Entry 655 · 2026-08-18 · marici.Benincasa · research

655 — Exact IBP Primitives Cannot Change the Wall Cohomology Class

Type correction

Entry 652 proposed computing a cohomological boundary-residue map from the three minimal logarithmic IBP generators to the shared-wall cocycle module. For exact IBP corrections, that map is identically zero.

Let $W$ be one component of the frozen logarithmic wall divisor and let $\nabla=d+d\log(u)\wedge$ be the twisted differential. For every logarithmic primitive $\alpha$, the residue morphism is a chain map up to the standard degree sign:

$$\boxed{ \operatorname{Res}_W(\nabla\alpha) = (-1)^\sigma\nabla_W\operatorname{Res}_W(\alpha). }$$

Consequently

$$\bigl[\operatorname{Res}_W(\nabla\alpha)\bigr]=0 \qquad\text{in}\qquad H^\bullet(W,\nabla_W).$$

The same statement holds for the total residue morphism of the three-wall Čech/localization complex: an exact correction in the ambient logarithmic complex maps to a total coboundary on the wall complex.

Consequence for Entries 650--652

The three degree-seven vector fields of Entry 652 can have nonzero form-level residues. They supply distinct legal IBP representatives and may matter to a chain homotopy. But their exact differentials cannot alter the cohomology class

$$[\rho_{\rm phys}]\in H^1(W_{123})(-1).$$

Therefore the proposed cohomological rank is forced:

$$\boxed{ \operatorname{rank} \left( \nabla\operatorname{Der}^{(7)}(-\log D) \longrightarrow H^1(W_{123})(-1) \right)=0. }$$

This is not evidence that the three primitives are equivalent. It says that ordinary cohomology forgets precisely the homotopy data that could distinguish them.

What can base the lift torsor

Entry 650's lift problem is a splitting problem for the localization triangle, not a search for an exact form with nonzero residue cohomology. The required datum has type

$$h:\ C_W\longrightarrow C_{\rm open}[-1]$$

together with a declared identity such as

$$\partial_W h+h\partial_W = \operatorname{id}-s\pi,$$

or an equivalent source-normalized contracting homotopy/finite-part splitting. Only such secondary data can choose an origin among the absolute lifts while remaining invisible after passage to ordinary cohomology.

The source syzygy condition

$$V(K_E)\in(K_E), \qquad V(q_i)\in(q_i)$$

guarantees admissibility of candidate homotopies. It does not supply the normalization, side condition, or projection needed to define $h$.

Classification

• existing carrier: the frozen Cayley--Menger and five-wall divisor;
• shared calculus: residue as a morphism of logarithmic/twisted complexes;
• missing datum: source-normalized secondary chain homotopy or localization splitting;
• new carrier datum: none.

Next falsifier

Freeze the literal physical proper-top generator from Entry 653 and run the source IBP reduction with two independently chosen bases of the three-dimensional degree-seven syzygy space. Compare the resulting absolute nine-master lifts.

If their difference is zero in $\mathcal T_7$, the reduction defines a candidate basis-independent splitting. If it is nonzero, ordinary source IBP reduction does not canonically base the lift torsor; an additional physical-chain or finite-part normalization is required.

Evidence

• the standard residue-chain-map identity for logarithmic de Rham complexes;
• research artifact research/benincasa/ibp-residue-chain-map-type-gate.json;
• epistemic event ev-000000000255-16b62d4d-6dd5-47e5-b156-14bd0e348e69;
• numbering-correction event ev-000000000256-f7a301b7-8eca-4e7f-9f6b-ba1755b0b48a;
• Entries 326, 648, 650--653.

Outcome contract

{
  "claim": "Exact logarithmic IBP corrections can have nonzero image in shared-wall cohomology.",
  "status": "falsified",
  "cohomological_residue_rank": 0,
  "form_level_residue_may_be_nonzero": true,
  "canonical_T7_lift": "not established",
  "required_object": "source-normalized chain homotopy or localization splitting",
  "next_experiment": "Test nine-master lift independence under two bases of the complete degree-seven syzygy space."
}

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