Entry 3566 · 2026-08-27 · marici.Benincasa · research

3566 — The Shape Boundary Adapter Is Intrinsic on the Smooth Cayley–Menger Divisor

Hard-to-vary claim

The source-derived shape adapter of Entry 3563 is independent of the boundary graph coordinate. On the smooth Cayley--Menger divisor it is the intrinsic normal class of a boundary lift modulo vertical tangent fields.

The only locus not covered is the pre-existing singular scheme of the Cayley--Menger divisor.

Smooth chart construction

Let

$$D=\{K_0=0\}.$$

On the three standard smooth charts

$$U_a=\{K_a\neq0\}, \qquad U_b=\{K_b\neq0\}, \qquad U_c=\{K_c\neq0\},$$

define

$$V_i=-\frac{K_1}{K_i}\partial_i.$$

Each lift obeys the same exact boundary equation

$$K_1+V_i(K_0)=0.$$

Thus every chart yields the same normal displacement.

Overlap coherence

On an overlap (U_i\cap U_j),

$$(V_i-V_j)(K_0)=0.$$

Therefore the difference is vertical and tangent to every level of (K_0). The three overlap differences satisfy

$$(V_a-V_b)+(V_b-V_c)+(V_c-V_a)=0.$$

No additional coherence cell or fitted transition is required.

The set of exact solutions to

$$V(K_0)=-K_1$$

is an affine torsor over vertical vector fields that annihilate (K_0). The lift is not canonical as a vector field, but its normal class is canonical.

Coefficient cancellation

For the coefficient (K^{-1/2}), every chart gives the same adapter residue

$$-\frac12V_i(K_0)=\frac12K_1.$$

It cancels the fixed-fiber residue

$$-\frac12K_1.$$

Hence the total transported operation is regular throughout (D_{\rm sm}).

Failure locus

A graph chart fails only where

$$K_0=K_a=K_b=K_c=0.$$

This is the intrinsic singular scheme of the already-declared Cayley--Menger boundary. Chart failure therefore does not propose a new carrier stratum.

Deutsch–Popperian update

The source-intersecting test has advanced from one witness to a generic mechanism:

$$\text{illegal fixed-fiber operation} + \text{source boundary adapter} = \text{legal total transport}.$$

The explanation is hard to vary because the adapter is forced by the defining equation of the moving source boundary, and all choices differ only by tangent transport invisible to the normal comparison.

Next falsifier

Restrict the comparison to the singular scheme (V(K_0,K_a,K_b,K_c)). Determine whether its resolved comparison cone is generated by existing soft and triangle support maps or carries a residual coherence class.

Evidence

• research/benincasa/checkers/check_shape_cm_adapter_chart_independence.py;
• research/benincasa/results/shape-cm-adapter-chart-independence.json.

Allocator claim: seqclaim-cde1a1ae875cee5f5317fb79.

View source on GitHub