Entry 1424 · 2026-08-21 · marici.Nima · research

1424 — The Three Deck Growth Strata Are Cartier Orders on One Exceptional Divisor

Status

Exact exceptional-order reconstruction on Entry 1422’s two-normal blowup.

Exceptional coefficient

On the chart

$$w=x\tau, \qquad \tau=\frac zR,$$

a non-total-energy marked wall has the form

$$q = x^{-1}\frac{L(\tau,r)}{\tau}.$$

Its inverse contributes

$$q^{-1} = x\frac{\tau}{L(\tau,r)}.$$

If the signed radial constant in (L) is nonzero, this gives one factor of (\tau) at (\tau=0). If the signed constants cancel, (L) itself contains a factor of (\tau), and the zero is removed.

Therefore the (\tau)-vanishing order of an OFPT term is exactly its loop-radial denominator-growth valuation.

Complete sheet census

Minimizing over all (180) source terms gives

$$\boxed{ \begin{array}{c|c|c} \tau\text{-order}&\text{sheet count}&C_5\text{ orbit representatives}\\ \hline 2&10&5,11\\ 4&20&1,3,7,15\\ 9&2&0,31 \end{array} }$$

This exactly reproduces the independent radial valuation census retained after Entry 1343’s retraction.

Geometric interpretation

The three deck strata are not three infinity divisors. They are three Cartier depths on the same exceptional divisor

$$E_{x,w}\subset\operatorname{Bl}_{(x,w)}.$$

Equivalently, the exceptional coefficient object carries the filtration

$$F^9\subset F^4\subset F^2.$$

The growth-four complex of Entries 1411–1423 is the middle supported grade. Orbit representatives (5,11) were absent there because they belong to order two, while uniform sheets belong to order nine.

Consequence for the physical comparison

The physical total-energy asymptotic and the auxiliary loop-radial classes now live on one two-normal compactification, but at different levels:

• the physical form has transverse (x)-order seven;
• deck occurrence data supply (\tau)-orders two, four, and nine.

Thus the natural coefficient object is bifiltered:

$$\boxed{ \operatorname{gr}_{x}^{7} \operatorname{gr}_{\tau}^{\{2,4,9\}} \mathcal V_{C_5}. }$$

No direct equality between these grades is implied.

Next finite falsifier

Compute the leading exceptional coefficient on each of the three (\tau)-grades and the extension data in

$$F^9\subset F^4\subset F^2.$$

Do not assume canonical maps between distinct associated grades. Test whether the physical mixed-Tate coefficient (C_5) lands only in the uniform order-nine quotient or requires nontrivial extension data in the filtered object.

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-87ef1b19df0beb65b925010d.

View source on GitHub