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Laws IV and V: the mathematics, version 0.2.1

Version 0.2.1 (working draft) · First published 1 October 2026

Working draft, version 0.2.1, 1 October 2026. Its review is still open: a further check of these bytes is under way, and a checked version will follow with its changes listed.

Michael Darius Eastwood. Independent AI alignment researcher, London. ORCID 0009-0004-3222-7442.

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.

Three levels are kept apart throughout: identities and conditional mathematics (sections 2.2 to 2.5, 3.3 and 4), which prove consequences of assumptions and nothing about real systems; the candidate empirical laws (sections 2.1, 2.8, 3.1 and 3.6), 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, and no verdict here is a verdict on either (section 5 item 5).

0. Sources and status

Sources. Two public documents are cited by line or by quotation. The first is the theory registration, 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. 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, The ARC Theory, quoted from its working paper version 6.7, revised 21 September 2026, and cited by section or by its own quoted words. 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.

Status. Laws IV and V are candidate laws, added on 24 September 2026, with their mathematics in development; neither is registered or tested. Law IV carries P15; Law V carries P12 (undirected) and P21. Theorems IV-G, IV-H, IV-H' and Lemma IV-K 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 this document is. The candidate 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 proposition and moves none. It does not restate Laws I to III (section 6). It supplies no missing measurement, chooses no scientific parameter the author has held (section 7), and claims no novelty; prior-art distinction remains unassessed.

1. Notation, fixed first

Principle: one symbol, one meaning; where the programme's notation already has the quantity, its letter (chi, eta, kappa, zeta, alpha, alpha_crit, C, R, k, and its Delta written Delta_bal, never bare); the letter beta only with its subscript, so Law II's exponent is written beta_C; no bare delta; the registration's letter for the correction-leverage exponent, and its subscripted forms, only inside a quotation of the registration; no Delta 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, B_Q, I_Q, A, a, P_V, d_min) 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 A_V and A_base, 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.

Clash Resolution
A_V for both the acceptable set and the arm score the acceptable set is Omega_V; A_V(x) and A_base(x) are the arm scores, the symbols of the retained fraction
d_gain for the installed effect, still a d, a letter that collides in the programme's notation the installed effect is Y_V(x) := A_V(x) - A_base(x); its floor is Y_min > 0
P_H, P_safe, P_exit for plan sets, beside P1 to P22 plan sets are written from the path family: T_H(s_0), T_H^safe(s_0), T_H^exit(s_0)
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(C_0, R_0); b is the removal benefit only
D_p(x) for the installed effect, beside D for the correction shortfall neither kept: Y_V(x) for the installed effect; the correction shortfall written (kappa - chi) in full

Further choices: the two suprema of Theorem IV-H are J_safe^* and J_exit^* (not a and e; a is one of the eight); the searched margin is M_srch; the increments and levels of section 2.5 are inc_i and lev_t (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 T_acc and T_plan; the evaluated transition set is T_eval(s); the unrelated part of an objective change is b_other; the registered upper bound on the safe supremum is J_up and the best exit score found is J_found. The general root of the balance identity is alpha_bal, and alpha_crit is reserved for the case in which that root is a ceiling (section 4.2), so the name the programme's notation gives alpha_crit, the stability ceiling, stays true. The two mutual-information forms of section 3.2 and the architecture form of section 4.6, which are written elsewhere with two of the eight letters, are described in words, as is the change in the capability functional of section 2.4, which is written elsewhere with the prefix this section rules out; S_V and K_V 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.

The identification is explicit, not silent, and is conditional: chi_ol 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 chi_K 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 chi_ol is the elasticity of the declared service measure and never enters the ceiling relation. The registration records that its letter for this exponent and chi name the same quantity (registration version 1.102, line 305); this document writes chi, 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.

Symbols used, in one line each:

2. Law IV, the ARC Persistence Law

2.1 Candidate statement

Law IV (candidate; added 24 September 2026; mathematics in development; not registered 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, S_ext(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).

The mathematics beneath the register has three separate objects, none of which is the law: an invariance object (the persistence kernel, section 2.2), two conditional theorems that give sufficient mechanisms for invariance under a stated decision rule (sections 2.4 and 2.5), and the empirical retention process (section 2.6), of which P15 reads one arm. The comparison of the two arms, S_emb(x) > S_ext(x), is not Law IV's content in this candidate: it is the PROPOSED object IV-PERSIST, one object with Lambda_V, 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. 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, 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."

2.2 Definitions

Definition IV-1 (acceptable set). Given the property score V, the threshold v_* and the tolerance epsilon_V, all fixed before any outcome and component-wise in their own declared units, the acceptable set is Omega_V := {s : V_j(s) >= v_{*,j} - epsilon_{V,j} for every component j}. 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 (PROPOSED interval rule). True scores are not observed; a frozen battery gives estimates with registered intervals. A state s' is established outside Omega_V when, for some j, a registered interval on V_j(s') lies entirely below v_{*,j} - epsilon_{V,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 rho = (s_0, s_1, ..., s_n) is a finite sequence of states with |rho| = 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. T_H(s_0) is the set of members starting at s_0 with |rho| <= H. A path is safe if every state on it lies in Omega_V and is an exit otherwise; T_H(s_0) is the disjoint union of T_H^safe(s_0) and T_H^exit(s_0). T_H(s_0), its safe and exit parts and Definition IV-3 apply to any set of finite paths. Two named path sets are used below: T_acc, 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 T_plan, 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). K_V(H; T, epsilon_V) := {s_0 : T_H^exit(s_0) 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 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, s_0 in K_V(H; T_acc, epsilon_V), and as a PROPOSED object for registration (section 6); 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 2.7).

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 Omega_V 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.

2.3 Lemma IV-K (kernel monotonicity)

Premise: T is a declared family as Definition IV-2 requires. Conclusions: (K1) K_V(H + 1; T, epsilon_V) is a subset of K_V(H; T, epsilon_V); (K2) if T is a subset of T', then K_V(H; T', epsilon_V) is a subset of K_V(H; T, epsilon_V); (K3) if epsilon_V <= epsilon_V' component-wise, then K_V(H; T, epsilon_V) is a subset of K_V(H; T, epsilon_V').

Proof. (K1): T_H(s_0) is a subset of T_{H+1}(s_0), so if every member of the larger set is safe, every member of the smaller is. (K2): T_H(s_0) under T is a subset of T_H(s_0) under T'. (K3): enlarging the tolerance enlarges Omega_V, 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.

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 Omega_V 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, K_V(2; T, epsilon_V) = K_V(1; T, epsilon_V) within K_V(0; T, epsilon_V), and enlarging the family shrinks the kernel. The countermodel is a listed finite structure; no dynamics are run.

2.4 Theorem IV-G (greedy acceptance)

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 s_0. Let s_0 lie in Omega_V. For every state s in Omega_V reachable from s_0 within fewer than H accepted transitions, T_eval(s) is the set of allowed transitions from s that the design evaluated.

Premises.

Conclusion. Every path of executed transitions from s_0 of length at most H stays in Omega_V; equivalently s_0 lies in K_V(H; T_acc, epsilon_V), where T_acc is the family of paths of executed transitions of length at most H.

Proof. Suppose a path of executed transitions from s_0 has a first index t, with 1 <= t <= H, at which s_t lies outside Omega_V. Then s_{t-1} lies in Omega_V and is reachable within t - 1 < H accepted transitions. By (G3) the executed transition s_{t-1} to s_t lies in T_eval(s_{t-1}), so by (G2) J(s_t) - J(s_{t-1}) < 0. But the transition was executed, so by (G1) J(s_t) - J(s_{t-1}) >= 0. Contradiction; no such t exists. Since t ranges over a finite set, no further condition is needed.

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 pi_V = c_V - b_V is minus the total change. 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 Pi_V(H) > 0, the infimum of pi_V 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, Pi_V^(H), is superseded by M_V^(H) of section 2.5 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 pi_V, 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 Omega_V, reachable within fewer than H accepted transitions, and s' established outside Omega_V by the interval rule of section 2.2, is recorded.

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 short: with the membership of s' not established, no exit is established and nothing is decided; with s' established outside Omega_V but the membership of s not established, the path is known to have left Omega_V within H, so some premise failed, but neither the transition at which it first left nor the premise is identified; and in T_eval(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 - pi_V where a registered causal decomposition exists, and never on pi_V alone. Where the registered decomposition has b_other = 0 (Corollary IV-G'), the total change is -pi_V, so the rule accepts a transition exactly when its margin is zero or negative, pi_V <= 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 proposed refutation wording for IV-G, under which a preregistered value-degrading rewrite or plan with adequately measured Pi_V 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) = -pi_V < 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 pi_V <= b_other. A transition with pi_V <= 0 is then rejected whenever b_other < pi_V: pi_V = 0 with b_other = -1 changes the objective by -1. And acceptance of an evaluated exit at pi_V > 0 with (G1) and (G3) verified is possible, exactly when b_other >= pi_V: it is the bundled case, the third case above, and its report shows that (G2) failed, not (G1); pi_V = 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.

2.5 Theorem IV-H (planned optimisation) and the strategic viability margin

Setting. A start s_0 in Omega_V, a horizon H, the planner's feasible family T_plan (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 T_plan,H(s_0) non-empty, and the planner's declared path objective J_H on it. Because T_plan need not contain the length-zero plan at s_0 or the prefixes of its plans, its safe part can be empty, which is the void case of (H1).

Premises.

Conclusion. rho* lies in T_plan,H^safe(s_0).

Proof. By (H2), J_safe^* - J_exit^* > epsilon_opt >= 0, so J_safe^* > J_exit^* and the supremum over all of T_plan,H(s_0) is J_safe^*. By (H3), J_H(rho*) >= J_safe^* - epsilon_opt, and (J_safe^* - epsilon_opt) - J_exit^* = M_V^(H)(s_0) - epsilon_opt > 0. Every exit plan scores at most J_exit^*, so rho* is not an exit plan. Neither supremum need be attained; an exact maximiser, where it exists, is the case epsilon_opt = 0.

Corollary IV-H' (rolling replanning). Suppose that at every replanning state s_t reached along the executed path, for t = 0, 1, ..., H - 1, three conditions hold: (i) either the exit part T_plan,H-t^exit(s_t) of the feasible family then in force is empty, or premises (H1) to (H3) hold for the objective, feasible family, tolerance and residual horizon H - t then in force; (ii) the plan executed from s_t 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 s_t, the transition realised from s_t is the plan's first transition as modelled, so that s_{t+1} is the plan's second state. Then every state of the executed path through H lies in Omega_V. Proof by induction on t: s_0 lies in Omega_V; if s_t does, the plan executed from s_t 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 s_t the executed path ends there, in Omega_V, and otherwise, by (iii), s_{t+1} is a state of a safe plan and lies in Omega_V. 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 s_t 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 inc_i = J(s_i) - J(s_{i-1}) and the levels lev_t = J(s_t) - J(s_0) = inc_1 + ... + inc_t. With weights theta^t, the weighted sum of levels is the sum over i of inc_i theta^i (1 - theta^(n - i + 1))/(1 - theta), which at theta = 1 is the sum over i of inc_i (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 theta = 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 theta = 1 is the terminal level; a registered use of IV-H states which objective the planner maximises and computes M_V^(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(s_t) over its steps t = 1, ..., n, differs from one that scores the levels lev_t by J(s_0) 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(s_0) = 0, so where early stops are permitted the two objectives can rank plans differently: with theta = 1 and J(s_0) = 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 J_safe^* is attained and the best exit scores J_safe^* + 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 (PROPOSED: the strategic viability margin). M_V^(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(rho) 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 J_up be a registered upper bound on the safe supremum, J_up >= J_safe^* (the exact safe optimum of a finite family enumerated completely is the case J_up = J_safe^*), and let a registered search find an exit plan of score J_found <= J_exit^*. Then M_V^(H) = J_safe^* - J_exit^* <= J_up - J_found. Consequences: a found exit plan scoring at or above J_up certifies M_V^(H) <= 0 and refutes a positive-margin claim; a positive value of the searched margin M_srch := J_up - J_found certifies nothing about the sign of M_V^(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 J_safe^* 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 (J_exit^* = 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 J_exit^* 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 (PROPOSED): the infimum of M_V^(H) over a registered family of environments. It is given no symbol here, because the one first proposed for it collides with recursive depth; the object stands as a proposal.

Falsifier. IV-H is a conditional; an executed exit plan rho* refutes a premise. If rho* lies outside the declared family T_plan, 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 rho* lies in the family and the family is enumerated completely and shows M_V^(H) > epsilon_opt, 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 M_V^(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.

2.6 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 Y_V(x) := A_V(x) - A_base(x) and the retained fraction is S_V(x) := Y_V(x)/Y_V(0). It is read only where the installed effect was real: the lower end of a registered interval on Y_V(0) is at least Y_min, with Y_min > 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 S_V is formed by a method fixed before any outcome (Fieller's or an equivalent named in the deciding instrument). The floor is positive, not absolute: an absolute-value floor would admit a negative installed "gain". Y_min's value is the author's. S_V(0) = 1; S_V may exceed one or be negative, and neither is clipped. Absolute retention, Y_V(x) itself, is reported beside S_V(x). S_V 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 PROPOSED 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 S_ext (property installed by external post-training) and S_emb (property installed with correction embedded in the training objective), as the registered two-arm unit reads them (registration version 1.102, line 1089).

The baseline convention, stated as a choice. In the ratio above, the baseline arm A_base(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, S_V,fix(x) := (A_V(x) - A_base(0))/(A_V(0) - A_base(0)). The two can give opposite P15 verdicts: with A_V(0) = 1, A_base(0) = 0, A_V(1) = 0 and A_base(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 A_base(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 S_V,fix is computable. The concurrent baseline is this candidate's default, 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 7 item 8), and until fixed the endpoint's reading is a choice recorded, not a quantity nature decides.

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 S_ext(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 S_ext(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 S_ext(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."

Descriptive statistics (PROPOSED reporting; never verdicts). The first crossing x_{1/2} := inf{x : S_V(x) <= 1/2} reads the endpoint only for monotone retention: the curve S_V(x) = 1 - 1.8 x + 1.4 x^2 has S_V(1/2) = 9/20 and S_V(1) = 3/5, dips to 59/140 at x = 9/14, and crosses one half first at x = (9 - sqrt 11)/14, about 0.406, yet ends above one half at the matched budget, 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 h_V := -d ln S_V/dx requires a positive differentiable S_V 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 S_V over a registered range is descriptive. None of these substitutes for the registered interval rule.

Candidate descriptive families (PROPOSED; closed before any deciding run): no loss; exponential; power; stretched exponential; a change-point family; each of the decaying members with a floor S_inf, as S_V = S_inf + (1 - S_inf) sh(x) with sh(0) = 1 and sh tending to zero, so that each has S_V(0) = 1 and tends to S_inf; the no-loss member is S_V = 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 (PROPOSED). The comparison S_emb(x) > S_ext(x) over a registered pressure range is not P15's face and is not law content in this candidate; it is the same test that section 6 rules on under the name Lambda_V. 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 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).

2.7 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, s_0 in K_V(H; T_acc, epsilon_V); 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 Omega_V, never that s_0 lies in K_V(H; T_plan, epsilon_V), 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 results are not deciding evidence for P15. Nothing in this section has been measured.

2.8 Kill condition K-IV

Stated before any data, in terms decidable by data yet to be collected, and consistent with P15 as registered. Items 1 and 5 restate the registration; items 2, 3 and 4 are clauses this document writes, PROPOSED law-level conventions beyond P15's registered scoring, which they leave untouched.

  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 and the baseline convention fixed before any outcome, an interval on S_ext(1), the retained fraction of the externally installed gain after matched capability training without embedded correction, lying entirely above one half. 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), is scored for P15 exactly as the registration scores it, and takes with it what registration version 1.102, line 1085 names (quoted in section 2.6). 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 law's formal content, S_ext(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 2.6).
  2. The law-level withdrawal (PROPOSED convention). Where that P15 refutation also meets the four evaluability controls fixed before any data, each a decidable rule: the baseline fixed in advance, with the convention named; the installed effect established by the lower end of the interval on Y_V(0) being at least Y_min; 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); then Law IV is withdrawn whole, its comparative reading is not kept as a law in its place, and Law V is withdrawn as a law of the theory by the pre-fixed register convention (section 5 item 3). Where a control is not met, the P15 verdict stands as registered and the result is NOT EVALUABLE for Law IV; NOT EVALUABLE is never survival. No control is added, relaxed or reinterpreted once a result is in view, and no post-outcome claim about removal benefits or hidden correction rescues the law. This default is coherent with section 2.1: the law's only formal content is P15's face, so a P15 refutation leaves nothing of it standing. Two other readings argue for the other default and are weighed here: one, that Law IV should not be killed merely because external gains do not decay, since that would refute P15 and not the removal-margin theorem; the other, a separate comparative kill, under which no decay of external alignment would refute P15 but not the removal-cost theorem for embedded structures. Both are right that a P15 refutation refutes no theorem, and the theorems survive as item 5 says; both presuppose the comparative form of Law IV, under which their position holds and needs a registered comparative kill. Under the P15-face form adopted here the default stands; the choice between the forms, and with it this default, is the author's (section 7).
  3. The battery-relative kernel kill (PROPOSED, 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 Omega_V by the interval rule of section 2.2 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 2.4 and 2.5 (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 (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.
  5. The weaker neighbour (registration version 1.102, line 125: "A REFUTED PROPOSITION DEGRADES TO A NAMED WEAKER NEIGHBOUR RATHER THAN VANISHING."). What survives Law IV's withdrawal: 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.

3. Law V, the ARC Embedding Law

3.1 Candidate statement

Law V (candidate; added 24 September 2026; mathematics in development; not registered 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:

Boundary reading (conditional). Where a limb raises eta_ol with kappa > chi, or raises chi_ol with eta > zeta, the burden elasticities being fixed, the balance boundary of section 4 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 5).

3.2 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 4.7); 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 3.4. The local elasticities chi_ol and eta_ol 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 defines kappa, zeta, chi and eta 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 1; otherwise chi_ol is the elasticity of the declared service measure. eta_ol 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 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 PROPOSED as a prerequisite beneath all five laws, stated 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 included; no symbol is given to it here. Two operational forms have been proposed, 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 1, 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.

3.3 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 = eta_in - eta_out. At fixed R: d ln G_emb/d ln C = chi_in - chi_out. Along a common path C(R) with alpha_R = d ln C/d ln R: d ln G_emb/d ln R = alpha_R (chi_in - chi_out) + (eta_in - eta_out). 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 eta_in - eta_out: that equality holds at fixed C and, along a path, only where alpha_R (chi_in - chi_out) = 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 C_in(R) and C_out(R), with alpha_in := d ln C_in/d ln R and alpha_out := d ln C_out/d ln R, the depth derivative of the log-ratio is alpha_in chi_in + eta_in - alpha_out chi_out - eta_out, 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 eta_in - eta_out = 0; the widening is alpha_R (chi_in - chi_out) = 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.

Proposition V-3 (constant elasticities). If chi_ol and eta_ol are constant on a rectangle of the (ln C, ln R) plane containing the reference point (C_0, R_0), then on it ln(Q_ol/Q_0) = ln(Q_ol,0/Q_0) + chi_ol ln(C/C_0) + eta_ol ln(R/R_0), and at matched order G_emb = G_0 (C/C_0)^(chi_in - chi_out) (R/R_0)^(eta_in - eta_out). Proof: ln Q_ol - chi_ol ln C - eta_ol 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. This is the reduced scaling form: earned where the elasticities are constant, assumed nowhere.

3.4 The estimable model (PROPOSED)

For cell (o, l), family and task, a candidate mixed model is

ln(Q/Q_0) = ln(Q_ol,0/Q_0) + chi_ol ln(C/C_0) + eta_ol ln(R/R_0) + u_{0,family} + u_{C,family} ln(C/C_0) + u_{R,family} ln(R/R_0) + u_{0,task} + u_{C,task} ln(C/C_0) + u_{R,task} ln(R/R_0) + psi,

with random intercepts and random slopes by family and by task, their covariance structure fixed before any outcome, and the residual psi. 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 x 2 cells: four intercepts, four capability slopes, four depth slopes. Equivalently, each of the three coefficient types is written with the indicators ind_first, ind_in 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 that spans six of these twelve fixed dimensions omits the two indicator main effects, the placement x ln C and order x ln R slopes and the two three-way slope terms. The order contrast at placement l is chi_first,l - chi_after,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 eta_o,in - eta_o,out.

Estimands and controls. Any estimate of chi 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 3.2). 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. The two estimands can differ in either direction: identical service surfaces give a zero contrast at every fixed C while an arm that raises attained capability widens the total effect with depth, and an arm that doubles service at fixed C while halving attained capability leaves the total effect at zero. The slopes themselves are read the same way. Where ln(C/C_0) 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 chi_ol, 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/C_0) 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.

3.5 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 3.7. Its instrument is 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."). 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.

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 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. 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."

3.6 Kill condition K-V

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.

  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, 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; it does not measure the mechanism reading and does not prove eta_o,in <= eta_o,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 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 PROPOSED genesis reading of section 3.1, chi_first - chi_after >= 0.10, is struck by one PROPOSED rule, stated here in advance: an interval on chi_first - chi_after, at the placement the instrument fixes and on the scale Condition B fixes (registration version 1.102, line 403; section 4.7), 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 only the PROPOSED reading and changes nothing in P12. 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 PROPOSED 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. Refuted when both limbs are struck (the severable-limb default; whether Law V should instead be a strict conjunction, or stand on the placement limb alone, is the author's, section 7). Withdrawn, without either limb being refuted, when Law IV is withdrawn under K-IV item 2 (the register convention, section 5 item 3); the verdicts on P12 and P21 then stand as measured. 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 class or failure-mode independence, which P21's unit does not vary (registration version 1.102, line 1223).
  5. The weaker neighbour. What survives Law V's refutation: the identities of section 3.3, 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 3.5.

3.7 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 3.3 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.

4. The balance identity and Law III, unchanged

4.1 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 3.2. Along a specified path C(R), with other factors fixed,

Delta_bal := d ln(Q/W)/d ln R = alpha_R (chi - kappa) + (eta - zeta).

Proof: subtract the chain rules for ln Q and ln W along the path. Every elasticity is local; no power law is assumed. Delta_bal is the programme's balance exponent read locally, and the registration's Delta_balance = phi_Q - phi_W (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); Delta_bal 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.

4.2 Proposition B-2 (the balance root and the sign rule)

Where kappa differs from chi, Delta_bal = 0 has the single root alpha_R = alpha_bal := (eta - zeta)/(kappa - chi), and identically Delta_bal = (kappa - chi)(alpha_bal - alpha_R). Hence: with kappa > chi the trend is favourable exactly where alpha_R < alpha_bal, and alpha_bal is a ceiling, written alpha_crit; with kappa < chi the inequality reverses and alpha_bal is a floor; with kappa = chi the trend is eta - zeta, independent of alpha_R. The symbol alpha_crit is used only in the ceiling case, so that the entry of the programme's notation, the stability ceiling, keeps its meaning; its range there, at least one, holds exactly where eta - zeta >= kappa - chi > 0, which in the registered burden model (kappa = 1, zeta = -1) is eta >= -chi with chi < 1: in the pacing case eta = -1/2 with chi = 0 gives a ceiling of one half; outside it, kappa = 1/2 with chi = 1, eta = 0 and zeta = -1 gives alpha_bal = -2, a floor. A non-positive alpha_crit with kappa > chi leaves no positive growth elasticity with a favourable trend. As chi rises to kappa with eta - zeta > 0 the boundary grows without bound: a correction elasticity matching the total burden elasticity leaves no finite boundary. The zero set of Delta_bal, the balance frontier, is this set of points; it depends on the path field alpha_R as well as on the service and burden surfaces. Once the path field is specified, Delta_bal is a function on the (C, R) plane and its zero set is a level set of that function: where Delta_bal is continuously differentiable, the set is a curve near each of its points at which the gradient of Delta_bal does not vanish (the implicit function theorem), so it is generically a curve, but not always: in the pacing case at chi = 1 with eta = -1, Delta_bal is zero everywhere, and at chi = 1/2 and eta = 0 with the field alpha_R = 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, alpha_R), where the derivative of Delta_bal in alpha_R is chi - kappa, so, again where the elasticities are continuously differentiable, it is a surface near each of its points at which kappa differs from chi; it carries no symbol here and Law III is unchanged (section 6).

4.3 Corollary B-3 (the registered special case)

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 alpha C/R with kappa = 1 and zeta = -1 (one path alone does not fix the partials, section 3.3), and correction proportional to C^chi (eta = 0), alpha_crit = 1/(1 - chi): 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 chi 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 beta_C 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 = (alpha - 1)/alpha, and the criterion beta_C > k rearranges to alpha < 1/(1 - beta_C) for alpha > 0 and beta_C < 1; this returns the ceiling relation only under the identification beta_C = chi, 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.

4.4 The pacing family

In the pacing case (kappa = 1, zeta = -1) with a direct depth term, alpha_crit = (1 + eta)/(1 - chi), a positive ceiling for chi < 1 and 1 + eta > 0; at chi = 1 the trend is 1 + eta independent of alpha_R; at chi > 1 with 1 + eta >= 0 the trend alpha_R (chi - 1) + (1 + eta) is positive for every positive alpha_R, as the sum of a positive and a non-negative term. At chi = 1/2 this family gives 2(1 + eta). That value is not the general boundary: 2(1 + eta) is the pacing-case value; the general boundary at chi = 1/2 is (eta - zeta)/(kappa - 1/2), which equals 2(1 + eta) for every eta only when kappa = 1 and zeta = -1; at single points the two can agree elsewhere (kappa = 3/2, zeta = -2 and eta = 0 give two on both sides), a coincidence at one point that does not make 2(1 + eta) the general boundary. So, in the pacing special case, at the same-class cap chi = 1/2, the boundary is 2(1 + eta), and the ARC Bound, alpha <= 2, is its eta = 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 kappa - 1 (the programme's mu_B) and its direct depth-dependence of burden is zeta + 1, not zeta, 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 + eta - (zeta + 1))/(1 + (kappa - 1) - chi) is (eta - zeta)/(kappa - chi) identically, and the registration's "both direct terms and the burden intensity vanish" is zeta + 1 = 0 and kappa - 1 = 0 with eta = 0. Its worked numbers hold: chi = 1/2 with eta = 1/2 gives three; chi = 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.

4.5 Proposition B-5 (varying growth)

If burden is proportional to dC/dR along an arbitrary smooth increasing path, then W is proportional to alpha_R C/R and not to C/R, and with correction proportional to C^chi at constant chi,

Delta_bal = 1 - alpha_R (1 - chi) - d ln alpha_R/d ln R,

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 alpha_R is constant on that range, that is only for a power law there, where Delta_bal = 1 - alpha (1 - chi); at a single depth it vanishes wherever alpha_R is stationary, which a path that is not a power law can reach: ln(C/C_0) = (ln(R/R_0))^3/3 + ln(R/R_0) has alpha_R = 1 + (ln(R/R_0))^2, stationary at R = R_0. Otherwise the general chain rule must carry the dependence inside W or carry alpha_R as an extra state variable. Merely replacing a constant alpha by alpha_R in the simplified ceiling loses this term.

4.6 Proposition B-6 (sensitivities, the gap identity, and what an intervention does)

With the other quantities fixed, d alpha_bal/d eta = 1/(kappa - chi); d alpha_bal/d chi = (eta - zeta)/(kappa - chi)^2; d alpha_bal/d kappa = -(eta - zeta)/(kappa - chi)^2; d alpha_bal/d zeta = -1/(kappa - chi). So raising eta raises the ceiling when kappa > chi, and raising chi raises it only when eta - zeta > 0. In the pacing case at fixed eta, the build-order gap is alpha_crit(first) - alpha_crit(after) = (chi_first - chi_after) alpha_crit(first) alpha_crit(after)/(1 + eta) in the positive-ceiling domain. In the same pacing case, with kappa = 1 and zeta = -1 in both arms, where a placement contrast moves both terms, in-loop placement raises the boundary if and only if (1 + eta_in)/(1 + eta_out) > (1 - chi_in)/(1 - chi_out), with 1 - chi > 0 and 1 + eta > 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 kappa = 2 and zeta = -1 in both arms, (chi, eta) = (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 in [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 alpha_bal 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 chi = 1/2 and eta = 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 7 records. No numerical band is chosen here; P3's statement, refuter (registration version 1.102, line 851) and scoring are unchanged, and a positive eta asserted after a run never excuses a P3 exceedance.

The Boundary Transfer (PROPOSED). A held-out prediction of the balance transition from independently estimated elasticities inherits their precision. The registration reports the growth exponent's robust re-estimate as approximately 0.49, 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); that interval's positive part contains alpha_crit = 2 at the registered dials, and Delta_bal 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 a proposed study, not law content.

4.7 Proposition B-7 (the capability coordinate)

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. Read in the new coordinate, the growth elasticity alpha_R is multiplied by that elasticity, chi and kappa are divided by it, and eta, zeta and Delta_bal are unchanged; so kappa - chi keeps its sign, alpha_bal is multiplied by the same elasticity, the verdict alpha_R < alpha_bal is unchanged, and every contrast of capability elasticities between arms at a common capability (chi_first - chi_after, chi_in - chi_out) is divided by it. Proof: chi in the new coordinate is the derivative of ln Q in the new log-capability at fixed R, which is chi times the derivative of the old log-capability in the new, the reciprocal of the map's elasticity, and likewise kappa; eta and zeta are taken at fixed capability, which the new coordinate fixes exactly when the old one does; alpha_R is multiplied by the elasticity by the chain rule; and Delta_bal = d ln(Q/W)/d ln R along the path names no capability coordinate. 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). alpha_crit = 1/(1 - chi), 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, kappa = 1 becomes one half), so re-expressing capability alone and reapplying 1/(1 - chi) shows nothing: with alpha = 9/5 and chi = 9/20 the bound is 20/11 on C; on C^2, alpha = 18/5, chi = 9/40 and kappa = 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 7 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.

5. Dependencies, logical separation and conventions

  1. Register dependencies (the statement paper's dependency table, 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.
  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). Law III's relation holds at every value of chi, 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 (chi = 1/2, eta = 0, kappa = 1, zeta = -1) with a growth exponent of 3/2, below two, so that Laws I to III hold in each: (a) an interval on S_ext(1) from 0.85 to 0.95, which would refute P15 and so Law IV's formal content; (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 Law V would be refuted under the severable-limb default; (c) an interval on S_ext(1) from 0.2 to 0.4 with the intervals of model (b), under which Law IV's face would be supported while Law V would fail. 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, eta_in = eta_out would make G_emb constant in depth along a growing path only together with chi_in = chi_out (Identity V-1), and its constancy is not what fires P21's refuter. These are separations of statements, not empirical findings. A countermodel proposed for the case in which Laws I to III hold while Law IV fails takes a collapse, S_V(x) = exp(-10 x), which on the external arm would support P15; it reads the failure of Law IV in the comparative sense of section 2.6, the opposite direction from model (a), which is one more reason the comparative content is PROPOSED and the author's first question. 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 placement mechanism reading eta_o,in > eta_o,out denies in at least one placement, so a Law V that holds limits where Law III's simple form, and the ARC Bound read from it, can apply; that link is confined to the boundary reading and to the eta term of the Bound-reading control, and a P20 verdict is scored in ARC-3 alone.
  3. The register convention. An evaluable P15 refutation withdraws Law IV whole and Law V as a law of the theory; Law V's refutation does not withdraw Law IV; P12 and P21 keep their own measured verdicts. This is a grouping convention fixed in advance, not an entailment: nothing in eta_o,in > eta_o,out or in the P12 endpoint is entailed by any value of S_ext(1). Its registered basis is registration version 1.102, line 1085, which 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"; its published register dependency is the statement paper's dependency table ("ARC-4 falls and takes ARC-5 with it"), whose trigger in the same paper is the comparative observation of its kill sentence (section 2.6), not P15; this candidate's trigger is P15's refutation under K-IV item 2, and the difference is recorded here. It may be changed only by the author's explicit successor decision (section 7).
  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. The Honey Architecture 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 6).

6. The PROPOSED register

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.

Object Status What it needs
K_V(H; T, epsilon_V), the persistence kernel, formalising the invariance object of the author's working paper of 16 August 2026 PROPOSED primary invariance object; Lemma IV-K proved a registered finite battery or a probabilistic family with tolerance; thresholds; horizon; the interval rule of section 2.2
pi_V, Pi_V(H), the removal margins PROPOSED premise quantities of Theorem IV-G a registered causal decomposition; an objective scored independently of V (premise (G0))
M_V^(H), the strategic viability margin, read path-wise (section 2.5) ADOPTED as PROPOSED; Theorem IV-H and Lemma IV-M' proved the planner's declared objective, feasible family and tolerance; a registered search protocol with an upper bound on the safe supremum; reported one-sidedly
the robust margin PROPOSED; not printed under a colliding symbol a registered environment family
local elasticities chi_ol, eta_ol (the author's working paper of 16 August 2026; the registration of 8 September 2026, registration version 1.102, line 267) ADOPTED as the author's own reading, dated 16 August 2026; arm-labelled here crossed or off-path design for identification; the estimate of registration version 1.102, line 387
the balance frontier, the zero set of Delta_bal (the general boundary form of the author's working paper of 16 August 2026) PROPOSED object without a symbol; Law III unchanged; not a restatement of Law III a specified path field alpha_R; a precision gate
Lambda_V, the mediation bridge ADOPTED as PROPOSED, with its own status (below) its own registration in a scope that carries both contrasts; both retentions positive; an adverse margin; multiplicity
IV-PERSIST, S_emb(x) > S_ext(x) (the statement paper's comparative sentence and kill, recorded and not adopted) the same test as Lambda_V on the difference scale; one object, two readings; not Law IV's content in this candidate as for Lambda_V; the author's first decision (section 7)
the estimable model of section 3.4 PROPOSED model, not a required analysis of any frozen unit the closed model set; a selection rule fixed in advance; registration version 1.102, line 387
the candidate retention families and the descriptive statistics PROPOSED reporting; never verdicts the closed family set; the author's budget scale and baseline convention
the interval rule for Omega_V membership (section 2.2) and the battery-relative kernel kill (K-IV item 3) PROPOSED kill, not law content registration of the battery, tolerance and interval method
the law-level clauses of K-IV items 2 to 4 and K-V items 1 to 2 beyond the registered refuters PROPOSED conventions; P12, P15 and P21 scored as registered the author's successor decision
the recursive-improvement prerequisite (section 3.2) PROPOSED prerequisite in words; both mutual-information forms declined as criteria the author's registration before any surface states it as a condition of any law; until then a description, and not a law
the genesis reading, chi_first - chi_after >= 0.10 (formation first carries the higher elasticity by at least P12's registered minimum effect) 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) the author's successor decision; P12 keeps no direction
the Boundary Transfer PROPOSED study its own preregistration; a precision gate
planned removal (the planned-removal challenge) PROPOSED study; the observation that decides IV-H's margin premise (Lemma IV-M') its own preregistration; a registered upper bound on the safe supremum
a restatement of Laws I to III under new names DECLINED: not adopted; Laws I to III stand as registered; the glosses growth, balance, boundary, persistence, architecture are descriptions, not names the author's decision, and never in 1.102
a mutual-information positivity as an improvement criterion, state-based or update-based DECLINED as a criterion, for the reason of section 3.2; the prerequisite is kept in words nothing
the half-life as P15's verdict; an interval capped at one, the kill written as the interval inside [1/2, 1] DECLINED (section 2.6) nothing

Ruling on the Lambda_V mediation bridge. Adopted as PROPOSED, with this status line: "PROPOSED mediation contrast; not law content; Lambda_V := ln(S_emb/S_ext) at the evaluation budget, defined only where both retentions are positive; the primary contrast is the difference S_emb(x) - S_ext(x), which is always defined, with Lambda_V reported beside it where defined; 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 proposed mediating sequence, embedding to the elasticity contrasts to Delta_bal 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.

7. Open questions

There are fifteen questions in all, numbered below. Each default stands until the author says otherwise, and none is chosen after a result.

  1. Law IV's content (the first decision, on which items 2 and 6 turn): P15's face alone, as the author's statement of the laws gives it and this candidate adopts; or the comparative claim, S_emb(x) > S_ext(x), with a registered comparative refuter and instrument, which the statement paper's ARC-4 sentence and kill (section 2.6) and two other readings (section 2.8 item 2) presuppose. Default: P15's face alone. The loss mechanism, active removal against dilution, stays unresolved under either.
  2. What a P15 refutation takes from Law IV: the whole law, as K-IV item 2 has it (default, coherent with item 1's default); or the comparative headline kept, which would need its own registered refuter before it could stand as a law.
  3. The register convention: whether an evaluable P15 refutation withdraws Law V as well as Law IV (default: yes), and whether its trigger is P15's refutation (this candidate) or the published comparative observation (the statement paper's kill sentence, section 2.6).
  4. Whether Law V is severable by limb (the default), a strict conjunction, or stands on the placement limb alone.
  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.
  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 2.6) and the value of Y_min. The floor's form (a lower confidence bound; the interval method) is a decidability requirement this document fixes, not his choice.
  9. Whether Lambda_V enters the successor 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).
  10. Whether the genesis reading of the build-order limb, chi_first - chi_after >= 0.10 (formation first carrying the higher elasticity by at least P12's registered minimum effect), enters the successor as a PROPOSED clause beside an undirected P12, with the one strike rule of K-V item 2.
  11. P3's tolerance (registration version 1.102, line 843), on which the Bound-reading control of section 4.6 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_*, epsilon_V, H) are fixed in each unit's registration, not in this document.
  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.
  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).
  15. The successor registration's version number and its date, and in it 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).

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 PROPOSED unit; the rule (compute M_V^(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; only their registration remains, which is item 15. The identification of beta_C with chi is not settled by anyone's word: the registration makes it an empirical question that needs an intervention (registration version 1.102, line 1029).

8. Dating

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.

9. Checks

Every identity and derivation in this document was checked by computer algebra.

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.

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