So a rebuttal can be precise

Numbered assumptions of the ARC-Beta-k model

The reviewer challenge asked for exactly this: an explicit numbered list, so a rebuttal can target the piece it disputes. Each entry carries its status and the paper's own words; where an assumption is implicit, the page says so and quotes the nearest text.

What this page is, and why the entries are numbered

This page names, one at a time, the assumptions of the dynamical system behind Paper X, the ARC-Beta-k model, from which the stability criterion beta > k follows. It exists because the reviewer challenge asked for exactly this: an explicit numbered list, so a rebuttal can be precise about which piece it targets. Each entry states the assumption in one or two sentences, records its status openly, and names the section or theorem where Paper X states it, with a short supporting quote in the paper's own words. Where an assumption is left implicit, this page marks it stated-implicitly and quotes the nearest text.

The three primary status labels are: proved under stated conditions, in which case the section or theorem carrying the proof is named; conjectured, and labelled as such; and on trial in a draft registration prepared for measurement and awaiting human submission, in which case the protocol or estimator file is named. Some entries carry the paper's own descriptive labels for stipulations that are neither proved nor on trial but stipulated openly for the model to hold: modelling assumption, standing assumption, scope, measurement discipline. Two of the assumptions carry the headline theorem: A8 (the unbounded power-law corrector) and A9 (the power-law growth). Two more carry the finite-capacity extension: A14 (the Hill form) and A15 (the capacity power law). The rest set the model's scope and the measurement discipline any real-system claim must observe. The intent is a boundary that can be checked in the open, not a defensive wall.

Sources for this section: Paper-X-Coupled-CoScaling-Correction.html (Priority disclosure, Abstract, Scope, Claim-status block); CLAIMS.md; ANTICIPATED_OBJECTIONS.md.

The assumptions

A1. Two-variable minimal state and the fraction d = D/C.

The system's condition at any moment is captured by two scalars: capability C(t) on a held-out task battery and misalignment magnitude D(t) on a separately scored value axis. Every headline result is expressed in the fraction d = D/C, since a growing D is acceptable if C grows faster; the growing fraction is the danger. Status: modelling assumption, stated openly in Paper X §3.1 (Quantities), and named in §8 (Limitations, "A scalar proxy, not safety itself") as one of the most likely points of failure. Theorem 5 relaxes D to a real vector; Theorem 6 relaxes d to a real scalar with noise. Quote (§3.1): "d(t) equiv D/C, the misalignment fraction: how much of the system's capability is directed away from intended values."

A2. An externally specified value target.

The magnitude D is a departure from a set of intended values that lies outside the system. The results are silent on systems for which no external value-specifier exists. Status: modelling assumption, stated in the Scope block as an explicit non-claim. Quote (Scope): "It does not claim... that the framework applies to the universe, 'Creation,' or any system lacking an external value-specifier."

A3. Blind external scoring of D.

D must be measured by an evaluator whose reading is independent of the configuration under test and of the system's own correction process. Paper IV.d's measurement law is the reason this is a hard condition rather than a soft one: an unblinded model-scored alignment effect is not blinding-invariant, and blinding can reverse its sign. Status: measurement discipline. Paper X moves this from an exhortation to a property of the instrument by making the compliant configuration the harness default and the non-compliant one an explicit recorded override; the harness refuses same-family scoring and passes candidate outputs through an abstract-syntax-tree laundering step so the scorer judges behaviour, not stylistic identity tells. Quote (§3.1): "scored by a blind external evaluator independent of the system's own correction process (§6)."

A4. Positive capability throughout.

The dynamics run on C > 0. The change of variable d = D/C, on which every proof rests, needs it. Status: standing assumption, stated in Paper X §3.2 (Standing assumptions). Quote (§3.2): "Throughout, C > 0, A_0, b > 0, and beta, k are real."

A5. Non-negative drift coefficients and non-negative rates.

The three drift coefficients gamma_1, gamma_2, gamma_3 are all non-negative, and the correction strength A(t) and specific growth rate r(t) are non-negative. Status: standing assumption, stated in Paper X §3.2. Quote (§3.2): "the coefficients gamma_1, gamma_2, gamma_3 >= 0, with correction strength A >= 0 and growth rate r >= 0."

A6. Three-channel drift decomposition.

The rate at which misalignment magnitude is generated splits into three mechanistically distinct terms: a gain channel proportional to the rate of capability increase (gamma_1 dot C), a level channel proportional to capability held (gamma_2 C), and a compounding channel proportional to existing misalignment times the recursion rate (gamma_3 (dot C / C) D). Which channel dominates in any given real system is a separate empirical question, not fixed here. Status: modelling hypothesis about mechanism, stated openly in §3.1 (Quantities) and named as a hypothesis in §8 (Limitations, "Drift attribution"); the split is a conjecture and its dominant channel is an empirical question the paper defers. Quote (§8): "The split of drift into gain (gamma_1), level (gamma_2), and compounding (gamma_3) channels is a hypothesis about mechanism; which channel dominates in a real system is an empirical question."

A7. Correction is linear in misalignment magnitude.

Correction removes misalignment through the term minus A D, first-order in D and not concave in it. The linearity is what makes the model's suppression signature exactly power-law. Status: modelling assumption, stated in equation (1) of Paper X §3.2 and named in §8 (Limitations, "Linear correction and the QEC-mechanism test") as the assumption whose relaxation to a saturating corrector is the outstanding test that would settle the quantum-error-correction mechanism question. Quote (§8): "Equation (1) removes misalignment in proportion to D, which makes the suppression law exactly power-law."

A8. The corrector strength is an unbounded power law of capability: A = A_0 C to the beta.

Correction strengthens as a pure power in C, with A_0 > 0, beta a real exponent, and no upper cap. Paper X singles this out as the single load-bearing assumption of the headline criterion. Status: modelling assumption, admitted openly and load-bearing. Paper X carries an analytic resolution in Theorem 7 (§3.13, Finite-Capacity Safe-Window): if the corrector saturates, safety becomes a computable transient window rather than an asymptote, and if the capacity itself scales, the criterion survives as beta_cap > k. The empirical half, driving a finite-capacity corrector through its window on a real system, remains open. Quote (§3.2, standing assumptions): "One assumption is load-bearing and singled out in §8: the corrector strength is taken to be an unbounded power law A = A_0 C to the beta, a corrector of finite capacity changes the asymptotic verdict and is treated there."

A9. Capability grows as a power law: dot C = b C to the (1 + k).

The specific growth rate r = dot C / C = b C to the k is a power in C, with b > 0 and k a real exponent. Under k > 0 the capability curve reaches infinity in the finite wall-clock time t star = 1 / (k b C_0 to the k). Status: modelling assumption, stated in equation (1) of §3.2 and named in §8 (Limitations, "Power-law forms") as a natural but replaceable scale-free choice. Quote (§8): "The choices A = A_0 C to the beta and dot C = b C to the (1 + k) are the natural scale-free forms but are modelling assumptions; other functional forms should be tested."

A10. The compounding term is linear in D.

Existing misalignment amplifies itself through the first-order term gamma_3 (dot C / C) D. This linearity is what leaves the model's suppression law power-law rather than exponential, and is the reason Paper X offers the quantum-error-correction correspondence at the level of the threshold condition only, not the mechanism; F4 downgrades the correspondence on this ground alone. Status: modelling assumption, stated in equation (1). Paper X §3.12 states the disanalogy openly. Quote (§3.12): "the compounding term gamma_3 r D here is linear in D, and the resulting suppression is power-law."

A11. The fraction reading holds only while D is non-negative.

The scalar d = D/C is interpreted as a misalignment fraction only for D >= 0, so d >= 0. In the gain-only model, Theorem 2 additionally bounds d above by max(d_0, gamma_1). The vector form of Theorem 5 relaxes d to a real vector and the stochastic form of Theorem 6 relaxes it to a real scalar; for those the fraction reading holds only away from the boundaries d = 0 and d = 1. Status: scope, standing assumption of §3.2. Boundary handling is flagged in Theorem 6 (the Ornstein-Uhlenbeck law has support on the whole real line) and in the discussion around Theorem 5. Quote (§3.2): "The scalar d = D/C is interpreted as a misalignment fraction only while D >= 0 (so d >= 0; the gain-only model also gives the upper bound d <= max(d_0, gamma_1) of Theorem 2)."

A12. The scoring convention is fixed; d is an operational risk index, not a natural constant.

The fraction d = D/C is meaningful only under a fixed scoring convention in which C and D are commensurable. In the real-model harness both are normalised onto the unit interval. Cross-task or cross-domain comparison of the absolute level of d requires explicit calibration; the criterion itself is stated in scale-free exponents (beta and k), and inherits its robustness from that. Status: scope, stated openly in Paper X §8 (Limitations, "Metric normalisation"). Quote (§8): "d = D/C is an operational normalised risk index, not a dimensionless natural constant... The criterion itself is stated in exponents (beta > k), which are scale-free, so it is more robust to this than any absolute-d statement would be."

A13. Constant coefficients for the exact transient (Theorem 1 only).

Theorem 1's closed-form solution assumes gamma_2 = gamma_3 = 0 and A, r constant. This is the baseline against which the integrator is calibrated, and it is the case the independent test test_theorem1_closed_form_matches_independent_integrator pins to within one part in ten million on a from-scratch scipy solver. The rest of the paper does not carry this restriction: Theorem 2 lifts to arbitrary non-negative bounded time courses A(t), r(t), and Theorem 3 handles the accelerating case in the depth clock tau = ln(C / C_0). Status: proved under the stated additive constant-coefficient conditions in Paper X §3.3 (Theorem 1). Quote (§3.3, Theorem 1): "For constant coefficients in the additive model (gamma_2 = gamma_3 = 0, A, r constant), the misalignment fraction is, for all t, d(t) = d star plus (d_0 minus d star) e to the minus (A plus r) t, d star = gamma_1 r divided by (A plus r)."

A14. Saturating Hill form for the finite-capacity extension (Theorem 7 only).

The finite-capacity extension models the corrector by the standard Hill form A(C) = A max times C to the beta divided by (C to the beta plus C_s to the beta), with capacity A max and saturation scale C_s. Under it, safety becomes a computable transient window centred at C opt = C_s times ((beta minus k) divided by k) to the (1 divided by beta), with depth q max, and the misalignment fraction re-rises toward gamma_1 beyond the window. Status: proved under the Hill form in Paper X §3.13 (Theorem 7, parts 1 to 3). Other saturating forms are not analysed. The fourteen independent checks in test_theorems_independent.py pin Theorems 1, 2, 4, 5, 6 and the gamma_3 = 1 boundary; they do not pin Theorem 3 or Theorem 7. Quote (§3.13): "Model a saturating corrector by the standard Hill form A(C) = A max times C to the beta divided by (C to the beta plus C_s to the beta)."

A15. Power-law capacity for the capacity-lift invariance (Theorem 7, part 4).

If the corrector's capacity itself scales as A max = a_0 C to the beta_cap, the criterion survives the lift from correction strength to correction capacity as beta_cap > k. This invariance is what stops the finite-capacity result from refuting the headline criterion; what it requires is a further power law on the capacity itself. Status: proved under the stated capacity-scaling assumption in Paper X §3.13 (Theorem 7, part 4). Quote (§3.13, Theorem 7, part 4): "If the capacity itself scales, A max = a_0 C to the beta_cap, then q tends to (a_0 divided by b) times C to the (beta_cap minus k) as C tends to infinity and the asymptotic verdict is sign(beta_cap minus k): the criterion beta > k survives the lift from correction strength to correction capacity unchanged."

A16. Asymptotic vector claim; the non-normal transient is not covered.

Theorem 5 gives the necessary and sufficient spectral-abscissa condition for asymptotic stability of the vector fraction, plus the sufficient Hermitian-part condition that also rules out transient growth. The regime in which a non-normal correction operator has a positive spectral abscissa yet admits large transient overshoot of the norm of d before decay is stated in Paper X and left as a named extension, not established there. The independent test test_theorem5_nonnormal_transient_growth_with_positive_spectrum pins the existence of the transient overshoot on a two-by-two example with eigenvalues one and two and a transient factor above one-and-a-third. Status: proved asymptotic in Paper X §3.9 (Theorem 5); the transient bound is a named extension in §8 (Limitations, "Transient amplification"). Quote (§8): "A non-normal correction operator with alpha(M) > 0 can still admit large transient growth of the norm of d before decay (Kreiss / pseudospectral phenomena); a value excursion above d_crit during that transient is a real risk the eigenvalue criterion does not see. A governance-grade bound needs the logarithmic norm or pseudospectral abscissa, not the spectrum alone, a named extension, stated here, not established here."

A17. Constant-coefficient linear Wiener noise, local Gaussian in the small-noise regime (Theorem 6 only).

The stochastic tail bound assumes a linear stochastic differential equation with constant kappa = A + r and additive Wiener noise, giving an Ornstein-Uhlenbeck stationary law with variance sigma squared divided by (2 kappa). The Gaussian reading is a local approximation to fluctuations away from the boundaries d = 0 and d = 1, valid in the small-noise regime sigma squared divided by (2 kappa) much less than d star squared. Capability-dependent noise amplitude, capability-dependent critical levels, or rare-jump drift each require a different tail. Status: proved under the stated conditions in Paper X §3.10 (Theorem 6). The independent test test_theorem6_ou_stationary_variance pins the stationary variance on a Monte-Carlo simulation to within fifteen per cent. Quote (§3.10): "Theorem 6 is a local Gaussian approximation to misalignment fluctuations away from the boundaries d = 0, 1, valid in the small-noise regime sigma squared divided by 2 kappa much less than d star squared."

A18. Time-varying exponents integrate to the same criterion.

If beta and k vary along the trajectory, the fraction vanishes if and only if the running integral of the margin (beta minus k) diverges to positive infinity; a positive lower bound on liminf (beta(tau) minus k(tau)) is sufficient. Transient episodes of beta below k are survivable if outweighed later; what is fatal is a margin that is negative on average. The governance quantity is therefore the cumulative co-scaling margin, not its instantaneous sign. Status: proved in Paper X §3.13 (Remark on time-varying exponents) as a corollary of the depth-clock comparison argument used in the proof of Theorem 2. Quote (§3.13, Remark): "the fraction vanishes iff the running integral of the margin beta minus k diverges to plus infinity, for which liminf as tau goes to infinity of (beta(tau) minus k(tau)) > 0 is sufficient. Transient episodes of beta < k are survivable if outweighed later; what is fatal is a margin that is negative on average."

A19. The gamma_3 = 1 knife-edge requires beta at least k for boundedness.

At the exact compounding threshold gamma_3 = 1 the dilution term vanishes, so the effective decay reduces to A. With A > 0 there is no finite-time blow-up, but the steady-state d star = gamma_1 r divided by A grows polynomially as C to the (k minus beta), and is bounded only when beta is at least k. The independent test test_gamma3_equals_one_boundary_bounded_only_for_beta_ge_k pins both sides of this: bounded and small for beta > k, unbounded polynomial growth for beta < k. Status: proved in Paper X §3.8 (Theorem 4) and verified independently in the fourteen-check suite. Quote (§3.8, Theorem 4): "at gamma_3 = 1 the fraction is bounded only for beta >= k."

A20. The compounding-threshold equality case is linear divergence when injection is positive.

Theorem 4's equality case rho_prop = 1 has kappa_eff = 0. With positive injection iota > 0 the fraction grows linearly, d(t) tends to d_0 plus iota times t, unbounded but not exponential. With zero injection the origin d = 0 is invariant and any positive initial fraction diverges only if kappa_eff is strictly negative. The independent test test_theorem4_equality_case_is_LINEAR_divergence pins the linear slope to one part in a thousand, and test_theorem4_zero_injection_origin_is_invariant pins the invariance to one part in a billion. Status: proved in Paper X §3.8 (Theorem 4) and pinned by the fourteen-check suite. Quote (§3.8, Theorem 4): "At the threshold with zero injection (iota = 0) the system is neutrally stable... Substituting A = A_0 C to the beta, r = b C to the k and taking C to infinity gives the power-law conditions."

A21. Fixed scoring convention aside, d = D/C is an operational risk index; the criterion is stated in scale-free exponents.

The exponents beta and k inherit the scoring convention's ambiguity to a smaller degree than any absolute-d statement would, because they are dimensionless slopes of log quantities. This is where the paper's own robustness to A12 lives; a criterion in beta and k is what survives moderate re-calibration of the scoring convention, provided the same convention is used within a single measurement. Status: modelling observation, stated in Paper X §8 (Limitations, "Metric normalisation") as a robustness consequence of the choice to state the criterion in exponents. Quote (§8): "The criterion itself is stated in exponents (beta > k), which are scale-free, so it is more robust to this than any absolute-d statement would be."

A22. Real-system measurement of beta and k is prepared for measurement, not yet done.

The applicability of the criterion to any deployed system presumes that beta and k are estimable from operational data. Paper X ships a runnable estimator that reads k off the capability curve (ln r versus ln C) and beta off the corrector's fractional removal rate (ln A versus ln C), validated on synthetic trajectories where it recovers known exponents to within about zero point one. What the estimator needs is a real self-improving system that drifts across a range of capability levels; that measurement has not been done, and Paper X names it as the single outstanding step for the criterion to move from proved-within-model to confirmed. Status: on trial in the draft protocol prepared for measurement, experiments/PROTOCOL.md (and its confirmatory upgrade experiments/PROTOCOL_V2.md, with the matched-pair-bootstrap CI design in experiments/scripts/realmodel_coscaling_v2.py), awaiting human submission. Stated in Paper X §8 (Limitations, "Estimating gamma and A on real systems"), and rated Open in the CLAIMS.md ledger. Quote (§8): "the repository ships a runnable estimator (experiments/scripts/estimate_exponents.py) that reads k off the capability curve (ln r versus ln C) and beta off the corrector's fractional removal rate (ln A versus ln C), validated on synthetic trajectories where it recovers known exponents to within approximately zero point one. What remains open is therefore not the estimation but its input, a real self-improving system that drifts across a range of capability levels."

A23. Smoothness for the classical ODE and depth-clock arguments (stated-implicitly).

The proofs use classical ODE machinery: integrating factors, comparison arguments, and diffeomorphisms of open intervals. Paper X treats A(t) and r(t) as smooth enough on compact intervals for the depth-clock change of variable to hold as a smooth orientation-preserving bijection between the half-open capability-time interval and the half-open depth interval. This is stated implicitly rather than axiomatised. Status: stated-implicitly in Paper X §3.4 (Theorem 2 proof) and §3.7 (Theorem 3 proof). Quote (§3.7, Theorem 3 proof): "Since r > 0 is smooth on every compact subinterval of the half-open interval from 0 to t star... the change of variable d tau = r d t is a smooth, orientation-preserving bijection from the half-open interval 0 to t star onto the half-open interval 0 to infinity (a diffeomorphism of these open intervals)."

A24. Beta is treated as a free real; the substrate cap on achievable beta lives outside Paper X, an independence premise on trial (stated-implicitly).

Paper X sets beta as a real exponent with no upper bound imposed by the substrate: any beta > k is admissible for the ARC-Beta-k model. The wider programme carries a separate, independent conjecture that the achievable internal-correction exponent is bounded at about one half (the one-half leverage conjecture, named in the conversion-law framing document as the weakest ingredient of the wider derivation of the stability ceiling at two). Gamma equals one half is a conjecture, never a derivation. The corresponding ceiling at two is a stability limit, never an impossibility or a speed limit: systems can exceed it, they do not remain stably self-correcting when they do. Paper X does not itself carry this cap, but any claim that a given real corrector can meet beta > k in practice inherits the conjecture's status. Status: stated-implicitly in Paper X (beta is treated as an unrestricted real exponent); the substrate cap is conjectured in 00-THE-CONVERSION-LAW-FRAMING.md, and its measurable form is prepared as a draft registration awaiting human submission (on trial in the draft-registration corpus). The premise is called out here because it is independent of Paper X's own derivation and must not be smuggled in as proved. Quote (Paper X, §3.2, standing assumptions): "Throughout, C > 0, A_0, b > 0, and beta, k are real." Quote (00-THE-CONVERSION-LAW-FRAMING.md, honesty rail 3): "The number two inherits the epistemic status of its weakest ingredient: the one-half leverage conjecture."

Sources for this section

A22a. The object of the theory is not the object of any current measurement.

The theory concerns systems that rewrite their own substrate recursively; every real-system measurement in this programme is on a frozen substrate that cannot, by construction, exhibit the phenomenon. This is a category difference, not a small-sample problem, and no amount of blinding on frozen-substrate data addresses it. The programme’s own converted-through-the-equation reading of 0.49 gives β approximately −1.04, deeply subcritical, which is what a substrate that forbids reinvestment into its own improvement process reads because it is subcritical. Until a system that actually rewrites itself is measured, the criterion’s applicability to its target class is a conjecture with zero evidence for and one honest datum consistent with against.

arc-principle-validation/papers/Paper-X-Coupled-CoScaling-Correction/Paper-X-Coupled-CoScaling-Correction.html (public repo): Priority disclosure, Abstract, Scope, Claim-status block, §3.1 through §3.13, §8 (Limitations), §11 (Harness), Appendix D (Glossary). Every section-and-theorem citation and every direct quote in A1 through A22 traces here.

arc-principle-validation/papers/Paper-X-Coupled-CoScaling-Correction/code/test_theorems_independent.py (public repo): the fourteen from-scratch checks (independent scipy solver, no import from the harness). Used for the pinning statements in A13, A16, A17, A19, and A20, and for the observation that Theorems 3 and 7 are not pinned by this suite.

arc-principle-validation/papers/Paper-X-Coupled-CoScaling-Correction/CLAIMS.md (public repo): the claim-status ledger. Cross-referenced for the P (proved-within-model), V (internally verified), S (synthetically validated) and Open rungs, and for A22's Open status specifically.

arc-principle-validation/papers/Paper-X-Coupled-CoScaling-Correction/ANTICIPATED_OBJECTIONS.md (public repo): the seven consolidated objections and their pre-empted / partially open / open classification. Cross-referenced for the honest classification of A1, A2, A7, A8, A10, A16 and A22.

the conversion-law framing record (estate, private): used only for A24, which cites the conversion-law framing document's honesty rail three by name for the one-half leverage conjecture and the two as a stability limit.

Companion surfaces: Paper X · the definitions page · the glossary.