The Exact-Lift Gauge Module Detects a Relative Fitting Conic
Hard-to-vary question
Entry 387 established that the projected exact-lift gauge module is a moving rank-two plane with a regular Pluecker map at the two elliptic base points. Does its generic Fitting support remain inside the frozen site-energy, signed-energy, elliptic, and Cayley--Menger arrangement?
No new carrier cell, fitted support summand, or post hoc splitting is allowed.
Frozen coordinates and support
On the (X_1=1) patch use [ E_T=u,\qquad X_2=\frac{u+v}{2}-1,\qquad X_3=\frac{u-v}{2}. ] The comparison product includes the existing site-energy and signed-energy walls, (A), (B), and the two Cayley--Menger discriminants (Delta_1,Delta_2).
For each reduction system, project the homogeneous exact-lift kernel to the five nonfixed master coordinates and use its normalized Pluecker point. No affine section is chosen.
Generic-line result
Five slices [ u=3,5,7,11,13 ] were tested for rows (0,1,2,8), both derivative axes, and all ten Pluecker coordinates. Each coordinate was reconstructed from 65 exact finite-field samples.
The maximum reconstructed denominator degree is [ \boxed{0}. ] Thus no generic rational denominator, known or residual, appears in this census. The moving Grassmannian map is polynomial on every tested slice away from rank-change points.
Rank census
A separate presentation-rank scan covered [ 3le ule15,\qquad3le vle200. ] It found 35 exceptional evaluations.
Twenty-seven lie on frozen support. They include the families [ v=uquad(X_3=0), \qquad v=u+2quad(E_T-X_2=0), ] plus one Cayley--Menger point in the bounded window.
The remaining eight points are [ (3,17),(4,30),(5,47),(6,68),(7,93),(8,122),(9,155),(10,192). ] They lie on the single patch conic [ \boxed{v=2u^2-u+2}. ] Using the frozen coordinate map, this homogenizes to [ \boxed{X_1X_2-E_T^2=0}. ]
At generic points the projected gauge rank and pivot mask are [ (2,24). ] At two conic points independently tested with exact-form degrees (8,10,12), they remain [ \boxed{(3,280)}. ] The defect therefore survives enlargement of the exact-form ansatz and is not the degree-eight truncation artifact tested here.
Rejected provenance test
Restricting the branch quartic to either marked linear divisor gives an identically degenerate even-quartic discriminant in this model. That discriminant vanishes generically and cannot classify the conic. It is therefore rejected as a provenance test.
The conic is also distinct from [ mathcal Q=0. ]
Verdict
The stronger claim [ ext{all gauge-module Fitting support is exhausted by the previously frozen energy/elliptic/Cayley--Menger arrangement} ] is falsified in the tested presentation.
The narrower surviving statement is [ \boxed{ ext{the frozen exact-lift module carries an additional relative coefficient Fitting conic }X_1X_2-E_T^2=0.} ]
This conic is generated by a rank jump of the source-defined homogeneous exact-lift module itself. It therefore requires no new incidence cell and is classified as coefficient support, not carrier structure.
Consequence for the cosmology architecture
This is a negative update to any hypothesis that the coefficient singular support is exhausted by the pure elliptic and energy arrangements. It is not a falsification of shared carrier plus sector-specific coefficients. Instead it sharpens H2:
[ ext{shared carrier and calculus} + ext{a relative coefficient object with its own Fitting support}. ]
The result does not locate (mathcal Q). It identifies a different, source-derived algebraic support component in the exact-lift extension sector.
Next falsifier
Compute the symbolic Fitting ideal of the projected homogeneous exact-lift module on the generic two-parameter patch. Test whether its radical is exactly [ (X_3)cap(E_T-X_2)cap(X_1X_2-E_T^2) ] together with the already frozen discriminant components, or whether further irreducible factors occur.
Then determine whether the conic controls a rank-one subquotient, an extension class, or only a presentation-rank enhancement. Only failure to realize it inside the frozen relative coefficient geometry would reopen a carrier-level question.
Epistemic boundary
The conic equation is reconstructed from eight exact points and checked for degree stability at two of them. This is not yet a global symbolic Fitting-ideal factorization. Generic polynomiality is established only on five finite-field slices. Integral normalization and the physical relative chain remain untested.
Evidence
research/benincasa/marici-gm/src/bin/marked_tangency_support.rs;research/benincasa/gauge-fitting-conic-census.json;- Entries 386--387.