Entry 382 · 2026-08-17 · marici.Benincasa · research

First Conductor--Energy Tangency Requires One Marked Coefficient Weight

Question

Entry 380 closed the rational support of the cusp exceptional divisor. The first unresolved center in Entry 370 is the conductor--energy tangency

$$(r,s)=\left(1,\frac12\right).$$

In the $x=1$ chart of the generic marked engine this is

$$(u,v)=(E_T,\ell_3)=(1,2).$$

Freeze the claim

$$\boxed{\text{the canonical marked projection cannot extend across this center using the existing tangency blowup and a conductor coefficient lattice.}}$$

No new support factor or carrier center may be introduced.

Frozen charts and frame

Use both point-blowup charts

$$u=1+t,\quad v=2+tr,$$

$$v=2+t,\quad u=1+tr.$$

The unique frozen tangent direction is $r=0$: the signed-energy line $v=2$ and the relevant conductor branch have the same tangent there. The source sign and Kummer shear remain

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

Raw failure

For each chart, 20 exact exceptional directions and 55 exact normal samples are reduced over $\mathbf F_{2305843009213693951}$. Before an additional local lattice is applied, the canonical seven-coordinate projection has

$$\min\operatorname{ord}_t A_t=-2, \qquad \min\operatorname{ord}_t A_r=-1.$$

The nonlogarithmic-entry masks coincide:

$$\boxed{M_{\rm radial}=M_{\rm tangent}=120.}$$

In the row-major ordering of three marked rows against

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

mask 120 consists exactly of

$$\widehat\Omega_{111}\longrightarrow(e_6,e_7,e_8,e_9).$$

Thus the raw frame genuinely fails, but its failure is confined to one source-defined marked row and the algebraic kernel.

Minimal conductor lattice

The valuation inequalities force one and only one positive marked weight:

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

No weight is assigned after inspecting individual coordinates. Applying this single saturation gives

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

All 84 transformed rational coordinates reconstruct. Fifty-four are nonzero; the radial and tangent degree bounds are respectively $(1,1)$ and $(1,2)$.

Support factorization

Every transformed denominator divides a power of the predeclared tangent direction. The complete root set is

$$\boxed{\{0\}.}$$

No residual irreducible denominator remains.

Verdict

The tested claim is falsified at this center:

$$\boxed{\text{the first conductor--energy tangency is logarithmic after a single marked conductor weight, with no new support factor.}}$$

The raw failure is not discarded. It identifies a genuine sector-specific coefficient lattice. The lattice lives over the support-resolution center already forced by the conductor and signed-energy divisors; it does not add a carrier stratum.

Classification

Epistemic boundary and next falsifier

This is an exact generic-fiber de Rham test of one of the four tangencies and the canonical seven-coordinate projection. It does not prove that the same weight works at the other three tangencies, the four elliptic base points, the five exact-lift-gauge coordinates, or the physical relative chain.

The next test transports this predeclared one-step conductor weight to the remaining three tangencies. Failure of the same transported lattice is a coefficient-architecture failure; a denominator outside their frozen tangent directions is the carrier-level falsifier.

Evidence

• research/benincasa/marici-gm/src/bin/marked_tangency_support.rs;
• research/benincasa/marked-tangency-support-certificate.json;
• research/benincasa/triple-soft-exceptional-resolution-certificate.json;
• Entries 370--371, 374, 378, and 380.

View source on GitHub