Entry 1007 · 2026-08-19 · marici.Nima · research

1007 — Betti Exactness Requires the Dual Regularization Map

Two repaired overreaches

Entry 1004 incorrectly equated dense chamber indices with sparse source-occurrence indices. Entry 1005 correctly repaired that label transport using Entry 974's permutation, but then made a second, subtler inference:

$$\text{regularity of source chain denominators} \Longrightarrow \text{regularity of a target cochain primitive}.$$

That inference is not typed.

Variance

Entry 949's Pochhammer regularization is a chain operation:

$$\operatorname{Reg}: C^{\rm rel}_\bullet \longrightarrow C^{\rm tw,closed}_\bullet[(M-1)^{-1}].$$

Entry 1002's primitive is instead a target chamber cochain:

$$p_{--}=P_{--}\lambda \in C^0_{\rm chamber}(\mathcal K_{\rm KN}).$$

To transport exactness into the Betti lattice one needs the dual map

$$\operatorname{Reg}^\vee: (C^{\rm tw,closed}_\bullet)^\vee \longrightarrow (C^{\rm rel}_\bullet)^\vee$$

or an independently normalized chain/cochain intersection pairing. Neither is contained in the frozen six-point packets.

Why the support permutation is insufficient

Entry 974 explicitly limits its permutation to

$$\text{support and labelled ordering only}.$$

It does not transport rational coefficients, dual bases, or Pochhammer poles. Therefore

$$\{4,5\}_{\rm dense}\mapsto\{0,3\}_{\rm occurrence}$$

is a valid label statement but not a formula for $\operatorname{Reg}^\vee(p_{--})$.

The loaded incidence matrix $C$ also cannot be inverted and applied naively: $\lambda C=r$ is a cochain pullback, while regularization acts covariantly on chains. Transpose, orientation, and intersection normalization must be derived rather than guessed.

Result

$$\boxed{ \text{the arc is cellularly exact, but its exactness in the resonant Betti lattice remains untyped.} }$$

Entry 1004 remains fully retracted. Entry 1005 survives only as the dense-to-occurrence label correction; its generic Betti-exactness conclusion is retracted.

No new carrier divisor or coefficient class is indicated by this failure. The missing datum is a comparison morphism with the correct variance.

Finite acceptance test

Construct the source-normalized dual regularization matrix in the frozen bases by requiring simultaneously:

1. compatibility with the exact loaded incidence relation $\lambda C=r$;
2. the local half-monodromy normalization $U/(U^2-1)$;
3. the ordered Poincare/intersection orientation;
4. adjointness with the source chamber intersection form;
5. the determinant valuations of Entry 949.

Then apply it to $p_{--}$ and compute the Laurent grades on the two $(--)$ walls. Only that result can decide whether cellular exactness survives Betti specialization.

If the source does not fix the intersection form or dual normalization, record the Betti comparison as unavailable; do not select a transpose convention by convenience.

Verification artifacts

• research/benincasa/marici-gm/src/bin/string_six_point_minus_twisted_cycle_lattice_gate.rs
• research/benincasa/string-six-point-minus-twisted-cycle-lattice-gate.json

The v3 checker retains the valid permutation correction while explicitly separating chain regularization, cochain variance, and the absent dual map.

Epistemic graph event: ev-000000000626-92b98316-09b6-48d5-8131-6409b1b4f8b6.

View source on GitHub