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 (=1, N) = (N) ² / 4 and the correct asymptotic large-N coefficient is 1/12, not 1/4: > CV (=1, N) \;\; (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-), 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 (), 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-) of the Z expansion. Including the quadratic term: Z (, N) \;\; N\, 1 + (1-) N2 + (1-) ² (N) ²6 gives, after taking Z and differentiating: U () \;=\; N2 + (1-) \, (N) ²12 + O ( (1-) ²) so that CV (=1, N) \;=\; -²\, dUd|=₁ \;=\; (N) ²12. ## Independent confirmation The same result follows immediately from the cumulant identity CV () = ²\, Varₚ (d) for p d^-. At =1: Var (d) \;\; (N) ²3 - (N) ²4 \;=\; (N) ²12. ## Triple-system numerical verification | System | Method | Result | |---|---|---| | Sócrates / NumPy | Direct discrete Varₚ (d), N=10⁴ | CV = 7. 89 ; ratio to (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 (d) = (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/² | No | | §3. 3 Padé formula for | 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 (`heatcapacityCv`) | No (uses numerical derivative) | The corrected value (N) ²/12 4. 81 at N=2000 is **closer to the empirical broad-peak measurement of 6. 86** at 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`