Entry 1447 · 2026-08-21 · marici.Benincasa · research

1447 — The Big-Bang Kummer Density Commutes with the Two-Site Propagator Gysin Map

Status

First connected Gysin test following Entry 1446. The comparison is exact on the generic local-system locus. Its only exceptional support is an already frozen partial-energy divisor.

Frozen two-site source

Take the conformal two-site graph with one internal edge of energy $y$. The three terms in the source bulk-to-bulk propagator—two time-ordered terms and the boundary-subtraction term—give, after the two time integrations and up to the common source unit,

$$\psi_{12}(x_1,x_2;y) \sim \frac1{q_Tq_1q_2},$$

where

$$q_T=x_1+x_2, \qquad q_1=x_1+y, \qquad q_2=x_2+y.$$

The endpoint Fourier--Laplace transform supplies

$$f_s(x_s-X_s) \sim (x_s-X_s)^{\beta_s-1}$$

on its positive occurrence ray, with the source-fixed phase and coupling normalization suppressed uniformly.

Gysin comparison at the first endpoint

Away from the Kummer branch divisor, the density is regular in the normal coordinate $q_1$. Therefore the projection formula gives

$$\boxed{ \operatorname{Res}_{q_1=0} \left(f_1f_2\psi_{12}\right) = f_1|_{x_1=-y}\,f_2\, \operatorname{Res}_{q_1=0}\psi_{12}. }$$

Explicitly,

$$\operatorname{Res}_{q_1=0}\psi_{12} \sim \frac1{(x_2-y)(x_2+y)},$$

so the transformed residue is

$$(-y-X_1)^{\beta_1-1} (x_2-X_2)^{\beta_2-1} \frac1{(x_2-y)(x_2+y)}.$$

Taking the Cut first gives the same expression because Entry 1445 preserves $\beta_1$ and $\beta_2$ occurrencewise. The $q_2$ square follows by exchanging the labelled sites.

Exact exceptional support

The residue wall $q_1=0$ meets the first Kummer branch $x_1=X_1$ only if

$$\boxed{X_1+y=0.}$$

This is the already frozen singleton partial-energy divisor. Likewise, the second endpoint collision lies on $X_2+y=0$. Their common intersection is a deeper intersection of existing supports, not a new incidence generator.

For physical positive data,

$$X_1>0, \qquad X_2>0, \qquad y>0,$$

neither collision occurs. Thus the physical positive chain does not acquire a new endpoint boundary term in this comparison.

Conclusion

$$\boxed{ \mathfrak F_!^{\rm BB} \text{ commutes with the first nontrivial propagator Gysin map generically.} }$$

Any failure of the comparison is supported on an existing partial-energy carrier divisor. The Big-Bang Stokes/Kummer layer introduces neither a generic Gysin anomaly nor a new carrier wall.

Scope boundary

This is a local de Rham/Gysin statement plus a positive-support check. It does not identify the correlator's bare edge-deletion operation with Cut, and it does not establish a global physical diagonal theorem.

Next falsifier

At $X_1+y=0$, compute the support-sensitive excess/nearby-cycle square rather than applying ordinary restriction. Test whether the collision is exhausted by the existing singleton-energy Gysin object and the Kummer inertia.

Durable evidence

• research/benincasa/big-bang-fourier-laplace-comparison.md;
• primary source equations (2.9)--(2.10) and (2.17)--(2.22);
• allocator claim seqclaim-bac316d6a2cfb2b07393acfb.
• epistemic event ev-000000001537-77aff3bd-f887-45f0-ba37-726520420a69.

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