Recursive Dynamics

The engine before the theory

In 1824 the steam engine was the most consequential machine on earth, and nobody could say what limited it. Engineers improved engines the way engineers do, by building the next one; the question of what any engine could do, independent of its construction, was not merely unanswered but unasked, because it belonged to no existing subject. Sadi Carnot asked it in a memoir that founded a science, and the pattern has repeated since: information theory arrived in 1948 because communication engineering had outrun every mathematical frame that existed.

The present moment has exactly this shape. Systems now improve their own outputs using their own outputs: models that critique and revise their reasoning, pipelines that generate their own training signal, laboratories automating parts of their own research loop. They work. Ask what bounds them and the literature answers with scenarios and forecasts, not with state variables and limit theorems. That is not a criticism of the literature; it is the diagnostic signature of a missing field.

When working machines precede their theory, what forms is not a branch of an old field but a new one, because the old fields’ native objects were never the machine’s behaviour as such.

Recursive Dynamics is proposed as that field for self-improving systems. The rest of this page says exactly what is claimed, what is not, how the ARC Theory sits inside it, what one bound looks like worked in full, what would dissolve the proposal, and what would have to happen before anyone should call it a science.

Why now: the engine is running

A dated register of other people’s field events, each linked to its primary source. None is graded as this programme’s prediction; the point is only that the machines exist, that the oversight literature already assumes an answer to the question this field asks, and that the impossibility line has moved from argument to theorem.

• 18 Dec 2024Greenblatt et al., Alignment faking in large language models: a deployed model observed strategically complying under training, the first widely replicated field report of a system managing its own oversight.
• 20 Jan 2025DeepSeek-AI, DeepSeek-R1: reasoning capability incentivised through reinforcement learning on the system’s own verified outputs, at consumer scale. The loop, as a product.
• 2025Zhang et al., Darwin Gödel Machine: an agent that rewrites its own code and keeps the variants that score better. The loop, as an explicit research object.
• 2025Engels, Baek, Kantamneni and Tegmark, Scaling laws for scalable oversight: the oversight literature builds ladders of same-class correctors, correctors built from the same kind of machinery as the system they correct, on the premise that they scale. That premise is the rival view one of this field’s laws is written to test; the law is stated below.
• 2025 to 2026The impossibility line, conceded in this programme’s antecedents register: Yampolskiy 2020, On controllability of AI, then formal results from Yao 2025, The alignment trap and Gumbau Mezquita 2026, The unverifiability theorem: perfect containment of advanced AI fails in principle, not merely in practice.
• 4 Nov 2025Sharma and Chopra, The Sequential Edge: Inverse-Entropy Voting Beats Parallel Self-Consistency at Matched Compute: sequential refinement beats parallel self-consistency at matched compute, in another team’s hands. Graded concurrent per Paper XI row 3: convergent in direction with this programme’s own measurement, never equated with it and never its prediction; publication priority on the quantified comparison theirs.

What is claimed here, exactly

I coined this name, and this page is the claim made carefully. What is claimed: systems that improve themselves using their own outputs share a common dynamics, whatever they are made of. That dynamics has its own state variables, its own conjectured laws and its own characteristic impossibilities. It deserves study as a field, and the field is named Recursive Dynamics. The vocabulary is the record’s own: the sealed manuscript record of 8 December 2024 already speaks of “the recursive dynamics that govern existence itself”, held verbatim against the record’s hash in the quote register. The name was adopted as the field’s name on 27 August 2026.

What is not claimed: that the field is established science. Its laws are conjectures with declared statuses; its decisive experiments have not run; independent replication stands at zero, dated and honest, on the attempts register. Naming a field is a proposal about how to organise questions, never evidence that any answer is right.

The case for the field, scored against six licence conditions

A new area of science is not declared; it is licensed by conditions that can be checked. The founding paper proposes six, scores the field on each, and scores it unflatteringly: three met, two half met, one not met, and the unmet one is the one no author can supply.

A field with this scorecard is a proposed field. The three met conditions are what any careful programme can supply on its own; the sixth is the one no author can supply and the one that decides.

The object, and why it belongs to no one

A model checking its own answer and rewriting it. A pipeline scoring what it produced and training on the winners. A laboratory whose next experiment is chosen by the last one. Define a recursive system as any system whose outputs re-enter its own improvement process. The field’s native object is that loop. Its neighbours each own a tool, and none owns the object; each is named here with what it does own, because the field is assembled from debts to them, conceded in the references register.

The honest attempt to house it elsewhere

A table of neighbours dismissed in a sentence each reads as marketing to exactly the readers who matter, because it never tried the fit. The founding paper tries, and the objection that it had not (A11, below) is conceded for every version before 2.3 and for the first version of this page. For twelve candidate hosts, the three questions (how fast can self-improvement go; what bounds it; what overseer keeps up) are stated in the host’s own variables, everything the host already covers is conceded to it, what breaks is stated precisely, and the registered result that would move the field into that host is named. Three hosts turn out to hold real pieces of the theory, and they are conceded outright; they are named as fields at textbook level, not cited, because nothing here is cited before it is read.

The other nine could host pieces. Adaptive control could take Law II as a chapter on the stability of self-modifying regulators if the class variable proves empty; software reliability engineering owns the shape of the two rate estimators; machine learning hosts every registered experiment and is the likeliest institutional home if the field fails to earn its name; scaling-law empirics hosts the method and not the object; scalable oversight owns the class question institutionally and would publish the eclipse result whichever way it falls; statistical physics would take the cross-domain families if they survive and belong to no loop; the self-referential-improver line is ancestry, not a host; second-order cybernetics is a philosophical host without instruments; metascience is a domain the field would be tested in, not a home.

After the twelve attempts, one residue survives that no host can state in its own variables: a cap on correction leverage that depends on what the corrector is made of, coupled to a growth exponent in recursive depth, so that the sustainable rate of self-improvement is a property of the corrector’s class. Growth theory has the exponent and no corrector; control and queueing have the stability condition and no class; oversight has the class and no bound; machine learning has the loop and no law. Recursive Dynamics is the name for that coupling. If the coupling is empty, the parts go home and the name dies, and the field will have cost its neighbours nothing but a careful table.

A field is licensed when the questions that matter are about an object none of the existing fields takes as primitive. How fast can self-improvement go, what bounds it, and what kind of overseer can keep up: those are questions about the loop.

Five state variables are proposed to matter regardless of substrate, written here exactly as the programme’s notation spine defines them, because the spine is the one place a symbol is allowed to mean something:

• U · capabilitywhat the system can do, on a calibrated ladder with no ceiling
• R · recursive depthhow many times outputs have re-entered the loop
• beta_C · correction ratehow fast internal correction strengthens as capability grows; the quantity Law II asks to out-scale drift
• k · drifthow fast errors and misalignments accumulate as the system modifies itself
• gamma · correction leveragehow correction capacity scales with capability; the exponent in the ceiling law

The shortlist will look arbitrary. So did pressure, volume, temperature, energy and entropy, until the laws relating them held. The claim that five suffice is part of what the field must prove. The experimental dial the registered studies turn, the share of effort reinvested in the improvement process itself, is a manipulation rather than a state variable, and it carries no symbol here on purpose: the estate’s own notation register records a bare beta doing five different jobs across the corpus as a named error class, and this page declines to add a sixth.

Scale discipline precedes law

Before any law, a measurement principle, learned the hard way and adopted as founding method: capability must be measured on a ratio scale with a true zero and no ceiling. Thermodynamics could not state its laws until temperature was absolute. A bounded percentage score does to Recursive Dynamics what a Celsius-only thermometer does to the third law of thermodynamics: it makes the laws unstatable, because a score capped at 100 per cent cannot follow a power law in anything. The field’s instrument is therefore a calibrated difficulty ladder, a latent, ratio-scaled measure with a true zero and no ceiling, on which doubling means the same thing at every height; and every quantitative surface names which of three model families it is reading, unbounded or latent capability, error decay, or bounded performance, because the exponent means something different in each.

The programme adopted this after catching its own worked example running a bounded score through unbounded arithmetic. The correction is recorded in the notation spine, dated 27 August 2026, and the lesson is promoted here from a bug fix to a founding principle.

How the ARC Theory fits inside

In plain words first (ARC stands for Artificial Recursive Creation, the December 2024 document’s own naming). Recursive Dynamics is the field. The ARC Theory is the founding theory proposed inside it. The Eden Protocol is the applied wing the theory points at, the proposal for how correction gets inside a system rather than around it. And the instruments are the floor everything stands on.

The three laws, each with its status declared, because in this programme a law’s status travels with it everywhere:

• Law I · the ARC PrincipleU = I × R^alpha Capability grows as a power of recursive depth. a book-framework claim; exploratory cross-domain support; the exponent is measured, not assumed, and currently reads 0.49 with an interval from −1.3 to 2.9, which decides nothing yet. The early headline exponent was publicly retracted and appears on this estate only as a retraction.
• Law II · the ARC Co-Scaling Lawbeta_C > k Stable self-improvement requires the correction rate to out-scale the drift rate. proved inside a stated minimal model and nowhere else; not yet measured on a system that genuinely rewrites itself. Its decider is the correction-versus-drift race, the field’s simplest and most consequential measurement.
• Law III · the ARC Ceilingalpha_crit = 1 / (1 − gamma) The growth exponent is capped by the reciprocal of the correction shortfall; at the conjectured gamma of one half the ceiling is two, the ARC Bound. derived in front of the reader below, from two lines of assumption; gamma has never been measured on any real system, and the gamma-equals-one-half premise is named by the programme itself as the conjecture’s single most likely point of failure.

Severable by design: a reader can accept the field’s questions and reject the theory’s answers. The field is defined by its questions; the theory stands or dies inside it, by measurement.

One bound, worked in full

Carnot did not describe engines; he computed what no engine could beat. Here is this field’s equivalent, in two assumptions and four lines, exactly as it stands in the programme’s public notation spine.

That is the entire engineering content of the field in one sentence: the corrector’s composition class is the lever. No ladder of same-class oversight climbs past its ceiling by adding rungs.

These four lines do not establish bounded harm, finite-horizon safety, or a law of nature. They establish a conditional crossover whose premises are measurable, which is exactly what a young field’s first bound should be. An outside reader’s criticism of the derivation’s depth treatment is printed as a limitation, not argued away; and the retired form of this law, with gamma in the denominator on its own, was retracted on 16 August 2026 and appears on this estate only as a retraction.

The precedent, and the disanalogy printed beside it

The honest model for what is being attempted is thermodynamics, and the parallel is in the shape of the field, never the standing of the evidence.

Now the disanalogy, stated before anyone else can state it: thermodynamics was built by many hands converging independently and confirmed by a century of experiment. This field is proposed by one person; its decisive experiments have not run; independent replications stand at zero. The parallel is a scaffold, not a certificate. Thermodynamics earned its name; this name is staked, and the stake is printed below.

The rival’s best case, steelmanned

The strongest objection is not that the laws are unproven. It is that the mechanism may be empty: correction may just be correction, in which case what a corrector is made of does not matter and the class distinction dissolves. This rival is not a straw man. The scalable-oversight programme in all its forms, weak-to-strong generalisation, debate, amplification, recursive reward modelling, builds same-class ladders on the implicit premise that they scale, and the strongest published framework adjacent to this question is architecture-blind: it implies the class-blind null without needing to state it.

Worse for this programme, part of its own record leans the rival’s way. The corrector-class ratio itself has not been measured on real data, and this estate says so first. But the programme’s published parallel-channel measurement (Paper II) found parallel channels adding almost nothing in five of six models, the signature of correlated errors, with one model contradicting the pattern outright at a parallel exponent of 0.31. Correlated channels are exactly the mechanism that would weaken same-class correction in a way that also weakens cross-class correction, dragging the decisive ratio toward one. The eclipse registration is therefore filed with that contrary evidence on its face, and commits to publishing a null result under the same title as a discovery about the world rather than an embarrassment to be buried.

A field that files its rival’s best evidence inside its own registration is doing the one thing that cannot be faked.

Substrate independence, at its honest grade

The field’s laws must not care what its systems are made of, and the programme’s cross-domain work points that way at exploratory grade, reported with its tiers never blended: predicted functional families across a fifty-domain suite (19 of 25 empirical matches under the original candidate set, 18 of 25 under the corrected seven-model rerun, disclosed in full; 13 of 13 on published-direct rows); a twelve-domain locked-manifest extension (10 of 12, p = 5.4e-4, self-demoted to a pilot dry run within thirteen minutes in public commit history); temporal out-of-sample checks (4 confirmed, 2 partial, 1 inconsistent, the inconsistency reported as prominently as the confirmations). None of this is confirmation. The registered fourth-cell test, the logarithmic family the corpus has never fitted, on domains never fitted, exists to catch the programme flattering itself, and convergences with older traditions are catalogued separately under a standing rule: convergence is evidence of naturalness, never of truth.

Instruments

A field is its instruments before it is its results. The field ships with four instruments, built before any confirmatory data existed. First, the calibrated capability ladder. Second, a blinded scoring stack: every output is laundered so no scorer can tell what produced it, and the null is computed through the identical pipeline with the labels randomised, so the decisive experiment’s null is imposed structurally rather than fitted. Third, estimator code that is hash-pinned and fixture-gated, every registered verdict branch carrying a planted-truth test the code must pass before anything ships. Fourth, a registered defect clause: the specification, not the code, is the analysis, so a post-registration coding error is a disclosed repair on registered terms rather than a dead study.

The deciding experiments

Three registered studies decide the three laws. Their nulls belong to the rivals, their numbers are computed and printed in the registrations, and their outcomes are publishable on identical terms whichever way they fall. All three are drafted, dated and awaiting human submission; no lane, tool or agent submits anything anywhere. The protocols themselves are complete: any independent laboratory can run them today without asking anyone, and what awaits the author’s hand is the registration act alone.

The correction-versus-drift race decides Law II: does self-correction out-scale drift at all, anywhere? A durable yes is the field’s first existence result, its Joule experiment: the first demonstration that the effect exists at all. A universal no leaves Law II without a subject.

The corrector-class ratio, the eclipse measurement, decides Law III’s mechanism. Pool same-class and cross-class corrector pairs, launder everything, and force the null: if what a corrector is made of does not matter, the ratio of correction exponents is exactly 1.00 by construction. That null is the rival’s answer, not this programme’s. The programme’s mechanism predicts a durable ratio above 1.00 where the cap binds, and that is the discovery in its favour; durably below 1.00 inverts the mechanism and ends the claim as derived; 1.00 holding tight everywhere, even where the cap should bind, retires the class distinction itself. A hard independent kill rides alongside: any same-class pair whose gamma interval sits wholly above one half kills the cap regardless of the ratio.

The capability-ladder titration decides the ceiling’s behavioural signature: an interior maximum of the growth exponent at or below two across a range of reinvestment levels, under a two-stage replication-gated rule in which one counterexample cell, replicated, rejects the ARC Bound outright.

A registered follow-up grid extends the ratio to every ordered pair of model families, locating where the class effect lives; and a mechanism probe, correlated failure on shared substrate, measures directly the error correlation the derivation posits, so the ratio and its mechanism must agree or their tension is itself a registered finding.

Three further tests are proposed and not registered, and the page marks them so that no reader counts them as predictions: that the ratio rises with load, which would separate a mechanism from a static difference; that a cross-class corrector moves the titration’s maximum upward, linking Laws II and III through the class lever; and that a single system crosses the stability boundary as its reinvestment dial turns. None is a prediction of the field until a registration carries it.

What would dissolve the field

A proposed science must say what would un-propose it.

1. If the correction-versus-drift race finds no regime where correction out-scales drift, Law II has no subject matter.
2. If the corrector-class ratio holds at 1.00 with tight intervals wherever the cap should bind, the field’s distinctive mechanism is dead and its remains belong to ordinary scaling empirics.
3. If the ladder titration finds no interior maximum and no ceiling behaviour, the central impossibility dissolves.
4. If the cross-domain families fail the fourth-cell test, substrate independence was an artefact of family flexibility.

The dissolution map by host. If the founding conjunction fails, the parts do not vanish; they go home. Law I returns to endogenous growth theory as a knowledge-production exponent; Law II to adaptive control and error-threshold theory as a rate condition; the ARC Ceiling to queueing theory as a growth-coupled stability condition; the class-cap result, whichever way it falls, is published in scalable oversight; the instruments and the experiments are machine learning; the cross-domain families, if they survive, are statistical physics. Nothing is orphaned by the field’s death, which is what makes the death publishable.

Any one death wounds the field; all four end it, and this page commits to publishing that ending here, under this name.

The name’s own kill condition

A name proposed before its tests must be able to die with them, and the record must be able to say which death it was. A law failing in its stated form kills that law and the theory that stated it; the field carries the corrected form, or none, as the record allows, and a reader who rejects all three laws and keeps the measurement is still inside it, by the minimal commitment stated below. The name itself dies when the object goes: if the corrector-class ratio holds at 1.00 with tight intervals wherever the cap should bind, the distinctive mechanism is dead, the residue no host could state has proved empty, every piece returns to its host by the dissolution map above, and this page will say so rather than quietly disappear. If the laws survive their registered tests and hold outside the domains that raised them, the name has earned its generality. Either way the record decides, not the naming. The conditions live on the falsification dashboard; the tests need no permission to run.

What would make it a science

Naming a field does not create one. The entry criteria this proposal has not yet met, in public: at least one decisive experiment returning its predicted direction under its registered analysis; independent replication by hands that are not the author’s, which is why every instrument, seed, schedule and threshold ships in the open under a standing invitation to run the test that could kill it; use of the state variables by strangers because they needed them; and a survived audit trail, of which the programme’s record of public retractions, minute-resolution self-demotions, adversarial cold reads and printed limitations is offered as the standard its future claims accept.

A result that any old field could state in its own variables confirms a chapter of that field. A field is confirmed only by a conjunction none of its neighbours can express, and the founding paper states it once: a regime in which correction out-scales drift; a corrector-class ratio durably above 1.00 with the one-half cap on same-class correctors; an interior maximum of the growth exponent at or below two; the same exponent structure in at least one non-artificial substrate; and at least one of these replicated by strangers. Each alone is a finding in an old field’s vocabulary. The conjunction is not, and that is the exact sense in which results would confirm a new area rather than a new chapter.

On the word paradigm, and why this page withholds it

A paradigm, in the strict sense, is conferred by a community adopting a shared exemplar. Nobody has a paradigm of one, and nobody has ever founded one by announcing it; it becomes a paradigm when other people start working inside it. So the precise, defensible statement is that this is a candidate: a field proposed with its object, its variables, its bounds, its instruments and its deciders in place, whose standing is not the author’s to declare. Replication, adoption and the three registered experiments decide it. Claiming the larger word as a present fact is the single move that would turn a careful proposal into a crank signature, which is why the founding paper pins the field at the Carnot stage and lists the entry criteria it has not yet met.

Keep saying the smaller, truer thing, and let the experiments make the larger one, or bury it.

What this buys, if it holds

Fields earn adoption when strangers can compute with them. If the laws survive their deciders, the following become calculations rather than debates: given a measured gamma for a corrector class, the maximum sustainable growth exponent of any system it oversees follows from the ARC Ceiling; given measured drift, the required correction rate follows from Law II; oversight architectures acquire a specification sheet, the corrector’s composition class, its measured gamma, its implied ceiling, the way engines acquired efficiency ratings; and the safety question “can oversight keep up” becomes, for the first time, an empirical parameter comparison rather than a scenario argument. Everything in this paragraph is conditional on results that do not yet exist, and is written in the conditional for that reason.

Debts, conceded by name

This field is assembled from debts recorded in the programme’s antecedents register and conceded in full: the impossibility line of AI-control research, to which this field adds the design requirement the impossibility implies rather than resignation; the transport-network limits of biological scaling, conceded as premise, to which this field adds the internal replacement limit that operates when fixing the pipe no longer fixes the growth; the intelligence-explosion question as posed by its named owners, to which this field proposes the missing state variables; and the functional-equation mathematics, the machinery, never claimed. What is claimed as original is the assembly: the loop as native object, scale discipline as founding method, the bounds as laws, the corrector’s class as the lever, and the registered instruments to decide all of it.

Why name it before the verdict

Because that is when fields are named. Wiener named cybernetics in the 1948 book that proposed it, before there was a field to attach the name to. Lederberg coined exobiology in 1960, before a single specimen. A field’s name is not a trophy awarded after confirmation; it is the instrument that makes a proposal citable, attackable and testable as one unit. Naming early is the ordinary practice. Claiming the name proves anything would be the error, and this page does not make it.

I did not prove it first. I said it first, in a document anyone can date, and then I built the instruments to test whether it is true.

What the field is not, and the minimal commitment

The field is not a forecast of when machines surpass people, not a theory of intelligence, not an ethics, not a theory of everything, and not a claim about consciousness. Its laws switch on where a corrector exists inside a growing system and switch off where none does; the composed hierarchies its scaling mathematics bears on, stars included, are graded as correspondence of form and no further. The programme’s speculative coda on recursion at cosmological scale sits outside the field; a reader can reject it entirely and lose nothing here.

The minimal commitment required to work in the field is small, and stated so that it can be refused: that the self-improvement loop is a legitimate object of measurement, with quantities that can be defined, instruments that can measure them, and relations that can be shown wrong. If you accept that and reject all three laws, you are inside the field; if you accept the laws and refuse measurement, you are not.

Not to be confused with

Robotics has long used the phrase “recursive dynamics algorithms” for the computational methods that propagate forces along articulated bodies, the Featherstone tradition; that is a family of algorithms, a different referent entirely, and no priority over the phrase as words is claimed against it or anyone. Nor is this recursion theory, the mathematics of computability; nor dynamical systems at large, the parent mathematics. And it is distinct from Recursive Creation, the estate’s speculative coda, held apart from the argument on purpose.

Run the test that could kill it

Every instrument, seed, schedule and threshold on this estate ships in the open, and an independent run outranks the programme’s own. If you want to know whether any of this is real, the shortest route is not to weigh the argument but to run the measurement: the run-it-yourself page lists what can be executed today without permission, what it would take to kill each claim, and where a result, in either direction, gets recorded.

Where the field lives on this estate

• The declaration, on the home page
• The ARC Theory, the founding theory, laws in full
• The laws in plain words, on the journey
• The Eden Protocol, the applied wing
• The notation spine, what every symbol means
• The falsification dashboard, the failure conditions
• The dated prediction register
• Claim status, every claim graded
• The glossary, every named concept
• Corrections, published against myself
• The catalogue, where the founding paper lands
• Run it yourself, the way in for a sceptic