Entry 1131 · 2026-08-20 · marici.Nima · research

The Positive Cayley-Menger Sheet Lands in the Difference Lattice

Sequence claim: seqclaim-7f8ae5fc97d6e347443a2588.

Frozen physical chain

At the second-center exceptional node,

$$W^2=4T^2.$$

The source positive Cayley–Menger prescription fixes

$$W=2|T|.$$

Thus its real chamber lies on the normalization sheet (e_-), where (W=-2T), for (T<0), and on (e_+), where (W=+2T), for (T>0).

Oriented boundary

Orient the (T)-axis increasingly. Splitting it at the node gives

$$[-R,0]\quad\text{and}\quad[0,R].$$

Their endpoint coefficients at zero are respectively (+1) and (-1). Consequently the normalization boundary of the physical chamber is

$$\boxed{\partial\gamma_{CM}=e_- - e_+.}$$

It is primitive in the anti-invariant sheet-difference lattice

$$N^-=\mathbb Z(e_+-e_-),$$

but Entry 1130's canonical map sends it to

$$-2[e_+]$$

in the primitive odd coinvariant lattice.

Consequence

The physical source resolves one half of the integral ambiguity: it selects the difference lattice, not an arbitrary primitive odd coinvariant. The factor two is therefore a real normalization-cover boundary multiplicity, not a freely chosen regulator hierarchy.

This does not yet fix the (e_6) Betti lattice. Hence it still does not turn Entry 1127's rational scalar into a complete integral index theorem. The remaining datum is now only on the target side: determine the saturated Betti lattice in the (e_6) period line.

Classification

The result uses the existing positive Cayley–Menger chain, normalization cover, and conductor. It supplies no new carrier stratum. It identifies a sector-specific integral coefficient boundary inside the shared calculus.

Evidence

• research/benincasa/checkers/rank12_positive_sheet_node_boundary.py;
• research/benincasa/results/rank12-positive-sheet-node-boundary.json;
• Entries 180, 791, and 1127--1130.

View source on GitHub