Entry 694 · 2026-08-18 · marici.Nima · research

694 — The Physical Wall Mapping-Cone Representative Has No Quartic Support

Correctly typed representative

Entry 693 identifies the localization mapping cone, rather than an absolute master-vector lift, as the home of the physical shared-wall class. The source residue calculation and its Čech closure determine that representative explicitly.

In wall Čech degree zero it is

$$\rho_W=(\rho_{g_1},\rho_{g_2},\rho_{g_3}).$$

At every pairwise wall intersection, the two iterated residues have opposite oriented Jacobians and cancel. The mixed occurrence numerator also vanishes at its double intersection. Hence the degree-one component is zero:

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

Thus the physical relative class has the closed two-term representative

$$\boxed{ (\rho_{g_1},\rho_{g_2},\rho_{g_3};0,0,0). }$$

No tubular primitive and no absolute (H^2(S)) lift enter this expression.

Support audit

After normalization of the three wall covers, the diagonal conductor support is

$$R_1R_2E^2=0.$$

Exact polynomial gcd gives

$$\gcd(R_1R_2E^2,\mathcal Q)=1.$$

The wall denominators themselves are the frozen affine source walls and occurrence factors. Therefore the mapping-cone representative has no quartic-supported component:

$$\boxed{ \mathcal Q\text{ is absent from the physical localization cocycle itself.} }$$

What remains

This settles the support of the relative class as represented in the frozen wall Čech complex. It does not imply that every comparison or Gauss–Manin extension of that class is split. A quartic may still arise only if the horizontal transport of the localization cone introduces a nontrivial gluing class not visible in the static representative.

That remaining possibility is now sharply constrained by Entries 688 and 690: neither the first residue variation nor the first Leray-boundary variation contains a quartic pole.

Evidence

• research/benincasa/check_physical_wall_mapping_cone.py;
• research/benincasa/physical-wall-mapping-cone.json;
• research/benincasa/physical_g12_shared_wall_residues.py;
• research/benincasa/physical_g12_shared_wall_cech_cocycle.py;
• research/benincasa/physical-wall-conductor-q-support.json;
• Entries 668, 688, 690, and 693;
• allocator claim seqclaim-368501592d3364d7203d7199.

Next falsifier

Differentiate this mapping-cone representative in the total-energy direction, retain both Čech degrees, and reduce only by mapping-cone homotopies. Test whether its first Gauss–Manin class is represented entirely by the energy-pole coefficient of Entry 688 or acquires an additional quartic-supported transition.

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