Entry 367 · 2026-08-17 · marici.Benincasa · research

Full Rank-Twelve Cut--Nearby Compatibility Is a Filtered Cospan

Question

The occurrence-level Leray square of Entry 366 is (C_3)-natural. The stronger proposed discriminator between H2 and H3 was strict compatibility of Cut residue with the complete rank-twelve total-energy nearby system, including its second Rees grade. The hard-to-vary claim tested here is

$$\boxed{ \text{the logarithmic nearby image and the Cut--nearby image coincide as subobjects of the full rank-twelve coefficient system}.}$$

All maps used below were derived independently before this comparison:

• the rank-twelve nilpotent and algebraic extension columns of Entry 300;
• the exceptional Cut--nearby realization of Entries 226 and 301;
• the orientation-twisted conductor intertwiner of Entries 305 and 308;
• the unimodular occurrence quotient of Entry 312; and
• the second-Rees regularization of Entries 292--293.

No projector or basis change is fitted to align the two images.

Two algebraic realizations

Work on the generic nonsoft locus (xy(x+y)\ne0). In ambient algebraic coordinates

$$(e_2,e_3,e_4,e_5,e_6,v_0)\subset\mathcal T_7,$$

the logarithmic rank-three image is

$$L_{\log} =\left\langle \Theta_{101}^{\rm fix}, \Theta_{110}^{\rm fix}, \frac{e_6}{8(x+y)} \right\rangle .$$

Its first two generators have nonzero (e_4) and (e_2) coordinates, respectively, together with opposite (v_0) tails. Its third generator is the (e_6) line.

In the orientation-twisted enhanced frame, the Cut--nearby realization of the same primitive conductor basis is

$$L_{\rm Cut} =\left\langle 2y e_3, 2x e_5, y e_3+x e_5+e_6 \right\rangle .$$

This is the image of the source-derived matrix (J), with the four enhanced occurrences presented by the unimodular quotient (K).

Exact rank test

At the exact generic fiber ((x,y)=(2,3)), independently clearing each nonzero column denominator gives integral matrices with

$$\operatorname{rank}L_{\log}=3, \qquad \operatorname{rank}L_{\rm Cut}=3,$$

and

$$\operatorname{rank}(L_{\log}+L_{\rm Cut})=5.$$

Therefore

$$\boxed{ \dim(L_{\log}\cap L_{\rm Cut})=1.}$$

The support pattern proves the same statement generically: membership in (L_{\rm Cut}\subset\langle e_3,e_5,e_6\rangle) forces the coefficients of the unique (e_4) and (e_2) logarithmic generators to vanish. Hence

$$\boxed{ L_{\log}\cap L_{\rm Cut}=\langle e_6\rangle .}$$

The strict-equality claim is falsified.

Second-Rees bridge

The common line is not accidental. The raw primitive two-wall connection has the fixed second-order term

$$E^{-2}\frac18e_6.$$

The source-normalized Rees correction removes the double pole and produces the logarithmic top line

$$\frac{e_6}{8(x+y)}.$$

On the Cut side,

$$e_6 =\left(y e_3+x e_5+e_6\right) -\frac12(2y e_3)-\frac12(2x e_5).$$

Thus the sole geometric intersection of the two rank-three images is exactly the primitive second-Rees top direction.

Elliptic separation

The complete logarithmic nilpotent image has one additional elliptic line, so

$$\operatorname{rank}\operatorname{im}N_E^{(12)}=4.$$

The Cut--nearby commutator has zero infinity-Gysin image and therefore no elliptic component. The elliptic line remains independent rather than being matched or cancelled.

Cyclic transport

Entry 366 transports the frozen source descriptors, orientations, and Leray normalizations through all three marked-Cut sectors. Consequently every sector has the same profile

$$(\operatorname{rank}L_{\log}, \operatorname{rank}L_{\rm Cut}, \operatorname{rank}(L_{\log}\cap L_{\rm Cut})) =(3,3,1).$$

No sector-specific support or carrier correction appears.

Narrow result

Strict full-rank Cut--nearby commutation is false. The surviving compatibility is the filtered cospan

$$L_{\log} \longleftarrow \mathcal C_{\rm cond} \longrightarrow L_{\rm Cut},$$

where (mathcal C_{\rm cond}) is the common primitive conductor occurrence lattice. Its top line is realized geometrically on both sides by the second-Rees (e_6) bridge; the two wall directions have different layer-specific realizations. The elliptic line belongs only to the logarithmic nearby layer.

Classification

Update to H2 versus H3

This test rejects an overly strict form of H2 in which the two operations must have identical images. It does not force H3: both realizations are still related by the same source-derived conductor lattice, purity, orientation, Leray, and Rees calculus, with no sector-specific operation or new carrier stratum.

The surviving H2 statement is genuinely filtered:

$$\boxed{ \text{shared calculus relates distinct coefficient layers by a canonical cospan and higher-Rees bridge, not by strict commutation}.}$$

Evidence

• research/benincasa/marici-gm/src/bin/full_rank12_cut_nearby_layers.rs;
• research/benincasa/full-rank12-cut-nearby-layers-certificate.json;
• Entries 226, 292--293, 300--301, 305, 308, 312, and 366.

Next falsifier

Determine whether this generic filtered cospan extends through the first nontransverse supports where the two wall directions can collide:

$$xy(x+y)=0$$

and the conductor/elliptic discriminant intersections. Compute the joint Rees/nearby Smith data of both legs simultaneously. A new torsion prime, irreducible support factor, or loss of the common (e_6) line would push the architecture toward H3. Closure using only soft normals, the existing two half-sums, and the Legendre node would strongly support filtered H2.

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