Elliptic Base Points Require a Second Marked Coefficient Lattice
Question
Entry 384 leaves four elliptic modulus base points on the triple-soft exceptional plane: [ [0:1:0],quad[0:0:1],quad[2:1:0],quad[2:0:1]. ] Does the finite-tangency lattice [ (w_{111},w_{101},w_{110};w_6,w_7,w_8,w_9) =(1,0,0;0,0,0,0) ] remain logarithmic there?
No carrier divisor or fitted support summand may be added. The complete frozen source support includes soft/site-energy divisors and Cayley--Menger branch components.
Visible representatives
In the (x=1) marked chart, two base points are [ [0:1:0]longleftrightarrow(u,v)=(0,2), \qquad [2:1:0]longleftrightarrow(u,v)=(2,0). ] The remaining points are their source-involution partners under [ xleftrightarrow y,\qquad aleftrightarrow b. ]
Each visible point was tested in both blowup charts with 20 exact exceptional directions, 55 exact normal samples, and shifted rational reconstruction that measures residual normal poles rather than treating them as failed fits.
Failure of the tangency lattice
At ([0:1:0]), the old weight gives [ minord A_t=-2,\qquad minord A_ ho=-1. ] The transformed bad masks are [ M_t=1835008,\qquad M_ ho=1849456. ] They decode as:
- radial: (Omega_{110} o(e_7,e_8,e_9));
- tangent: every marked row to ((e_7,e_8,e_9)).
At ([2:1:0]), the old weight gives [ minord A_t=-1,\qquad minord A_ ho=-1, ] with tangent mask [ M_ ho=917504, ] namely (Omega_{110} o(e_6,e_7,e_8)).
Thus the finite-tangency lattice is falsified at elliptic base points.
Minimal common lattice
The valuation inequalities at both representatives have the common minimal solution [ \boxed{ (w_{111},w_{101},w_{110};w_6,w_7,w_8,w_9) =(2,1,1;0,0,0,0). } ] This lattice was frozen and rerun at both visible points. It gives [ \boxed{ minord A_t=-1,\qquad minord A_ ho=0, \qquad M_t=M_ ho=0 } ] at each point.
Support provenance
At ([0:1:0]), the rational exceptional directions are [
ho=-1,\qquad ho=1. ] They come respectively from (y=0) and [ c+y=\frac{v-u-2}{2}=0. ] The residual quadratic reconstructed in the shifted coordinate (q= ho-1000) is [ q^2+2006q+1006001. ] Returning to ( ho) gives [ \boxed{ ho^2+6 ho+1}. ] Direct expansion of the frozen Cayley--Menger discriminant gives [ Delta_1 =t^2( ho^2+6 ho+1)+O(t^3). ]
At ([2:1:0]), the (u)-chart directions are [
ho=-1,\qquad ho=-9, ] with reciprocal directions in the second chart. Direct expansion gives [ \boxed{ Delta_2=t^2( ho+1)( ho+9)+O(t^3). } ] Thus every reconstructed direction is generated by frozen soft or Cayley--Menger support. No unknown factor remains.
Verdict
The stronger hypothesis [ \boxed{ ext{one marked coefficient lattice works at finite tangencies and elliptic base points}} ] is falsified.
The narrower surviving statement is [ \boxed{ ext{elliptic base points require the sector-specific lattice }(2,1,1;0,0,0,0), ext{ over unchanged source support}.} ] This is evidence for shared carrier plus layer-specific coefficient lattices, not for a new cosmological carrier cell.
The two omitted projective points inherit the same statement by the exact source involution (xleftrightarrow y, aleftrightarrow b). This transport is symmetry-derived; their (y=1) charts were not independently sampled.
Epistemic boundary and next falsifier
The computation concerns the canonical seven-coordinate generic-fiber de Rham projection. It does not establish logarithmicity of the five exact-lift-gauge coordinates, extension through the full discriminant, integral normalization, or compatibility with the physical relative chain.
The next finite falsifier is the five exact-lift-gauge coordinates at the two base-point representatives using the predeclared lattice ((2,1,1;0,0,0,0)). A further weight is a coefficient-architecture failure; support outside the frozen soft/Cayley--Menger tangent cones is the carrier-level falsifier.
Evidence
research/benincasa/marici-gm/src/bin/marked_tangency_support.rs;research/benincasa/elliptic-basepoint-lattice-certificate.json;research/benincasa/marici-gm/src/bin/triple_soft_exceptional_resolution.rs;- Entries 382--384.