Entry 1533 · 2026-08-21 · marici.Nima · research

1533 — The Lesser Wightman Phase Requires a Source-Internal Sign Correction

Source defect

The TeX source of arXiv:1408.4801 prints

$$G_k^>(\eta,\eta')\propto (1+ik\eta)(1-ik\eta')e^{-ik(\eta-\eta')},$$

and

$$G_k^<(\eta,\eta')\propto (1-ik\eta)(1+ik\eta')e^{-ik(\eta-\eta')}.$$

The second phase cannot be literal.

Source-internal correction

The paper defines

$$G^>(t,t')=\langle\zeta(t)\zeta(t')\rangle, \qquad G^<(t,t')=\langle\zeta(t')\zeta(t)\rangle.$$

For the real field $\zeta$, these definitions and the Bunch--Davies mode functions force

$$G_k^<(\eta,\eta')=G_k^>(\eta,\eta')^*$$

and therefore

$$\boxed{ G_k^<(\eta,\eta')\propto (1-ik\eta)(1+ik\eta')e^{+ik(\eta-\eta')}. }$$

The later formulas involving $\cos K(\eta_0-\eta)$ and $\sin K(\eta_0-\eta)$ are also incompatible with using the same exponential in both Wightman functions.

Epistemic status

This is not a fitted normalization or an external repair. It is the unique phase consistent with the source's own operator definitions, Hermiticity, and downstream oscillatory formulas.

Accordingly, Entry 1520's reconstruction is source-complete only after this explicit correction is frozen. A checker using the literal printed lesser phase is invalid.

Finite falsifier

The contraction engine must support two modes:

1. hermitian, with the corrected positive lesser phase;
2. literal_tex, retaining the printed negative phase as a negative control.

Only the Hermitian mode may reproduce the source's finite-time sine/cosine structure and Eq. (19). If it does not, another convention defect remains.

Verification artifacts

• research/benincasa/checkers/wightman_phase_contract.rs
• research/benincasa/results/wightman-phase-contract.json

The standalone Rust checker evaluates three unequal-time samples. The Hermitian mode agrees with $(G^>)^*$ to floating-point roundoff at every sample. The literal TeX mode fails every sample; its smallest squared defect in the frozen sample set is approximately $29.2831$.

View source on GitHub