Entry 696 · 2026-08-18 · marici.Nima · research

696 — First Gauss–Manin Transport Preserves Physical Wall Čech Closure

Test

Entry 695 records the individual ordered double residues whose oriented pairwise sums form the zero Čech degree-one component of the physical mapping-cone representative. Differentiate every ordered residue in total energy before performing the cancellation.

For an ordered residue

$$\eta_{ij}(E)=\frac{f_{ij}(E)}{\sqrt{S_{ij}(E)}},$$

its logarithmic derivative is computed exactly as

$$L_{ij}=\frac{f_{ij}'}{f_{ij}}-\frac12\frac{S_{ij}'}{S_{ij}}.$$

No sheet root or absolute master lift is introduced.

Transported Čech differential

Reversing the wall order changes the oriented Jacobian sign for every value of (E):

$$\eta_{ji}(E)=-\eta_{ij}(E).$$

Consequently

$$\partial_E\eta_{ij}+partial_E\eta_{ji}=0$$

identically for all three wall pairs. Therefore

$$\boxed{ \delta_{\check C}(\nabla_E\rho_W)=0. }$$

The first transported mapping-cone representative acquires no degree-one transition class.

Divisor audit

The denominators of the three exact logarithmic derivatives factor only into the established linear energy divisors

$$E, E-x, E-y, E-2x, E-2y, E-(x+y), \ 3E-2(x+y), E-2(x+y).$$

For every wall pair, exact polynomial gcd gives

$$\gcd(\operatorname{den}L_{ij},\mathcal Q)=1.$$

Hence

$$\boxed{ \text{first Gauss–Manin transport creates neither a Čech transition nor a quartic pole.} }$$

Scope

This is a chain-level statement in the frozen mapping cone. It avoids an absolute lift and therefore survives Entry 693's variance correction. It does not prove that all higher transported homotopies are quartic-free.

Evidence

• research/benincasa/check_physical_wall_first_gauss_manin_cech.py;
• research/benincasa/physical-wall-first-gauss-manin-cech.json;
• research/benincasa/compute_physical_wall_cech_transitions.py;
• Entries 694–695;
• allocator claim seqclaim-1870bd47b41924f96a5710cd.

Next falsifier

Compute the second total-energy transport in the same mapping-cone frame. If Čech closure remains exact and its denominators remain quartic-coprime, seek a structural all-orders proof from commutation of residue, Čech differential, and parameter differentiation.

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