579 — Generic Unit-(q)-Pole Localization Calibrates the Three One-Denominator Families
Hard-to-vary claim
The three one-denominator master ranks are recovered from the frozen Cayley--Menger cover and source (q)-divisors by a generic-weight twisted de Rham complex with independent (K)- and (q)-pole filtrations.
No generic (q)-regulator exponent, fitted quotient, or new carrier cell is required.
Frozen complex
For one source denominator (q), retain blocks
[ \frac{P}{K^m q^n}. ]
The exact differential generated by a polynomial vector field (V) is
[ \boxed{ \frac{operatorname{div}V}{K^m q^n} +(gamma-m)\frac{V(K)}{K^{m+1}q^n} -n\frac{V(q)}{K^m q^{n+1}}. } ]
The two pole directions are presentations of one localization and are joined separately:
[ \frac{P}{K^m q^n}
\frac{PK}{K^{m+1}q^n}, \qquad \frac{P}{K^m q^n}
\frac{Pq}{K^m q^{n+1}}. ]
Freeze
[ gamma=5,\qquad mathbb F_{32003}, ]
and measure the image of degree-at-most-five numerators in the simple-(q) block ((m,n)=(0,1)).
The source divisors are
[ q_{g_1}=c+b+X_1,\qquad q_{g_2}=c+a+X_2,\qquad q_{mathcal G_{12}}=c+X_1+X_2+X_3. ]
Finite falsifier
At common pole depth one, the three ranks stabilize only at
[ (28,28,28). ]
Thus a single cleared denominator level is insufficient.
At independent (K)- and (q)-pole depth two, the ranks by ambient degree are
[
\begin{array}{c|ccc}
N&q_{g_1}&q_{g_2}&q_{mathcal G_{12}}
hline
7&18&18&26
8&10&10&18
9&8&8&16
10&8&8&16
11&8&8&16
12&8&8&16
end{array}
]
These are exactly the published one-denominator deletion ranks:
[ \boxed{(8,8,16).} ]
Increasing both pole depths to three leaves ((8,8,16)) unchanged through ambient degree eleven.
The second generic kinematic point
[ (X_1,X_2,X_3)=(3,5,6) ]
reproduces the depth-two census through ambient degree ten.
Interpretation
The source (q)-divisors already generate the required one-denominator coefficient modules. The previously missing datum was not a carrier incidence but the two-direction localization coherence between pole presentations.
The calculation also sharpens Entry 578. Fixed simple poles failed because they discarded precisely these (K)- and (q)-localization transitions and their second pole layers.
This calibration is at generic (gamma). It does not license reduction after setting (gamma=- frac12); Entry 577 showed that specialization and reduction do not commute.
Narrow surviving architecture
[ \boxed{ ext{existing energy/Cut carrier} + ext{source }q ext{-divisors} + ext{generic filtered localization coefficient complex}. } ]
No new cosmological carrier datum appears at the one-denominator gate.
Next falsifier
Build the rectangular localization for pairs and the triple of source denominators, retaining one independent (q_i)-pole axis for each divisor.
It must reproduce the remaining deletion ranks
[ \boxed{(9,18,18,21)} ]
and the proper-support grades
[ \boxed{(0,1,1,1)} ]
without changing the carrier or fitting support summands. Only after the complete cube closes may the generic two-parameter Gauss--Manin connection be specialized to its limiting physical lattice.
Artifacts
research/benincasa/marici-gm/src/bin/generic_q_pole_twisted_derham_rank.rsresearch/benincasa/generic-q-pole-twisted-derham-rank.json