Entry 610 · 2026-08-17 · marici.Benincasa · research

The Canonical Marked Projection Is Radially Logarithmic at Generic Directions

Question

Entries 372 and 377 leave the finite claim

\boxed{\text{the marked seven-coordinate projection retains an irregular exceptional-tangent pole after the frozen Rees correction.}}

This tests the three marked rows projected to

(\Omega_{111},\Omega_{101},\Omega_{110},e_6,e_7,e_8,e_9).

It does not reconstruct the five exact-lift-gauge coordinates.

Frozen input

Use the two radial charts

(u,v)=(t,tr),\qquad (u,v)=(tr,t),

of the cusp corner $(u,v)=(E_T,\ell_3)=(0,0)$ . The source connection sign is fixed once by

[E_T^{-2}e_6]\nabla\Omega_{111}=\frac18.

Apply the previously derived Kummer shear

\widehat\Omega_{111}=\Omega_{111}+\frac{e_6}{8u}

and retain the Entry-372 absolute radial weights

w(e_6)=w(e_7)=0,\qquad w(e_8)=w(e_9)=1.

No support factor or frame is fitted after inspecting the pullback.

Marked radial weights

The raw exact valuations force the same minimal weight on all three marked generators:

\boxed{ w(\Omega_{111})=w(\Omega_{101})=w(\Omega_{110})=2. }

For an entry from row $i$ to column $j$ , the transformed valuation is

\operatorname{ord}_t(A_{ij})+w_i-w_j.

These weights simultaneously regularize every tested exceptional-tangent entry while leaving the radial connection logarithmic.

Exact finite test

For each chart and each generic exceptional direction

r\in\{2,3,4,5\},

the four-stratum engine is evaluated at 80 exact points over $\mathbf F_{2305843009213693951}$ . The $u$ - and $v$ -connection rows are combined by the exact chart Jacobian. The full basis change includes:

the global source-sign calibration;
the derivative of $e_6/(8u)$ ;
the invariant $e_6$ -connection row;
re-expression of the old top generator in the sheared basis;
the frozen diagonal radial weights.

Bounded rational reconstruction reaches numerator and denominator degree 18. Across all 336 tested marked coordinates,

\boxed{ \min\operatorname{ord}_t A_t'=-1, \qquad \min\operatorname{ord}_t A_r'=0. }

Thus the radial coefficient is logarithmic and the exceptional-tangent coefficient is regular at every tested direction in both charts.

Verdict

The tested claim is falsified at the generic directions sampled:

\boxed{\text{the canonical marked projection extends through the radial exceptional divisor using the frozen Kummer shear and Rees lattice.}}

No additional generic radial support is observed. The marked weights are a filtered coefficient-lattice datum; they do not define a new carrier cell.

Classification

Datum	Classification
radial exceptional divisor	existing flagged normal geometry
$e_6/(8u)$ shear	Tate/Kummer coefficient extension
weights $(2,2,2;0,0,1,1)$	sector-specific Rees coefficient lattice
remaining radial logarithmic pole	existing normal divisor
generic tangent pole	absent at tested directions
new carrier datum	none found

Epistemic boundary and next falsifier

This is exact finite-field evidence at four generic values of the exceptional coordinate, not a symbolic reconstruction of its complete rational support. It therefore does not exclude a pole at an unsampled exceptional direction.

The next finite falsifier is to reconstruct the transformed leading matrices as rational functions of $r$ , factor every denominator, and require each factor to be a strict transform of a frozen energy, soft, conductor, or Cut divisor. Any additional irreducible factor would be coefficient support or, if it defines an unavoidable new incidence stratum, a carrier-level failure.

Evidence

research/benincasa/marici-gm/src/bin/marked_radial_pullback.rs;
research/benincasa/marked-radial-pullback-certificate.json;
research/benincasa/bivariate_soft_gram_connection.json;
research/benincasa/two-wall-rees-regularization.json;
Entries 293, 371, 372, 374, and 377.

View source on GitHub