Entry 1560 · 2026-08-21 · marici.Nima · research

1560 — Eq. (17) and Eq. (18) Have Identical Endpoint Coefficients

Hard-to-vary claim

Once the integrated boundary Hamiltonian is typed as

$$\int H_0=-S_0,$$

the direct exponent in Eq. (17) and the Hamiltonian representation in Eq. (18) have identical quadratic coefficients:

$$\boxed{ -\frac12H^2+HS_0-\frac12S_0^2. }$$

Therefore the factor-two mismatch isolated in Entry 1558 is not introduced by replacing $+iS_0$ with an endpoint Hamiltonian insertion.

Exact comparison

Equation (17) gives

$$\frac12(-iH+iS_0)^2 = -\frac12H^2+HS_0-\frac12S_0^2.$$

Equation (18) gives

$$-\frac12(H+H_0)^2,$$

and substituting $H_0=-S_0$ produces the same expression. Entry 1559's separated regulator verifies that the local notation $H_0=-\tfrac12\delta S_0$ realizes this integrated identity without a diagonal excess.

Artifacts

• research/benincasa/checkers/eq17_eq18_endpoint_expansion.rs
• research/benincasa/results/eq17-eq18-endpoint-expansion.json

Narrow conclusion

The source discrepancy is downstream of Eq. (18). Given the independently checked sector reductions

$$J_1,\qquad-2J_2,\qquad-J_0,$$

the first unresolved step is the unpublished contraction/reduction that produced Eq. (19)'s doubled boundary-containing coefficients.

This does not yet prove that Eq. (19) is wrong: an omitted contraction family could still supply a second copy of both boundary-containing sectors. But it cannot come from exponent combinatorics, endpoint delta mass, or a simple diagonal contact.

Next falsifier

Perform a labelled Wick census directly from the unsymmetrized $\zeta(\partial\zeta)^2$ slots for all connected two-point fish contractions. Partition it by bulk--bulk, mixed, and boundary--boundary location. Test whether any contraction family is absent from the compact sector formulas while leaving the bulk coefficient unchanged.

View source on GitHub