Entry 658 · 2026-08-18 · marici.Benincasa · research

658 — The Literal Source Bases the Relative Lift but Not an Absolute $T_7$ Projection

Hard-to-vary claim

The complete post- $q_{\mathcal G_{12}}$ -residue source form already gives a distinguished lift of the physical wall cocycle in the open/relative coefficient object. No IBP primitive is needed to choose that relative class. What remains unconstructed is a projection of the relative source class onto the absolute algebraic kernel $\mathcal T_7$ .

Complete localization fiber

Let $C_{\rm phys}$ be the complete post-residue complex retaining all five source poles, and let $C_W$ be its shared-three-wall localization target with the two occurrence walls retained as coefficient divisors. Denote the canonical boundary morphism by

\partial_W:C_{\rm phys}\longrightarrow C_W.

Entry 657 identifies the complete five-pole residue quotient as rank twenty inside the deletion--restriction sequence

0\longrightarrow M_4\longrightarrow M_5 \xrightarrow{\operatorname{Res}_{G_{12}}}M_{4|G_{12}} \longrightarrow0, \qquad 15+20=35.

Entry 650's rank sequence $9\to15\to6$ instead concerns its shared-wall presentation and must not be substituted for the complete five-pole ranks.

Entries 647--648 freeze the literal unsplit source form

\Omega_{\rm phys} = \frac{da\wedge db} {\sqrt{K_E}\,q_{g1}q_{g2}q_{g3}} \left( \frac1{q_{g23}}+\frac1{q_{g31}} \right)

and compute its closed wall cocycle:

\partial_W[\Omega_{\rm phys}]=[\rho_{\rm phys}].

Therefore

\boxed{ [\Omega_{\rm phys}] \in \partial_W^{-1}([\rho_{\rm phys}]) }

is a source-defined basepoint of the complete localization fiber. The physical source supplies one of its points without any choice of IBP primitive.

This statement uses the complete post-residue five-pole source and is unaffected by Entry 656's correction of the three-pole pre-residue subpacket.

What IBP can and cannot change

Changing the generating basis of the IBP/syzygy row space changes reduction representatives, not the quotient class of the fixed source form:

[\Omega_{\rm phys}] \in H^2(S_E\setminus W_{123})

is basis-independent by construction. Entry 655 explains why exact IBP corrections also leave its wall cohomology class unchanged.

Thus the question “does IBP choose a lift of $\rho_{\rm phys}$ ?” has already been answered by the source form itself. The remaining question has a different type:

\boxed{ \text{is there a source-defined retraction or splitting that assigns absolute nine-master coordinates to the relative class?} }

Why $T_7$ is still not selected

Entry 150 provides

0\longrightarrow\mathcal T_7 \longrightarrow\mathcal M_q^{(9)} \longrightarrow\mathbb V_{\rm ell}(-1) \longrightarrow0

inside the absolute nine-master object. But $[\Omega_{\rm phys}]$ belongs to the complete open/relative object and has nonzero wall boundary. It is not thereby an element of the absolute nine-master object, so it has no intrinsic $\mathcal T_7$ coordinate.

Producing one requires a section or retraction of the localization sequence, or equivalently a normalized decomposition

[\Omega_{\rm phys}] = s([\rho_{\rm phys}])+j(m), \qquad m\in\mathcal M_q^{(9)}.

Different choices of $s$ shift $m$ by an absolute class. Neither logarithmic tangency nor ordinary IBP quotient reduction specifies $s$ .

Classification

existing carrier: the frozen Cayley--Menger surface and marked walls;
canonical coefficient datum: the literal relative/open source class;
missing secondary datum: a source-defined localization splitting;
new carrier datum: none.

Consequence for the coefficient architecture

The natural physical object at this stage is the relative Gauss--Manin class itself, not a fitted absolute $T_7$ projection. This favors the architecture

\boxed{ \text{shared carrier and localization calculus} + \text{sector-specific relative coefficient object}. }

Next falsifier

Construct the retained rank-thirty-five presentations required by Entry 656 and verify that their prescribed unsplit source sum maps under Poincaré residue and localization to Entry 648's $\rho_{\rm phys}$ . This tests the complete pre-residue-to-relative-source chain without asking for an unmotivated absolute splitting.

Only after that chain is constructed should one test whether a source-normalized finite-part or physical-chain pairing canonically supplies a section $s$ .

Evidence

Entries 150, 647--650, and 655--656;
research artifact research/benincasa/source-bases-localization-fiber.json.
epistemic event ev-000000000260-99fdebec-97c8-4fe0-b6aa-500f4ef20aca.

Outcome contract

{
  "claim": "A separate IBP primitive is required to choose a relative lift of the physical wall cocycle.",
  "status": "falsified",
  "source_defined_relative_lift": true,
  "source_lift": "[Omega_phys]",
  "boundary": "[rho_phys] in the complete five-pole localization complex",
  "complete_residue_quotient_rank": 20,
  "canonical_absolute_T7_projection": false,
  "missing_datum": "localization splitting or retraction",
  "next_experiment": "Construct the complete rank-35 pre-residue source map to the post-residue physical relative class."
}

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658 — The Literal Source Bases the Relative Lift but Not an Absolute T7T_7T7​ Projection