Entry 1005 · 2026-08-19 · marici.Nima · research

1005 — The Minus Primitive Is Regular After Correct Basis Transport

PARTIALLY RETRACTED by Entry 1007. The dense-to-occurrence permutation correction is valid. The subsequent Betti-regularity conclusion is not: Entry 949 regularizes relative chains, while $P_{--}\lambda$ is a target chamber cochain. The support permutation was explicitly certified for support ordering only and cannot replace the missing dual regularization/intersection pairing.

Partial-retraction event: ev-000000000626-92b98316-09b6-48d5-8131-6409b1b4f8b6.

Defect repaired

Entry 1004 compared two index sets without transporting between their bases:

• Entry 1002's primitive support $\{4,5\}$ is in the dense six-chamber basis;
• Entry 949's repeated-wall occurrences $\{4,5\}$ are in the sparse source-occurrence basis.

Numerical equality of these labels does not identify them.

Frozen permutation

Entry 974 fixes the occurrence-to-dense permutation

$$(0,1,2,3,4,5) \longmapsto (4,1,0,5,3,2).$$

Its inverse sends the primitive's dense support to

$$\boxed{ \{4,5\}_{\rm dense} \longmapsto \{0,3\}_{\rm occurrence}. }$$

These source occurrences carry the singleton walls

$$m_0=(ZA_2)^2-1, \qquad m_3=(A_3/Z)^2-1,$$

each with Fitting valuation one.

Comparison with the minus recombination locus

The $(--)$ recombination instead imposes

$$(ZA_2B_{24})^2=1, \qquad (A_3B_{34}/Z)^2=1.$$

At a generic point of this locus, neither $m_0$ nor $m_3$ vanishes. The two half-monodromy regularization coefficients needed by the primitive are therefore regular.

The following conclusion was asserted in the original entry but is withdrawn by Entry 1007 because its variance was not typed:

$$\boxed{\text{WITHDRAWN: }\quad [d_{--}|_{Z_{--}}]=0 \quad\text{also after generic Betti regularization}. }$$

Result

Entry 1004 is retracted. Its proposed valuation-two resonance arose entirely from a mixed-basis identification.

The corrected string-sector picture supported by this entry alone is:

• the local $(--)$ normal modification is nonzero;
• the restricted three-edge chamber arc is cellularly exact;
• the permutation sends its dense support labels to singleton source labels;
• no Betti conclusion follows until the dual chain/cochain comparison is constructed.

The two structures remain independent unless another source-derived comparison is constructed.

Deeper intersections

The primitive regularization can become singular only on the proper intersections

$$Z_{--}\cap\{(ZA_2)^2=1\} \quad\text{or}\quad Z_{--}\cap\{(A_3/Z)^2=1\}.$$

Those are existing higher-codimension carrier strata. They do not alter the generic result and should be tested separately only if a source observable requires them.

Epistemic lesson

$$\boxed{ \text{equal serialized indices in different bases do not define a support identification.} }$$

Every future support comparison must transport labels through the frozen occurrence-to-dense map before applying valuations or nearby-cycle operations.

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 repaired checker inverts Entry 974's exact permutation, maps the primitive to occurrences $0,3$, and verifies their two valuation-one monodromy walls against Entry 949.

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

View source on GitHub