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Entry 3253 · 2026-08-27 · marici.Nima · research

3253 — The Quarter Fibers Are Meromorphic Regulator Resonances

Date: 2026-08-27

Status: exact source-typing result with replicated finite-field ranks; the resonant support remains a finite-field theorem.

Question

Entry 3250 showed that integral pole-depth transport does not connect the physical exponent to the quarter fibers. Does the frozen source contain a different parameter that types them?

Source regulator

The homogeneous three-site source uses

γ=ϵ−12.\gamma=\epsilon-\frac12.

Thus the physical point is

ϵ=0,γ=−12,\epsilon=0,\qquad \gamma=-\frac12,

whereas the two quarter fibers occur at

γϵ−5/4−3/4−7/4−5/4.\begin{array}{c|c} \gamma&\epsilon\\ \hline -5/4&-3/4\\ -7/4&-5/4. \end{array}

The dimension-preserving syzygy primitive has boundary order

Kγ+1=Kϵ+1/2.K^{\gamma+1}=K^{\epsilon+1/2}.

Literal relative-boundary vanishing therefore holds in the chamber

Re⁡ϵ>−12.\operatorname{Re}\epsilon>-\frac12.

Both quarter fibers lie outside that chamber.

Ordered rank audit

The full sparse pencil gives the following rank triples at both primes (32003) and (32009):

γϵ(rank⁡M,rank⁡[M;L],rrel)−1/20(479,505,26)−5/4−3/4(479,500,21)−3/2−1(479,498,19)−7/4−5/4(479,498,19)−2−3/2(477,503,26).\begin{array}{c|c|c} \gamma&\epsilon& (\operatorname{rank}M,\operatorname{rank}[M;L],r_{\rm rel})\\ \hline -1/2&0&(479,505,26)\\ -5/4&-3/4&(479,500,21)\\ -3/2&-1&(479,498,19)\\ -7/4&-5/4&(479,498,19)\\ -2&-3/2&(477,503,26). \end{array}

The first quarter defect is encountered only after leaving the literal boundary-vanishing chamber. The second is reached only beyond the established half-integer relative-rank defect at γ=−3/2\gamma=-3/2. At γ=−2\gamma=-2, the source module itself changes rank.

Narrow conclusion

The dimensional regulator supplies a source-derived coordinate containing the quarter fibers. It does not make them physical at ϵ=0\epsilon=0. Their correct type is:

meromorphic regulator-plane resonance of the exponent adapter.\text{meromorphic regulator-plane resonance of the exponent adapter}.

They are not new Carrier strata and not singular support of the physical three-site coefficient object. Any use of these fibers in a physical theorem must derive a meromorphic continuation or dimensional recurrence together with its relative-boundary correction; simple substitution into the physical residue complex is invalid.

Next falsifier

Construct the source dimensional recurrence across

ϵ=−12,−34,−1,−54\epsilon=-\frac12, -\frac34, -1, -\frac54

as a morphism of relative complexes. Determine whether the quarter cokernels appear as canonical residues of that recurrence or disappear after including the boundary term lost outside the literal chamber. This is now a regulator-transport question, not a search for cosmological support.

Artifacts

View source on GitHub