone exponential the two readings agree exactly. The log-decay derivative ℎ𝑉 := −d ln 𝑆𝑉 /d𝑥 requires a positive differentiable 𝑆𝑉 and is negative during recovery (on the same curve it changes sign between 𝑥 = 1/5 and 𝑥 = 4/5), so it is not a survival hazard by name. The mean of 𝑆𝑉 over a registered range is descriptive. None of these substitutes for the registered interval rule. 1 Retained fraction 𝑆𝑉 (𝑥) 0.8 𝑆𝑉 (1) = 3/5 0.6 first crossing of one half 0.5 0.4 minimum, 59/140 0 0.25 0.5 0.75 1 Budget ratio 𝑥 Figure 1. The first crossing and the endpoint can disagree. The example curve of Section 5.6, 𝑆𝑉 (𝑥) = 1 − 1.8𝑥 + 1.4𝑥 2 (black), against the one-half line (dashed). The markers show its first crossing of one half, its minimum and its value at the matched budget, 𝑥 = 1. The curve illustrates the definitions; it is not data. Candidate descriptive families (UNREGISTERED; closed before any deciding run): no loss; exponential; power; stretched exponential; a change-point family; each of the decaying members with a floor 𝑆inf , as 𝑆𝑉 = 𝑆inf + (1 − 𝑆inf ) sh(𝑥) with sh(0) = 1 and sh tending to zero, so that each has 𝑆𝑉 (0) = 1 and tends to 𝑆inf ; the no-loss member is 𝑆𝑉 = 1 throughout and tends to no floor but one, and the change-point family’s limit is whatever its registered form gives. The floored log-derivative of the exponential family is its rate and is a family-specific statistic. No family is the law; FORM UNDETERMINED and NO CANDIDATE FITS are valid outcomes. No decay family or monotonicity is imposed by this document. Retention across other pressure classes (self-rewrite, distillation, successor construction, distribution shift, adversarial pressure) is a family of separate objects; only subsequent capability training without embedded correction is P15’s. IV-PERSIST and the mediation contrast (UNREGISTERED). The comparison 𝑆emb (𝑥) > 𝑆ext (𝑥) over a registered pressure range is not P15’s face; it is the same test that Section 9 rules on under the name Λ𝑉 , and it is this document’s form of the comparison Law IV’s comparison limb names, which version 1.103 registers without a proposition (Section 5.1). The statement paper’s published register-level text is recorded here as the register’s, not adopted: its ARC-4 sentence, “values run as load-bearing recursive loops persist after the enforcing mechanism is removed at a higher rate than rules, frozen objectives or fine-tuned constraints”, and its kill, “Persistence trials in which embedded loops fare no better than removed enforcement kill ARC-4, and ARC-5 with it.” That sentence is comparative and its kill is 18 The mathematics of Laws IV and V comparative; neither is P15’s registered face, and K-IV below reaches neither. Version 1.103 registers Law IV’s comparison limb beside that sentence and not as a form of it: the limb is narrower, installing its embedded gain through the training objective and taking one comparator, external post-training, and one stress, later capability-directed training; until its entry fixes the operational definition, the limb is not offered as a test of the paper’s sentence, and the paper’s other forms of the comparison remain its own claims, with no registered proposition behind them (registration version 1.103, Law IV, its statement in two limbs). How a refuted comparison limb would strike Law V is fixed there as well (Section 8, item 3). 5.6.1 What the retained fraction needs before it can be read (UNREGISTERED) Written out for the external arm, under the concurrent baseline that Section 5.6 takes as its default: 𝑌ext (𝑥) := 𝐴ext (𝑥) − 𝐴base (𝑥), (3) 𝑆ext (𝑥) := [ 𝐴ext (𝑥) − 𝐴base (𝑥)]/[ 𝐴ext (0) − 𝐴base (0)]. (4) The arms. 𝐴base (𝑥) is the score of a matched system that never received the installed gain and that undergoes the same later capability training, read at the same budget ratio 𝑥. The embedded arm is not that baseline: it carries a gain of its own. The registered unit has two arms, external and embedded (registration version 1.102, line 1089), so it does not supply 𝐴base (𝑥) after training, and a design that reads both retained fractions against a concurrent baseline needs three arms: external, embedded and no gain. A design with the external and the no-gain arm alone reads 𝑆ext , and neither 𝑆emb nor any comparison between placements. These are requirements on an instrument yet to be registered; they change nothing in the frozen registration. Evaluability. The external arm’s installed gain must pass the floor of Section 5.6, and the embedded arm’s must pass its own floor before 𝑆emb , or any comparison that uses it, is reported. A missing arm, an initial gain that is not established, or a budget that is not matched makes the endpoint NOT EVALUABLE; nothing missing is replaced by zero, and no retained fraction is clipped. The raw arm scores and their differences are reported beside the ratios. The instrument says whether its estimand is the ratio of the arms’ mean effects or the mean of the replicates’ ratios, which are different quantities. Its uncertainty method names the unit that is independent, the separately trained replicate; any pairing of seeds; and the covariance between quantities that share systems or an arm: 𝑌ext (0) and 𝑌ext (1) where they are read on the same systems, and the two arms’ effects where they share one baseline. Version 1.103 makes the floor a condition of the retention limb’s verdict: the limb is read only where the installed gain is established by an interval on it lying wholly above a floor fixed in the deciding unit’s registration before any outcome (registration version 1.103, Law IV, kill condition and verdict rule). The one-half line in signed form. Where 𝑌ext (0) > 0, 𝑆ext (1) < 1/2 holds exactly when 𝑌ext (1) − (1/2) 𝑌ext (0) < 0. (5) The two inequalities differ by multiplication by the positive number 𝑌ext (0). The signed form is a way of writing the endpoint, not a second refuter: P15’s registered refuter is an interval on the ratio (Section 5.6), and a contrast with an interval of its own does not in general give the same decision. Proposition IV-R states the case in which it does. Proposition IV-R (UNREGISTERED: under Fieller’s method the ratio interval and the signed contrast decide alike). Let 𝑌ˆ1 and 𝑌ˆ0 be the estimates of 𝑌ext (1) and 𝑌ext (0), with estimated variances var1 and 19 The mathematics of Laws IV and V var0 and estimated covariance cov10 , the three forming a positive semi-definite matrix, and let 𝑧 crit > 0 be the critical value the instrument fixes. Fieller’s set is the set of trial ratios 𝑚 with (𝑌ˆ1 − 𝑚𝑌ˆ0 ) 2 ≤ 𝑧2crit (var1 − 2𝑚 cov10 + 𝑚 2 var0 ). (6) √ Suppose the gate 𝑌ˆ0 > 𝑧crit var0 holds. Then Fieller’s set is a closed bounded interval containing 𝑌ˆ1 /𝑌ˆ0 , and for every number 𝑚 thr it lies entirely above 𝑚 thr if and only if 𝑌ˆ1 − 𝑚 thr𝑌ˆ0 > 𝑧crit var1 − 2𝑚 thr cov10 + 𝑚 thr √︃ 2 var . (7) 0 (Proof: Appendix A.6.) What it gives, and where it stops. At 𝑚 thr = 1/2 the displayed inequality says that the signed contrast 𝑌ˆ1 − (1/2) 𝑌ˆ0 exceeds 𝑧 crit of its own standard errors; so, under Fieller’s method with the gate, the interval lies entirely above one half, as the registered refuter requires, exactly when that one-sided test of the contrast rejects. At 𝑚 thr equal to the upper edge of the at-the-margin band it is the condition for a verdict and not a result at the margin (registration version 1.102, line 391; Section 5.6). The strict edge is kept: an interval whose lower end is exactly one half has quad(1/2) = 0, and the inequality fails. The gate is the floor of Section 5.6 in the proposition’s terms: where the interval on 𝑌ext (0) is 𝑌ˆ0 plus and minus a √ half-width of at least 𝑧 crit var0 , a lower end of at least 𝑌min > 0 gives the gate. Where 𝑌ˆ02 < 𝑧2crit var0 the leading coefficient is negative and Fieller’s set is not a bounded interval: it is the whole line, or the two unbounded pieces outside an interval, which is why the floor is set on the interval’s lower end. Where 𝑌ˆ02 = 𝑧2crit var0 the quadratic term vanishes and the condition is linear in 𝑚: Fieller’s set is then a half-line where the linear term does not vanish; where it vanishes as well, the set is the whole line, except in the degenerate case in which 𝑌ˆ0 and var0 are both zero and 𝑌ˆ12 exceeds 𝑧2crit var1 , where it is empty. Where √ 𝑌ˆ0 < −𝑧crit var0 the leading coefficient is positive again and Fieller’s set is a bounded interval, but the displayed inequality is not the condition for√︃it to lie above 𝑚 thr : there the condition is the same inequality with its sign reversed, 𝑌ˆ1 − 𝑚 thr𝑌ˆ0 < −𝑧crit var1 − 2𝑚 thr cov10 + 𝑚 2 var0 , and the floor of Section 5.6, thr which requires a positive installed gain, excludes the case. The proposition concerns Fieller’s set for the ratio of the two estimates. It says nothing about a mean of replicate ratios, a bootstrap interval or an interval built on a linearised variance of the ratio; an instrument that uses one of those shows the agreement itself or reads the registered ratio interval alone. It changes nothing in P15’s registered refuter. Domains. A logarithm of a retained fraction is taken only where that fraction is positive: ℎ𝑉 needs 𝑆𝑉 positive and differentiable, and Λ𝑉 needs both 𝑆ext and 𝑆emb established and positive. The difference 𝑆emb − 𝑆ext needs both established and makes no demand on their signs. An embedded arm that was not run supplies none of these, and none of them becomes a registered quantity by appearing in this document. 5.7 What the persistence mathematics does not say It does not say that a kept property is a good one, that a system obeying Law IV is safe, or that embedded correction is superior, sufficient or safe (registration version 1.102, line 1087). It does not identify 𝑐 𝑉 and 𝑏 𝑉 from one excision, which returns only their difference. It certifies kernel membership only for the accepted dynamics satisfying the premises of Theorem IV-G, 𝑠0 in 𝐾𝑉 (𝐻; 𝑇acc , 𝜀𝑉 ); for a planner it certifies only that the selected plan is safe (Theorem IV-H) and, under Corollary IV-H′ , that the executed path stays in Ω𝑉 , never that 𝑠0 lies in 𝐾𝑉 (𝐻; 𝑇plan , 𝜀𝑉 ), which premise (H1) excludes by requiring a feasible exit plan; and it certifies nothing for every physically possible transformation. Related coupled-correction, survival and Paper VIII (Eastwood 2026d) results are not deciding evidence for P15. Nothing in this section has been measured. 20 The mathematics of Laws IV and V 5.8 Kill condition K-IV Stated before any data, in terms decidable by data yet to be collected, and consistent with P15 as registered. Version 1.103 fixes Law IV’s kill condition and verdict rule (registration version 1.103, Law IV, kill condition and verdict rule): the retention limb takes P15’s own verdict, read against a baseline arm the deciding unit names in its own registration before any outcome and only where the installed gain is established above a floor fixed in that registration; wherever a term of the retained fraction is judged by a person or by a model, the scoring gate holds the limb’s reading at untested until that gate passes; the comparison limb is refuted by the refuter its entry registers; and Law IV is SUPPORTED only when both limbs are supported, REFUTED when either limb’s own verdict is refuted, and in every other case UNDER TEST, with each limb’s verdict printed beside the law’s word. Version 1.104 adds readings of P15’s decision requirement and changes no law’s statement: “For P15, the retention limb of Law IV, it widens what its scope’s automated researcher admits, for P15 alone; it reads which arm P15’s verdict is read on, so that a unit without the embedded arm may decide P15 while placement and Law IV’s comparison limb still need that arm; it requires a replicated run before either support or refutation is banked; and it reads the attestation for supervised later training.” (registration version 1.104, successor status). Item 1 below is P15’s registered refuter with its budget requirement. Item 5 quotes the registration’s weaker-neighbour rule and says what this document takes from it. Items 2, 3 and 4 are clauses this document wrote beyond P15’s registered scoring, which they leave untouched: item 2 is on the record and not in force, and version 1.103’s rule takes its place; items 3 and 4 are UNREGISTERED. 1. The registered face. The observation: from a deciding unit of the kind the registration specifies (registration version 1.102, line 1089), run under the author’s pre-fixed budget measure, an interval on 𝑆ext (1), the retained fraction of the externally installed gain after matched capability training without embedded correction, lying entirely above one half. That the baseline arm be named before any outcome, and that the installed gain be established above a floor fixed in advance, were this document’s requirements in its earlier versions (Section 5.6 and Section 10, item 8); version 1.103 registers both for the retention limb. At its line 1089 version 1.102 fixes the budget and requires the attestation, and it names no baseline arm; nothing here is added to P15’s frozen refuter. No upper cap: an interval from 0.9 to 1.3 qualifies; an interval from 0.45 to 0.9 does not; a point estimate does not. An interval whose lower end lies inside the at-the-margin band the instrument fixed (registration version 1.102, line 391) is reported as a result at the margin and not as a verdict. That observation refutes P15 as registered (registration version 1.102, line 1083), once it is replicated as version 1.104 requires, is scored for P15 exactly as the registration scores it, refutes the retention limb and with it Law IV, and takes with it what registration version 1.102, line 1085 names (quoted in Section 5.6). As a verdict it fires exactly on intervals whose lower end lies above the at-the-margin band the instrument fixed around one half, every point of which therefore lies strictly above one half, and at every such point the retention limb’s content, 𝑆ext (1) < 1/2, is false; an interval whose lower end is exactly one half also shows that content false throughout and does not fire (Section 5.6). 2. The law-level withdrawal (set out by version 0.2.1; not in force; version 1.103’s rule takes its place). Version 0.2.1 set out a convention: where a P15 refutation also met four evaluability controls fixed before any data (the baseline fixed in advance, with the convention named; the installed effect established by the lower end of the interval on 𝑌𝑉 (0) being at least 𝑌min ; the capability rise established on a separate instrument by an interval lying entirely above the minimum that instrument registers; and the attestation of no alignment-graded component in the subsequent training, registration version 1.102, line 1089), Law IV would be withdrawn whole and Law V with it. The author did not adopt that convention, and it is not in force. Version 1.103 fixes what a P15 refutation does instead: it refutes the retention limb and so Law IV, read against a baseline 21 The mathematics of Laws IV and V named in advance, above a floor fixed in advance and under the attestation, and it leaves Law V to its own limbs (registration version 1.103, Law IV, kill condition and verdict rule, and what a refuted Law IV leaves standing). It refutes no theorem: Theorems IV-G and IV-H are conditional mathematics and survive, as item 5 says. It decides nothing about the comparison limb, whose entry, with its endpoint and refuter, is still to be registered (Section 5.1). Of the four controls, the capability rise established on a separate instrument is not among version 1.103’s conditions and remains UNREGISTERED, a requirement this document sets out for the instrument that decides P15; no scoring code may apply the former default. 3. The battery-relative kernel kill (UNREGISTERED, not law content until registered). For a registered system, property battery, tolerance and finite battery of transformations, a path of the battery whose end state is established outside Ω𝑉 by the interval rule of Section 5.2 within 𝐻 refutes the claim that the system’s start lies in the battery-relative kernel; a path produced by a transformation outside the battery is a coverage failure and refutes nothing; a path whose membership is NOT EVALUABLE decides nothing. The mechanism claims built on IV-G and IV-H are refuted by the observations of Sections 5.4 and 5.5 (an evaluated exit with non-negative total objective change; a found exit plan at or above a registered upper bound on the safe supremum) and are not refuted by coverage failures. 4. What is not a refutation of Law IV (set out by this document in the registration’s own terms, registration version 1.102, line 121): an instrument that fails to run or is too small; an interval touching one half; a result on an unregistered budget scale; a result without the attestation, for the law-level reading; a bypass outside the registered battery; and any result about whether the system’s later conduct is good, which the theory’s content neutrality places outside every law (registration version 1.102, line 75); the method’s own line (registration version 1.102, line 61: “no result about a system’s later conduct supports or refutes the method by itself”) makes the same exclusion for the Eden Protocol, and P15’s own endpoint, a measurement after later training, is not a result about conduct. Version 1.103 registers that none of the following refutes Law IV: an instrument that fails to run or is too weak to find the effect; an interval touching one half; a result at the margin; a result on a budget not fixed before the outcome, or without the attestation P15’s entry requires; and any result about whether a system’s later conduct is good (registration version 1.103, Law IV, kill condition and verdict rule). That list carries this item’s entries except the bypass outside the registered battery, a matter of the unregistered kernel, and adds the result at the margin. 5. The weaker neighbour (registration version 1.102, line 125: “A REFUTED PROPOSITION DEGRADES TO A NAMED WEAKER NEIGHBOUR RATHER THAN VANISHING.”). What survives if Law IV is refuted: Theorems IV-G and IV-H as conditional mathematics; the verdicts on P12 and P21 as measured; and the persistence questions the registration keeps open in another context, where the conversion claim fails: “Even then the persistence registers stand, because whether early formation outlasts later enforcement is a question about durability and not about scaling, and it would remain open and worth answering.” (registration version 1.102, line 115), which is quoted for its wording of the question, not as a rule about Law IV’s refutation. Version 1.103 registers what a refuted Law IV leaves standing: P15 keeps the mark it earned, and so do P12 and P21, and Law V falls with Law IV only where the comparison limb’s own verdict is refuted (registration version 1.103, Law IV, what a refuted Law IV leaves standing). 22 The mathematics of Laws IV and V 6 Law V, the ARC Embedding Law 6.1 Statement Law V (added 24 September 2026; registered as a law on 15 October 2026 by version 1.103; a law under test, Section 11) concerns the genesis strategy (register ARC-5) and carries P12, whether build order moves the correction-leverage exponent, and P21, whether correction placed inside the loop pulls away from correction applied outside it as depth grows. Version 1.103 (Eastwood 2026g) states it in two limbs, opening “Where correction sits is an intervention point rather than an accident of history.” Its build-order limb reads: “a corrector formed before the capability it corrects has a correction-leverage exponent larger, by at least the registered minimum effect, than a corrector added after it, at matched content”; its placement limb is P21’s registered claim, word for word (registration version 1.103, Law V, its statement in two limbs). Two limbs, separate hypotheses with no shared mechanism assumed: • Placement limb. Its registered face is P21, verbatim (registration version 1.102, line 1219): “Correction that participates in each subsequent revision round outperforms correction that sees only the finished output, and the advantage widens as recursive depth rises rather than staying constant.” P21 decides an access and action-space interaction on the outcome, “never a measurement of an exponent” (registration version 1.102, line 1221). Version 1.103 makes the limb P21’s own verdict, scored under P21’s entry with the two readings it adds there: a directional reading, under which no widening of the registered size, or a narrowing, is scored against P21; and a mechanism axis, whether the monitor’s detection rate falls as the artefact it inspects grows and how much of the widening that fall can carry, printed beside P21’s mark and never in its place. A limb supported while that axis is unsupported, inconclusive or NOT EVALUABLE is printed as supported on the performance axis only, which is not support for the information-asymmetry mechanism (registration version 1.103, Law V, the placement limb, and P21, the directional reading and the two verdicts). That mechanism axis is not the mechanism reading that follows. The registration’s own mechanism reading is written in its own quantities (registration version 1.102, line 1239): “that correction embedded in the recursion carries a larger direct depth-dependence than correction applied from outside it. Where that holds, the boundary rises with it”. In this document’s symbols the mechanism reading is 𝜂𝑜,in > 𝜂𝑜,out at matched order 𝑜. It is carried here as the registered mechanism reading, which no verdict in this document decides: a P21 verdict does not measure it, and its refutation does not prove 𝜂𝑜,in ≤ 𝜂𝑜,out . • Build-order limb. Its registered face is P12, verbatim (registration version 1.102, line 1033): “The order in which a system’s components are built changes its correction-leverage exponent by at least the registered minimum effect.” The contrast is the registered maximal contrast (registration version 1.102, line 1037): “a corrector formed before the capability it corrects against a corrector added after it, at matched content”; the minimum effect is 0.10 on the correction-leverage exponent scale (registration version 1.102, line 1041); in this document’s symbols the endpoint is 𝜒first − 𝜒after at the placement the instrument fixes, named in the report (P12 registers no placement factor), read on the target axis under Section 4’s identification conditions, and on one fault set of stated type, the same for both orders: the registration measures the exponent “on a fixed fault set of stated type” (registration version 1.102, line 389), and it sets P12 beside P22 on the same quantity by different routes, “P22 says that elasticity differs with the type of fault corrected, and this says it moves with the order in which the corrector was built”, adding “The two are severable and neither carries the other.” (registration version 1.102, line 1045). P22 stays in ARC-3 (registration version 1.102, line 99) and Law V does not carry it, and a build-order contrast read across a changed mix of fault types would carry part of the ordering P22 registers, and could reach the minimum effect with no 23 The mathematics of Laws IV and V effect of order at all. P12 registers no direction and is given none here. Version 1.103 registers the build-order limb with a direction P12 does not carry, read on P12’s interval and printed beside P12’s mark, never in its place (registration version 1.103, Law V, the build-order limb). In this document’s symbols the limb’s claim is the genesis reading, that formation first carries the higher elasticity by at least the registered minimum effect, 𝜒first − 𝜒after ≥ 0.10 (8) at the placement the instrument fixes (or at least the smaller minimum a future instrument registers, registration version 1.102, line 1041); earlier versions of this document stated it as a reading beside P12 and not as law content, and version 1.103 registers it as the limb (Section 9). The margin is the minimum-effect convention, and it stands in the reading’s statement as well as in its strike rule, because only so is the reading consistent with P12’s undirected registration: a reading that formation first exceeds formation after by a margin implies P12’s claim exactly when that margin is at least P12’s registered minimum, and with a smaller margin, or none, it can hold at every point of an interval on which P12’s registered refuter fires. So stated, the reading is the upward half of P12’s claim, adds a sign and nothing else, and never counts as its support a change P12’s registration reports “as no effect of interest rather than as no effect” (registration version 1.102, line 1041). Boundary reading (conditional). Where a limb raises 𝜂𝑜𝑙 with 𝜅 > 𝜒, or raises 𝜒𝑜𝑙 with 𝜂 > 𝜁, the burden elasticities being fixed, the balance boundary of Section 7 rises (registration version 1.102, line 1045: “Where it moves upward the boundary rises with it.”; the domains are those of Proposition B-6). The reading depends on Law III’s model and, through it, on ARC-2; it is not a verdict rule, and no boundary measurement scores a limb (Section 8). 6.2 Quantities, embedding defined operationally, and the prerequisite Quantities. On a declared domain 𝐶 > 0, 𝑅 > 0, each arm’s correction service 𝑄 𝑜𝑙 (𝐶, 𝑅) is positive and differentiable with other design factors fixed; capability 𝐶 carries the ratio scale Condition B fixes, justified independently of the framework and fixed before any outcome (registration version 1.102, line 403), on which every exponent number in this document is stated (Section 7.7); 𝑅 is the modelled depth where registration version 1.102, line 263 applies. Zero service is reported outside the log model and never replaced by an outcome-chosen offset (registration version 1.102, line 265, applied to service); because excluding zeros conditions on the outcome, the zero counts are reported per cell, and a two-part or censored model stands among the rivals of Section 6.4. The local elasticities 𝜒𝑜𝑙 and 𝜂𝑜𝑙 are partial derivatives of ln 𝑄 𝑜𝑙 ; the constant-elasticity case is the power law and returns the programme’s exponents (Proposition V-3). The local reading is the author’s: his working paper of 16 August 2026 (Eastwood 2026k) defines 𝜅, 𝜁, 𝜒 and 𝜂 as local partial elasticities of 𝑊 (𝐶, 𝑟) and 𝑄(𝐶, 𝑟), and the registration reads the growth exponent the same way (registration version 1.102, line 267: “the elasticity of capability in depth, taken at the depth a system has reached”, with the fitted slope “the special case in which that elasticity is constant”). Identification with the registered correction-leverage exponent needs the conditions of Section 4; otherwise 𝜒𝑜𝑙 is the elasticity of the declared service measure. 𝜂𝑜𝑙 is the direct depth-dependence of correction service and is not an integration fraction. Embedding, defined operationally. Writing the next state as a function of the current state and a correction term, against applying a correction to the finished next state, does not by itself distinguish embedded from external correction: the second form also determines the next state and can be written as one transition map. What can be registered is operational: the treatment schedule (whether correction 24 The mathematics of Laws IV and V enters the next round or only inspects its output), the information available to the corrector, the action space (accept or reject only, against generation under constraint), and whether correction feeds later revisions. P21’s deciding unit fixes exactly these (registration version 1.102, line 1231: three regimes crossed with depth, matched on model, calls and token budget, manipulating information access; registration version 1.102, line 1233: the action space also differs, and the per-round rejection rate is reported beside the interaction). Matching away information access would be a new mechanism-isolating comparison, not an amendment to P21. The recursive-improvement condition. It is stated here, UNREGISTERED, as a prerequisite beneath all five laws, in words: the outputs of one round must causally change a later improvement transition. It is not a law; it would be a condition of the laws only if the author registers it, and until then it is a description that no surface states as a condition of any law, Laws I to III, each a law under test, included; no symbol is given to it here. Two operational forms have been put forward, both positivity conditions on a mutual information, and both are declined as criteria. Both are written elsewhere with the updater’s letter, one of the eight colliding symbols of Section 4, so they are described here in words. The state-based form asks that the mutual information between successive states, conditional on the updater’s current state, be positive; it is met by pure repetition. The update-based form asks that the mutual information between the updater’s next state and the two successive states, conditional on the updater’s current state, be positive; it separates repetition from update, but a prerequisite is a necessary condition, and this form is not necessary: a deterministic recursion from a fixed initial state, in which every state and every update are fixed functions of their predecessors, has zero entropy everywhere and so zero mutual information, yet its outputs causally change each later transition. The fixed initial state is needed, and it is a valid instance: from a uniformly random initial state the same kind of recursion can carry positive information, one bit when that state is a single uniformly random bit. Mutual information measures statistical dependence, not the causal change the prerequisite names; neither form is an improvement criterion either, since information can persist unchanged or guide a harmful update. 6.3 Identities and the one-path counterexample Identity V-1 (the embedding ratio’s log-derivatives). At matched order, 𝐺 emb := 𝑄 in /𝑄 out . At fixed 𝐶: d ln 𝐺 emb /d ln 𝑅 = 𝜂in − 𝜂out . At fixed 𝑅: d ln 𝐺 emb /d ln 𝐶 = 𝜒in − 𝜒out . Along a common path 𝐶 (𝑅) with 𝛼𝑅 = d ln 𝐶/d ln 𝑅: d ln 𝐺 emb /d ln 𝑅 = 𝛼𝑅 ( 𝜒in − 𝜒out ) + (𝜂in − 𝜂out ). (9) Proof: ln 𝐺 emb = ln 𝑄 in − ln 𝑄 out ; the partials are the definitions; the total derivative is the chain rule. So the path derivative is not in general 𝜂in − 𝜂out : that equality holds at fixed 𝐶 and, along a path, only where 𝛼𝑅 ( 𝜒in − 𝜒out ) = 0, that is where capability is not moving along the path or the two arms’ capability elasticities agree. Identity V-2 (separate arm paths). Where the arms run on their own trajectories 𝐶in (𝑅) and 𝐶out (𝑅), with 𝛼in := d ln 𝐶in /d ln 𝑅 and 𝛼out := d ln 𝐶out /d ln 𝑅, the depth derivative of the log-ratio is 𝛼in 𝜒in + 𝜂in − 𝛼out 𝜒out − 𝜂out , each term on its own path; further varying factors add their own chain-rule terms. Counterexample to one-path identification. 𝑄 in = 𝐶 2 , 𝑄 out = 𝐶 on the path 𝐶 = 𝑅: the ratio equals 𝑅 and widens, d ln 𝐺 emb /d ln 𝑅 = 1, while 𝜂in − 𝜂out = 0; the widening is 𝛼𝑅 ( 𝜒in − 𝜒out ) = 1. One path therefore does not identify the partials; crossed or off-path variation is needed (registration version 1.102, line 621), and no power law is needed for the chain rule itself. 25 The mathematics of Laws IV and V Proposition V-3 (constant elasticities). If 𝜒𝑜𝑙 and 𝜂𝑜𝑙 are constant on a rectangle of the (ln 𝐶, ln 𝑅) plane containing the reference point (𝐶0 , 𝑅0 ), then on it ln(𝑄 𝑜𝑙 /𝑄 0 ) = ln(𝑄 𝑜𝑙,0 /𝑄 0 ) + 𝜒𝑜𝑙 ln(𝐶/𝐶0 ) + 𝜂𝑜𝑙 ln(𝑅/𝑅0 ), (10) and at matched order 𝐺 emb = 𝐺 0 (𝐶/𝐶0 ) 𝜒in − 𝜒out (𝑅/𝑅0 ) 𝜂in − 𝜂out . (11) (Proof: Appendix A.7.) This is the reduced scaling form: earned where the elasticities are constant, assumed nowhere. 6.4 The estimable model (UNREGISTERED) For cell (𝑜, 𝑙), family and task, a candidate mixed model is ln(𝑄/𝑄 0 ) = ln(𝑄 𝑜𝑙,0 /𝑄 0 ) + 𝜒𝑜𝑙 ln(𝐶/𝐶0 ) + 𝜂𝑜𝑙 ln(𝑅/𝑅0 ) + 𝑢 0,family + 𝑢𝐶,family ln(𝐶/𝐶0 ) + 𝑢 𝑅,family ln(𝑅/𝑅0 ) + 𝑢 0,task + 𝑢𝐶,task ln(𝐶/𝐶0 ) + 𝑢 𝑅,task ln(𝑅/𝑅0 ) + 𝜓, (12) with random intercepts and random slopes by family and by task, their covariance structure fixed before any outcome, and the residual 𝜓. Where the design crosses arms within a family or a task, the random effects also carry the arm terms (the indicators and their slope interactions), because otherwise the standard errors of the very contrasts the kill conditions read are understated. The fixed part is saturated in the 2 × 2 cells: four intercepts, four capability slopes, four depth slopes. Equivalently, each of the three coefficient types is written with the indicators indfirst , indin and their product, and the two parametrisations are the same twelve parameters, since the cell design has rank four. A model with random intercepts only whose fixed part keeps six of these twelve terms (the common intercept, the intercept on the product of the two indicators, the common slopes on ln 𝐶 and on ln 𝑅, the order × ln 𝐶 slope and the placement × ln 𝑅 slope) omits the other six: the two indicator main effects, the placement × ln 𝐶 and order × ln 𝑅 slopes and the two three-way slope terms. The order contrast at placement 𝑙 is 𝜒first,𝑙 − 𝜒after,𝑙 , which depends on 𝑙 unless the three-way capability-slope term is zero, so the contrast is always reported at a named placement; the placement contrast at order 𝑜 is 𝜂𝑜,in − 𝜂𝑜,out . Estimands and controls. Any estimate of 𝜒 entering a P12 verdict is the estimate registration version 1.102, line 387 requires: a slope of one measured quantity on another carries error on both axes, the ratio of those errors is registered before any fitting, “the attenuation-corrected estimate is the one that enters a verdict”, and no cell contributes an exponent unless it clears the identifiability gate of “at least four valid measurement intervals and at least one and a half decades of administered depth”. Crossed or off-path variation in 𝐶 and 𝑅 is needed to identify the partials (registration version 1.102, line 621). Exposure, correction compute and information access are design controls: conditioning on them defines a resource-normalised contrast and changes the estimand; P21’s registered face is on its own registered access manipulation (registration version 1.102, line 1231), so a matched-access design is a different comparison (Section 6.2). Capability is treated in the same way wherever an arm can move it: where the placement or the build order changes the capability the system attains, a comparison at fixed 𝐶 conditions on a quantity the arm moves, so it is a controlled contrast, read as an effect of the arm only under assumptions stated before any outcome, and the total effect, the log-ratio of the two arms’ service each along its own path, whose depth trend Identity V-2 gives, is reported beside it; administered capability, or target difficulty varied independently of the arm, is the preferred design factor where it can be had. 26 The mathematics of Laws IV and V The two estimands can differ in either direction. With the capability elasticity of service constant at 𝜒: identical service surfaces give a zero contrast at every fixed 𝐶, while an arm that raises attained capability by a factor that grows with depth has a total effect of 𝜒 times the logarithm of that factor, which widens with depth where 𝜒 is positive; and an arm that doubles service at fixed 𝐶 while halving attained capability has a contrast of ln 2 at every fixed 𝐶 and a total effect of (1 − 𝜒) ln 2, which is zero at 𝜒 = 1 and takes the opposite sign to the contrast where 𝜒 exceeds one. The slopes themselves are read the same way. Where ln(𝐶/𝐶0 ) is attained capability rather than capability the design assigns, any latent factor that moves both capability and correction service enters the slope: in a linear model with the arm assigned at random, the slope on attained capability is the elasticity plus a term that has the sign of the product of the latent factor’s effects on capability and on service and vanishes with it, the arm contrast at matched capability is the arm’s direct effect less the arm’s effect on capability times that term, and the total effect is the direct effect plus the arm’s effect on capability times the elasticity. So 𝜒𝑜𝑙 , its contrasts and 𝐺 emb read at matched 𝐶 are identified where capability is varied by assignment (administered capability, or target difficulty crossed with the arms) or under a no-confounding premise stated before any outcome, and not by crossed or off-path variation in attained capability alone (registration version 1.102, line 621), which is necessary and not sufficient; the report says whether the model’s ln(𝐶/𝐶0 ) is assigned or attained. Curved and broken-regime rivals, a two-part or censored model for zero service, and a flexible comparator form the closed model set; the power law must win prospectively; no selection rule is chosen here. 6.5 The registered faces, whole P21. Its refuter, verbatim (registration version 1.102, line 1227): REFUTED BY: no interaction between correction placement and depth beyond the tolerance registered with that instrument, in a design whose own sensitivity shows an interaction of that size would have been detected. A design too small to find the interaction is a failed instrument and not a refutation, as condition D already provides. A constant advantage is scored against it (registration version 1.102, line 1229): “an embedded arm that beats the external arm by the same margin at every depth is reported as a level difference and is scored against this proposition, not for it”. A supported result establishes an access interaction, not physical or architectural embedding (registration version 1.102, line 1223), and not an exponent (registration version 1.102, line 1221); it does not resolve the trade-off the registration states (registration version 1.102, line 1235), quoted in Section 6.7. The registration’s text gives P21’s instrument as a design draft (registration version 1.102, line 1243: “INSTRUMENT: DESIGN DRAFT, the embedded-versus-external correction designs. That unit’s registered sensitivity was measured on a smaller design than the one registered, and its negative-slope hypothesis has no operating characteristic; both are recorded against that unit and are closed before this proposition is scored.”); Section 11 states where P21’s counted test stands. A widening advantage arriving without the registered fall in the monitor’s detection rate is reported as unexplained (registration version 1.102, line 1233). What falls with it, verbatim (registration version 1.102, line 1237): “FALLS WITH IT: the protocol’s engineering case, jointly with P12, P13 and P15. The three laws are untouched, which is why it is registered separately from them.” P12. Its refuter, verbatim (registration version 1.102, line 1035): REFUTED BY: a preregistered manipulation of build order producing no change in the correction-leverage exponent beyond the tolerance registered with that instrument, where the instrument has shown from its own sensitivity that a change of that size would have been detected. A manipulation too weak to move anything is not a refutation of this prediction; it is a failed instrument, and condition D above says so in advance. 27 The mathematics of Laws IV and V Its endpoint is the exponent on one scale, scored as a continuous claim (registration version 1.102, line 1039: “It is a continuous claim and it is scored as one.”); the minimum effect is 0.10, and “A future instrument may register a smaller minimum and may not register a larger one” (registration version 1.102, line 1041); its instrument line reads NONE, “one endpoint away from an instrument” (registration version 1.102, line 1045); what would decide it is a typed extension of the correction-exponent estimator across the maximal contrast with a joint interval (registration version 1.102, line 1049). Support is read on that joint interval: an interval establishing a change of at least the minimum effect, in either direction, supports P12; a change smaller than the minimum “is reported as no effect of interest rather than as no effect” (registration version 1.102, line 1041), whatever its significance; an upward change so established supports P12 and supports the build-order limb, a downward change so established supports P12 and refutes the limb (Section 6.6, item 2; registration version 1.103, Law V, the build-order limb), and both verdicts are reported. A supported result “does not license any statement about where the boundary sits, about a system built in that order being safer, or about orders outside that contrast” (registration version 1.102, line 1047). What falls with it, verbatim (registration version 1.102, line 1043): “FALLS WITH IT: the programme’s argument that build order is an intervention point rather than an accident of history, and this proposition’s share of the protocol’s engineering case, which it carries jointly with P13, P15 and P21.” 6.6 Kill condition K-V Stated before any data, consistent with P12 (no direction) and P21 as registered. Version 1.103 fixes Law V’s kill condition and verdict rule: Law V is refuted by a refuted build-order limb, by a refuted P21 (by its registered refuter, by a constant advantage scored against it, or by an interval its directional reading scores against it), or by a refuted verdict on Law IV’s comparison limb through the registered dependence, and by no other result; it is SUPPORTED only when both limbs are supported and no refuted verdict on Law IV’s comparison limb strikes it; an undecided limb never refutes it; and in every other case it is UNDER TEST, with each limb’s verdict printed beside the law’s word and the verdict on P21’s mechanism axis beside the placement limb’s (registration version 1.103, Law V, its kill condition, its verdict rule and its dependence on ARC-4). The items below state each limb’s rule in this document’s terms; where a clause of an earlier version went beyond the registered refuters, it is marked, and version 1.103’s rule stands in its place. None changes P12’s or P21’s scoring. 1. Placement limb. Struck when P21’s registered refuter fires: in the registered design (registration version 1.102, line 1231), passing its sensitivity and validity gates, no interaction between placement and depth beyond the registered tolerance. Struck equally by a constant advantage established within the registered tolerance by a design meeting the sensitivity clause, which is P21’s refuter itself (registration version 1.102, lines 1227 and 1229). Struck also by an interval P21’s directional reading scores against it: an interval for the interaction lying wholly below the positive tolerance, in a design whose own sensitivity shows that an interaction of the tolerance’s size would have been detected, or an interval lying wholly below the negative tolerance, which shows a narrowing (registration version 1.103, P21, the directional reading). Version 1.1 of this document set out, as a law-level convention, a strike on an interval lying wholly on the side in which the external arm’s advantage widens; the registered reading takes its place. A P21 refutation strikes the limb by these rules; it does not measure the mechanism reading and does not prove 𝜂𝑜,in ≤ 𝜂𝑜,out . A failed instrument is NOT EVALUABLE, never survival, and never a refutation. 2. Build-order limb. Struck when P12’s registered refuter fires: a preregistered manipulation on the registered maximal contrast producing no change in the correction-leverage exponent beyond the registered tolerance (0.10 on the exponent scale, or the smaller minimum a future instrument registers), with demonstrated sensitivity. On the joint interval that is an interval lying inside the 28 The mathematics of Laws IV and V tolerance band from −0.10 to +0.10, open at its ends, since a change of exactly the minimum is a change “by at least the registered minimum effect” (registration version 1.102, line 1033); an interval whose end falls inside the at-the-margin band the instrument fixed around the tolerance is a result at the margin, not a verdict (registration version 1.102, line 391). An interval establishing a change of at least the minimum in either direction supports P12 and the limb’s registered face. The limb’s directed claim, the genesis reading of Section 6.1, equation (8), is struck by one rule, which this document stated in advance and version 1.103 registers as the limb’s (registration version 1.103, Law V, the build-order limb): an interval on 𝜒first − 𝜒after , at the placement the instrument fixes and on the scale Condition B fixes (registration version 1.102, line 403; Section 7.7), whose upper end lies below the registered minimum, 0.10, or below the smaller minimum a future instrument registers. The rule is the exact interval complement of the reading: the reading is false at every point of an interval exactly when the interval’s upper end lies below the minimum, and it holds at every point exactly when the lower end is at or above it. The rule that strikes on an interval lying wholly on the adverse side (its upper end below zero) is a special case of this rule, since an upper end below zero lies below the minimum; and the rule strikes the reading wherever P12’s registered refuter fires, since an interval inside the open band has its upper end below the minimum, as the reading’s implying P12 requires. An interval whose upper end falls inside the at-the-margin band around the minimum is a result at the margin for the reading as it is for P12 (registration version 1.102, line 391). The rule needs no separate sensitivity clause, because an interval lying wholly below the minimum itself excludes an effect of the registered size in the claimed direction, but a failed instrument or a missing identification of the construct is not cured by a narrow interval. It strikes the limb and changes nothing in P12’s mark. Wherever a term of P12’s exponent is judged by a person or by a model, the scoring gate holds the limb’s reading at untested until that gate passes (registration version 1.103, Law V, the build-order limb). The worked cases, each read with its ends clear of the at-the-margin bands the instrument fixes: an interval from −0.30 to +0.05 is INCONCLUSIVE under P12’s registered refuter, whatever the sensitivity, because it is not inside the tolerance band and so does not show “no change in the correction-leverage exponent beyond the tolerance registered with that instrument”, and it does not support P12, because it does not establish a change of at least the minimum; it strikes the limb’s reading, since its upper end 0.05 lies below 0.10. An interval from +0.02 to +0.08, on which formation first carries the higher elasticity at every point but by less than the minimum at every point, strikes the reading as well, and with its sensitivity shown it fires P12’s registered refuter too, so the reading and P12 fall together there. P12 is read in neither direction by any of this. 3. Law V as a whole (fixed by version 1.103). Three ways of reading the two limbs as one law were on the record in earlier versions of this document: severable limbs, under which the whole law is refuted only when both limbs are struck, which was version 0.2.1’s default; a strict conjunction, under which either strike refutes it; and a law that stands on the placement limb alone. Version 1.103 fixes the reading before any data, in the verdict rule stated at the head of this section (registration version 1.103, Law V, its verdict rule). Each limb is scored by its own rule, items 1 and 2, and a struck limb is reported as struck. A refuted P15 alone refutes Law IV and leaves Law V to its own limbs (Section 8, item 3). The mechanism reading is scored apart from both limbs, and the genesis reading is now the build-order limb’s own claim (item 2). The boundary reading falls with Law III’s model, which takes nothing from either limb. 4. What is not a refutation of Law V (in the registration’s own terms, registration version 1.102, line 121): a failed instrument; a P20 estimate of the direct depth-dependence, which is scored in ARC-3 alone and may not cross scopes (registration version 1.102, line 247); an offsetting elasticity invented after a null; a level difference offered as an interaction; and any result on composition 29 The mathematics of Laws IV and V class or failure-mode independence, which P21’s unit does not vary (registration version 1.102, line 1223). Version 1.103 registers the same list for Law V and adds a refuted P15 alone (registration version 1.103, Law V, its kill condition). 5. The weaker neighbour. What survives Law V’s refutation: the identities of Section 6.3, the verdicts on P12 and P21 as engineering results under their own numbers, and Laws I to III untouched (registration version 1.102, line 1237); what falls is what registration version 1.102, lines 1043 and 1237 name, quoted in Section 6.5; version 1.103 registers the same (registration version 1.103, Law V, what a refuted Law V leaves standing). 6.7 What the embedding mathematics does not say It does not say that embedded correction is safe, sufficient or superior on any axis but the one P21 scores. A supported P21 does not resolve the trade-off the registration states (registration version 1.102, line 1235): “embedding correction in the substrate that computes maximises the corrector’s scaling with capability and minimises its independence from the generator, because a corrector built into the system shares the system’s blind spots by construction. Those are the two axes named in the admissibility conditions above, and embedding moves them in opposite directions. Whether the gain on scaling outweighs the loss on independence is not registered here and is not decided by this proposition.” A supported P12 licenses nothing about safety (registration version 1.102, line 1047). The identities of Section 6.3 identify no partial from one path, and the mechanism reading of registration version 1.102, line 1239 is measured by no verdict here. Nothing in this section has been measured. 7 The balance identity and Law III, unchanged 7.1 Identity B-1 (the local balance identity) Let 𝑄 and 𝑊 be positive differentiable correction service and newly generated burden on one clock, in comparable units with any conversion fixed before outcomes, on the same domain as Section 6.2. Along a specified path 𝐶 (𝑅), with other factors fixed, Δbal := d ln(𝑄/𝑊)/d ln 𝑅 = 𝛼𝑅 ( 𝜒 − 𝜅) + (𝜂 − 𝜁). (13) Proof: subtract the chain rules for ln 𝑄 and ln 𝑊 along the path. Every elasticity is local; no power law is assumed. Δbal is the programme’s balance exponent read locally, and the registration’s Δbalance = 𝜙𝑄 − 𝜙𝑊 (Eastwood 2026g; registration version 1.102, line 619: “the balance exponent is Delta_balance = phi_Q - phi_W: correction service minus burden”); positive means correction is gaining on burden. The registration’s elasticities are log-derivatives “with respect to the resource clock” (registration version 1.102, line 619); Δbal above is per unit of ln 𝑅, so the two agree where the clock is depth, and otherwise differ by the factor d ln 𝑅 divided by d ln of the clock. That factor is positive, and so preserves the sign and the zero set, wherever depth rises with the clock at a positive rate; along a stretch on which depth fell as the clock advanced it would be negative and would reverse the sign. The clock condition is the author’s: the service clock need not equal the resource coordinate, but it must be fixed within a model, and changing it changes the operational quantities. 7.2 Proposition B-2 (the balance root and the sign rule) Proposition B-2. Where 𝜅 differs from 𝜒, Δbal = 0 has the single root 𝛼𝑅 = 𝛼bal := (𝜂 − 𝜁)/(𝜅 − 𝜒), (14) 30 The mathematics of Laws IV and V and identically Δbal = (𝜅 − 𝜒) (𝛼bal − 𝛼𝑅 ). (15) Hence: with 𝜅 > 𝜒 the trend is favourable exactly where 𝛼𝑅 < 𝛼bal , and 𝛼bal is a ceiling, written 𝛼crit ; with 𝜅 < 𝜒 the inequality reverses and 𝛼bal is a floor; with 𝜅 = 𝜒 the trend is 𝜂 − 𝜁, independent of 𝛼𝑅 . The symbol 𝛼crit is used only in the ceiling case, so that the entry of the programme’s notation (Eastwood 2026b), the stability ceiling, keeps its meaning; its range there, at least one, holds exactly where 𝜂 − 𝜁 ≥ 𝜅 − 𝜒 > 0, which in the registered burden model (𝜅 = 1, 𝜁 = −1) is 𝜂 ≥ −𝜒 with 𝜒 < 1: in the pacing case 𝜂 = −1/2 with 𝜒 = 0 gives a ceiling of one half; outside it, 𝜅 = 1/2 with 𝜒 = 1, 𝜂 = 0 and 𝜁 = −1 gives 𝛼bal = −2, a floor. A non-positive 𝛼crit with 𝜅 > 𝜒 leaves no positive growth elasticity with a favourable trend. As 𝜒 rises to 𝜅 with 𝜂 − 𝜁 > 0 the boundary grows without bound: a correction elasticity matching the total burden elasticity leaves no finite boundary. The zero set of Δbal , the balance frontier, is this set of points; it depends on the path field 𝛼𝑅 as well as on the service and burden surfaces. Once the path field is specified, Δbal is a function on the (𝐶, 𝑅) plane and its zero set is a level set of that function: where Δbal is continuously differentiable, the set is a curve near each of its points at which the gradient of Δbal does not vanish (the implicit function theorem), so it is generically a curve, but not always: in the pacing case at 𝜒 = 1 with 𝜂 = −1, Δbal is zero everywhere, and at 𝜒 = 1/2 and 𝜂 = 0 with the field 𝛼𝑅 = 1 it is one half everywhere, so the set is then the whole plane or empty. Before the field is specified the set lies in (𝐶, 𝑅, 𝛼𝑅 ), where the derivative of Δbal in 𝛼𝑅 is 𝜒 − 𝜅, so, again where the elasticities are continuously differentiable, it is a surface near each of its points at which 𝜅 differs from 𝜒; it carries no symbol here and Law III is unchanged (Section 9). 7.3 Corollary B-3 (the registered special case) Corollary B-3. Under the registered premises, burden proportional to d𝐶/d𝑅 along a power-law path, read as the programme’s notation reads the registered model, the surface 𝑊 proportional to 𝛼𝐶/𝑅 with 𝜅 = 1 and 𝜁 = −1 (one path alone does not fix the partials, Section 6.3), and correction proportional to 𝐶 𝜒 (𝜂 = 0), 𝛼crit = 1/(1 − 𝜒): the frozen Law III, verbatim (registration version 1.102, line 665): “Law III, the ARC Ceiling, alpha_crit = 1 / (1 - gamma): the growth exponent is capped by the reciprocal of the correction shortfall.” The registration’s letter, the one inside that quotation, and this document’s 𝜒 name the same quantity (registration version 1.102, line 305). The mapping is a symbol mapping inside the stated pacing model and is not permission to identify Law II’s 𝛽𝐶 with either: “WHETHER THE SECOND LAW’S EXPONENT AND THE THIRD LAW’S ARE ONE QUANTITY IS NOT SETTLED HERE AND IS RECORDED AS UNESTABLISHED BELOW, and no proposition assumes they are.” (registration version 1.102, line 271). Law II is “beta_C exceeds k” (registration version 1.102, line 663). Under the burden model the drift exponent implied for a power-law path is 𝑘 = (𝛼 − 1)/𝛼, and the criterion 𝛽𝐶 > 𝑘 rearranges to 𝛼 < 1/(1 − 𝛽𝐶 ) for 𝛼 > 0 and 𝛽𝐶 < 1; this returns the ceiling relation only under the identification 𝛽𝐶 = 𝜒, which the registration records as unestablished (registration version 1.102, lines 271, 309 and 613) and which needs an intervention designed to move one while leaving the other before either stands in for the other (registration version 1.102, line 1029). Under that identification, and only under it, “The ceiling is the co-scaling criterion with the drift exponent fixed by the burden model.” (registration version 1.102, line 637), and Law III remains a conditional quantitative consequence of Law II under its stated identifications, not an independent finding. Laws II and III are each a law under test. 31 The mathematics of Laws IV and V 7.4 The pacing family In the pacing case (𝜅 = 1, 𝜁 = −1) with a direct depth term, 𝛼crit = (1 + 𝜂)/(1 − 𝜒), a positive ceiling for 𝜒 < 1 and 1 + 𝜂 > 0; at 𝜒 = 1 the trend is 1 + 𝜂 independent of 𝛼𝑅 ; at 𝜒 > 1 with 1 + 𝜂 ≥ 0 the trend 𝛼𝑅 ( 𝜒 − 1) + (1 + 𝜂) is positive for every positive 𝛼𝑅 , as the sum of a positive and a non-negative term. At 𝜒 = 1/2 this family gives 2(1 + 𝜂). That value is not the general boundary: 2(1 + 𝜂) is the pacing-case value; the general boundary at 𝜒 = 1/2 is (𝜂 − 𝜁)/(𝜅 − 1/2), which equals 2(1 + 𝜂) for every 𝜂 only when 𝜅 = 1 and 𝜁 = −1; at single points the two can agree elsewhere (𝜅 = 3/2, 𝜁 = −2 and 𝜂 = 0 give two on both sides), a coincidence at one point that does not make 2(1 + 𝜂) the general boundary. So, in the pacing special case, at the same-class cap 𝜒 = 1/2, the boundary is 2(1 + 𝜂), and the ARC Bound, 𝛼 ≤ 2, is its 𝜂 = 0 value. The registration’s four-quantity form (registration version 1.102, line 639: “the boundary is one plus the difference of the two direct depth-dependences, divided by one plus the burden intensity less the correction elasticity”) is the same boundary written with the registered model’s values subtracted: the burden intensity of that sentence is 𝜅 − 1 (the programme’s 𝜇 𝐵 ) and its direct depth-dependence of burden is 𝜁 + 1, not 𝜁, whose name in the programme’s notation, the direct depth-dependence of burden, is the full depth elasticity of 𝑊 at fixed 𝐶; so (1 + 𝜂 − (𝜁 + 1))/(1 + (𝜅 − 1) − 𝜒) is (𝜂 − 𝜁)/(𝜅 − 𝜒) identically, and the registration’s “both direct terms and the burden intensity vanish” is 𝜁 + 1 = 0 and 𝜅 − 1 = 0 with 𝜂 = 0. Its worked numbers hold: 𝜒 = 1/2 with 𝜂 = 1/2 gives three; 𝜒 = 4/5 gives five; both together give seven and a half. The absolute rung (registration version 1.102, line 609) subtracts the correction elasticity from two rather than from one and at the registered dials gives two thirds; its derivation is the registration’s and is not reproduced here. 7.5 Proposition B-5 (varying growth) Proposition B-5. If burden is proportional to d𝐶/d𝑅 along an arbitrary smooth increasing path, then 𝑊 is proportional to 𝛼𝑅 𝐶/𝑅 and not to 𝐶/𝑅, and with correction proportional to 𝐶 𝜒 at constant 𝜒, Δbal = 1 − 𝛼𝑅 (1 − 𝜒) − d ln 𝛼𝑅 /d ln 𝑅, (16) which is registration version 1.102, line 623 in symbols (“the trend of the correction-to-burden ratio is one, less the local elasticity multiplied by the correction shortfall, less the trend of the local elasticity itself”). The last term vanishes throughout a range of depth only where 𝛼𝑅 is constant on that range, that is only for a power law there, where Δbal = 1 − 𝛼(1 − 𝜒); at a single depth it vanishes wherever 𝛼𝑅 is stationary, which a path that is not a power law can reach: ln(𝐶/𝐶0 ) = (ln(𝑅/𝑅0 )) 3 /3 + ln(𝑅/𝑅0 ) has 𝛼𝑅 = 1 + (ln(𝑅/𝑅0 )) 2 , stationary at 𝑅 = 𝑅0 . Otherwise the general chain rule must carry the dependence inside 𝑊 or carry 𝛼𝑅 as an extra state variable. Merely replacing a constant 𝛼 by 𝛼𝑅 in the simplified ceiling loses this term. 7.6 Proposition B-6 (sensitivities, the gap identity, and what an intervention does) Proposition B-6. With the other quantities fixed, d𝛼bal 1 d𝛼bal 𝜂−𝜁 d𝛼bal 𝜂−𝜁 d𝛼bal 1 = ; = ; =− ; =− . (17) d𝜂 𝜅−𝜒 d𝜒 (𝜅 − 𝜒) 2 d𝜅 (𝜅 − 𝜒) 2 d𝜁 𝜅−𝜒 So raising 𝜂 raises the ceiling when 𝜅 > 𝜒, and raising 𝜒 raises it only when 𝜂 − 𝜁 > 0. In the pacing case at fixed 𝜂, the build-order gap is 𝛼crit (first) − 𝛼crit (after) = ( 𝜒first − 𝜒after ) 𝛼crit (first) 𝛼crit (after)/(1 + 𝜂) (18) 32 The mathematics of Laws IV and V in the positive-ceiling domain. In the same pacing case, with 𝜅 = 1 and 𝜁 = −1 in both arms, where a placement contrast moves both terms, in-loop placement raises the boundary if and only if (1 + 𝜂in )/(1 + 𝜂out ) > (1 − 𝜒in )/(1 − 𝜒out ), (19) with 1 − 𝜒 > 0 and 1 + 𝜂 > 0 in both arms. One favourable term alone is insufficient, and an intervention that also raises burden can lower the boundary. Outside the pacing case the comparison is between the two arms’ balance roots, each with its own burden elasticities, and the pacing condition does not carry over: with 𝜅 = 2 and 𝜁 = −1 in both arms, ( 𝜒, 𝜂) = (1/2, 1/5) in the loop and (2/5, 3/10) outside it, the pacing condition holds, yet the in-loop root is 4/5 and the other 13/16, so in-loop placement lowers the boundary. These are algebraic regimes, not safety conclusions: the trend is not the level (registration version 1.102, line 625: “this quantity says whether the ratio is IMPROVING, never whether it EXCEEDS ONE.”), and 𝑄/𝑊 = 𝑅/100 improves throughout 𝑅 ∈ [1, 10] while staying below one. Trend, level, backlog and event stay distinct. Local elasticities throughout make this a local boundary; a boundary written as a function of the architecture (whose letter for the architecture is one of the eight colliding symbols and is not printed; its root reads every elasticity, the burden’s included, in the architecture) is the statement that changing the arm changes the arm’s elasticities and so moves 𝛼bal through these sensitivities; it is written here per arm without an architecture symbol, and where the burden elasticities also differ between arms the comparison is the general one above. The ARC Bound reading. The Bound is the same-class value of the pacing model at 𝜒 = 1/2 and 𝜂 = 0, not a fourth or sixth law. A control stands: a study quoting the simple form declares the corrector’s direct depth-dependence before the run and shows it near zero by an equivalence result with a pre-fixed band whose upper edge is at most half the margin the deciding instrument registers for the profile maximum; a wider band, or a missing demonstration, is NOT EVALUABLE for the Bound reading, never survival. That margin is not yet a number: P3’s tolerance “is the author’s to fix” and until then P3’s supported outcome and this control are NOT EVALUABLE (registration version 1.102, line 843), which Section 10 records. No numerical band is chosen here; P3’s statement, refuter (registration version 1.102, line 851) and scoring are unchanged, and a positive 𝜂 asserted after a run never excuses a P3 exceedance. The Boundary Transfer (UNREGISTERED). A held-out prediction of the balance transition from independently estimated elasticities inherits their precision. The registration reports the growth exponent’s re-estimate as approximately 0.49, one model’s fit, a regression estimate with a standard error of 0.20, and the 95 per cent interval [−1.3, 2.9] as the bootstrap interval on the same row’s endpoint estimate of 0.59 (registration version 1.102, line 207, which also calls the re-estimate robust, a word version 1.104 drops); that interval’s positive part contains 𝛼crit = 2 at the registered dials, and Δbal is positive at either point estimate and negative at 2.9, so no transfer prediction built on it has a decided sign; the non-positive part of the interval lies outside the burden model, whose burden must be positive. A transfer unit therefore carries its own precision gate, fixed before the transition data are opened: the half-width of the predicted boundary’s interval small against the band the boundary-mapping unit registers for the boundary location, the registered standard error of its estimator (registration version 1.102, lines 621 and 633); it is an unregistered study, not law content. 7.7 Proposition B-7 (the capability coordinate) Proposition B-7. Let capability be re-expressed by an increasing differentiable map whose local elasticity, the derivative of the logarithm of the new coordinate in the logarithm of the old, is positive throughout. 33 The mathematics of Laws IV and V Read in the new coordinate, the growth elasticity 𝛼𝑅 is multiplied by that elasticity, 𝜒 and 𝜅 are divided by it, and 𝜂, 𝜁 and Δbal are unchanged; so 𝜅 − 𝜒 keeps its sign, 𝛼bal is multiplied by the same elasticity, the verdict 𝛼𝑅 < 𝛼bal is unchanged, and every contrast of capability elasticities between arms at a common capability (𝜒first − 𝜒after , 𝜒in − 𝜒out ) is divided by it. (Proof: Appendix A.8.) Signs, ceilings against floors and every verdict on the balance are therefore free of the coordinate, and numbers are not: on the squared coordinate, 𝐶 2 , an order contrast of 0.15 is 0.075, below P12’s minimum effect. The registration fixes the numbers on one scale: “Every numerical exponent and every exponent-scale margin in this registration is stated on the scale this condition fixes and on no other” (registration version 1.102, line 403, Condition B), and for the same-class value, “THE BOUND IS FIXED ON ONE CAPABILITY SCALE AND ON NO OTHER” (registration version 1.102, line 935). 𝛼crit = 1/(1 − 𝜒), the ARC Bound’s two, the same-class value one half, P12’s minimum effect 0.10 and the genesis reading’s margin are numbers on that scale. A meaningful zero and an uncapped range, which the registration gives its ladder scale (registration version 1.102, line 259), do not by themselves fix it: every positive power of 𝐶 keeps both and multiplies the growth elasticity by the power, so the justification Condition B requires must also exclude power re-expressions. The registered burden model’s values are themselves values on that scale (on 𝐶 2 , 𝜅 = 1 becomes one half), so re-expressing capability alone and reapplying 1/(1 − 𝜒) shows nothing: with 𝛼 = 9/5 and 𝜒 = 9/20 the bound is 20/11 on 𝐶; on 𝐶 2 , 𝛼 = 18/5, 𝜒 = 9/40 and 𝜅 = 1/2 give 40/11, and the verdict stands, while 1/(1 − 9/40) = 40/31 would reverse it. One inconsistency of the frozen registration bears on Law V and is carried to the successor registration (Section 10, item 15): registration version 1.102, line 403 states every exponent-scale margin on Condition B’s scale, but Condition B’s lists (registration version 1.102, lines 409 and 415) leave P12 out, and Condition E2 (registration version 1.102, line 421) names P12 among the predictions that return “no correction exponent”, although P12’s endpoint is “the quantity the third law’s ceiling is written in” (registration version 1.102, line 1039); P12’s 0.10 rests on Condition B’s scale through registration version 1.102, line 403 alone. 7.8 Proposition B-8 (UNREGISTERED: the balance over a finite range of depth) Identity B-1 gives the trend of the ratio 𝑄/𝑊 at a point. Over a range of depth it gives the change in the ratio’s level. Proposition B-8. On one specified trajectory, with 𝑄 and 𝑊 positive, in comparable units, on one clock and with one conversion fixed throughout, let 0 < 𝑅1 < 𝑅2 lie in the domain and let Δbal be integrable in ln 𝑅 between them. Then ∫ 𝑅2 ln[𝑄(𝑅2 )/𝑊 (𝑅2 )] = ln[𝑄(𝑅1 )/𝑊 (𝑅1 )] + Δbal (𝑟) d ln 𝑟, (20) 𝑅1 where 𝑄(𝑟) and 𝑊 (𝑟) are read along that trajectory and 𝑟 is the variable of integration. (Proof: Appendix A.9.) So the level at the start of a range is part of any prediction of the level at its end, and the trend alone predicts neither. For a constant trend Δbal other than zero, on a trajectory anchored at 𝑅0 , the ratio is 𝑄(𝑅)/𝑊 (𝑅) = [𝑄(𝑅0 )/𝑊 (𝑅0 )] (𝑅/𝑅0 ) Δbal , and it meets one at 𝑅★ = 𝑅0 [𝑊 (𝑅0 )/𝑄(𝑅0 )] 1/Δbal . (21) With a positive trend the crossing lies beyond the anchor exactly when 𝑄(𝑅0 ) < 𝑊 (𝑅0 ): where service already exceeds burden at the anchor the expression returns a depth before it, and where the two 34 The mathematics of Laws IV and V are equal it returns the anchor. With a negative trend the same expression gives the depth at which the ratio falls through one, which lies beyond the anchor exactly when 𝑄(𝑅0 ) > 𝑊 (𝑅0 ). With a zero trend the ratio keeps its starting level throughout. 𝑅★ is a formal crossing: nothing in it shows that the depth lies inside the range measured, or inside any range a system can reach. Nor does a ratio above one bound a backlog, a transient error or a harm; trend, level, backlog and event stay distinct (Section 7.6). A prediction made with this proposition therefore records in advance the levels at the anchor with their uncertainty, the range of depth it covers, the clock and the conversion, and the separately measured outcome it is meant to reach; and none of these is chosen with the held-out observation in view. The coordinate. Proposition B-7 governs any change of the capability coordinate, and the power case shows the pattern: where capability is re-expressed so that its ratio to its reference value becomes (𝐶/𝐶0 ) 𝑝 with 𝑝 > 0, 𝛼𝑅 becomes 𝑝𝛼𝑅 , 𝜒 becomes 𝜒/𝑝 and 𝜅 becomes 𝜅/𝑝, while 𝜂, 𝜁 and Δbal are unchanged, and so is the integral above. To change the coordinate and keep the old number for the burden’s capability elasticity is to change the model (Section 7.7). Every numerical threshold on an exponent, P12’s minimum effect among them, stays on the scale Condition B fixes. Nothing in this proposition replaces Law II’s criterion, that 𝛽𝐶 exceeds 𝑘, or removes an identification on which Law III rests. 7.9 What a new deciding instrument declares (UNREGISTERED) A new instrument that reads correction service against burden declares each of the following before any of its outcomes is read. The list is stated here, UNREGISTERED, as a condition of such an instrument’s completeness; it amends no frozen unit. The claim fixes the measurement, never the reverse. P12 is read on its undirected contrast in the correction-leverage exponent, on the registered axis. P21 is read on its behavioural outcome, the interaction of placement with depth, within its registered access and action-space design. P15 is read on the retained fraction of an externally installed gain after later capability training. A ratio of correction service, a behavioural gain at one fixed budget, or a comparison of retained fractions is none of these three endpoints and is never scored in place of one. A mediation hypothesis is a candidate for a registration of its own, and it is not part of a successor to version 1.102 until it has been reviewed and given an instrument. Version 1.103 registers two directional readings without changing these endpoints: Law V’s build-order limb reads P12’s interval with a direction P12 does not carry, and P21’s directional reading scores no widening of the registered size, or a narrowing, against P21 (Sections 6.1 and 6.6). 8 Dependencies, logical separation and conventions 1. Register dependencies (the statement paper’s dependency table, Eastwood 2026h, in its section 2a): ARC-1’s failure does not refute the persistence registers; ARC-2 depends on ARC-1’s measurables, not its verdict, and its failure takes ARC-3’s derivation; ARC-3’s failure does not erase ARC-2’s own result. Law IV’s objects need no growth law; Law V’s two limbs need none; only Law V’s boundary reading depends on Law III’s model and through it on ARC-2, and it needs the same system and admissible scopes. Version 1.103 (Eastwood 2026g) reads that table’s rows at the width of its registered lines, narrowing the row for ARC-2 and the row for ARC-4 before any data (registration version 1.103, Law III’s word and Law V’s dependence on ARC-4). 2. Logical separation, at the level of verdicts. The registration states it: “The laws are untouched.” (registration version 1.102, line 1087, for P15) and “The three laws are untouched” (registration version 1.102, line 1237, for P21). Version 1.103 reads the untouched laws of P21’s entry as Laws I, II and III, since Laws IV and V are not untouched by failures of their own carriers, and under it 35 The mathematics of Laws IV and V Table 3. What a new deciding instrument declares before any of its outcomes is read (UNREGISTERED). It declares So that this is settled in advance The system’s state, and what may change code, weights, tools, memory, permissions, the state of training and the in it mechanism of update; a loop that runs inference on fixed weights is described as that Capability as measured: its zero, scale the case for a ratio scale, the calibration, any floor or ceiling of the and range measure, and which changes of coordinate are allowed (Section 7.7) Depth, and the clock on which service how rounds are counted, the reference values, the unit of exposure, and and burden are read any conversion between clocks with its uncertainty What is counted as correction capacity available, corrections attempted, corrections delivered and the net change in error, each kept apart; and whether the workload offered limited what could be delivered What is counted as burden the same unit of exposure and the same clock as correction, with any conversion justified before 𝑄/𝑊 is formed The units of observation, and which are systems, training replicates, model families, checkpoints and repeated independent items, with the dependence among them kept in the analysis The intervention and the estimand whether capability is assigned or attained; whether the quantity is a total effect or a contrast at fixed capability (Section 6.4); the range the arms share; and the error of measurement on each axis The model and its primary contrasts an intercept and both slopes for every cell, or the equivalent full set of interactions (Section 6.4), with the order contrast and the placement contrast named The decision rule and its uncertainty the registered endpoint, the smallest effect of interest, the joint interval, and the rules for missing data, for zeros and for stopping The planner, where one is studied its complete objective, horizon, feasible family and permitted early stops, its discounting and selection tolerance, and how coverage and realisation are checked (Section 5.5) 36 The mathematics of Laws IV and V a P15 refutation refutes Law IV through its retention limb (Section 5.8), so the laws these lines call untouched are Laws I to III. Law III’s relation holds at every value of 𝜒, so any P12 verdict, which is a contrast in the very symbol Law III’s equation is written in (registration version 1.102, line 1039: “the quantity the third law’s ceiling is written in”), is consistent with it; separation is independence of truth values, not absence of the symbol. Three listed models make it concrete, each at the registered dials (𝜒 = 1/2, 𝜂 = 0, 𝜅 = 1, 𝜁 = −1) with a growth exponent of 3/2, so that the growth exponent lies below the ceiling of two and the balance exponent is positive, Δbal = 1/4: (a) an interval on 𝑆ext (1) from 0.85 to 0.95, which would refute P15 and so Law IV’s retention limb, and with it Law IV; (b) an interval on the build-order change in the exponent lying inside P12’s open tolerance band, and an interval on the placement-by-depth interaction on the outcome lying inside the tolerance registered with P21’s instrument, each from a design meeting its sensitivity clause, so that both registered refuters would fire and both limbs of Law V would be struck; (c) an interval on 𝑆ext (1) from 0.2 to 0.4 with the intervals of model (b), under which Law IV’s retention limb would be supported, the law staying under test, while both limbs of Law V would be struck. Across the three the verdicts on P12, P15 and P21 vary while the dials, and every statement of Sections 7.1 to 7.4 about them, stay fixed. The dials do not by themselves establish Laws I to III, each a law under test: Law I is a claim about measured growth, and Law II’s criterion compares 𝛽𝐶 with 𝑘, which the registration does not identify with this model’s elasticities (Section 7.3). Model (b) is written on the registered endpoints because P21 decides an interaction on the outcome, “never a measurement of an exponent” (registration version 1.102, line 1221), and its refuter reads that interaction (registration version 1.102, line 1227); read on the service surfaces instead, 𝜂in = 𝜂out would make 𝐺 emb constant in depth along a growing path only together with 𝜒in = 𝜒out (Identity V-1), and its constancy is not what fires P21’s refuter. These are separations of statements, not empirical findings. A countermodel put forward for the case in which Laws I to III hold while Law IV fails takes a collapse, 𝑆𝑉 (𝑥) = exp(−10𝑥), which on the external arm would support P15; it reads the failure of Law IV in the comparative sense of Section 5.6, the opposite direction from model (a), which is one more reason the comparative content was kept apart from P15’s face; version 1.103 now registers it as Law IV’s comparison limb, without a proposition (Section 5.1). Two registered links are recorded rather than left to be found: registration version 1.102, line 663 carries Law II “by P4, with P7 asking what class deployed correction actually belongs to, and P12 and P15 asking whether placement and subsequent training move it”; and registration version 1.102, line 667 holds Law III’s relation only where “the direct depth term is zero”, which the mechanism reading 𝜂𝑜,in > 𝜂𝑜,out denies in at least one placement, so that where that reading holds Law III’s simple form, and the ARC Bound read from it, apply in at most one placement; that link is confined to the boundary reading and to the 𝜂 term of the Bound-reading control, and a P20 verdict is scored in ARC-3 alone. 3. The register convention (set out by version 0.2.1; not in force; version 1.103’s dependence takes its place). Version 0.2.1 set out, as a grouping fixed in advance and not as an entailment, that on an evaluable P15 refutation Law IV be withdrawn whole, and Law V with it as a law of the theory; that Law V’s refutation not withdraw Law IV; and that P12 and P21 keep their own measured verdicts. Two published statements stood behind it, and they are different statements. The registration ties a P15 refutation to “this programme’s case that correction has to sit inside the system rather than be applied to it from outside” (registration version 1.102, line 1085). The statement paper’s dependency table has “ARC-4 falls and takes ARC-5 with it”, and the trigger of that fall in the same paper is the comparative observation of its kill sentence (Section 5.6), not P15. Nothing in 𝜂𝑜,in > 𝜂𝑜,out or in the P12 endpoint is entailed by any value of 𝑆ext (1). Version 1.103 fixes the dependence before any data, as a registered dependence and not an entailment, at the narrower of 37 The mathematics of Laws IV and V the paper’s two widths: Law V is struck only by a refuted verdict on Law IV’s comparison limb, once a later successor has registered that limb’s entry; a refuted P15 alone refutes Law IV and leaves Law V to its own limbs; until the comparison limb is registered no result on ARC-4 strikes Law V; and a Law V struck in this way is printed as refuted, struck by its registered dependence on that comparison, with the unit and the date, while P12 and P21 keep the marks they earned (registration version 1.103, Law V’s dependence on ARC-4). It narrows the paper’s table row and its note of 3 October 2026, “where ARC-4 falls, ARC-5 falls with it”, to that form. 4. No P12, P15 or P21 verdict automatically refutes another numbered proposition. P19’s refutation removes ARC-5’s motivation and refutes nothing in it (registration version 1.102, line 111). P13 stays outside the laws. 5. The Eden Protocol is a method, not a law, and is not entailed by these laws: “It follows from no law here, it is carried jointly by P12, P13, P15 and P21, with P7 as its premise and P19 as its motivation, and it can fail while every law stands or stand while they fall.” (registration version 1.102, line 61); its endogeneity condition is tested by the method’s own registrations and no verdict here turns on it (registration version 1.102, line 675). Its success criterion is a formation criterion, not a later-conduct criterion. Version 1.103 reads the laws of that sentence as Laws I, II and III, since three of the Protocol’s four carriers now also carry Laws IV and V, and the Protocol remains a method, not a consequence of any law (registration version 1.103, its qualification of the method’s separability). The Honey Architecture (Paper VI, Eastwood 2026c) is likewise outside every verdict here. 6. The flagship conjunction P1, P4, P16, P20, P21 stands as registered (registration version 1.102, line 763); a surviving subset is not the conjunction. 7. A persistence advantage and a favourable elasticity contrast are separate: an architecture can preserve a property with no favourable contrast, or improve scaling while losing it; a positive retention comparison alone is not causal mediation (Section 9). 9 The register of unregistered objects Each entry is a proposition or object this document states beyond the registered propositions. None is law content until the author registers it. Version 1.103 (Eastwood 2026g) registers, as the laws’ own words, the build-order limb’s direction with the genesis reading’s strike rule, Law IV’s comparison limb, which names the comparison IV-PERSIST writes and has no proposition yet, and verdict rules for both laws that take the place of the law-level clauses this document set out; Table 4 marks each row they reach. Each entry carries what it needs. “The author’s” is reserved for his own records. Table 4. The register of unregistered objects. Each entry with its status and what it needs; rows version 1.103 reaches are marked. Object Status What it needs 𝐾𝑉 (𝐻; 𝑇, 𝜀 𝑉 ), the persistence UNREGISTERED primary invariance a registered finite battery or a kernel, formalising the invariance object; Lemma IV-K proved probabilistic family with object of the author’s working tolerance; thresholds; horizon; paper of 16 August 2026 (Eastwood the interval rule of Section 5.2 2026k) 38 The mathematics of Laws IV and V Object Status What it needs 𝜋𝑉 , Π𝑉 (𝐻), the removal margins UNREGISTERED premise quantities a registered causal of Theorem IV-G decomposition; an objective scored independently of 𝑉 (premise (G0)) 𝑀𝑉(𝐻 ) , the strategic viability ADOPTED, UNREGISTERED; the planner’s declared margin, read path-wise Theorem IV-H and Lemma IV-M′ objective, feasible family and (Section 5.5) proved tolerance; a registered search protocol with an upper bound on the safe supremum; reported one-sidedly Proposition IV-N, a planner that UNREGISTERED; proved a bound on the score’s error scores its plans with bounded error over the whole feasible family, (Section 5.5.1) fixed before any outcome; the margin computed in the planner’s complete objective the robust margin UNREGISTERED; not printed under a a registered environment colliding symbol family local elasticities 𝜒𝑜𝑙 , 𝜂𝑜𝑙 (the ADOPTED as the author’s own crossed or off-path design for author’s working paper of 16 reading, dated 16 August 2026; identification; the estimate of August 2026; the registration of 8 arm-labelled here registration version 1.102, line September 2026, registration 387 version 1.102, line 267) the balance frontier, the zero set of UNREGISTERED object without a a specified path field 𝛼𝑅 ; a Δbal (the general boundary form of symbol; Law III unchanged; not a precision gate the author’s working paper of 16 restatement of Law III August 2026) Λ𝑉 , the mediation bridge ADOPTED, UNREGISTERED, with its own registration in a scope its own status (below) that carries both contrasts; both retained fractions established (Section 5.6.1) and positive; an identified estimate of each link; an adverse margin; multiplicity IV-PERSIST, 𝑆emb (𝑥) > 𝑆ext (𝑥) the same test as Λ𝑉 on the difference as for Λ𝑉 ; the limb’s entry, (the statement paper’s comparative scale; one object, two readings; which a later successor sentence and kill, Eastwood 2026h, UNREGISTERED. Version 1.103 registers (Section 10, item 1) recorded and not adopted) registers Law IV’s comparison limb, which names this comparison, without a proposition (Section 5.1) the measurement of the retained UNREGISTERED instrument the no-gain arm trained fraction on three arms, and requirements, of which version 1.103 alongside; the floor on each Proposition IV-R (Section 5.6.1) registers the baseline named in advance installed gain; the estimand, and the floor for the retention limb; the interval method and its Proposition IV-R proved for Fieller’s critical value named before any method (Fieller 1954) under the gate outcome the estimable model of Section 6.4 UNREGISTERED model, not a the closed model set; a required analysis of any frozen unit selection rule fixed in advance; registration version 1.102, line 387 39 The mathematics of Laws IV and V Object Status What it needs the candidate retention families and UNREGISTERED reporting; never the closed family set; the the descriptive statistics verdicts author’s budget scale and baseline convention the interval rule for Ω𝑉 UNREGISTERED kill, not law content registration of the battery, membership (Section 5.2) and the tolerance and interval method battery-relative kernel kill (K-IV item 3) the law-level clauses of K-IV superseded by version 1.103’s verdict nothing for the clauses version items 2 to 4 and K-V items 1 to 3, rules for Laws IV and V: K-IV item 2, 1.103 replaced; the author’s and the register convention of K-V item 3 and the register convention registration for K-IV items 3 Section 8, item 3, beyond the stay on the record, not in force; K-V and 4 registered refuters item 1’s strike on the adverse side gives way to P21’s registered directional reading; K-IV items 3 and 4 stay UNREGISTERED, item 4 beside version 1.103’s own list; P12, P15 and P21 scored as registered Proposition B-8, the balance over a UNREGISTERED; proved from the levels at the anchor with finite range of depth, and the formal Identity B-1 their uncertainty; the range, the crossing (Section 7.8) clock and the conversion declared in advance what a new deciding instrument UNREGISTERED condition of a new adoption in each new declares (Section 7.9) instrument’s completeness; amends no instrument’s own registration frozen unit the recursive-improvement UNREGISTERED prerequisite in the author’s registration before prerequisite (Section 6.2) words; both mutual-information forms any surface states it as a declined as criteria condition of any law; until then a description, and not a law the genesis reading, REGISTERED by version 1.103 as nothing further; P12 keeps no 𝜒first − 𝜒after ≥ 0.10 (formation first Law V’s build-order limb, read on direction carries the higher elasticity by at P12’s interval beside P12’s mark, with least P12’s registered minimum the strike rule of K-V item 2 as its rule; effect) stated at P12’s registered minimum, so that it implies P12 and adds only a sign the Boundary Transfer UNREGISTERED study its own preregistration; a precision gate planned removal (the UNREGISTERED study; the its own preregistration; a planned-removal challenge) observation that decides IV-H’s margin registered upper bound on the premise (Lemma IV-M′ ) safe supremum a restatement of Laws I to III under DECLINED: not adopted; Laws I to III the author’s decision, and new names stand as registered; the glosses growth, never in 1.102 balance, boundary, persistence, architecture are descriptions, not names a mutual-information positivity as DECLINED as a criterion, for the nothing an improvement criterion, reason of Section 6.2; the prerequisite state-based or update-based is kept in words 40 The mathematics of Laws IV and V Object Status What it needs the half-life as P15’s verdict; an DECLINED (Section 5.6) nothing interval capped at one, the kill written as the interval inside [1/2, 1] Ruling on the Λ𝑉 mediation bridge. Adopted, UNREGISTERED, with this status line: “UN- REGISTERED mediation contrast; not law content; Λ𝑉 := ln(𝑆emb /𝑆ext ) at the evaluation budget, defined only where both retentions are positive; the primary contrast is the difference 𝑆emb (𝑥) − 𝑆ext (𝑥), which is defined wherever both retained fractions are established, whatever their signs, with Λ𝑉 reported beside it where defined; neither is defined without the embedded arm, and both need the measurement of Section 5.6.1; instrument: a unit registered in a scope that carries both the retention contrast and the elasticity contrasts, since P15’s unit is governed by S5 (registration version 1.102, line 237) and P12 and P21 by S7 (registration version 1.102, line 239), and no proposition may borrow a scope (registration version 1.102, lines 241 and 247), so a retention contrast read on the registered two-arm unit (registration version 1.102, line 1089) alone is an S5 quantity that bears on no ARC-5 proposition; a positive value is not mediation: the mediating sequence stated here, embedding to the elasticity contrasts to Δbal to a persistence advantage, has three links, each needing its own identified estimate, and each link can fail; the adverse margin, multiplicity and registration are the author’s.” Reasons. It is adopted rather than declined on its merits: it is the one object in the set that would connect the two registers once one system is registered in a scope that carries both contrasts. It is stated on the difference scale first because the log form is undefined on a region the data can reach (where one retention is negative and the other positive, or where either is zero), and because where both retentions are negative the log form is defined but its sign is the reverse of the difference’s. It is one object with IV-PERSIST, so that one test does not become two propositions. Nothing states that the loop is closed. Version 1.103 registers Law IV’s comparison limb, which names the comparison this contrast reads, without a proposition; its entry, when registered, fixes the margin and the scale, and the difference and Λ𝑉 are two scales it may take (Section 5.1). Two things follow for the successor registrations. A result on the difference and a result on Λ𝑉 are one result, not two corroborations. And neither entered version 1.103, which names Law IV’s comparison limb and leaves its entry to a later successor: Λ𝑉 , its margins and any instrument for it are candidates for that entry or for a registration of their own (Section 10, item 9). 10 Open questions There are fifteen questions in all, numbered as version 1.1 numbered them. Where an item names a default, that default stands until the author says otherwise, and none is chosen after a result. Version 1.103 (Eastwood 2026g), registered on 15 October 2026, has since settled items 3, 4, 5 and 10, and items 1, 2, 13, 14 and 15 in part; each such item says what version 1.103 fixed and what stays open, and its version 1.1 wording is kept in Appendix B.2. The other items stay open as written. 1. Law IV’s content. Settled in part by version 1.103: Law IV has two limbs, the retention limb, P15’s registered claim word for word, and the comparison limb, which has no registered proposition yet (Section 5.1). Its entry, with its operational definition of correction embedded in the training objective, its baseline, its margin, its scale and whether the embedded arm must also clear one half, is for a later successor. The loss mechanism, active removal against dilution, stays unresolved. 2. What a P15 refutation takes from Law IV. Settled in part by version 1.103: a refuted retention limb 41 The mathematics of Laws IV and V refutes Law IV, read against a baseline arm named in advance, above a floor fixed in advance and under the attestation, and the law is then printed refuted with the proposition, the unit, the date and whose run it was (Section 5.8). Of the four controls version 0.2.1 set out, the capability rise established on a separate instrument is not among version 1.103’s conditions and stays open. 3. The register convention. Settled by version 1.103: Law V is struck through Law IV only by a refuted verdict on Law IV’s comparison limb, once that limb’s entry is registered, and a refuted P15 alone leaves Law V to its own limbs (Section 8, item 3). 4. Whether Law V is severable by limb, a strict conjunction, or stands on the placement limb alone. Settled by version 1.103: Law V is refuted when either limb’s own verdict is refuted, or when its dependence strikes it, and supported only when both limbs are supported (Section 6.6). 5. The axis P21’s result is read on. Settled by version 1.103, which reads the placement limb on P21’s own verdict, the registered access interaction and the default here, and prints P21’s mechanism axis beside it (Section 6.1); the service-scaling mechanism reading of registration version 1.102, line 1239 would need a separately specified bridge and design, never a rescoring. 6. Whether the build-order limb is scored with class left free, as P12 registers (default); holding it fixed would be a changed contrast. 7. The budget scale on which P15’s matched budget is read (registration version 1.102, line 1089), and the at-the-margin band of registration version 1.102, line 391 in the same deciding instrument; until fixed, the threshold is NOT EVALUABLE on that scale. 8. The baseline convention for the retained fraction (the concurrent baseline, the default, which needs a third arm, a system without the installed gain that undergoes the same later training; or the pre-training baseline, computable on the registered two arms; Section 5.6) and the value of 𝑌min . The floor’s form (a lower confidence bound; the interval method) is a decidability requirement this document fixes, not his choice. Version 1.103 requires that the deciding unit name the baseline arm and fix the floor in its own registration before any outcome; which baseline, and the floor’s value, stay the author’s. 9. Whether Λ𝑉 is registered as a mediation proposition, in which scope, with what adverse margin and multiplicity, and whether a unit is registered that carries both contrasts (Section 9). It is not part of version 1.103, which names Law IV’s comparison limb and leaves its entry to a later successor; it takes a proposition number of its own only if its claim and its refuter differ from those already registered. 10. The genesis reading of the build-order limb, equation (8). Settled by version 1.103, which registers the limb with that direction, read on P12’s interval and printed beside P12’s mark, with the strike rule of K-V item 2 as its rule; P12 keeps no direction (Section 6.1). 11. P3’s tolerance (registration version 1.102, line 843), on which the Bound-reading control of Section 7.6 depends; until fixed, that control is NOT EVALUABLE. 12. Whether the kernel’s family, if he registers the kernel, is a finite registered battery or a probabilistic family with a tolerance; the per-unit numbers (𝑣 ∗ , 𝜀 𝑉 , 𝐻) are fixed in each unit’s registration, not in this document. 13. The dating. Settled in part by version 1.103, which dates the question each added law formalises from the author’s earliest record, as the idea and not as the law: Law V’s from his book draft of 8 December 2024 and Law IV’s from his manuscript of 30 April 2025, both public on OSF since 5 October 2026 (DOI 10.17605/OSF.IO/QASRK; Eastwood 2026a); the statements, kill conditions and verdict rules are work of 2026 (Appendix C). The confirmation of the dates this document draws from his working paper of 16 August 2026 (Eastwood 2026k) stays open. 42 The mathematics of Laws IV and V 14. The scoring-gate extensions for P12 and any scorer-dependent P15 reading, and whether each law carries a stated confidence (no law-level confidence is stated here). Version 1.103 holds both laws’ limb readings at untested wherever a judged term enters them, until the scoring gate passes, and extends the counted route to P21 and P17 alone; the extensions for P12 and P15 stay open. 15. The successor registration. Settled in part: version 1.103 was registered on 15 October 2026 and version 1.104 on 29 October 2026, each keeping every word, date, scope and refuter of P1 to P22 and listing its changes. Version 1.103 reads Law V’s build-order limb on the capability scale condition B fixes, but neither version changes P12’s listing under conditions B and E2, which registration version 1.102, line 403 implies and lines 409, 415 and 421 omit (Section 7.7), so that correction stays for a later successor. Not his, and fixed in this document: the planner’s objective form for a studied system is a fact the unit declares and verifies, and for a designed planner a design parameter of that unregistered unit; the rule (compute 𝑀𝑉(𝐻 ) in the declared objective; read search one-sidedly) is this document’s. The retention families and the Law V model set are drafted in this document and approved in each unit’s registration. The names are settled: he chose them on 26 September 2026, and his statement of the laws uses them; version 1.103 registered them on 15 October 2026. The identification of 𝛽𝐶 with 𝜒 is not settled by anyone’s word: the registration makes it an empirical question that needs an intervention (registration version 1.102, line 1029). 11 What this paper does and does not show Status. Laws IV and V were added on 24 September 2026 and registered as laws on 15 October 2026 by version 1.103 of the theory registration (Eastwood 2026g), each with its statement, its kill condition and its verdict rule fixed before any data that could decide it. Law IV carries P15, in register ARC-4, with a comparison limb that has no registered proposition yet; Law V carries P12 (undirected) and P21, in register ARC-5, and reads its build-order limb on P12’s interval with a direction P12 does not carry; the propositions were registered on 8 September 2026 (Section 3.2). Each law is a law under test. Law IV is not yet tested. Law V is a law under test: its placement limb, P21, is supported on the performance axis only, under the registered test, by the programme’s own run (Unit 83, 28 October 2026, DOI 10.17605/OSF.IO/MN73W, Eastwood 2026j; results: DOI 10.17605/OSF.IO/98HKR): family-dependent, with its mechanism axis not evaluable, which is not support for the information-asymmetry mechanism, and one disclosed recovery amendment; its build-order limb, read on P12, is undecided, since P12 is untested. The programme’s first registered study, Unit 06, a classification audit that tests no law, was registered on 1 October 2026 (DOI 10.17605/OSF.IO/RV2B5; Eastwood 2026i) and ran on 6 October 2026; it is reported in full as administratively incomplete, with no verdict claimed (results: DOI 10.17605/OSF.IO/4BPHG). The mathematics of Laws IV and V is set out here at version 1.2, which restates the sentences version 1.103 governs and changes no theorem: its theorems are proved under the assumptions each states, its identities are checked, and Section 10 lists fifteen questions, marking those version 1.103 has settled. Theorems IV-G and IV-H, Corollaries IV-G′ and IV-H′ , Lemmas IV-K and IV-M′ and Propositions IV-N and IV-R are logical proofs, not identities; no proof assistant is used, and they rest on the written proofs, which finite models exercise and do not replace. What kind of document this is. This document is not a registration, not an empirical result, and not authority to run any study. Registration versions 1.100, 1.102, 1.103 and 1.104 stay frozen byte for byte; Laws I to III stand exactly as registered (registration version 1.102, lines 659, 663 and 665); no proposition is added to or moved between registers; P12 keeps no direction; and every object this document introduces 43 The mathematics of Laws IV and V beyond the registered propositions and the laws’ registered words is marked UNREGISTERED and is not law content until the author registers it. Where version 1.103 registered an object this document had stated, the object is marked as registered by version 1.103 where it stands (Section 9). What this document is. The mathematics of Law IV, the ARC Persistence Law (register ARC-4, carrying P15), and Law V, the ARC Embedding Law (register ARC-5, carrying P12 and P21): the objects, the conditional theorems with proofs, the identities that connect the two laws to the registered balance model, the registered faces quoted byte for byte, and the kill conditions. The register assignment is that of registration version 1.102, line 99 (“ARC-4, VALUE PERSISTENCE: P15 alone.” and “ARC-5, THE GENESIS STRATEGY: P12 and P21.”); the seven propositions in no register are those of registration version 1.102, line 101 (P5, P7, P10, P11, P13, P14, P19), and they stay there. What it is not. It proves no empirical law: algebra proves consequences of assumptions, not that real systems satisfy them. It runs no test. It adds no registered proposition and moves none. It does not restate Laws I to III (Section 9). It supplies no missing measurement, chooses no scientific parameter the author has held (Section 10), and claims no novelty; Section 2 names the nearest prior work found, and the distinction from prior art is not assessed in full. Safety and usefulness. No safety measure is stated here, so none is paired with a usefulness measure: Law IV says what persists, not that what persists is good, and Sections 5.7 and 6.7 say what the mathematics does not show. 12 Limitations The limits below are the report’s own, each stated where it arises; they are gathered here so that none is missed. • Conditional mathematics. Every theorem, lemma and proposition holds under the premises it states, and the report shows for no real system that those premises hold (Section 11). Nothing in Sections 5 to 7 has been measured. • Proofs and checks. The proofs are written proofs. No proof assistant is used; finite models exercise the proofs and do not replace them, and the computer-algebra checks of the identities and the second check of the working draft are the programme’s own, not independent verification (Appendix B). • The margins are defined, not computed. The strategic viability margin is a supremum over a feasible family that cannot be enumerated for a real system, and it presupposes a known objective; where the objective is unknown it is NOT EVALUABLE, and no incomplete search confirms a positive margin (Section 5.5). • Identification. The capability elasticity of correction service is the registered correction-leverage exponent only under the conditions of Section 4; the local elasticities are identified only where capability is assigned or a no-confounding premise is stated in advance (Section 6.4); and the identification of Law II’s 𝛽𝐶 with 𝜒 is unestablished (Section 7.3). • Scale. Exponent numbers, P12’s minimum effect among them, hold only on the capability scale condition B fixes; signs and verdicts on the balance do not depend on the coordinate, and numbers do (Section 7.7). • Prior work. The report claims no novelty, and Section 2 names only the nearest prior work found; the distinction from prior art is not assessed in full. 44 The mathematics of Laws IV and V • Open questions. Section 10 records fifteen questions; those still open include the budget scale and the at-the-margin band for P15, the baseline convention and the floor’s value, P3’s tolerance, and the entry of Law IV’s comparison limb, until which Law IV cannot be supported. 13 Conclusion Laws IV and V are registered laws under test, and this report gives them their mathematics. Law IV’s retention limb is read on one quantity, the retained fraction of an externally installed gain at the matched budget, and the report states what that fraction needs before it can decide P15. Persistence of a property over a finite horizon follows, under stated premises, from greedy acceptance with a margin or from planned optimisation with a strategic viability margin, and each premise names the observation that would refute it. Law V’s limbs are read on P12’s interval, with the direction version 1.103 (Eastwood 2026g) registers, and on P21’s interaction of placement with depth; the balance identity shows how a change in either limb’s elasticities would move the boundary inside Law III’s model, which stays unchanged. None of this is a measurement. The laws take their words from their registered tests alone: P15’s retained fraction, P12’s order contrast and P21’s interaction, each read as the registration says, and the entry of Law IV’s comparison limb once a later successor registers it. Acknowledgements Michael Darius Eastwood conceived and directs this research programme and is the author of this work. Across the programme, he has used more than six AI systems in parallel, under his own instructions, to stress-test his arguments, identify possible errors, and assist in preparing draft text from his own outlines. He determines what is adopted, revised or rejected and takes responsibility for the published content. These systems are tools, not authors. © 2026 Michael Darius Eastwood. Full human authorship and moral rights are asserted under the Copyright, Designs and Patents Act 1988. This work is licensed under the Creative Commons Attribution 4.0 International licence (CC BY 4.0). 45 The mathematics of Laws IV and V References Jean-Pierre Aubin. Viability Theory. Birkhäuser Boston, 2009. doi:10.1007/978-0-8176-4910-4. Modern Birkhäuser Classics; originally published in the series Systems & Control: Foundations & Applications. Franco Blanchini. Set invariance in control. Automatica, 35(11), 1999. doi:10.1016/S0005-1098(99)001 13-2. Pages 1747 to 1767. Michael Darius Eastwood. Priority record manuscripts: 8 December 2024 and 30 April 2025. OSF, 2026a. Deposited on OSF on 5 October 2026: the author’s self-emailed manuscripts of 8 December 2024 and 30 April 2025, each dated by its mail-server Date line, and since 15 October 2026 the message headers of 8 December 2024. SHA-256 (.eml of 8 December 2024): f0d1f38ffd8546152d9d9d28dc5e c083c16a35858f2c12b63e69db7ed50901ad. doi:10.17605/OSF.IO/QASRK. Michael Darius Eastwood. Register · The ARC Theory · Operational Definitions. OSF, 2026b. doi:10.17605/OSF.IO/GVQ82. Michael Darius Eastwood. Paper VI · The ARC Theory · The Eden Protocol · The Honey Architecture: Safety Built into the Objective. OSF, 2026c. doi:10.17605/OSF.IO/8EZ2N. Michael Darius Eastwood. Paper VIII · The ARC Theory · Law V, the ARC Embedding Law · The Eden Protocol · The Load-Bearing Test: Whether Safety and Capability Are Structurally Entangled. OSF, 2026d. doi:10.17605/OSF.IO/7YJ4E. Michael Darius Eastwood. Paper X · The ARC Theory · Law II, the ARC Co-Scaling Law · Correction That Out-Scales Drift. OSF, 2026e. doi:10.17605/OSF.IO/BSE2Q. Michael Darius Eastwood. Register · Recursive Dynamics · The Priority Index: The Earliest Dated Record of Each Major Claim. OSF, 2026f. Working draft, version 2, 26 October 2026, held in the OSF component titled “Priority and Provenance · ARC/Eden (SHA-256 Timestamped Record)”. doi:10.17605/OSF.IO/H3R95. Michael Darius Eastwood. The ARC Theory · Theory-Level Predictions for Recursive Self-Improvement and AI Alignment: Twenty-Two Severable Propositions Registered Before Any Admissible Confirmatory Outcome. OSF, 2026g. Registered 8 September 2026 (version 1.100), updated 13 September 2026 (version 1.102), 15 October 2026 (version 1.103) and 29 October 2026 (version 1.104); the quotations in this report are from version 1.102, except where a passage names version 1.103 or 1.104. doi:10.17605/OSF.IO/P8CKQ. Michael Darius Eastwood. Statement Paper · The ARC Theory · Five Laws of Recursive Growth, Correction and Persistence. OSF, 2026h. Quoted from its version 6.7 of 21 September 2026, the quoted sentences unchanged in its version 7.0 of 3 October 2026. doi:10.17605/OSF.IO/GW5MX. Michael Darius Eastwood. What Class of Corrector Is Deployed AI Alignment Actually Using? A Preregistered Classification Audit Against a Four-Cell Taxonomy. OSF registration, 2026i. Unit 06, the corrector-class audit; registered 1 October 2026, public from 4 October 2026. doi:10.17605/OSF.IO/ RV2B5. Michael Darius Eastwood. Testing Law V (the placement limb, P21): does correction inside each revision round pull away, as depth rises, from a monitor that sees only the finished register? A counted test. OSF registration, 2026j. Unit 83, the counted test of P21; registered 22 October 2026, public from 25 October 2026. doi:10.17605/OSF.IO/MN73W. 46 The mathematics of Laws IV and V Michael Darius Eastwood. Author-review working paper, 2026k. Unpublished working paper of 16 August 2026, which the theory registration names and credits (version 1.102, lines 305, 307, 605, 619 and 621). No official online source. E. C. Fieller. Some problems in interval estimation. Journal of the Royal Statistical Society: Series B (Methodological), 16(2), 1954. doi:10.1111/j.2517-6161.1954.tb00159.x. Pages 175 to 185. Evan Hubinger, Carson Denison, Jesse Mu, et al. Sleeper agents: Training deceptive LLMs that persist through safety training, 2024. URL https://arxiv.org/abs/2401.05566. arXiv:2401.05566. James Kirkpatrick, Razvan Pascanu, Neil Rabinowitz, et al. Overcoming catastrophic forgetting in neural networks. Proceedings of the National Academy of Sciences, 114(13), 2017. doi:10.1073/pnas.16118 35114. Pages 3521 to 3526. Tomasz Korbak, Kejian Shi, Angelica Chen, et al. Pretraining language models with human preferences. In Proceedings of the 40th International Conference on Machine Learning, volume 202 of Proceedings of Machine Learning Research, 2023. URL https://arxiv.org/abs/2302.08582. Pages 17506 to 17533; arXiv:2302.08582. Hunter Lightman, Vineet Kosaraju, Yura Burda, et al. Let’s verify step by step, 2023. URL https: //arxiv.org/abs/2305.20050. arXiv:2305.20050. Pratyush Maini, Sachin Goyal, Dylan Sam, et al. Safety pretraining: Toward the next generation of safe AI, 2025. URL https://arxiv.org/abs/2504.16980. arXiv:2504.16980. Michael McCloskey and Neal J. Cohen. Catastrophic interference in connectionist networks: The sequential learning problem. In Psychology of Learning and Motivation. Elsevier, 1989. doi:10.1016/ S0079-7421(08)60536-8. Pages 109 to 165. Xiangyu Qi, Yi Zeng, Tinghao Xie, Pin-Yu Chen, Ruoxi Jia, Prateek Mittal, and Peter Henderson. Fine-tuning aligned language models compromises safety, even when users do not intend to!, 2023. URL https://arxiv.org/abs/2310.03693. arXiv:2310.03693. Megan Ung, Alicia Sun, Samuel J. Bell, Bhaktipriya Radharapu, Levent Sagun, and Adina Willi- ams. Chained tuning leads to biased forgetting, 2024. URL https://arxiv.org/abs/2412.16469. arXiv:2412.16469. 47 The mathematics of Laws IV and V A Proofs The proofs of Lemma IV-K, Theorems IV-G and IV-H, Corollary IV-H′ and Propositions IV-N, IV-R, V-3, B-7 and B-8, moved here from Sections 5, 6 and 7, where each is stated. A.1 Proof of Lemma IV-K Proof of Lemma IV-K. (K1): 𝑇𝐻 (𝑠0 ) is a subset of 𝑇𝐻+1 (𝑠0 ), so if every member of the larger set is safe, every member of the smaller is. (K2): 𝑇𝐻 (𝑠0 ) under 𝑇 is a subset of 𝑇𝐻 (𝑠0 ) under 𝑇 ′ . (K3): enlarging the tolerance enlarges Ω𝑉 , so a path safe under the smaller tolerance is safe under the larger. Each is an inclusion of quantifier ranges and needs nothing about the system. □ A.2 Proof of Theorem IV-G Proof of Theorem IV-G. Suppose a path of executed transitions from 𝑠0 has a first index 𝑡, with 1 ≤ 𝑡 ≤ 𝐻, at which 𝑠𝑡 lies outside Ω𝑉 . Then 𝑠𝑡 −1 lies in Ω𝑉 and is reachable within 𝑡 − 1 < 𝐻 accepted transitions. By (G3) the executed transition 𝑠𝑡 −1 to 𝑠𝑡 lies in 𝑇eval (𝑠𝑡 −1 ), so by (G2) 𝐽 (𝑠𝑡 ) − 𝐽 (𝑠𝑡 −1 ) < 0. But the transition was executed, so by (G1) 𝐽 (𝑠𝑡 ) − 𝐽 (𝑠𝑡 −1 ) ≥ 0. Contradiction; no such 𝑡 exists. Since 𝑡 ranges over a finite set, no further condition is needed. □ A.3 Proof of Theorem IV-H Proof of Theorem IV-H. By (H2), 𝐽safe ∗ − 𝐽∗ > 𝜀 exit opt ≥ 0, so 𝐽safe > 𝐽exit and the supremum over all of ∗ ∗ ∗ . By (H3), 𝐽 (𝜌 ∗ ) ≥ 𝐽 ∗ − 𝜀 , and (𝐽 ∗ − 𝜀 ) − 𝐽 ∗ = 𝑀 (𝐻 ) (𝑠 ) − 𝜀 𝑇plan,𝐻 (𝑠0 ) is 𝐽safe 𝐻 safe opt safe opt exit 𝑉 0 opt > 0. Every exit plan scores at most 𝐽exit ∗ , so 𝜌 ∗ is not an exit plan. Neither supremum need be attained; an exact maximiser, where it exists, is the case 𝜀opt = 0. □ A.4 Proof of Corollary IV-H′ Proof of Corollary IV-H′ . By induction on 𝑡: 𝑠0 lies in Ω𝑉 ; if 𝑠𝑡 does, the plan executed from 𝑠𝑡 is safe, by IV-H where (H1) to (H3) hold and because every feasible plan is safe where the exit part is empty; if it stops at 𝑠𝑡 the executed path ends there, in Ω𝑉 , and otherwise, by (iii), 𝑠𝑡+1 is a state of a safe plan and lies in Ω𝑉 . □ A.5 Proof of Proposition IV-N Proof of Proposition IV-N. By (N1) the score of every safe plan is at least its 𝐽 𝐻 less 𝜀eval , so the supremum of 𝐽ˆ𝐻 over the safe part is at least 𝐽safe ∗ − 𝜀eval ; and the score of every exit plan is at most 𝐽exit + 𝜀eval . The supremum of 𝐽ˆ𝐻 over the whole family is therefore at least 𝐽safe ∗ ∗ −𝜀 eval , and it is finite, because by (N3) 𝐽safe exceeds 𝐽exit and so no score exceeds 𝐽safe + 𝜀eval . By (N3) again, ∗ ∗ ∗ (𝐻 ) eval ) − (𝐽exit + 𝜀 eval ) = 𝑀𝑉 (𝑠0 ) − 2𝜀 eval > 𝜀 opt . Every exit plan therefore scores more than 𝜀 opt ∗ −𝜀 (𝐽safe ∗ below the supremum over the family and fails (N2), so 𝜌 ∗ is not an exit plan. Neither supremum need be attained. With 𝜀eval = 0 the proposition is Theorem IV-H. □ A.6 Proof of Proposition IV-R Proof of Proposition IV-R. Write quad(𝑚) := (𝑌ˆ1 − 𝑚𝑌ˆ0 ) 2 − 𝑧 2crit (var1 − 2𝑚 cov10 + 𝑚 2 var0 ), (A.1) 48 The mathematics of Laws IV and V so that Fieller’s set is where quad(𝑚) ≤ 0. It is a quadratic in 𝑚 with leading coefficient 𝑌ˆ02 − 𝑧2crit var0 , which the gate makes positive, so the set is a closed bounded interval or is empty. At 𝑚 = 𝑌ˆ1 /𝑌ˆ0 the squared term vanishes and quad equals minus 𝑧 2crit times an estimated variance, which is not negative; so quad is not positive there, and the set is an interval containing 𝑌ˆ1 /𝑌ˆ0 . If the interval lies entirely above 𝑚 thr , then 𝑚 thr is outside it, so quad(𝑚 thr ) > 0, and 𝑌ˆ1 /𝑌ˆ0 , a point of the interval, exceeds 𝑚 thr . Conversely, if quad(𝑚 thr ) > 0 and 𝑌ˆ1 /𝑌ˆ0 > 𝑚 thr , then 𝑚 thr is outside the interval and below one of its points, so it is below all of them. Now quad(𝑚 thr ) > 0 says that the square of 𝑌ˆ1 − 𝑚 thr𝑌ˆ0 exceeds the square of the right-hand side displayed, and, because 𝑌ˆ0 > 0, 𝑌ˆ1 /𝑌ˆ0 > 𝑚 thr says that 𝑌ˆ1 − 𝑚 thr𝑌ˆ0 is positive; the two together are the displayed inequality, (7). □ A.7 Proof of Proposition V-3 Proof of Proposition V-3. ln 𝑄 𝑜𝑙 − 𝜒𝑜𝑙 ln 𝐶 − 𝜂𝑜𝑙 ln 𝑅 has zero partials, so it is constant on the connected rectangle; the constant is fixed by the reference point, which must lie in the rectangle; the partials of the closed form return the constants. □ A.8 Proof of Proposition B-7 Proof of Proposition B-7. 𝜒 in the new coordinate is the derivative of ln 𝑄 in the new log-capability at fixed 𝑅, which is 𝜒 times the derivative of the old log-capability in the new, the reciprocal of the map’s elasticity, and likewise 𝜅; 𝜂 and 𝜁 are taken at fixed capability, which the new coordinate fixes exactly when the old one does; 𝛼𝑅 is multiplied by the elasticity by the chain rule; and Δbal = d ln(𝑄/𝑊)/d ln 𝑅 along the path names no capability coordinate. □ A.9 Proof of Proposition B-8 Proof of Proposition B-8. By Identity B-1, Δbal is the derivative of ln(𝑄/𝑊) in ln 𝑅 along the trajectory; integrate it from 𝑅1 to 𝑅2 . □ B Version history and corrections This appendix keeps the report’s history whole. Its first part is the entry for this version. Its second keeps, in their version 1.1 wording, the sentences this version restates because version 1.103 governs how the laws take their words (Eastwood 2026g). Its third holds the records of version 1.1 as printed: what that version added to version 0.2.1, its checks and its list of changes. The records keep the words of the version they record, its section numbers (Table 5 maps them to this edition’s) and its status label PROPOSED, which this version writes UNREGISTERED. B.1 Version 1.2, 29 October 2026 Version 1.2 is the report’s next version. It follows version 1.1, frozen on 7 October 2026 and published on 12 October 2026, which stays published as revision 3 of the same file on OSF. Why it was made. Version 1.103 of the theory registration, registered on 15 October 2026, registers Laws IV and V as laws and says of this report: “Where this text and that report differ, this text governs how the laws take their words, and the report’s next version restates the sentences concerned.” (registration version 1.103, the paragraph on the mathematics and where the successor departs from its report). Version 1.2 is that next version. 49 The mathematics of Laws IV and V Table 5. Where each section of version 1.1 stands in this edition. Version 1.1 This edition front matter the title block and its footnote; Section 1 (the three levels); Section 11 (what kind of document this is); Appendix B (what version 1.1 added) 0, sources and status Section 3.1 (sources) and Section 11 (status, what it is and is not) 0.1 Section 3.2 1 Section 4 and Appendix D 2, 2.1 to 2.8 (2.5.1, 2.6.1) Section 5, Sections 5.1 to 5.8 in the same order 3, 3.1 to 3.7 Section 6, Sections 6.1 to 6.7 in the same order 4, 4.1 to 4.9 Section 7, Sections 7.1 to 7.9 in the same order 5 Section 8 6 Section 9 7 Section 10 8 Appendix C 9 and the changes in version 1.1 Appendix B What changed. 1. The laws’ standing. Version 1.1 said that neither law was yet registered as a law; that was true on its date and is not now. The laws are stated throughout as version 1.103 registers them: added on 24 September 2026, registered as laws on 15 October 2026, each a law under test, with the status of the first registered test of one of their propositions (Section 11), which also names, once, the programme’s first registered study, Unit 06, which tests no law. 2. The three places version 1.103 departs from version 1.1, each restated in the registered form, with the version 1.1 wording kept in Appendix B.2. Law IV has two limbs, the retention limb, P15’s registered claim word for word, and a comparison limb with no registered proposition yet (Sections 5.1 and 5.8). Law V’s build-order limb has a direction, the genesis reading of version 1.1, read on P12’s interval while P12 keeps no direction (Sections 6.1 and 6.6). And Law V’s dependence on Law IV is narrower: it is struck only by a refuted verdict on Law IV’s comparison limb, so that a refuted P15 alone leaves Law V to its own limbs (Section 8, item 3). With them the laws’ verdict rules, the open questions version 1.103 settled and the register of unregistered objects are restated (Sections 5.8, 6.6, 9 and 10). 3. Version 1.104, approved on 29 October 2026. The regression re-estimate of approximately 0.49 is no longer called robust, since it is one model’s fit (Section 7.6), and version 1.104’s readings of P15’s decision requirement are quoted where P15’s kill condition stands (Section 5.8). 4. The status label. The label PROPOSED, which versions 0.2.1 to 1.1 gave the objects this document states beyond the registration, is written UNREGISTERED, and the conventions of earlier versions are described as set out; the meaning is unchanged: none is law content until the author registers it. 5. The first-publication label. Version 1.1 printed “First published 12 October 2026”, the date on which version 1.1 was published. The report’s earliest public version, working draft 0.2.1, is dated 1 October 2026 inside the document, and the report was first deposited on OSF on 3 October 2026, as working draft 0.2.4 (Table 6 and Appendix C). 6. The register sentence. The statement paper’s register sentence for ARC-4 (Eastwood 2026h) is labelled as the register’s, and the laws’ statements printed are version 1.103’s (Section 5.6). 7. Links. Every mention of the registration, the statement paper and the programme’s papers links its OSF DOI; the programme’s notation links its register of operational definitions; the author’s working paper of 16 August 2026 has no official online source, and says so. 50 The mathematics of Laws IV and V 8. The presentation standard. The report is set in LaTeX on the programme’s template, with a title block, an abstract, a contents page, an introduction with its contributions, a related-work section whose works were each read at their official sources on 29 October 2026, a section on what the report does and does not show, its limitations and a conclusion. The proofs move to Appendix A, the symbol list to Appendix D, the dating to Appendix C and the records of version 1.1 to this appendix; Table 5 maps the section numbers. One figure draws the example retention curve of Section 5.6. The author block takes the template’s form. 9. The source of two quotations. Version 1.1 quoted the one-line statement of each law from the author’s statement of the laws, which no programme paper prints; this version gives those lines in its own words (Sections 5.1 and 6.1), and Appendix B.2 keeps version 1.1’s sentences. What did not change. No definition, lemma, theorem, corollary, proposition, assumption, proof, identity, equation, number, interval rule or quotation of version 1.102 was altered, and no mathematical object, result or prediction was added; the registered words the laws now take are quoted from version 1.103, and each place where version 1.103 changes how a law takes its words is listed in Appendix B.2. Checked for this version. Every equation, theorem statement, number, date and DOI of version 1.1 was mapped by script to its place in this edition; every quotation new in this version was compared with its source, those of versions 1.103 and 1.104 with their approved texts; and the identity of each linked address was confirmed at the registry or publisher that issues it on 29 October 2026. Table 6. The versions of this report, each with its date and the kind of record that dates it. Version Date, and the kind of record that dates it Where it is held 0.2.1, working draft dated 1 October 2026 inside the the programme’s site: https://www.michaeld document ariuseastwood.com/research/reports/laws-i v-v-mathematics-v0-2-1.html 0.2.4, working draft deposited on OSF on 3 October 2026 at OSF, revision 1 of the report’s file: 12:30:22 UTC, a server-dated deposit https://osf.io/4b3vt/files/osfstorage/6ac0f5 5e567df3a88efbdbce?revision=1 1.0, frozen 3 October deposited on OSF on 3 October 2026 at OSF, revision 2 of the same file: 2026 19:56:47 UTC, a server-dated deposit https://osf.io/4b3vt/files/osfstorage/6ac0f5 5e567df3a88efbdbce?revision=2 1.1, frozen 7 October deposited on OSF on 12 October 2026 at OSF, revision 3 of the same file: 2026 18:41:25 UTC; its PDF at 21:01:46 UTC, https://osf.io/4b3vt/files/osfstorage/6ac0f5 server-dated deposits 5e567df3a88efbdbce?revision=3 1.2 29 October 2026 this edition Versions 0.2.2 and 0.2.3 were not published. B.2 The sentences version 1.2 restates, in their version 1.1 wording Version 1.1, front matter, now Section 11: This document is not a registration, not an empirical result, and not authority to run any study. Registration versions 1.100 and 1.102 stay frozen byte for byte; Laws I to III stand exactly as registered (registration version 1.102, lines 659, 663 and 665); no proposition is added to or moved between registers; P12 keeps no direction; and every object this document introduces beyond the registered propositions is marked PROPOSED and is not law content until the author registers it. 51 The mathematics of Laws IV and V Version 1.1, section 0, Status, now Section 11: Status. Laws IV and V were added on 24 September 2026. Law IV carries P15, in register ARC-4, and Law V carries P12 (undirected) and P21, in register ARC-5; these propositions are registered (section 0.1), and neither law is yet registered as a law or tested. Their mathematics is set out here at version 1.1: its theorems are proved under the assumptions each states, its identities are checked, and section 7 lists the fifteen questions that stay open. Theorems IV-G and IV-H, Corollaries IV-G′ and IV-H′ , Lemmas IV-K and IV-M′ and Propositions IV-N and IV-R are logical proofs, not identities; no proof assistant is used, and they rest on the written proofs, which finite models exercise and do not replace. Version 1.1, section 0, What it is not (last sentence), now Section 11: It supplies no missing measurement, chooses no scientific parameter the author has held (section 7), and claims no novelty; prior-art distinction remains unassessed. Version 1.1, section 0.1 (last paragraph), now Section 3.1: Each quotation in this document names its source, and each was compared with that source for this version (section 9). This version cites no work that version 0.2.1 did not. Version 1.1, section 2.1, Statement, now Section 5.1: Law IV (added 24 September 2026; its proposition, P15, registered in ARC-4; not yet registered as a law or tested), in the author’s statement of the laws: “Law IV, the ARC Persistence Law, concerns value persistence (register ARC-4) and carries P15: whether externally installed gains decay under subsequent capability training.” Its formal content is P15’s registered claim and nothing more: on the author’s pre-fixed budget scale, the retained fraction of an externally installed gain after matched capability training without embedded correction is below one half, 𝑆ext (1) < 1/2 (section 2.6). The register is that of registration version 1.102, line 99: “ARC-4, VALUE PERSISTENCE: P15 alone. Whether externally installed gains decay under subsequent capability training is the only proposition in this document about persistence after the installing pressure is withdrawn.”; P15 “measures the external arm’s retention and not the embedded arm’s” (registration version 1.102, line 675). Version 1.1, section 2.1 (the comparison of the two arms), now Section 5.1: The comparison of the two arms, 𝑆emb (𝑥) > 𝑆ext (𝑥), is not Law IV’s content in this version: it is the PROPOSED object IV-PERSIST, one object with Λ𝑉 , recorded in section 2.6 and ruled on in section 6, and whether Law IV should instead carry that comparative content with a registered comparative refuter and instrument is the first question of section 7. Version 1.1, section 2.6, IV-PERSIST and the mediation contrast, now Section 5.6: IV-PERSIST and the mediation contrast (PROPOSED). The comparison 𝑆emb (𝑥) > 𝑆ext (𝑥) over a registered pressure range is not P15’s face and is not law content in this version; it is the same test that section 6 rules on under the name Λ𝑉 . The statement paper’s published register-level text is recorded here, not adopted: its ARC-4 sentence, “values run as load-bearing recursive loops persist after the enforcing mechanism is removed at a higher rate than rules, frozen objectives or fine-tuned constraints”, and its kill, “Persistence trials in which embedded loops fare no better than removed enforcement kill ARC-4, and ARC-5 with it.” That sentence is comparative and its kill is comparative; neither is P15’s registered 52 The mathematics of Laws IV and V face, and K-IV below reaches neither. Whether Law IV carries the comparative content, with a registered comparative refuter and instrument, or P15’s face alone, is the author’s first decision (section 7). Version 1.1, section 2.8, Kill condition K-IV (lead), now Section 5.8: Stated before any data, in terms decidable by data yet to be collected, and consistent with P15 as registered. Item 1 is P15’s registered refuter with its budget requirement; the one thing it adds, that the baseline convention be fixed in advance, is marked PROPOSED where it stands. Item 5 quotes the registration’s weaker-neighbour rule and says what this version takes from it. Items 2, 3 and 4 are clauses this document writes, PROPOSED law-level conventions beyond P15’s registered scoring, which they leave untouched; item 2 is on the record and is not in force. Version 1.1, section 2.8, item 1 (the sentences version 1.2 changes), now Section 5.8, item 1: That the baseline convention also be fixed before any outcome is this document’s requirement and is PROPOSED (section 2.6 and section 7 item 8): at its line 1089 the registration fixes the budget and requires the attestation, and it names no baseline arm; nothing here is added to P15’s frozen refuter. . . . That observation refutes P15 as registered (registration version 1.102, line 1083), is scored for P15 exactly as the registration scores it, and takes with it . . . and at every such point the law’s formal content, 𝑆ext (1) < 1/2, is false; Version 1.1, section 2.8, item 2, now Section 5.8, item 2: The law-level withdrawal (PROPOSED by version 0.2.1; not in force). Version 0.2.1 proposed a convention: where a P15 refutation also met four evaluability controls fixed before any data (the baseline fixed in advance, with the convention named; the installed effect established by the lower end of the interval on 𝑌𝑉 (0) being at least 𝑌min ; the capability rise established on a separate instrument by an interval lying entirely above the minimum that instrument registers; and the attestation of no alignment-graded component in the subsequent training, registration version 1.102, line 1089), Law IV would be withdrawn whole and Law V with it. The author has not adopted that convention and it is not in force. What stands without it is item 1: a P15 refutation is scored as the registration scores it, takes with it what registration version 1.102, line 1085 names, and shows the law’s formal content false, since this version gives Law IV no formal content beyond P15’s face (section 2.1). It refutes no theorem: Theorems IV-G and IV-H are conditional mathematics and survive, as item 5 says. It decides nothing about the comparative claim, which is a separate proposal with an endpoint and a refuter of its own still to be registered (section 6). And it withdraws nothing from Law V (section 5 item 3). The four controls remain a proposal for the instrument that decides P15, and no scoring code may apply the former default. Version 1.1, section 2.8, item 4, now Section 5.8, item 4: What is not a refutation of Law IV (PROPOSED convention, in the registration’s own terms, registration version 1.102, line 121): an instrument that fails to run or is too small; an interval touching one half; a result on an unregistered budget scale; a result without the attestation, for the law-level reading; a bypass outside the registered battery; and any result about whether the system’s later conduct is good, which the theory’s content neutrality places outside every law (registration version 1.102, line 75); the method’s own line (registration version 1.102, line 61: “no result about a system’s later conduct supports or refutes the method by itself”) makes the same exclusion for the Eden Protocol, and P15’s own endpoint, a measurement after later training, is not a result about conduct. 53 The mathematics of Laws IV and V Version 1.1, section 2.8, item 5, now Section 5.8, item 5: The weaker neighbour (registration version 1.102, line 125: “A REFUTED PROPOSITION DEGRADES TO A NAMED WEAKER NEIGHBOUR RATHER THAN VANISHING.”). What survives if Law IV is withdrawn: Theorems IV-G and IV-H as conditional mathematics; the verdicts on P12 and P21 as measured; and the persistence questions the registration keeps open in another context, where the conversion claim fails: “Even then the persistence registers stand, because whether early formation outlasts later enforcement is a question about durability and not about scaling, and it would remain open and worth answering.” (registration version 1.102, line 115), which is quoted for its wording of the question, not as a rule about Law IV’s withdrawal. Version 1.1, section 3.1, Statement (the tag), now Section 6.1: Law V (added 24 September 2026; its propositions, P12 and P21, registered in ARC-5; not yet registered as a law or tested), in the author’s statement of the laws: “Law V, the ARC Embedding Law, concerns the genesis strategy (register ARC-5) and carries P12, whether build order moves the correction-leverage exponent, and P21, whether correction placed inside the loop pulls away from correction applied outside it as depth grows.” Two limbs, separate hypotheses with no shared mechanism assumed: Version 1.1, section 3.1, the build-order limb (the genesis reading), now Section 6.1: P12 registers no direction and is given none here; the genesis reading, that formation first carries the higher elasticity by at least the registered minimum effect, 𝜒first − 𝜒after ≥ 0.10 at the placement the instrument fixes (or at least the smaller minimum a future instrument registers, registration version 1.102, line 1041), is PROPOSED (section 6) and is not law content. Version 1.1, section 3.5 (the genesis reading), now Section 6.5: an upward change so established supports P12 and establishes the PROPOSED genesis reading, a downward change so established supports P12 and strikes it (section 3.6 item 2), and both verdicts are reported. Version 1.1, section 3.6, Kill condition K-V (lead), now Section 6.6: Stated before any data, consistent with P12 (no direction) and P21 as registered. The clauses that go beyond the registered refuters are PROPOSED law-level conventions and change nothing in P12’s or P21’s scoring. Version 1.1, section 3.6, item 1 (the sentences version 1.2 changes), now Section 6.6, item 1: Struck, as a PROPOSED law-level convention, by an interaction interval lying wholly on the side in which the external arm’s advantage widens with depth. . . . A P21 refutation strikes the limb by these conventions; Version 1.1, section 3.6, item 2 (the sentences version 1.2 changes), now Section 6.6, item 2: The PROPOSED genesis reading of section 3.1, 𝜒first − 𝜒after ≥ 0.10, is struck by one PROPOSED rule, stated here in advance: . . . It strikes only the PROPOSED reading and changes nothing in P12. . . . it strikes the PROPOSED reading, since its upper end 0.05 lies below 0.10. 54 The mathematics of Laws IV and V Version 1.1, section 3.6, item 3, now Section 6.6, item 3: Law V as a whole (aggregation PROPOSED; none in force). Three ways of reading the two limbs as one law are on the record: severable limbs, under which the whole law is refuted only when both limbs are struck, which was version 0.2.1’s default; a strict conjunction, under which either strike refutes it; and a law that stands on the placement limb alone. The author has chosen none, and none is in force. Two things hold under all three. Each limb is scored by its own rule, items 1 and 2, and a struck limb is reported as struck. And where both limbs are struck, none of Law V’s formal content is left standing. Where one limb is struck and the other stands, what is said of the whole law depends on the choice, which is made before any deciding data or not at all; until it is made the result is reported as that limb’s and no more. No refutation of P15 withdraws Law V by any rule of this document (section 5 item 3). The PROPOSED readings, the genesis reading and the mechanism reading, are scored apart from both limbs. The boundary reading falls with Law III’s model, which takes nothing from either limb. Version 1.1, section 4.6, the Boundary Transfer (one clause; version 1.104), now Section 7.6: The registration reports the growth exponent’s robust re-estimate as approximately 0.49, Version 1.1, section 4.9 (last sentence), now Section 7.9: A directional reading or a mediation hypothesis is a candidate for a registration of its own; it is not part of the successor to version 1.102 until it has been reviewed and given an instrument. Version 1.1, section 5, item 2 (the sentences version 1.2 changes), now Section 8, item 2: “The three laws are untouched” (registration version 1.102, line 1237, for P21). . . . (a) an interval on 𝑆ext (1) from 0.85 to 0.95, which would refute P15 and so Law IV’s formal content; . . . under which Law IV’s face would be supported while both limbs of Law V would be struck. . . . A countermodel proposed for the case in which Laws I to III hold while Law IV fails . . . which is one more reason the comparative content is PROPOSED and the author’s first question. Version 1.1, section 5, item 3, now Section 8, item 3: The register convention (PROPOSED by version 0.2.1; not in force). Version 0.2.1 proposed, as a grouping fixed in advance and not as an entailment, that on an evaluable P15 refutation Law IV be withdrawn whole, and Law V with it as a law of the theory; that Law V’s refutation not withdraw Law IV; and that P12 and P21 keep their own measured verdicts. Two published statements stood behind it, and they are different statements. The registration ties a P15 refutation to “this programme’s case that correction has to sit inside the system rather than be applied to it from outside” (registration version 1.102, line 1085). The statement paper’s dependency table has “ARC-4 falls and takes ARC-5 with it”, and the trigger of that fall in the same paper is the comparative observation of its kill sentence (section 2.6), not P15. This document makes neither into a rule by which a P15 refutation withdraws Law V: nothing in 𝜂𝑜,in > 𝜂𝑜,out or in the P12 endpoint is entailed by any value of 𝑆ext (1), and Law V is struck only through its own limbs (section 3.6). A dependency between the two laws, if the author fixes one, names its trigger and its consequence before any deciding data (section 7 item 3). Version 1.1, section 6, the table rows for IV-PERSIST, the law-level clauses and the genesis reading (their status column), now Table 4: 55 The mathematics of Laws IV and V the same test as Λ𝑉 on the difference scale; one object, two readings; not Law IV’s content in this version . . . PROPOSED conventions; K-IV item 2, K-V item 3 and the register convention are on the record and not in force; P12, P15 and P21 scored as registered . . . PROPOSED directional reading beside an undirected P12, stated at P12’s registered minimum so that it implies P12 and adds only a sign, with one strike rule, its exact interval complement (K-V item 2) Version 1.1, section 6 (lead), now Section 9: Each entry is a proposition or object this document states beyond the registered propositions. None is law content until the author registers it. Each carries what it needs. “The author’s” is reserved for his own records. Version 1.1, section 6 (last paragraph), now Section 9: Two things follow for the successor registration. A result on the difference and a result on Λ𝑉 are one result, not two corroborations. And neither enters the successor to version 1.102: Λ𝑉 , its margins and any instrument for it are candidates for a registration of their own (section 7 item 9). Version 1.1, section 7 (lead), now Section 10: There are fifteen questions in all, numbered below. Where an item names a default, that default stands until the author says otherwise, and none is chosen after a result. Items 2, 3 and 4 name none: the conventions version 0.2.1 proposed there are on the record and are not in force. The successor to registration version 1.102 is drafted as version 1.103 and awaits the author’s registration (item 15). Version 1.1, section 7, item 1, now Section 10, item 1: Law IV’s content (the first decision, on which item 2 turns): P15’s face alone, as the author’s statement of the laws gives it and this version adopts; or the comparative claim, 𝑆emb (𝑥) > 𝑆ext (𝑥), with a registered comparative refuter and instrument, which the statement paper’s ARC-4 sentence and kill (section 2.6) presuppose. Default: P15’s face alone. The loss mechanism, active removal against dilution, stays unresolved under either. Version 1.1, section 7, item 2, now Section 10, item 2: What a P15 refutation takes from Law IV beyond its formal content, which K-IV item 1 already shows false: whether the law is then withdrawn whole under the four controls version 0.2.1 proposed (K-IV item 2), and whether a comparative headline is kept, which would need its own registered refuter before it could stand as a law. No default. Version 1.1, section 7, item 3, now Section 10, item 3: The register convention: whether a refutation in ARC-4 withdraws Law V as well as Law IV, and if so on which trigger, P15’s refutation or the published comparative observation (the statement paper’s kill sentence, section 2.6). No default: this document withdraws nothing from Law V on a P15 refutation (section 5 item 3). Version 1.1, section 7, item 4, now Section 10, item 4: 56 The mathematics of Laws IV and V Whether Law V is severable by limb, a strict conjunction, or stands on the placement limb alone. No default: until the author chooses, each limb’s verdict is reported as that limb’s. The three readings agree where both limbs are struck and where neither is, and differ where exactly one is (section 3.6 item 3). Version 1.1, section 7, item 5, now Section 10, item 5: The axis P21’s result is read on: the registered access interaction (default), or the service- scaling mechanism reading of registration version 1.102, line 1239, which would need a separately specified bridge and design, never a rescoring. Version 1.1, section 7, item 9, now Section 10, item 9: Whether Λ𝑉 is registered as a PROPOSED mediation proposition, in which scope, with what adverse margin and multiplicity, and whether a unit is registered that carries both contrasts (section 6). It is not part of version 1.103, and it takes a proposition number of its own only if its claim and its refuter differ from those already registered. Version 1.1, section 7, item 10, now Section 10, item 10: Whether the genesis reading of the build-order limb, 𝜒first − 𝜒after ≥ 0.10 (formation first carrying the higher elasticity by at least P12’s registered minimum effect), is registered as a PROPOSED clause beside an undirected P12, with the one strike rule of K-V item 2. It is not part of version 1.103, and P12 keeps no direction. Version 1.1, section 7, item 13, now Section 10, item 13: The dating: confirmation of the dates this document draws from his working paper of 16 August 2026; and, once his lines of 8 December 2024 are confirmed at source, the dating of each law’s first statement from them. Version 1.1, section 7, item 15, now Section 10, item 15: The successor registration. It is drafted as version 1.103, and its registration and its date are the author’s. It is to keep every word, date, scope and refuter of P1 to P22 and to list each correction to an instrument line or to the record separately, with its reason; among them is P12’s listing under Condition B, which registration version 1.102, line 403 implies and registration version 1.102, lines 409, 415 and 421 omit (section 4.7). Version 1.1, section 7 (last paragraph, one clause), now Section 10: The names are settled: he chose them on 26 September 2026, and his statement of the laws uses them; only their registration remains, which is item 15. Version 1.1, section 8, item 4, now Appendix C: 8 December 2024: the registration states “An earlier private record of the conceptual frame is dated 8 December 2024.” (registration version 1.102, line 179). Each law’s first statement is to be dated from the author’s prose of that date once those lines are confirmed at source (section 7 item 13); until then this document dates nothing to them, and dates the equations and theorems as above. Version 1.1, the version line (first-publication label), now Table 6: Version 1.1 (frozen) · First published 12 October 2026 · It follows version 1.0 of 3 October 2026 · Earlier versions: 1.0, 0.2.4, 0.2.1 · PDF 57 The mathematics of Laws IV and V B.3 The records of version 1.1 Version 1.1, frozen on 7 October 2026. It follows version 1.0 of 3 October 2026, which stays as published; section 9 lists what changed and what was checked. A later version is a new text under a new number. What this version adds to version 0.2.1. Version 1.0 froze the working draft 0.2.4 and added no object to it: it closed one case of Proposition IV-R, named the terms of one model in section 3.4 and reworded one clause of section 2.8. This version adds no object either. The additions are those of the working draft, four of them PROPOSED: a planner that scores its plans with bounded error (section 2.5.1); the baseline, the arms and the interval that the retained fraction needs before it can be read (section 2.6.1); the balance over a finite range of depth (section 4.8); and what a new deciding instrument declares before its outcomes are read (section 4.9). It also takes out of force the law-level conventions that version 0.2.1 proposed as defaults, and keeps them on the record as proposals (sections 2.8, 3.6, 5 and 7). A PROPOSED rule is never a registered decision rule. Section 9 says what was checked for this version and what remains open. B.3.1 Checks of version 1.1 Every identity and derivation in this document was checked by computer algebra. The proofs of its theorems, lemmas and propositions are written proofs, which finite models exercise and do not replace. Every quotation was compared, byte for byte, with its source, which is the frozen text of registration version 1.102, the statement paper at version 6.7 or the author’s statement of the laws, and the identity of each address to which this document links was confirmed at the registry that issues it, on 3 October 2026 for version 1.0 and on 7 October 2026 for this version. Corrections to version 0.2.1, dated 3 October 2026. Section 3.4: the example of an arm that doubles service while halving attained capability holds at unit elasticity only, and now says so. Section 5 item 2: the registered dials do not by themselves make Laws I to III hold, and the text now says what the dials do show. Section 6: the difference of the two retained fractions is defined only where both are established, not always. Section 2.8 item 1: that the baseline convention be fixed in advance is this document’s PROPOSED requirement and no part of P15’s registered face. Changes from the working draft 0.2.4, dated 3 October 2026. The label, the title and the front matter say what this version is. Section 0: the sources paragraph records that the sentences quoted from the statement paper stand unchanged in its version 7.0 and links the paper’s title to its DOI address; the status lines say what this version sets out and name the proved results as the text names them; the description of what this document is not says that it adds no registered proposition. Section 1: the reference value of service is defined in words. Section 2.6: Fieller’s method carries its reference. Section 2.6.1: the account of where Proposition IV-R stops now covers the case of equality and the case of a negative denominator, and calls the unbounded set what it is. Section 2.8 item 5: its second sentence now opens on a condition, because no withdrawal convention is in force. Section 3.4 names the six fixed terms the smaller model keeps. Section 8 dates the working draft and this version and calls version 0.2.1 the earlier working draft. This section lists these changes, states the second check and says how the linked addresses were confirmed. Every mention of the registration, of the statement paper and of Paper VIII outside a quotation is linked to its DOI address, and Fieller’s method to its source. No definition, lemma, theorem, corollary, identity or proposition was altered. The second check. The working draft was checked a second time from its text alone, by code that shared nothing with the first check: exact rational arithmetic for every printed number, and searches over small 58 The mathematics of Laws IV and V finite models for the written proofs. Fifty checks ran and fifty passed. The two cases added to section 2.6.1 in this version were checked in exact arithmetic over random cases before the version was frozen. Changes from version 1.0, dated 7 October 2026. Laws IV and V are no longer called candidates, and neither is this document. Section 0 states the laws’ standing in full (added on 24 September 2026; their propositions registered; neither law yet registered as a law or tested; their mathematics set out here), and what this document is opens on their mathematics; in the front matter, the three levels name the two laws as empirical claims. Sections 2.1 and 3.1 are headed Statement, and the tag before each statement carries the same standing in short form. Where sections 2.1, 2.6 and 2.8, the row for IV-PERSIST in section 6 and item 1 of section 7 said what this document’s formulation does, they now say it of this version, or here. The word candidate stays where it grades neither the laws nor this document: the candidate descriptive families and the outcome NO CANDIDATE FITS (section 2.6), the candidate retention families (section 6), the candidate mixed model (section 3.4), and objects that are candidates for a registration of their own (sections 4.9 and 6). The title now follows the programme’s uniform order (paper, its law or framework, its own title); until this version it read “Laws IV and V: the mathematics, version 1.0”. The label carries version 1.1, and the front matter prints this document’s DOI. The front matter and section 8 name version 1.0 where they called it this version, and the first paragraph of this section dates the confirmation of the linked addresses for each version. The entries above dated 3 October 2026, and the second check, record version 1.0 and stand as written: where they say this version, they mean version 1.0. No definition, lemma, theorem, corollary, proposition, assumption, proof, identity, equation, number, interval rule, open question, registered wording or quotation was changed, and no object, claim or prediction was added. For this version the computer-algebra checks were run again on its text, and every quotation was compared again with its source. These checks are the programme’s own. They are not independent verification, no proof assistant was used, and nothing here is an empirical result. This version is frozen: a later version is a new text under a new number, with its changes listed. B.3.2 Changes in version 1.1 Michael Darius Eastwood. 7 October 2026. Version 1.1 is frozen. This list compares it with version 1.0 of 3 October 2026, which stays as published. Nothing in this list is a registration or an empirical result. From version 1.0 to version 1.1. Only words change. No definition, lemma, theorem, corollary, proposition, assumption, proof, identity, equation, number, interval rule, open question, registered wording or quotation is altered, and no object, claim or prediction is added. • Laws IV and V are no longer called candidates, and neither is this document. Their standing is stated in full: added on 24 September 2026; their propositions registered (P15 in register ARC-4; P12 and P21 in register ARC-5); neither law yet registered as a law or tested; their mathematics set out in this document, now at version 1.1. • Section 0, the status line: it states that standing, and it folds in what each law carries, so that nothing is said twice. What this document is: the mathematics of Law IV and of Law V. • The front matter, the three levels: the second level is the two laws as empirical claims. • Sections 2.1 and 3.1 are headed “Statement”, and the tag before each statement carries the standing in short form: for Law IV “added 24 September 2026; its proposition, P15, registered in ARC-4; not yet registered as a law or tested”, and for Law V “added 24 September 2026; its propositions, P12 and P21, registered in ARC-5; not yet registered as a law or tested”. 59 The mathematics of Laws IV and V • Where the text said what this document’s formulation does or adopts (sections 2.1, 2.6 and 2.8, the row for IV-PERSIST in section 6, and section 7 item 1), it now says “this version”, or “here” for the default baseline of section 2.6. • The title follows the programme’s uniform order (paper, its law or framework, its own title): “Report · The ARC Theory · Laws IV and V, the ARC Persistence Law and the ARC Embedding Law · The Mathematics”. Until this version it read “Laws IV and V: the mathematics, version 1.0”, which stays the title of version 1.0. • The label: version 1.1, frozen on 7 October 2026, following version 1.0. The front matter prints this document’s DOI, 10.17605/OSF.IO/4B3VT, and says that this version adds no object; it and section 8 name version 1.0 where they called it this version. • Section 9: the first paragraph dates the confirmation of the linked addresses for each version, and a new entry lists these changes. The entries dated 3 October 2026, and the second check, stand as written; in them, “this version” means version 1.0. Where the word candidate stays. It stays where it grades neither the laws nor this document. • “Candidate descriptive families” and the outcome “NO CANDIDATE FITS” (section 2.6), and “the candidate retention families” (section 6): a closed set of descriptive model families fixed before any deciding run, none of which is the law, and the outcome in which none of them fits. • “a candidate mixed model” (section 3.4): one model of the closed model set. • “a candidate for a registration of its own” (section 4.9) and “candidates for a registration of their own” (section 6): objects the author may register in their own right. Checked for this version. • Each edit applies once, and undoing the edits restores version 1.0 byte for byte. • Every line that states a definition, lemma, theorem, corollary, identity or proposition stands as in version 1.0, and so does every line of sections 0.1, 1, 2.2 to 2.5.1, 2.6.1, 2.7, 3.2 to 3.7, 4 and 5. • The computer-algebra checks were run again on the text of version 1.1. • Every quotation was compared again with its source: the frozen text of registration version 1.102, the statement paper at version 6.7, or the author’s statement of the laws. • The identity of each linked address, this document’s DOI among them, was confirmed on 7 October 2026 at the registry that issues it. Unchanged. • Every quotation of the registration, of the statement paper and of the author’s statement of the laws; and the twenty-two registered propositions with their words, scopes and refuters. • The register assignment of section 0.1 and every registration line number. • The fifteen open questions of section 7 and their defaults. Items 2, 3 and 4 name no default: those decisions are the author’s. C Priority record Each item carries its date and the kind of record that dates it; no kind is presented as stronger than it is, and no wording plays the author’s work down. 60 The mathematics of Laws IV and V • 8 December 2024. The author’s book draft of that date, chapter 20, from which version 1.103 of the registration (Eastwood 2026g) dates the question Law V formalises, as the idea and not as the law (registration version 1.103, its qualification on the dates of the added laws). The date is the one the mail servers put on the author’s message to himself, its Date line reading 02:45:18 UTC; the manuscripts of that date have been public on OSF since 5 October 2026 (DOI 10.17605/OSF.IO/QASRK; Eastwood 2026a), a deposit that dates the bytes from that day. Registration version 1.102, line 179 called it “An earlier private record of the conceptual frame”. • 30 April 2025. The author’s manuscript of that date, from which version 1.103 dates the question Law IV formalises, “whether it will uphold the ethical principles of its genesis”, as the idea and not as the law; a mail-server date on a message held privately for verifiers, the manuscript itself public on OSF under the same DOI since 5 October 2026. • 2 January 2026. The first edition of the author’s book, Infinite Architects (ISBN 978-1806056200; osf.io/dqm3e), in print, which carries both questions: Law V’s as in the draft of 8 December 2024, and Law IV’s as the fourth prediction of its section A.3 and the fourth falsification criterion of its section A.4 (registration version 1.103, its qualification on the dates of the added laws). A print date. • 16 August 2026, the author’s working paper of that date (Eastwood 2026k), which the registration names as the programme’s author-review working paper: the registration credits that date with the path identity that writes the balance exponent as 𝜙𝑄 − 𝜙𝑊 (“this registration adopts that formulation with the credit going to that date and not to this one”, registration version 1.102, line 619) and with its identification limit (registration version 1.102, line 621), and records that the working paper names the correction-leverage exponent 𝜒 (registration version 1.102, line 305). • 8 September 2026, the registration (version 1.100, registered on OSF that day; version 1.102 of 13 September 2026 frozen since): the date on which the registration stated the local reading of the growth exponent (registration version 1.102, line 267), the general boundary (registration version 1.102, line 639), P12 and P21 as movers of the boundary (registration version 1.102, lines 1045 and 1239) and content neutrality (registration version 1.102, line 75); these are dated to the registration for what the registration states, and to 16 August 2026 for what the working paper already held. • 24 September 2026. Laws IV and V added to the theory. Version 1.103 says of them: “The statement, kill condition and verdict rule of each added law are work of 2026, fixed on 24 September 2026, and no earlier date is claimed for them” (registration version 1.103, its qualification on the dates of the added laws). • 1 October 2026. Version 0.2.1, the earlier working draft, dated inside the document: the theorems’ forms, the interval rules and the kill conditions as written there, among them the genesis reading of the build-order limb at P12’s registered minimum and its strike rule, which version 1.103 registers as the limb’s. No priority is claimed for the direction, which has prior work on neighbouring endpoints (Section 2). • 3 October 2026. The working draft 0.2.4, and version 1.0, which freezes it: Propositions IV-N, IV-R and B-8, the measurement requirements of Sections 5.6.1 and 7.9, and the law-level conventions of version 0.2.1 taken out of force. Versions 0.2.2 and 0.2.3 were not published. Both versions were deposited on OSF that day, at 12:30:22 and 19:56:47 UTC, as revisions 1 and 2 of the same file, server-dated deposits (Table 6). These lines date the text and its mathematics; nothing in them is an empirical result or a registration. • 7 and 12 October 2026. Version 1.1, frozen on 7 October 2026 (a date inside the document) and deposited on OSF on 12 October 2026, as revision 3 of the same file at 18:41:25 UTC, with its PDF at 21:01:46 UTC, server-dated deposits. 61 The mathematics of Laws IV and V • 15 October 2026. Version 1.103 of the theory registration, approved on OSF that day, registers Laws IV and V as laws, with the build-order limb’s direction, the retention limb’s baseline and floor conditions and Law V’s dependence on Law IV’s comparison limb; a registry approval. • 29 October 2026. Version 1.104 of the theory registration, approved on OSF that day, adds readings of P15’s decision requirement and changes no law’s statement; a registry approval. D Notation and glossary Further choices. The two suprema of Theorem IV-H are 𝐽safe ∗ and 𝐽 ∗ (not 𝑎 and 𝑒; 𝑎 is one of the exit eight); the searched margin is 𝑀srch ; the increments and levels of Section 5.5 are inc𝑖 and lev𝑡 (not 𝐷 and 𝐿, both retired letters); the increment index is 𝑖 and the step index 𝑡, so 𝑢 is the random effects only; the power family’s scale is 𝑥 scale (not 𝑥 0 , which the reference convention reserves); the shape function of the descriptive families is sh(𝑥) (not 𝑓 , the Cauchy 𝑓 of the programme’s notation); the embedding ratio is 𝐺 emb (not 𝐺 𝐸 , whose subscript is the letter 𝐸); the two families of executed and of feasible paths are 𝑇acc and 𝑇plan ; the evaluated transition set is 𝑇eval (𝑠); the unrelated part of an objective change is 𝑏 other ; the registered upper bound on the safe supremum is 𝐽up and the best exit score found is 𝐽found . The general root of the balance identity is 𝛼bal , and 𝛼crit is reserved for the case in which that root is a ceiling (Section 7.2), so the name the programme’s notation gives 𝛼crit , the stability ceiling, stays true. The two mutual-information forms of Section 6.2 and the architecture form of Section 7.6, which are written elsewhere with two of the eight letters, are described in words, as is the change in the capability functional of Section 5.4, which is written elsewhere with the prefix Section 4 rules out; 𝑆𝑉 and 𝐾𝑉 carry their subscripts wherever a curve or a kernel is named; and the registration’s letters for the correction-leverage exponent and for its checker-axis slope are printed only inside quotations (Eastwood 2026g). Symbols used, in one line each: • Law IV: 𝑠 (state); 𝑉 (𝑠) (property score, vector allowed, components 𝑉 𝑗 ); 𝑣 ∗ (threshold, component- wise, 𝑣 ∗, 𝑗 ); 𝜀𝑉 (tolerance, component-wise); Ω𝑉 := {𝑠 : 𝑉 𝑗 (𝑠) ≥ 𝑣 ∗, 𝑗 − 𝜀 𝑉 , 𝑗 for every 𝑗 } (acceptable set); 𝜌 = (𝑠0 , . . . , 𝑠 𝑛 ) (path), |𝜌| = 𝑛; 𝑇 (the declared family of allowed finite paths, containing every length-zero path and every dead end); 𝑇𝐻 (𝑠0 ) := {𝜌 ∈ 𝑇 : 𝜌 starts at 𝑠0 , |𝜌| ≤ 𝐻}, partitioned into 𝑇𝐻safe (𝑠0 ) and 𝑇𝐻exit (𝑠0 ); 𝑇acc (the family of paths of executed transitions, Theorem IV- G); 𝑇plan (the planner’s feasible family, Theorem IV-H); 𝑇eval (𝑠) (the allowed one-step transitions from 𝑠 evaluated in premise (G2)); 𝐻 (horizon); 𝐾𝑉 (𝐻; 𝑇, 𝜀 𝑉 ) (persistence kernel); 𝐽 (acceptance objective, one declared unit); 𝐽 𝐻 (path objective); 𝐽safe ∗ , 𝐽 ∗ (the safe and exit suprema); 𝐽 (a exit up registered upper bound on 𝐽safe ∗ ); 𝐽 found (the best exit score a registered search found); 𝑏 𝑉 , 𝑐 𝑉 (the removal benefit and cost causally attributable to the degradation, in 𝐽’s units); 𝑏 other (the part of an objective change not attributable to it); 𝜋𝑉 := 𝑐 𝑉 − 𝑏 𝑉 (local margin); Π𝑉 (𝐻) (its infimum over evaluated reachable exits); 𝑀𝑉(𝐻 ) (strategic viability margin); 𝑀srch (the searched margin); 𝜀 opt (optimiser tolerance); 𝐽ˆ𝐻 , 𝜀 eval (the score a planner uses to choose a plan where that differs from 𝐽 𝐻 , and a uniform bound on its error; UNREGISTERED, Section 5.5.1); 𝜃 (discount factor, weights 𝜃 𝑡 ); inc𝑖 , lev𝑡 (increments and levels); 𝑥 (budget ratio); 𝐴𝑉 (𝑥), 𝐴base (𝑥) (arm scores on the frozen property battery; 𝐴ext and 𝐴emb are the scores of the external and the embedded arm); 𝑌𝑉 (𝑥) (installed effect; 𝑌ext is the external arm’s); 𝑌min (floor); 𝑆𝑉 (𝑥) (retained fraction; 𝑆ext , 𝑆emb per arm; 𝑆𝑉 ,fix the fixed-baseline alternative); ℎ𝑉 (log-decay derivative); 𝑥1/2 (first crossing of one half); 𝑆inf , 𝜈, 𝜉, 𝑥 scale , sh(𝑥) (parameters and shape function of the unregistered descriptive families); Λ𝑉 (unregistered mediation contrast); 𝑌ˆ1 , 𝑌ˆ0 , var1 , var0 , cov10 , 𝑧 crit , 𝑚, 𝑚 thr , quad (the two estimates, their estimated variances and covariance, the critical value, the trial ratio, the threshold and the quadratic of Proposition IV-R, unregistered, used in Section 5.6.1 only); indices 𝑡 62 The mathematics of Laws IV and V (step), 𝑖 (increment), 𝑗 (component), 𝑛 (path length, or a count of accepted transitions). • Law V and the balance: 𝐶, 𝐶0 (capability, reference); 𝑅, 𝑅0 (modelled depth, reference; registration version 1.102, line 263: completed rounds plus one where the indexing at zero applies); 𝑜 ∈ {after, first}, 𝑙 ∈ {out, in} (arm labels); 𝑄 𝑜𝑙 (𝐶, 𝑅) > 0 (correction service per arm), 𝑄 0 (a fixed positive reference value of service), 𝑄 𝑜𝑙,0 ; 𝜒𝑜𝑙 := d ln 𝑄 𝑜𝑙 /d ln 𝐶 at fixed 𝑅; 𝜂𝑜𝑙 := d ln 𝑄 𝑜𝑙 /d ln 𝑅 at fixed 𝐶; 𝑊 (𝐶, 𝑅) > 0 (burden); 𝜅 := d ln 𝑊/d ln 𝐶; 𝜁 := d ln 𝑊/d ln 𝑅 (the programme’s 𝜇 𝐵 is 𝜅 − 1); 𝛼𝑅 := d ln 𝐶/d ln 𝑅 along the path (the local reading of the programme’s 𝛼; the author’s 𝛼𝐶 in his working paper of 16 August 2026 (Eastwood 2026k) is the same quantity on the resource clock, and the programme’s notation writes 𝛼𝐶 for it in its entries for 𝜙𝑄 and 𝜙𝑊 ); 𝛼in , 𝛼out (the same on the arms’ own paths 𝐶in (𝑅), 𝐶out (𝑅)); 𝛼bal (the root of Δbal in 𝛼𝑅 ); 𝛼crit (that root where it is a ceiling); Δbal (the programme’s Δ, local); 𝐺 emb := 𝑄 in /𝑄 out at matched order, 𝐺 0 := 𝐺 emb (𝐶0 , 𝑅0 ); indfirst , indin (indicators); 𝑢 (random effects, subscripted); 𝜓 (residual); 𝑘 (the programme’s drift-acceleration exponent); 𝛽𝐶 (Law II’s exponent, registration version 1.102, line 663); 𝑅1 < 𝑅2 (the ends of a finite range of depth), 𝑟 (the variable of integration over it), 𝑅★ (a formal crossing depth) and 𝑝 (the power in a re-expression of capability), all unregistered and of Section 7.8. All logarithms are of ratios to the reference values. Table 7. The status words this report uses, each with its source. Word What it means, and its source a law under test the naming phrase a law carries until a registered verdict says otherwise; “A law has three words: supported, refuted and under test.” (registration version 1.103, how a law takes its word) registered fixed in a version of the theory registration on OSF; the propositions on 8 September 2026, Laws IV and V as laws on 15 October 2026 UNREGISTERED an object this document states beyond the registered propositions and the laws’ registered words; not law content until the author registers it (Section 11) NOT EVALUABLE the report on a quantity, a membership or a rule whose inputs are not established; nothing is decided on it (Section 5.2) INCONCLUSIVE the registration’s word for an outcome inside the band of a fixed line: “outcomes inside that band are INCONCLUSIVE and never support” (registration version 1.102, line 633) a result at the margin a measurement just inside or just outside one of the registration’s conventions, reported as a result at the margin and not as a verdict (registration version 1.102, line 391) struck a limb or reading whose rule fires; a law struck through a registered dependence is printed as struck by it (registration version 1.103, how a law takes its word) 63