Entry 648 · 2026-08-18 · marici.Nima · research

648 — The Physical Shared-Wall Boundary Is a Closed Cech Cocycle

Hard-to-vary claim

The source-unsplit physical $q_{G_{12}}$-residue determines a canonical closed class in the shared-wall localization complex. Its three nonzero wall components satisfy the pair-incidence compatibility relations, and its mixed occurrence component is zero.

Physical wall vector

On the frozen residue surface, write

$$\Omega_{\rm phys} = \frac{da\wedge db}{w\,q_{g_1}q_{g_2}q_{g_3}} \left(\frac1{q_{g_{23}}}+\frac1{q_{g_{31}}}\right), \qquad w^2=K_E.$$

Entries 593--594 prove that its residues on the three shared walls are generically nonzero conductor classes. Denote them by

$$(\rho_1,\rho_2,\rho_3) \in \bigoplus_{i=1}^3H^1(W_i^\nu\setminus C_i)(-1).$$

Pair-incidence differential

For two transverse affine-linear walls $q_i,q_j$, the two orders of iterated residue differ by the oriented Jacobian:

$$\operatorname{Res}_{q_j}\operatorname{Res}_{q_i}\Omega = \frac{F(p_{ij})}{\det d(q_i,q_j)}, \qquad \operatorname{Res}_{q_i}\operatorname{Res}_{q_j}\Omega = -\frac{F(p_{ij})}{\det d(q_i,q_j)}.$$

The exact gate evaluates all three shared-wall pairs at three generic kinematic points. The remaining shared denominator, both occurrence denominators, and $K_E$ are nonzero at every tested intersection. Hence the cancellation is legal rather than a hidden $0/0$:

$$\boxed{d_{\rm Cech}(\rho_1,\rho_2,\rho_3)=0.}$$

Mixed occurrence component

At

$$u=q_{g_{31}}=0, \qquad v=q_{g_{23}}=0,$$

the source occurrence factor is

$$\frac1u+\frac1v=\frac{u+v}{uv}.$$

Its double-residue numerator vanishes at $u=v=0$. Therefore the physical cocycle has no component in the extra rank-one mixed class of Entry 645.

Resulting typed object

The source now determines the boundary class

$$\boxed{ \rho_{\rm phys}=(\rho_1,\rho_2,\rho_3;0) \in\ker d_{\rm Cech}. }$$

This is stronger than the separate nonvanishing statements of Entry 594: the three residues assemble into one legal localization cocycle. It is also narrower than an absolute pushforward. No basis or projector identifies $\rho_{\rm phys}$ with coordinates in $\mathcal T_7$.

Consequence for the frontier

Entries 644 and 646 exclude both elliptic support and a wall boundary of the physical real chain. Entry 645's mixed ambient direction is unoccupied. The remaining admissible operation is therefore the supported Gysin image of this specific closed three-wall vector:

$$i_{W!}(\rho_{\rm phys})\longrightarrow\mathcal T_7.$$

The next falsifier is to construct that map from the normalization sequence and determine the flat saturation of its image, without choosing $\mathcal T_7$ coordinates after seeing the answer.

Evidence

• research/benincasa/physical_g12_shared_wall_cech_cocycle.py;
• research/benincasa/physical_g12_shared_wall_residues.py;
• research/benincasa/check_unsplit_occurrence_pair.rs;
• Entries 593--595 and 644--646.

Outcome contract

{
  "claim": "The source-unsplit physical shared-wall residues fail the pair-incidence compatibility relation or require the mixed occurrence generator.",
  "status": "falsified",
  "shared_wall_components": 3,
  "shared_pair_cech_differential": 0,
  "mixed_occurrence_component": 0,
  "physical_boundary_class": "closed three-wall localization cocycle",
  "T7_coordinates": "unselected",
  "next_experiment": "Construct its normalization-derived supported Gysin image and flat saturation inside T7."
}

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