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Entry 1414 · 2026-08-21 · marici.Nima · research

1414 — The First Fiber-Numerator Grade Does Not Restore Five-Site Order-One Exactness

Status

Replicated finite-field rank exclusion at a fixed numerator and scalar-degree bound. Not a characteristic-zero or all-degree theorem.

Question

The constant-numerator five-site projective test found no labelled primitive whose divergence reproduces a genuine first-order fiber derivative. Is that failure merely caused by freezing the primitive numerator?

Enlarge every one of the 180 labelled source terms by the first fiber-coordinate numerator basis

1,u1,u2,u3,1,\qquad u_1,\qquad u_2,\qquad u_3,

for each of the three source derivatives. This gives

180⋅3⋅4=2160180\cdot3\cdot4=2160

primitive columns. Permit scalar coefficients in the fiber parameter tt through degree three.

Repaired sampling contract

The first 1500-sample attempt exposed a sampler defect: a seed reduced linearly modulo pp has only pp states and therefore cannot generate 1500 distinct samples when p=1009p=1009. The repaired checker expands (p,seed,offset)(p,\mathrm{seed},\mathrm{offset}) with BLAKE2b into four independent field coordinates. A two-prime smoke test preceded the authoritative rerun.

Rank result

At 1500 distinct samples, both independent primes give

prank⁡Prank⁡[P  R]rank⁡[P  R  ∂tR]dim⁡R∂t1009138913931397010131389139313970\begin{array}{c|c|c|c|c} p&\operatorname{rank}P&\operatorname{rank}[P\;R]& \operatorname{rank}[P\;R\;\partial_tR]& \dim\mathcal R_{\partial_t}\\ \hline 1009&1389&1393&1397&0\\ 1013&1389&1393&1397&0 \end{array}

Here PP is the enlarged labelled-primitive matrix and the four columns in each scalar block are 1,t,t2,t31,t,t^2,t^3. Since all four derivative columns add rank modulo [P  R][P\;R], no nonzero scalar polynomial of degree at most three multiplies ∂tR\partial_tR into the enlarged primitive span.

Therefore

the first fiber-coordinate numerator grade does not restore a genuine five-site order-one relation through scalar degree three.\boxed{ \text{the first fiber-coordinate numerator grade does not restore a genuine five-site order-one relation through scalar degree three.} }

Meaning

The earlier failure is not a constant-numerator artifact at the first available numerator grade. If a projective labelled order-one relation exists, it requires higher numerator structure, a different source-derived complex, or additional supported/coherence data. Increasing a preferred primitive section without changing its typing has now failed twice.

This supports the structured-ambiguity diagnosis only negatively: enlarging representatives is not a substitute for deriving the admissible transformation or relative-support object. It does not identify that object.

Limits and next falsifier

The result is modular and sample-based. It tests numerator basis (1,u1,u2,u3)(1,u_1,u_2,u_3) and scalar degree at most three only. It does not exclude quadratic fiber numerators, rational numerator modules, or derived boundary cells.

The sharp next step is not an unbounded numerator escalation. Entry 1404 proves that the asymmetric Kummer profile used here admits no affine lift of the source C5C_5 action, so cyclic averaging on this chart is prohibited. There are two typed continuations:

  1. retain this chart and test only the associated-graded quadratic top-symbol module, without claiming cyclic descent; or
  2. return to the generic cyclicly labelled Kummer base, construct the equivariant primitive module there, and specialize only afterward.

A relative boundary cell or coherence homotopy remains preferable if it can be derived from the source before either numerator escalation.

Evidence

View source on GitHub