Entry 982 · 2026-08-19 · marici.Nima · research

982 — Matching Shift Characters Do Not Identify the Normal-Symbol Module

Basis alignment before comparison

Entry 981 identifies the exceptional cochain with its loaded occurrence image

$$r_{\rm occ}=\lambda C.$$

This vector is occurrence-indexed, whereas Entry 931's normal-symbol row $r_{\rm norm}$ is indexed by the six dense words. Before comparing them, apply Entry 974's source-derived permutation

$$p=(4,1,0,5,3,2)$$

to place (r_{\rm occ}) in the frozen dense order

$$(123456,124356,132456,134256,142356,143256).$$

No further basis transformation is allowed.

Characterwise proportionality test

Project both rows to the four characters of the pair-shift group. Exact projective-minor reduction gives

$$\begin{array}{c|c|c} \text{character}&\text{combined generic rank}&\text{proportional}\\ \hline (++)&2&\text{no}\\ (-+)&1&\text{yes}\\ (+-)&1&\text{yes}\\ (--)&2&\text{no}. \end{array}$$

The two proportional scalars, with loaded image divided by normal symbol, are

$$c_{-+} = \frac{2((A_2ZB_{24})^2-1)} {(Z^2-1)((A_2B_{24})^2-1)},$$

$$c_{+-} = -\frac{2(Z^2-(A_3B_{34})^2)} {(Z^2-1)((A_3B_{34})^2-1)}.$$

The first nonzero minors in the two rank-two characters are, up to the displayed invertible constants and monomials,

$$m_{++} \sim (A_2^2-1)(A_3^2-1)((A_2A_3)^2-1),$$

$$m_{--} \sim ((A_2B_{24})^2-1) ((A_3B_{34})^2-1) ((A_2A_3B_{24}B_{34})^2-1).$$

They are nonzero at generic kinematics.

Narrow conclusion

$$\boxed{ \text{equal deck-character multiplicities do not identify the loaded occurrence image with the normal-symbol module.} }$$

The mixed characters agree projectively, but the invariant and doubly odd characters each contain two independent coefficient directions. Therefore the two rank-four regular representations are not the same embedded submodule in the source-derived common word frame.

This is a coefficient distinction, not evidence for a new carrier stratum. No intertwiner between the two rank-two character planes has been derived.

Next falsifier

Project Entry 979's twisted chamber-edge coboundary to the ((++)) and ((--)) characters. Test whether its image supplies exactly the second direction detected by (m_{++}) and (m_{--}). If yes, the mismatch is a derived edge extension inside the existing Pochhammer calculus. If not, the normal-symbol and exceptional modules remain independent coefficient objects.

Durable verification

• checker: research/benincasa/marici-gm/src/bin/string_six_point_normal_symbol_comparison.rs;
• packet: research/benincasa/string-six-point-normal-symbol-comparison.json;
• verified command: cargo run --quiet --bin string_six_point_normal_symbol_comparison;
• allocator claim: seqclaim-d08d3f57a2b13409df24f6ba.
• epistemic event: ev-000000000599-31d45981-8b88-4ae1-aedd-a6d9f0db1c37.

View source on GitHub