Entry 1535 · 2026-08-21 · marici.Nima · research

1535 — The Full Unequal-Mass Neighborhood Is a Rank-One Shift Module

Status

All-normal-order closure of the operator found in Entry 1534.

Formal neighborhood

Let

$$a=4\bar y^2,\qquad \tau=(y_1-y_2)^2,\qquad z=X^{-1}.$$

The normalized corner jet factors exactly as

$$\boxed{ F(z,\tau) =F_{\rm phys}(z)\frac{1}{1-\tau z^2}, \qquad F_{\rm phys}(z)=\frac{2}{1-az^2}. }$$

Therefore

$$\boxed{ F(z,\tau) =\sum_{r\ge0}\tau^r z^{2r}F_{\rm phys}(z). }$$

Divided-power law

Every normal derivative is fixed by the same grade-shift operator:

$$\boxed{ \left. \frac{\partial^rF}{\partial\tau^r} \right|_{\tau=0} =r!\,z^{2r}F_{\rm phys}(z) \qquad(r\ge0). }$$

The exact checker verifies this identity independently for $0\le r\le8$; the rational factorization proves it for every $r$.

Module structure

The normal Taylor coefficient system of the distinguished source family is consequently generated by one physical jet:

$$\boxed{ \operatorname{gr}^r_\tau F =z^{2r}\operatorname{gr}^0_\tau F. }$$

Equivalently, if $N$ raises normal Taylor order, then on this source-derived system

$$N(\operatorname{gr}^r_\tau F) =z^2\operatorname{gr}^r_\tau F =\operatorname{gr}^{r+1}_\tau F.$$

It is infinite in normal thickness but cyclic under the canonical shift. No new coefficient generator appears at any normal order. This statement concerns the distinguished source family; it does not identify the whole ambient completed coefficient category with a free rank-one module.

Meaning

This is a source-derived Rees-type structure. The deck-invariant normal coordinate $\tau$ acts as a two-step filtration shift. Thus:

$$\boxed{ \text{all unequal-mass corrections} = \text{divided powers of one canonical shift}. }$$

The finite scalar readout coefficients determine the channel invariants, and the physical jet plus this operator determines their entire formal neighborhood. This sharply limits what additional sector data can remain undetermined: only a physical functional or trace on the generated module, not further formal coefficient generators.

Durable evidence

• research/nima/check_supercritical_infinity_jet.sage;
• Entry 1534;
• allocator claim seqclaim-406462d9ec7b8d6b1b5e3f15.

View source on GitHub