Definitions, precisely

The declaration, first. What this programme calls Laws are named conjectures under registered, adversarial test: named like laws, held as hypotheses. Recursion and intelligence, as this estate uses them, are not established scientific quantities in the way mass, charge or temperature are, and no page here should ever leave the opposite impression.

What is established, with sources. Recursion as mathematics is as settled as anything humanity knows: recursion theory is a foundation of logic and computation, and recursion depth is a precise, countable quantity inside any defined procedure. Scaling regularities are established within single domains: metabolic scaling in biology (West, Brown and Enquist, Science, 1997, with its contested details part of the record, White and Seymour, 2003), urban scaling (Bettencourt and colleagues, PNAS, 2007), and error suppression below threshold in quantum hardware (Acharya and colleagues, Nature, 2024). Each is peer-reviewed science about its own domain, and none of them asserts a shared cross-substrate law.

What is stipulated, so it can be measured. There is no SI unit of recursion and no instrument that reads recursive depth across arbitrary systems the way a thermometer reads temperature. R here is an operational protocol quantity: rounds of a specific, frozen improvement loop, counted under a stated protocol per substrate, never recursion in the Kleene sense. Intelligence has no agreed scientific measure anywhere, in humans a century-contested construct, in machines openly fought-over proxies; this programme therefore never measures intelligence directly, only defined capability on task banks of measured difficulty. A stipulated definition is a lens, not a discovery, and it earns its keep only if measurements made through it reproduce in other hands.

What is proposed, which is the part not established anywhere. That those domain regularities are instances of one compositional law; that its exponent is measurable on the same footing in machines; and that a correction exponent sets a ceiling on stable self-improvement. Law on this estate means what it means in Kleiber’s law, Moore’s law or the Reynolds number: a named regularity proposed for a domain and held to measurement, never legislation of nature. Temperature itself was an operational protocol number for two centuries before thermodynamics grounded it; the Reynolds number was located empirically and trusted because it kept working in other people’s hands. That is the road; these three names are at its beginning: one exponent measured once under blinding, the ceiling conjectured inside a stated model, the correction exponent never measured by anyone.

Why this is said in the author’s own voice, first. A sophisticated reader’s first attack on named laws is reification: conjectures dressed in the costume of laws. The declaration above is the answer, made before the attack: a statement of method, never of doubt. The registered programme exists to earn the standing of a law by measurement, replication and survived refutation, or to lose it in public.

What it is: a composition law. Two senses, both live: in the theory’s current statement the ARC Principle names the three laws together, the way Newton’s laws of motion carry one collective name; historically it names the founding composition conjecture that became Law I, the ARC Equation. The lineage is dated: U = IR, unsquared, in the December 2024 sealed manuscript (in words, Universe = Intelligence × Recursion); the squared form in the private record from 30 April 2025 and in print from 2 January 2026 as the α ≈ 2 simplification; the papers operationalised α as a measured parameter. The square is evidence of priority, not of derivation, and the word’s literal far end is held apart at its own rung, never in the empirical register. Where a paper title says ARC Principle, its masthead plate states which sense it carries. Capability is base intelligence amplified by recursion: U = I × g(R), with the amplification exponent written α = 1/(1−β), where β is the self-referential coupling measured from data. It describes how recursive self-improvement composes, in any system that feeds its own output back into its own improvement. Symbol discipline, and it matters more since the correction of 16 August 2026, not less: this coupling parameter is a different quantity from the correction-leverage exponent γ on the law page, whose ceiling is written α_crit = 1/(1−γ). Before that correction the ceiling was written 1/γ, and the two expressions agreed at exactly one half, which is where both are evaluated. They now share the same reciprocal-shortfall shape 1/(1−x) with different arguments, so the shape no longer distinguishes them at all and only the argument does. No theorem connecting the two quantities exists, so the shared form stands on the record as an open question, never as a unification.

What it is: the proposed stability limit on self-correctable capability scaling. A self-improving system stays correctable only while its internally generated correction keeps pace with its capability; the best internal correction is proposed to strengthen at most like the square root of what it has accumulated; the reciprocal of one half is two. So quadratic is the proposed boundary at which the best brakes a system can build from itself break even with the engine. It is a scaling limit, not a speed limit: it bounds how much growth internal self-correction can carry as recursion deepens, not how fast anything happens in time.

The ruling, on the record: the author states the ARC Bound as a stable limit, not a hard limit. His words, dated 8 August 2026: "Papers I to III were written prior to changing what the ARC Bound was. I propose it as a stable limit, not a hard limit." Crossing it is therefore not impossible; it is supercritical. Above the boundary the framework predicts the loss of guaranteed internal self-correction: correction lag, accumulating deviation, drift out of alignment. Growth does not hit a wall; control does.

What would kill it: a system that runs supercritical, an exponent above 2, sustained without loss of stability. A system that exceeds the boundary and does not destabilise refutes the ARC Bound. This is strictly harder to satisfy than a bare measurement above 2, so stating it honestly makes the claim stronger, not weaker. The live condition sits in the falsification register.

What it is: the operative safety quantity, from Paper X. Under accelerating self-improvement, the misalignment fraction vanishes asymptotically if and only if the correction exponent β exceeds the drift-acceleration exponent k. The load-bearing question is not a fixed number but a race: does correction co-scale faster than drift accelerates?