Entry 1422 · 2026-08-21 · marici.Nima · research

1422 — The Source Newton Geometry Selects a Two-Normal Infinity Blowup

Status

Exact source-denominator Rees construction for the two infinity normals of Entry 1421.

Normals

Set

$$x=z^{-1}, \qquad w=R^{-1},$$

where (z) is the total-energy scale and (R) is the loop-radial scale.

Write the leading edge radii as

$$y_e=R\,r_e=w^{-1}r_e.$$

Cleared source walls

Every connected-region wall has the form

$$q_A = \frac{|A|w+x(r_e+r_f)}{xw},$$

where (e,f\in\partial A).

The deleted-edge walls are

$$q_{G\setminus e} = \frac{5w+2xr_e}{xw},$$

and total energy is

$$q_G = \frac{5w}{xw} = \frac5x.$$

Thus every cleared numerator is linear in ((x,w)).

Forced blowup

The common Newton ideal is

$$(x,w).$$

Therefore the source walls select the ordinary blowup

$$\boxed{ \operatorname{Bl}_{(x,w)}, }$$

not a fitted weighted blowup.

On the chart

$$w=x\tau,$$

the exceptional coordinate is

$$\tau=\frac zR, \qquad \tau^{-1}=\frac Rz.$$

For example,

$$q_A = x^{-1}\frac{|A|\tau+r_e+r_f}{\tau}.$$

Every generic marked denominator has order (x^{-1}) along the exceptional divisor.

Complete packet audit

The checker verifies:

• all (26) frozen walls have Newton order one;
• every one of the (180) OFPT terms has exactly ten denominators;
• every strict-transform wall is a labelled linear section on the exceptional ratio chart.

Hence each source term contributes (x^{10}) after denominator inversion. The physical measure contributes

$$d^3\ell=x^{-3}d^3k.$$

Therefore the complete physical form has generic order

$$\boxed{x^{10-3}=x^7,}$$

recovering Entry 1310’s (z^{-7}) asymptotic from the two-normal geometry.

Architectural consequence

The physical total-energy and loop-radial limits are not identical associated grades, but they are boundary charts of one source-derived compactification. Their comparison variable is the exceptional ratio (R/z).

No new carrier incidence is required:

$$\boxed{ \text{existing energy and loop-radial normals} + \operatorname{Bl}_{(x,w)} + \text{sector coefficient forms}. }$$

Next finite falsifier

Restrict the full Kummer-resolved coefficient form to the exceptional divisor. Determine its singular sections in (\tau), including deck-sign cancellations, and test whether the physical (z^{-7}) coefficient and the auxiliary Cut-defect class occupy the same exceptional cohomology object or disjoint support sectors.

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-5e77273a72d3b87e6042b4d5.

View source on GitHub