Erratum: Predicting How Transformers Attend §5.2 Theorem 5.2 — Heat capacity coefficient at γ=1

# Erratum — Predicting How Transformers Attend §5.2 Theorem 5.2 ========================================================= **Author**: Carles Marín — `transformerkmarin@gmail.com` **Date**: 2026-05-02 **Original paper**: [Marín 2026, Predicting How Transformers Attend (Zenodo 19826343)](https://doi.org/10.5281/zenodo.19826343) ========================================================== ## TL;DR §5.2 Theorem 5.2 of the paper states > CV(γ=1, N) = (log N)² / 4 and the correct asymptotic large-N coefficient is $1/12$, not $1/4$: > CV(γ=1, N) \;→\; (log N)² / 12 as N → ∞ The error is **a factor of three**. It originates in the truncation of the partition-function expansion at first order in (1-γ), missing a quadratic term that contributes the correction. The qualitative conclusion of §5.2 (CV finite at finite N, sharp in N → ∞) is **unaffected**. The TAF Agent diagnostic tool ([HF Space](https://huggingface.co/spaces/karlexmarin/taf-agent)) computes CV via numerical derivative of U(γ), not via the analytic formula, and therefore returns the corrected value automatically. ## Where the proof goes wrong The original proof retained only the linear term in (1-γ) of the Z expansion. Including the quadratic term: Z(γ, N) \;≈\; log N\,[1 + (1-γ)log N/2 + (1-γ)² (log N)²/6] gives, after taking log Z and differentiating: U(γ) \;=\; log N/2 + (1-γ)\,(log N)²/12 + O((1-γ)²) so that CV(γ=1, N) \;=\; -γ²\,dU/dγ|γ=1 \;=\; (log N)²/12. ## Independent confirmation The same result follows immediately from the cumulant identity CV(γ) = γ²\,Varₚ(log d) for p ∝ d-γ. At γ=1: Var(log d) \;→\; (log N)²/3 - (log N)²/4 \;=\; (log N)²/12. ## Triple-system numerical verification | System | Method | Result | |---|---|---| | Sócrates / NumPy | Direct discrete Varₚ(log d), N=10⁴ | CV = 7.89 ; ratio to (log N)²/4 = 0.37 | | Sage docker | Exact rational arithmetic for N ∈ \10, 50, 100\ | ratio/12 → 1 from above ; ratio/4 → 1/3 | | SymPy | Symbolic integration of cumulant identity | Var(log d) = (log N)²/12 exactly | Three independent computer-algebra systems agree. The hand-derivation above produces the same coefficient. ## Impact | Item | Affected? | |---|---| | §5.2 qualitative conclusion (crossover, not divergence) | No | | §5.2 numerical example N=2000 → 14.4 | Yes — corrected value ≈ 4.81 | | §5.3 Fisher–metric identity I = CV/γ² | No | | §3.3 Padé formula for γ | No | | §4 MaxEnt power-law derivation | No | | §6 critical-layer formula | No | | §7 KV-compression theorems | No | | §8+ attention grammar / RLHF / fingerprinting | No | | TAF Agent (`heat_capacity_Cv`) | No (uses numerical derivative) | The corrected value ~ (log N)²/12 ≈ 4.81 at $N=2000$ is **closer to the empirical broad-peak measurement of 6.86** at γ ≈ 1.235 reported in the paper, removing the previously unexplained factor-3 discrepancy between theory and the paper's own measurement. ## How the error was caught This erratum was generated by the **Sócrates Audit Framework**, a methodology for self-falsifying scientific claims, applied recursively to its own author's published work. The framework's three-stage algebraic / numerical / triangulation workflow flagged the §5.2 claim within minutes of being run on the paper. It is the framework's first published real-world impact. The full audit log, the Sage script that triangulated the result, and the SymPy symbolic derivation are reproducible from the framework's repository. ------------------------------------------------------------------------------------------------------------ **Related fix**: TAF Agent v0.5.3 (KV-compression bug for Phase B models, simultaneous deploy) **Contact**: `transformerkmarin@gmail.com`