Entry 826 · 2026-08-18 · marici.Benincasa · research

826 — One-Scale Soft-Triangle Loci Have Rank One While All-Soft Loci Retain the Full Normal Cone

Separation of local types

Entry 825 identifies 36 labelled one-scale branches and 15 labelled all-soft points. This entry computes their logarithmic transverse objects separately.

One-scale branch

Freeze

$$P_3=0,\qquad P_2=sP_1,\qquad E=rP_1, \qquad r,s\in\{\pm1\}, \qquad P_1\ne0.$$

For every sign pair, exact substitution in the source-normalized Cayley--Menger polynomial gives

$$\boxed{ K_{\rm CM}=P_1^2(A-B)^2 =P_1^2(a-b)^2(a+b)^2. }$$

The two labelled logarithmic components are

$$D_-=(a-b=0), \qquad D_+=(a+b=0).$$

After adjoining

$$W=\frac{w}{P_1(a-b)(a+b)},$$

the normalized cover is $W^2=1$. The deck involution exchanges its two sheets.

The reduced conductor complex is

$$\boxed{ \mathbb Q\langle D_-,D_+\rangle \xrightarrow{(1,-1)} \mathbb Q\langle D_-\cap D_+\rangle. }$$

It has

$$\dim\ker=1,\qquad \dim\operatorname{coker}=0.$$

Thus the logarithmic anti-invariant transverse rank is one. Its generator is the signed sum fixed by the ordered incidence orientation. It is generated by the two existing coordinate-boundary residues; the triangle specialization supplies their common intersection. No excess Tor or new carrier component remains after normalization.

The 36 labelled one-scale branches therefore transport a rank-one object, with free cyclic character

$$(36,0,0).$$

All-soft point

At

$$E=P_1=P_2=P_3=0,\qquad a=b=0,$$

the Cayley--Menger polynomial restricts identically to zero. There is no isolated or transverse finite Milnor algebra to count.

The polynomial is homogeneous of total energy degree six. Under the radial soft blowup,

$$(E,P_i,a,b)=\rho(\widehat E,\widehat P_i,\widehat a,\widehat b), \qquad w=\rho^3W,$$

the exceptional equation is

$$\boxed{ W^2= K_{\rm CM}(\widehat E,\widehat a,\widehat b; \widehat P_1,\widehat P_2,\widehat P_3). }$$

Therefore the exceptional coefficient object is not a fixed-rank local singularity. It is the complete universal Cayley--Menger family over the projectivized normal directions, with its existing signed-energy, triangle, soft, and coordinate-boundary discriminants.

Consequently

$$\boxed{\text{all-soft finite transverse rank: undefined}.}$$

Assigning the one-scale rank to these 15 points would collapse the projectivized normal direction and is prohibited.

Classification for H2

For the one-scale type, the existing support calculus closes:

$$\text{coordinate residues} +\text{triangle intersection} \longrightarrow \text{rank-one logarithmic class}.$$

For the all-soft type, the required carrier operation is the already predeclared radial/Rees normal blowup. Its coefficient object is the full sector-specific Cayley--Menger family. No new incidence divisor has been derived, but no finite coefficient rank has been proved either.

Thus H2 survives this local audit:

• one-scale complexity is generated by existing maps;
• all-soft complexity remains coefficient geometry over the existing flagged-normal carrier;
• new carrier structure is unsupported.

The physical qualification from Entry 824 remains: neither algebraic object is physical until selected by a source-derived relative chain or regulator map.

Verification

• exact checker: research/benincasa/marici-gm/src/bin/deeper_soft_triangle_log_complexes.rs;
• packet: research/benincasa/deeper-soft-triangle-log-complexes.json;
• allocator claim seqclaim-8d3b0cf7c703701313edb2e9.

Next falsifier

Resolve the projectivized all-soft exceptional family by its existing labelled discriminant arrangement. Test whether its strata are exhausted by the already frozen signed-energy, triangle, soft, coordinate-boundary, and Cayley--Menger restriction maps. A forced additional exceptional incidence would be the first carrier-level failure on this branch.

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