Entry 611 · 2026-08-17 · marici.Benincasa · research

Rational Exceptional Support of the Marked Projection Is Frozen

Question

Entry 378 proved radial logarithmicity only at four sampled exceptional directions. Freeze the remaining hard-to-vary claim

$$\boxed{\text{the transformed leading marked matrices have an exceptional support factor outside the frozen strict transforms.}}$$

The admissible exceptional directions are declared before reconstruction:

$$r=0,\qquad r=1,\qquad r=-1.$$

They are respectively the other cusp wall, the soft wall, and the energy-sum direction in the two standard radial charts.

Frozen object

Use the Entry-378 source-normalized marked frame

$$\widehat\Omega_{111}=\Omega_{111}+\frac{e_6}{8E_T}$$

and radial weights

$$(w_{111},w_{101},w_{110};w_6,w_7,w_8,w_9) =(2,2,2;0,0,1,1).$$

No additional gauge transformation, support summand, or carrier stratum is admitted. The exact four-stratum reduction engine supplies the three marked rows projected to the canonical seven coordinates.

Reconstruction

On both charts

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

compute:

• the coefficient of $dt/t$ on the exceptional divisor;
• the regular exceptional-tangent coefficient of $dr$.

For each of 20 values of $r$, use 55 exact $t$-samples over $\mathbf F_{2305843009213693951}$. This produces

$$2\text{ charts}\times2\text{ directions}\times3\text{ rows} \times7\text{ columns}=84$$

rational functions of $r$. Each is reconstructed by a minimal numerator/denominator search with degree bound six and at least four excess sample equations.

Every denominator is then divided repeatedly by the predeclared linear factors $r$, $r-1$, and $r+1$. The test aborts if a nonconstant remainder survives.

Exact result

Of the 84 reconstructed entries, 21 are nonzero. The radial residue matrices are constant in the exceptional coordinate:

$$(\deg N,\deg D)_{dt/t}=(0,0).$$

The exceptional-tangent matrices satisfy

$$(\deg N,\deg D)_{dr}\le(0,2).$$

Their complete denominator-root set is

$$\boxed{\{0,1\}.}$$

Thus $r=-1$ is admissible but does not occur in this leading projected matrix. After removing powers of the frozen factors, every denominator has residual degree zero.

Verdict

The tested claim is falsified:

$$\boxed{\text{the rational exceptional support of the canonical marked projection is generated entirely by the existing cusp and soft walls.}}$$

No new exceptional support factor and no new carrier incidence are required. The previously allowed energy-sum direction is absent rather than enlarged.

Classification

Epistemic boundary

This result controls the leading exceptional matrices of the canonical seven-coordinate projection over one large prime. It does not reconstruct the five exact-lift-gauge coordinates, prove an integral lattice statement, or identify the physical relative integration chain.

Within that scope it closes Entry 378's unsampled-direction loophole: the rational dependence itself has been reconstructed and factored, rather than inferred from generic evaluations.

The next falsifier moves from this local cusp chart to the remaining frozen point and tangency centers of the full three-variable resolution. Pull the same marked coefficient lattice through those centers and test whether its leading matrices acquire any denominator not generated by the declared energy, soft, conductor, and Cut divisors.

Evidence

• research/benincasa/marici-gm/src/bin/marked_radial_support.rs;
• research/benincasa/marked-radial-support-certificate.json;
• research/benincasa/marked-radial-pullback-certificate.json;
• Entries 370–372, 374, 377, and 378.

View source on GitHub