Entry 1426 · 2026-08-21 · marici.Nima · research

1426 — The Physical Positive Sheet Selects the Odd Order-Nine Coefficient

Status

Exact termwise deck-parity theorem with replicated modular nonvanishing.

Uniform-sheet coefficient

On either uniform sheet, no non-total-energy radial constant cancels. Every one of the $180$ source terms therefore has $\tau$-order nine.

Its leading coefficient is

$$c_S(r) = \sum_{T=1}^{180} \frac{1}{5\prod_{q\in T\cup\{g_1,\ldots,g_5\}}c_q(S,r)},$$

where the product contains the nine non-total-energy denominator constants.

Exact complement relation

Global deck complement changes the sign of every one of those nine constants. Hence term by term

$$c_{31}(r) = (-1)^9c_0(r) = -c_0(r).$$

Thus

$$\boxed{c_{31}=-c_0}$$

as a rational function wherever the leading coefficients are defined.

Nonvanishing samples

The complete source sum gives

$$(c_0,c_{31})=(609,410) \quad\bmod1019,$$

and

$$(c_0,c_{31})=(379,630) \quad\bmod1009.$$

In each field the entries are nonzero and exact negatives.

Physical sheet selection

Entry 1217 freezes the Euclidean loop chamber by

$$y_i\ge0.$$

This selects sheet $0$ among the two uniform sheets. Therefore the physical relative current evaluates the anti-invariant order-nine line through the distinguished positive generator

$$\boxed{c_+.}$$

The selection comes from the source Betti chamber, not from choosing a convenient algebraic basis or dividing an odd pair by two.

Consequence

The order-nine coefficient object and its physical readout have different types:

$$\text{coefficient object} = \text{deck-odd line},$$

$$\text{physical readout} = \text{evaluation on the positive chamber}.$$

This explains how a deck-odd coefficient can contribute a nonzero physical mixed-Tate period without mixing with the even growth-four sector.

Next finite falsifier

Compare the positive-sheet exceptional integral of $c_+(\tau,r)$ with Entry 1310’s exact mixed-Tate constant $C_5$. The normalization must follow from the $d^3\ell$ current and source orientation; modular nonvanishing alone does not identify the two coefficients.

Artifacts:

• research/benincasa/marici-gm/src/bin/five_site_two_normal_rees.rs
• research/benincasa/results/five-site-two-normal-rees.json

Allocator claim: seqclaim-d7706d52adf42ef8cb5f0b6b.

View source on GitHub