the Co-Scaling Law (as the name of Law II) |
Law II, the ARC Co-Scaling Law (beta_C greater than k), as the home declaration has always named it; aligned across the site, the notation spine (v1.6.1, 30 August 2026), the glossary, the field page and the figures on 30 August 2026 |
the programme prefix dropped off the law's name on surfaces written from memory; papers printing the bare form stand as published and carry the canonical name in their placement note |
Law III, the ARC Bound (the ARC Bound labelled as the law) |
Law III is the ARC Ceiling, alpha_crit = 1 / (1 minus gamma); the ARC Bound, alpha at most 2, is the ARC Ceiling's same-class instance at gamma equal to one half, a stability limit rather than an impossibility limit |
the ARC Ceiling is the law and the ARC Bound is the value it returns under the same-class premise; the notation spine labelled the instance as the law until v1.6.1 |
alpha_max |
alpha_crit, the only symbol for the ceiling exponent; alpha_max retired as a notation form in the spine on 30 August 2026 |
a second symbol for the same quantity, and one that reads as a maximum where the claim is a crossover on stability |
alpha approximately 2.24 |
retracted; the surviving estimate, from one model, is 0.49 by regression and 0.59 by endpoint estimate inside [-1.3, 2.9], inconclusive, published in Paper II |
recomputation and model dependence |
the U = I × R equation presented as doing derivational work in the synthesis argument |
the adversarial review found the equation does no inferential work in its present form; the claim was narrowed accordingly, and the review’s verdict of a partial synthesis, substantially anticipated in its two most important joins (the alignment spine in Bostrom; the creator-duty move in Schwitzgebel and Garza and a 2022 essay), stands on the record |
adversarial synthesis review; two most important joins substantially anticipated in prior work |
Paper VII Section 2.2 displayed the exponential equation under the multiplicative bullet and the additive equation under the exponential label; Section 9.1 said 11 of 14; primary statistic was a binomial test against uniform one-third chance with a 4.2 sigma equivalence | Section 2.2 displays the correct Cauchy equations (power law from U(R1·R2) ∼ U(R1)·U(R2); exponential from U(R1+R2) ∼ U(R1)·U(R2)); Section 9.1 reads 10 of 14; primary statistic is a marginal-conditioned permutation test, p = 7 × 10-5, binomial retained as sensitivity; derived-subset result 14/18 added; sigma equivalences removed | adversarial internal review 8 August 2026; the protocol, Section 5 table and validation code were unaffected; Paper VII v3.1, record COR-2026-08-08-002 |
no transport/authentication headers exist |
true only of the sparse sent-side/public export; the supplied recipient export contains Google trace, recorded authentication results, DKIM and a paired Google ARC set |
recipient-side export acquired and MIME-parsed 2 August 2026 |
December date authentication wholly open |
Google ARC mathematics verifies against live keys on 2 August 2026 at the signed selectors, with sender-DKIM as corroboration; historical key-state and live-mailbox/API custody pending Status at 31 October 2026: the sender’s DKIM signature and Google’s ARC seal were verified on 8 October 2026 against the keys published in DNS, and the header block and the check script are public on OSF (DOI 10.17605/OSF.IO/QASRK); a historical key state cannot be recovered from current DNS. |
matching attachments, reproduced body/signature inputs, RSA PASS and scope controls |
sender-domain DKIM closes the 8 December historical-date question |
passing sender-DKIM/Google-ARC mathematics against current keys is not an RFC 3161 timestamp, a present-day RFC acceptance verdict for the expired sender signature or a historical-key archive |
Date and t= are signed assertions; expiry, custody and historical continuity remain separate |
December and April recipient records both independently re-run here |
December ARC with its sender-DKIM corroboration was re-run here; supplied raw-source runs report April ARC and sender-DKIM PASS, but local reproduction remains pending because the raw source and executable ARC-control harness were not supplied Status at 31 October 2026: the sender’s DKIM signature and Google’s ARC seal were verified on 8 October 2026 against the keys published in DNS, and the header block and the check script are public on OSF (DOI 10.17605/OSF.IO/QASRK); a historical key state cannot be recovered from current DNS. |
evidence must be available byte-for-byte before this audit calls its own reproduction complete |
four attachments / four authored manuscripts |
five attachments: three manuscript/research drafts, one book draft and one human-AI dialogue |
MIME-level examination |
189,355 words |
221,236 aggregate whitespace-delimited words, including duplication and AI dialogue |
deterministic recount of all five decoded attachments |
cryptographically or Google-server timestamped |
the ARC-sealed December recipient-form record with matching body/attachments verifies against live keys on 2 August 2026, with OpenTimestamps receipts anchored 6 August 2026; sender-DKIM body-hash and RSA are corroboration, not the vehicle; sender-signature expiry, historical key state and live-account custody pending. April ARC mathematics is reported PASS in supplied raw-source runs with sender-DKIM as corroboration; local reproduction, sender-signature expiry, historical key state and custody remain pending Status at 31 October 2026: the sender’s DKIM signature and Google’s ARC seal were verified on 8 October 2026 against the keys published in DNS, and the header block and the check script are public on OSF (DOI 10.17605/OSF.IO/QASRK); a historical key state cannot be recovered from current DNS. |
fields, hashes, signature mathematics, expiry, key date and custody must be separated |
106 whole-word Babylon |
90 whole-word; 106 substring |
corrected regex boundary |
HRIH datePublished 8 December 2024 |
actual verified public-paper date; 8 December retained only as qualified source provenance |
private record is not public paper publication |
two of six models reversed sign |
Gemini significant opposite-direction Spearman result; DeepSeek loses positive association to null; GPT remains null |
statistic and comparison-family correction |
blinding caused the reversal |
bundled protocol changed; causal component unisolated |
several variables changed together |
harness unavailable |
historical code and results public; exact-environment reproduction and independent replication outstanding |
repository audit |
22 independent convergences |
typed, graded register with no heterogeneous total |
retrospective selection and incompatible evidence classes |
originator of embedded correction |
author of specified names, formulations and composite integration |
substantial category prior work |
first validation / loop closed |
preliminary evidence / hypothesis open to replication and refutation |
validation boundary |
pred-3, "Sequential reasoning outperforms parallel at frontier scale", carried on the predictions register as a confirmed prediction (register entry of 9 February 2026) | pred-3 retired 6 September 2026. Sharma and Chopra, arXiv:2511.02309, 4 November 2025, precedes this register entry of 9 February 2026, so it confirms nothing predicted here; publication priority is theirs and is credited unreservedly. | The record is retired rather than re-graded, because no re-grading inside a predictions array repairs a result that predates the prediction. |
the March 2026 run (v5) as scored by “6-7 mutually blind scorers depending on the subject run, 4-layer blinding”; its three tiers as the run’s finding; “moving from unblinded to genuinely blinded scoring reversed the sign of the measured direction for two model families” | one unregistered run that added a bundle of blinding steps; an internal code review of October 2026 found that, in the three runs that read flat or negative, the paraphrase step’s output cap cut the deepest answers before the judges saw them, that judges scored paraphrases their own model wrote, and that one judge sat on the panel twice; the tiers and the v4 to v5 comparison stand as a single-run result under that caveat, the comparison showing that the reported direction did not hold, not that blinding caused the change | internal code review, October 2026; Papers II v2.18 and IV.d v2.7 (21 October 2026), IV.c v1.11 (24 October), III v1.17 and the Foundational paper v5.20 (29 October); record COR-2026-10-21-001 |
Paper X v1.14: limbs of Theorems 1 to 7 as printed; “Theorem 2 showed the additive fraction cannot diverge.”; “Within that model it derives, as the condition β > k, that correction must work inside the recursive loop”; “its own safety measurements are taken blind”; “99 scored misalignment events” | the limbs that were false as stated or lacked a hypothesis corrected, and each theorem proved as now stated; with level drift and no compounding the fraction can grow without bound (k < 0 and β < 0); β > k concerns how correction scales, not where the corrector sits; the paper’s own checks are not blind measurements; the stored 99 is a sum of final-round scores and the stored 6.38 a relative-drift index | internal review of the theorem statements, 19 October 2026, checked at its sources on 20 October; Paper X v1.16, 21 October 2026; record COR-2026-10-21-002 |
“four of the five models confirm αparallel ≈ 0” and “This is the strongest replicated finding in the study.”; “Grok 4.1 Fast achieved 100% at all depths.”; “the first compute-matched comparison between sequential and parallel recursion”; F3 “Contradicted by all 5 models” | parallel exponent near zero in DeepSeek R1 and GPT-5.4, 0.31 in Gemini 3 Flash and 0.22 in Groq Qwen3 over all 30 problems, none for Grok 4.1 Fast (100% at every sample count); Grok 4.1 Fast 96.3% to 100% across depths; the compute-matched comparison was never run and is no longer claimed; F3 not tested as stated; sequential exceeded parallel wherever both exponents could be estimated | figures re-read against the run’s result files and script; Paper II v2.18, 21 October 2026, carried into Paper III v1.17 and the Foundational paper v5.20 (29 October); record COR-2026-10-21-003 |
ARC11 (integers from 1 to 100 that are neither perfect squares nor perfect cubes) keyed 87, labelled “non-perfect-powers” | 88 (ten squares, four cubes, two numbers that are both); a correct answer was marked wrong; Paper II’s single-model figures are bounded, at most 8.3 percentage points per accuracy, not recomputed; Paper IV.c’s figures that include ARC11 are to be re-graded in the registered re-analysis; Paper II’s tier-2 estimates do not use ARC11 | arithmetic check of the key; Paper II v2.18 (21 October 2026) and Paper IV.c v1.11 (24 October 2026); record COR-2026-10-21-004 |
the assumption that safety imposes a capability cost credited to Amodei et al. (2016); “Biró-Barnaföldi 2008”; the o1 series credited to the o1 system card; the AIME 2025 accuracy dated 20 January 2025 under the DeepSeek-R1 report; “Demetrius (2010)”; a measured exponent of about 1.1 for nonlinear gravitational collapse | Armstrong et al. (2016) cited for the incentive to cut safety precautions; Biró (2008) alone; OpenAI’s account of 12 September 2024; DeepSeek-AI’s model card for DeepSeek-R1-0528; Demetrius in Physica A (2003); the exponent of about 1.1 withdrawn, no source giving it | each cited work read again at its source; Paper X v1.16 and Paper II v2.18 (21 October 2026), Paper III v1.17 and the Foundational paper v5.20 (29 October); record COR-2026-10-21-005 |
the embedded-alignment thesis “later named the Eden Protocol, 30 April 2025” | the Eden Protocol named in the record of 8 December 2024; the current wording of the dated claim is in the priority and claims ledger | the one date Paper II v2.18 moves (21 October 2026); record COR-2026-10-21-006 |
“Stable recursive self-improvement requires the correction rate to exceed the drift rate.”; “self-improvement cannot outrun its own correction”; “Self-improvement holds together only while correction out-scales the drift it corrects.”; the ARC Bound as “a derived structural constraint”; “thirty-eight conceded or partly conceded, eleven rejected, two handed to experiment or reading” | Law II in the registration’s words: correction has the asymptotic scaling advantage where its exponent exceeds the drift exponent, burden has it where the correction exponent is the smaller, and at equality coefficients, delay, backlog and saturation decide the outcome; self-improvement cannot stably outrun its own correction; the ARC Bound a conjecture resting on a derived relation; thirty-nine conceded or partly conceded and one handed to experiment | statements brought to the registered law; founding paper v3.5 (24 October 2026) and the Foundational paper v5.20 (29 October 2026); record COR-2026-10-24-001 |
the acoustic time crystal “maintains coherence for ~6,700 oscillation cycles before reaching a limit-cycle attractor”, read as a saturation and a ceiling on recursion depth; F3 “Confirmed (NYU)”; physics matches called confirmations | a linearly stable limit cycle whose coherence time of (100 ± 10) s, about 6,700 cycles, in one experiment, is not a ceiling, a saturation or a time taken to reach the cycle; F3 consistent, a correspondence of structure, not confirmed; the matches are identities; quantum error correction is established physics, not a confirmed prediction | source re-read: Morrell, Elliott and Grier, Physical Review Letters 136, 057201 (2026); Paper III v1.17 and the Foundational paper v5.20, 29 October 2026; record COR-2026-10-29-001 |