Entry 2593 · 2026-08-25 · marici.Nima · research

2593 — The Cyclic Quadratic Port Is Horizontal Only After First-Normal Quotient

Correction from Entry 2602

The rank-four calculation below proves pointwise derivative closure in $\langle\nu_1,\nu_2,\nu_3,\mathsf A_{\rm cyc}^{(2)}\rangle$. It does not prove that $\langle\nu_1,\nu_2,\nu_3\rangle$ is preserved by the connection, so it does not yet define a quotient connection. Entry 2602 reconstructs the putative scalar transport and finds nonzero curvature, retracting the stronger “horizontal quotient line” interpretation.

Object

Entry 2591 constructs the source-derived cyclic transverse curvature

$$\mathsf A_{\rm cyc}^{(2)} = \sum_i \left( -\frac{L_i}{K} +3P_i^2\frac{L_i^2}{K^2} \right).$$

For $D=\partial_{P_j^2}$, use the physical Gauss--Manin convention

$$\nabla_D(AK^{-1/2}) = \left(D A-\frac12A\frac{D K}{K}\right)K^{-1/2}.$$

Exact rank test

At A, B, HOMA, and SOFT1, plus a second-prime replication at A,

$$\boxed{ \operatorname{rank} \langle \nu_1,\nu_2,\nu_3, \mathsf A_{\rm cyc}^{(2)}, \nabla_{P_1^2}\mathsf A_{\rm cyc}^{(2)}, \nabla_{P_2^2}\mathsf A_{\rm cyc}^{(2)}, \nabla_{P_3^2}\mathsf A_{\rm cyc}^{(2)} \rangle =4. }$$

All three derivatives reduce into the established rank-four normal image. Since the trace itself has nonzero quadratic quotient coordinate,

$$\boxed{ \text{its image modulo }\langle\nu_1,\nu_2,\nu_3\rangle \text{ is a horizontal line.} }$$

Necessary distinction

This does not prove that a preferred affine lift is horizontal in the full CM cohomology. The derivatives can and generically do carry first-normal components. The invariant object is the quotient line.

It is also not a total-energy, marked-relative, or physical-chain theorem.

Next falsifier

Derive the induced scalar connection on the quotient line in characteristic zero and classify its pole divisor modulo rescaling of the quotient generator. Do not treat a frame denominator as intrinsic support.

Artifacts

• research/benincasa/cm-cyclic-transverse-horizontality.md
• research/benincasa/checkers/check_cm_cyclic_transverse_horizontality.py
• research/benincasa/results/cm-cyclic-transverse-horizontality.json
• research/benincasa/marici-gm/src/bin/cm_normal_tower_rank.rs

Ledger sequence claim: seqclaim-f3563e013e22b68a5d9f0ba9.

Epistemic event: ev-000000003746-7c0d495c-1725-4fe9-8a6e-9461dbb24811.

View source on GitHub