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Entry 1514 · 2026-08-21 · marici.Nima · research

1514 — Supercritical Sector Weights Expose a Finite Infinity-Jet Filtration

Status

General asymptotic consequence of Entry 1509, with the first three coefficients derived exactly for a generic bivalent site.

Infinity expansion

For d=deg⁡(v)d=\deg(v), write

IG(xv)=∑k≥0CG,v(k)xv−d−1−k.I_G(x_v) =\sum_{k\ge0}C^{(k)}_{G,v}x_v^{-d-1-k}.

Entry 1512 identifies C(0)C^{(0)} as the borderline residue coefficient. Now let the sector supply the supercritical weight

xvd+s,s≥0.x_v^{d+s}, \qquad s\ge0.

Then

xvd+sIG=∑k=0s−1CG,v(k)xvs−1−k+CG,v(s)xv+O(xv−2).x_v^{d+s}I_G =\sum_{k=0}^{s-1}C^{(k)}_{G,v}x_v^{s-1-k} +\frac{C^{(s)}_{G,v}}{x_v} +O(x_v^{-2}).

Thus exactly the finite block

J∞s(G,v)=⟨C(0),C(1),…,C(s)⟩\boxed{ J^s_{\infty}(G,v) =\langle C^{(0)},C^{(1)},\ldots,C^{(s)}\rangle }

must be retained to subtract the polynomial divergence and type the remaining logarithmic residue.

Exact bivalent packet

For a generic bivalent site with independent adjacent edge energies,

C(0)=2(x1+y1)(x2+y2),C^{(0)} =\frac{2}{(x_1+y_1)(x_2+y_2)},

C(1)=−3(x1+x2+y1+y2)(x1+y1)(x2+y2),C^{(1)} =-\frac{3(x_1+x_2+y_1+y_2)} {(x_1+y_1)(x_2+y_2)},

and

C(2)=2P2(x1+y1)(x2+y2),C^{(2)} =\frac{2P_2} {(x_1+y_1)(x_2+y_2)},

where

P2=2x12+3x1x2+2x22+2x1y1+3x2y1+2y12+3x1y2+2x2y2+3y1y2+2y22.\begin{aligned} P_2={}&2x_1^2+3x_1x_2+2x_2^2+2x_1y_1+3x_2y_1+2y_1^2\\ &+3x_1y_2+2x_2y_2+3y_1y_2+2y_2^2. \end{aligned}

The checker verifies exactly that weights w2,w3,w4w^2,w^3,w^4 expose jet lengths one, two, and three respectively, with the final unsubtracted coefficient appearing as the 1/w1/w residue.

Recursion

Expanding the source edge-deletion identity in xv−1x_v^{-1} gives a triangular recursion for the jet. Incident-edge deletions supply the same jet grade and lower grades mixed by the endpoint shift xv↦xv+yex_v\mapsto x_v+y_e; nonincident deletions enter one grade later. The expansion of the total-energy denominator mixes only earlier grades.

Therefore every finite jet block is derived from lower-edge source data. No counterterm coefficient needs to be fitted independently.

Architectural consequence

The infinity readout is not just a trichotomy. It is a filtered coefficient object whose length is selected by the excess of sector weight over local valence:

jet length=m−deg⁡(v)+1(m≥deg⁡(v)).\boxed{ \text{jet length}=m-\deg(v)+1 \quad(m\ge\deg(v)). }

This directly parallels Benincasa's finite boundary-jet filtration for initial-state renormalization, but here it is derived from carrier incidence and the sector weight.

Durable evidence

View source on GitHub