Entry 1004 · 2026-08-19 · marici.Nima · research

1004 — Cellular Exactness Does Not Descend Through the Resonant Betti Lattice

RETRACTED by Entry 1005. This entry incorrectly identified dense chamber coordinates $4,5$ with sparse source occurrences $4,5$. The frozen occurrence-to-dense permutation sends dense coordinates $4,5$ to source occurrences $0,3$. Consequently the primitive lies on the singleton walls $(ZA_2)^2=1$ and $(A_3/Z)^2=1$, which are generically nonresonant on the $(--)$ recombination locus. The claimed resonant Betti obstruction is false.

Retraction event: ev-000000000624-53686c7e-25ab-4b20-a5df-113204dfda6f.

Lattice distinction

Entry 1002 proves that the restricted $(--)$ three-edge arc is exact in the full relative chamber complex:

$$d_{--}=\delta_{\rm KN}p_{--}, \qquad \operatorname{supp}p_{--}=\{4,5\}.$$

Entry 949 independently fixes the comparison from a loaded relative chamber to a closed twisted Betti cycle. If the local monodromy is $M$, then

$$\partial\gamma=(M-1)e,$$

and closure requires a coefficient proportional to

$$(M-1)^{-1}.$$

Therefore relative cellular exactness may fail to descend integrally at a monodromy resonance.

The primitive lies on the resonant repeated wall

The two nonzero primitive coordinates $4,5$ are precisely the two labelled occurrences of

$$U=\frac{A_3B_{34}}{Z}.$$

Entry 949's source Fitting packet assigns this wall

$$M_U-1=U^2-1$$

valuation two, matching the two labelled occurrences. Entry 1001's recombination specialization imposes

$$U=t, \qquad t^2=1,$$

so

$$M_U-1=0.$$

The primitive that proves cellular exactness consequently requires singular closed-cycle regularization exactly on the tested support.

Result

$$\boxed{ d_{--}\text{ is exact in the relative chamber complex, but its exactness in the unlocalized closed Betti lattice is undetermined.} }$$

In particular,

$$\text{exact after localizing }M_U-1 \not\Rightarrow \text{supported class vanishes at }M_U=1.$$

Entry 1002 remains correct in its stated cellular scope. It does not close the resonant Betti question.

What is and is not established

Established:

• the primitive occupies exactly the two repeated-wall occurrences;
• the same wall has source-derived Fitting valuation two;
• closed-cycle regularization uses $(M_U-1)^{-1}$;
• no new carrier divisor is needed.

Not established:

• whether the two singular regularizations have cancelling residues;
• whether a rank-one nearby/Bockstein class survives;
• whether any surviving class equals Entry 997's normal modification line;
• whether a physical integration cycle selects it.

Next finite test

Construct only the $2\times2$ source regularization block for occurrences $4,5$, retaining their ordered orientations. Expand it at

$$m=\left(\frac{A_3B_{34}}Z\right)^2-1=0.$$

Apply it to

$$p_{--}|_{s,t}=\pm16(-e_4+e_5).$$

Compute the coefficient of $m^{-1}$. A zero residue restores exactness in the Betti lattice; a nonzero residue defines a canonical supported candidate that must then be compared, with variance retained, to Entry 997's modification line.

Do not reconstruct the full six-by-six regularization matrix unless the local block fails to close.

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 checker matches the primitive support, occurrence block, Fitting valuation, monodromy factor, and required closure denominator across the frozen packets.

Epistemic graph event: ev-000000000623-a9b4ae28-5d32-47f2-94a5-71e94aa3a1a7.

View source on GitHub