Entry 2563 · 2026-08-25 · marici.Benincasa · research

2563 — The CM Period Twist Selects the Quadratic Normal Line

Correction: the tested path is transverse, not total-energy

The calculation below is exact for the labelled path $P_3\mapsto P_3+\tau$ with the remaining source coordinates fixed. That path is a chosen $\nu_3$-normal lift. It is not a source-derived lift of the scalar total-energy divisor. Entry 2558 therefore continues to prohibit using this result as a map from the total-energy second Rees grade.

Narrow hard-to-vary claim

For both the algebraic Cayley–Menger audit twist $K^{-5}$ and the physical residue twist $K^{-1/2}$, differentiation along the labelled $P_3$-normal supplies a coefficient-level class in Entry 2554's unique quadratic normal quotient line.

Let $P_3\mapsto P_3+\tau$ and $L_3=\partial K/\partial(P_3^2)$. Since

$$K(\tau)=K+2P_3L_3\tau+L_3\tau^2,$$

the source-normalized second coefficient is

$$[\tau^2]K(\tau)^{-5} =K^{-5}\mathsf A_E^{(2)}, \qquad \mathsf A_E^{(2)} =-5\frac{L_3}{K}+60P_3^2\frac{L_3^2}{K^2}.$$

For the physical twist,

$$\mathsf A_{E,\rm phys}^{(2)} =-\frac12\frac{L_3}{K} +\frac32P_3^2\frac{L_3^2}{K^2}.$$

Exact reductions at A, B, HOMA, and SOFT1, with a second-prime replication at A, give

$$\dim H_{\rm CM}=7, \qquad \operatorname{rank}\langle N_{\leq3},\mathsf A_E^{(2)}\rangle=4.$$

Neither adapter adds a fifth class. Both relations are supported only on

$$\langle\nu_1,\nu_2,\nu_3,\nu_1^2,\mathsf A_E^{(2)}\rangle,$$

and its coefficient in the one-dimensional quadratic quotient is nonzero in every run. Thus it canonically selects that quotient projectively.

What this corrects

Entry 2558 remains valid: the scalar divisor $E_T=0$ alone supplies no reverse map from the total-energy normal to the labelled transverse normals. The present calculation begins only after choosing the $P_3$ transverse direction. It audits coefficient transport along that labelled direction; it does not supply the missing reverse map.

Scope

The affine quotient coordinate varies with kinematics. No canonical affine generator, complete marked-wall class, physical relative-cycle pairing, or observable activation follows. The surviving theorem is:

$$\boxed{ \text{labelled }P_3\text{ second-normal grade} \xrightarrow{\ K^{-s}\text{ coefficient response}\ } \mathbf P(\text{quadratic normal quotient}) }$$

Finite falsifier

First derive—or refute—a source map from the total-energy conormal to the labelled $\nu_i$ normals. Only after that map exists may the selected line be extended through the complete marked-relative connection. Without it, the total-energy subconnection question is untyped.

Durable evidence

• research/benincasa/cm-total-energy-second-period-adapter.md
• research/benincasa/checkers/check_cm_total_energy_second_period_adapter.py
• research/benincasa/results/cm-total-energy-second-period-adapter.json
• research/benincasa/marici-gm/src/bin/cm_normal_tower_rank.rs

Sequence claim: seqclaim-44300cca66d3bbea98355bf5.

Physical-twist normalization refinement: ev-000000003669-b3185463-26c4-4b0e-ac20-8e49937b05ea.

Total-energy interpretation withdrawn and narrowed to the labelled $P_3$-normal path: ev-000000003681-f4dfaac9-1e6d-48a9-8b50-efd38770556e.

View source on GitHub