The Gauge Fitting Conic Is an Intrinsic Rank-One Special-Fiber Enhancement
Question
Entry 388 detected the residual conic [ C_{ m fit}:\qquad X_1X_2-E_T^2=0 ] from a bounded rank census. Is this merely a failure of the projection to master coordinates, or does the full frozen reduction presentation itself change rank there?
Full-rank comparison
At generic points, exact-form degree (8) gives full matrix rank (117), nullity (255), projected gauge rank (2), and pivot mask (24).
At tested conic points, the same presentation gives [ 116,\qquad256,\qquad3,\qquad280. ] Thus [ \boxed{Deltaoperatorname{rank}A=-1,\qquad Deltadimker A=+1,\qquad Deltaoperatorname{rank}G=+1.} ]
The pattern persists at exact-form degrees (10) and (12). The conic is therefore not only a bad choice of Pluecker chart or master projection: one additional homogeneous exact-lift class appears in the special fiber and its projection is nonzero.
For comparison, the signed-energy wall has a two-dimensional enhancement, while the ordinary soft wall tested here has the same one-dimensional enhancement pattern as the conic. This comparison is about ranks, not an identification of their geometric origins.
Uniformity sweep
On the (X_1=1) patch, the conic is [ v=2u^2-u+2. ] For every [ u=3,ldots,100 ] and at exact-form degrees (8) and (10), the conic fiber and the two transverse neighbors (vpm1) were evaluated exactly over the working finite field.
Results: [ \boxed{196/196} ] conic fibers have full rank drop one, and [ \boxed{196/196} ] gain exactly one projected gauge direction. All [ \boxed{392/392} ] neighboring fibers retain the generic full rank, projected rank, and pivot mask. No exception occurred.
Verdict
In the tested family, [ \boxed{C_{ m fit} ext{ is an intrinsic codimension-one rank-one special-fiber enhancement of the frozen exact-lift presentation}.} ]
This strengthens Entry 388's coefficient-support classification. The additional direction belongs to the kernel of the source-defined relative reduction map and survives projection to the master block. No new carrier cell is required.
The result does not yet prove that a chosen maximal minor contains (X_1X_2-E_T^2) with multiplicity one, nor that the entire global Fitting radical has no further components.
Architectural update
The surviving H2 form is now more specific: [ ext{shared carrier and six-functor calculus} + ext{sector-specific relative coefficient presentations} ] whose special fibers may acquire rank-one classes on internal Fitting divisors.
This is incompatible with a coefficient architecture determined solely by the pure elliptic rank-two quotient. It remains compatible with a relative Gauss--Manin extension over the unchanged carrier.
Next falsifier
Construct a generic maximal-rank minor and its transverse first jet along (C_{ m fit}). Test whether [ det M=(X_1X_2-E_T^2),U ] with (U) a unit generically on the conic. Equivalently, pair the extra right kernel class and left cokernel class with the transverse derivative of the presentation matrix.
A nonzero pairing establishes simple Fitting multiplicity. A zero pairing requires higher normal order or a different minor. Afterward compute the remaining radical factors and determine whether the new class enters a rank-one subquotient or only the extension data.
Epistemic boundary
The sweep is exact but finite-field and evaluation based. It does not replace a symbolic Fitting-ideal calculation, establish integral normalization, or identify the physical integration-chain image.
Evidence
research/benincasa/marici-gm/src/bin/marked_tangency_support.rs;research/benincasa/gauge-fitting-conic-uniformity.json;- Entry 388.