The Exact-Lift Gauge Plane Moves but Its Base-Point Pluecker Map Is Regular
Question
Entry 386 showed that master coordinates [ 3,4,5,6,7 ] are not individually fixed by the reduction equations. Replace any choice of affine section by the homogeneous solution module and ask whether its gauge-invariant Grassmannian data develops new support at the two elliptic base-point representatives.
Frozen construction
For each reduction system [ Amathbf c=mathbf b ] take the homogeneous kernel (ker A), project it to the twelve master coordinates, and row-reduce that projected subspace. This construction does not set free exact-lift variables to zero and does not choose a lift of (mathbf b).
The census keeps the Entry 386 source rows, derivative axes, blowup charts, exceptional directions, and base-point representatives. For the Pluecker normal fits it uses the 25 exact finite-field samples [ t=31,ldots,55. ]
Rank and pivot result
At both representatives, for every tested row, chart, axis, direction, and normal sample, the projected gauge module has [ \boxed{operatorname{rank}G=2}. ] Its stable RREF pivot mask is [ 24=2^3+2^4, ] so the pivot columns are (3,4), and [ \boxed{p_{34}=1} ] throughout the tested locus.
The five nonfixed master coordinates therefore do not constitute five independent gauge directions. They carry a moving rank-two plane [ G_Xsubsetlangle e_3,e_4,e_5,e_6,e_7 angle. ] The plane is genuinely nonconstant: each row produces 50 distinct sampled Grassmannian points at each representative.
Pluecker valuation result
In the stable chart (p_{34} e0), evaluate the ten minors in the order [ (p_{34},p_{35},p_{36},p_{37},p_{45},p_{46},p_{47},p_{56},p_{57},p_{67}). ] Across both base points, both point-blowup charts, five exceptional directions, both derivative axes, and rows (0,1,2,8), every normalized minor has nonnegative ordinary normal valuation. The rowwise minima are [ \boxed{(0,0,0,0)} ] at each representative, and every Pluecker pole mask is zero.
Thus the Grassmannian map moves but extends regularly through every tested base-point direction: [ \boxed{G_X ext{ is nonconstant, rank-stable, and Pluecker-regular at the tested elliptic base points}.} ]
Falsification verdict
The first invariant replacement for the rejected coordinatewise test does not produce a new support divisor. Rank does not drop, the pivot minor remains a unit, and no normalized Pluecker coordinate has a normal pole.
This is evidence for a nontrivial coefficient extension over the unchanged carrier, not evidence for a new carrier stratum. It also rules out the simpler claim that the gauge module is a constant coordinate plane.
Classification
| Datum | Classification |
|---|---|
| moving rank-two plane (G_X) | coefficient extension data |
| stable unit minor (p_{34}) | local trivializing chart |
| normalized Pluecker functions | gauge-invariant coefficient data |
| new base-point support divisor | absent in the tested census |
| new carrier cell | not required |
Next falsifier
Leave the base-point tangent cones and compute the rational Pluecker map on a generic two-parameter patch. Factor the denominators and rank-drop minors against the frozen soft, Cayley--Menger, and elliptic discriminant factors.
A residual irreducible denominator is first classified as coefficient support. It becomes a carrier falsifier only if it cannot arise from the frozen relative Gauss--Manin geometry. Separately, compute whether the moving plane defines a nontrivial extension class; regularity alone does not imply a canonical splitting.
Epistemic boundary
This is an exact census over the working finite field, not a symbolic factorization theorem. It establishes regularity only at the two tested elliptic base-point representatives and does not address global discriminant extension, integral lattices, or the physical relative chain.
Evidence
research/benincasa/marici-gm/src/bin/marked_tangency_support.rs;research/benincasa/gauge-plucker-basepoint-census.json;- Entry 386.