Title: The ARC Equation Measured: Blinded Cross-Architecture Replication and the Retraction of a Super-Linear Estimate (Paper II)
Author: Michael Darius Eastwood
Publication date: 2026-01-22
Version: v2.7
Revised: 2026-08-23
OSF DOI: 10.17605/OSF.IO/8FJMA
Canonical URL: https://www.michaeldariuseastwood.com/research/papers/paper-ii-experimental-validation.html

Abstract
--------
This paper presents experimental validation of the ARC Principle (Artificial
Recursive Creation), a mathematical framework proposing that error rates in
intelligent systems decrease according to a power law with recursive depth. The
principle, first articulated in Infinite Architects (Eastwood, December 2024)
and formalised in Paper I (Eastwood, 17 January 2026), predicts that the form of
recursion determines the scaling regime: sequential recursion should yield
super-linear error suppression (scaling exponent $\alpha > 1$), while parallel
recursion should yield sub-linear suppression ($\alpha < 1$).

We conducted controlled experiments in two phases. Phase 1 (the initial
single-model study) used DeepSeek R1 with visible reasoning tokens on 12
competition-level mathematics problems, finding $\alpha_{\text{sequential}}
\approx 2.24$ (95% CI: 1.5-3.0) and $\alpha_{\text{parallel}} \approx 0.0$.
Phase 2 (the six-model study) extended testing to six frontier models on 18
AIME/Putnam-level problems (n=54 per depth per model) with bootstrap confidence
intervals and 4-layer cross-verification.

Cross-architecture replication: The original $\alpha \approx 2.24$ (quadratic)
does not replicate across architectures. Only Gemini 3 Flash produced clean,
monotonic scaling data: $\alpha_{\text{seq}} = 0.49$ (regression, $r^2 = 0.86$,
SE = 0.20, boot CI [−1.3, 2.9]). DeepSeek R1 reached ceiling (94.4%-100%).
GPT-5.4 exhibited a binary step function (50% → 100%). Grok 4.1 Fast achieved
100% at all depths. Groq Qwen3 showed no trend (~50%).

Parallel scaling at or near zero in every model, with one measured exception:
$\alpha_{\text{parallel}} \approx 0$ for every model except Gemini 3 Flash,
which returns $\alpha_{\text{par}} = 0.31$ at $r^2 = 0.93$. That exception
matters, because Gemini 3 Flash is the model this paper treats as its most
reliable estimate. The replicated finding is therefore that parallel scaling is
at or near zero in every model measured, with one measured exception, not that
it is universally zero. Sequential outperformed parallel for every model where
both were measurable.

Revised parameter estimate: The most robust cross-architecture estimate is
$\alpha_{\text{sequential}} \approx 0.49$ (sub-linear, from Gemini 3 Flash),
substantially below the initial single-model estimate of 2.24. Under the
Intelligence Formula $\alpha = 1/(1 - \beta)$ from Paper I, $\alpha < 1$ places
current models in the physical regime (multiplicative composition through
finite-dimensional networks) rather than the intelligence regime (recursive
self-reference with $\alpha > 1$). The fundamental inequality
$\alpha_{\text{sequential}} > \alpha_{\text{parallel}}$ remains confirmed.

Key findings (quotable)
-----------------------
- This paper presents experimental validation of the ARC Principle (Artificial Recursive Creation), a mathematical framework proposing that error rates in intelligent systems decrease according to a power law with recursive depth.
- Phase 1 (the initial single-model study) used DeepSeek R1 with visible reasoning tokens on 12 competition-level mathematics problems, finding $\alpha_{\text{sequential}} \approx 2.24$ (95% CI: 1.5-3.0) and $\alpha_{\text{parallel}} \approx 0.0$.
- Phase 2 (the six-model study) extended testing to six frontier models on 18 AIME/Putnam-level problems (n=54 per depth per model) with bootstrap confidence intervals and 4-layer cross-verification.
- Cross-architecture replication: The original $\alpha \approx 2.24$ (quadratic) does not replicate across architectures.
- Only Gemini 3 Flash produced clean, monotonic scaling data: $\alpha_{\text{seq}} = 0.49$ (regression, $r^2 = 0.86$, SE = 0.20, boot CI [−1.3, 2.9]).
- "alpha ~ 2.24 (quadratic) does not replicate across architectures" - the initial single-model estimate is retracted; the robust cross-architecture estimate is alpha ~ 0.49 (sub-linear).

Priority claims relevant to this paper
--------------------------------------
- PC-025 (2026-01-22): Paper II - experimental validation across a six-model
frontier panel (Claude, DeepSeek, Gemini, Grok, Groq Qwen, GPT) - was first
published on 22 January 2026 under OSF DOI 10.17605/OSF.IO/6C5XB (sister DOI
10.17605/OSF.IO/8FJMA).
- PC-026 (2026-01-22): The parallel-recursion sub-linear scaling exponent
alpha_parallel of approximately zero was first reported across the six-model
panel in Paper II on 22 January 2026. Single-lab result; awaits external
independent replication.
- PC-027 (2026-01-22): The initial single-model sequential exponent estimate
alpha_sequential of approximately 2.24 (95 per cent CI 1.5 to 3.0) reported in
Paper II is now formally retracted. Never cite as a current finding; always pair
with the retraction. Correct framing: Eastwood published, then self-corrected,
an early single-model estimate.
- PC-028 (2026-01-22): The revised robust sequential exponent alpha_sequential
of approximately 0.49 (r-squared 0.86, SE 0.20; Gemini 3 Flash) is a sub-linear
point estimate with a wide bootstrap CI whose lower bound extends below zero
(approx. -1.3), so the effect is not statistically distinguishable from zero at
the reported CI. alpha > 1 remains an open prediction, not an observed finding.
- PC-029 (2026-01-22): The empirical observation alpha_sequential >
alpha_parallel across the tested six-model panel was first published in the
specific formulation of Paper II on 22 January 2026. The site's own ledger
records this as concurrent with Sharma and Chopra (arXiv:2511.02309, 4 November
2025); concurrent work must be cited whenever this claim is used.
- PC-030 (2026-01-22): The six-frontier-model empirical panel (Claude, DeepSeek,
Gemini, Grok, Groq Qwen, GPT) was first published as this specific comparative
study on 22 January 2026. Priority is over this specific study on this date, not
over panel-testing methodology in general.
- PC-011 (2026-01-02): The ARC Principle equation - book form U = I x R squared,
paper form U = I x R^alpha - was first published as a conjecture in Infinite
Architects on 2 January 2026 and formally in Paper I on 17 January 2026. Framing
is conjecture throughout. The early alpha of approximately 2.24 is retracted;
the robust v13 estimate is approximately 0.49 with interval [-1.3, 2.9]
(sub-linear); alpha > 1 remains an open prediction. Bare d/(d+1) form is prior
work.

Citation
--------
Michael Darius Eastwood (2026). The ARC Equation Measured: Blinded
Cross-Architecture Replication and the Retraction of a Super-Linear Estimate
(Paper II). The ARC Theory (the Theory of Artificial Recursive Creation) ·
ARC/Eden experiments, OSF DOI 10.17605/OSF.IO/8FJMA.
https://www.michaeldariuseastwood.com/research/papers/paper-ii-experimental-validation.html

Notes
-----
Hardware framings held at the site's already-published two-sentence concept
ceiling: safety constraints at hardware level through cryptographic tokens in
silicon, TRL 0-1, no prototype. No enabling implementation detail is stated in
this companion file. Balanced-ternary computing is prior work (Setun 1958);
recursion as a structural concept is prior work (evolutionary theory,
self-modifying computation, quantum error correction).

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