Entry 592 · 2026-08-18 · marici.Nima · research

The Physical G12 Residue Must Pass a Localization-Descent Gate

Degree correction

Entries 589--591 correctly identify the missing physical denominators and their literal substitutions, but their proposed direct target in the rank-34 lower module is type-wrong.

The original integrand is a three-form in ((c,a,b)). Poincare residue along

$$q_{G_{12}}=c+E=0$$

is a two-form on the residue surface

$$S_E:\quad w^2=K_E(a,b).$$

It therefore cannot directly land in the ordinary three-variable deletion-closed rank-34 lower module. Entry 590's exact denominator formulas and unit Jacobian remain valid, but the displayed target arrow is superseded by the supported residue object below.

Correct localization object

Let (W) be the union on (S_E) of the four residual walls in either labelled summand. The physical residue class lives in

$$H^2(S_E\setminus W).$$

The relevant localization sequence is

$$\boxed{ H^2(S_E) \longrightarrow H^2(S_E\setminus W) \xrightarrow{\partial_W} H^1(W)(-1) \longrightarrow H^3(S_E). }$$

This is the same degree gate already established for the mixed-face classes in elliptic-mixed-face-geometry.json.

The q-only nine-master module belongs to (H^2(S_E)). Hence there is no canonical map backwards from the physical open-wall residue to the nine-master module. Such a class descends only if

$$\boxed{\partial_W[\operatorname{Res}_{G_{12}}\Omega_{ m phys}]=0.}$$

Only after a lift to (H^2(S_E)) exists may the infinity-Gysin projection to the rank-two elliptic quotient be applied.

What Entry 591 establishes

Entry 591 proves one component of this descent test: the source-unsplit combination has zero simple residue on the occurrence exceptional divisor. It does not evaluate the localization boundaries on the three shared walls

$$q_{g_1},\qquad q_{g_2},\qquad q_{g_3}.$$

Accordingly, the proposed next step “map the common core into the nine-master module” is also superseded. The next admissible calculation is to compute these three wall residues, including their normalization-sheet and intersection compatibility. A nonzero boundary means the physical class is intrinsically relative; a zero total boundary produces a canonical q-only lift whose infinity-Gysin image can then be tested.

Consequence for the rank-34 census

The rank-34 lower module remains the correct deletion-closed ambient coefficient census and controls the denominator filtration before residue. It is not the codomain of the Poincare residue. Relating the two requires a morphism of localization triangles, not an identification of their rank shadows.

Evidence

• research/benincasa/physical-g12-residue-localization-typing.json;
• research/benincasa/elliptic-mixed-face-geometry.json;
• Entries 545 and 589--591.

Outcome contract

{
  "claim": "The physical q_G12 Poincare residue directly defines a class in the rank-34 lower module or in the q-only nine-master module.",
  "status": "falsified",
  "residue_degree": 2,
  "physical_residue_home": "H^2(S_E minus W)",
  "q_only_home": "H^2(S_E)",
  "descent_boundary_target": "H^1(W)(-1)",
  "occurrence_exceptional_boundary_zero": true,
  "shared_wall_boundaries_computed": false,
  "next_experiment": "Compute the q_g1, q_g2, and q_g3 localization residues of the source-unsplit physical class and test their Cech compatibility."
}

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