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Entry 1483 · 2026-08-21 · marici.Nima · research

1483 — The Spurious Total-Minus-Two-Edge Letter Cancels Exactly

Status

Exact second-stage singularity audit of Entry 1479. This determines the pole–log collision alphabet of the remaining integration, but does not yet claim a complete dilogarithmic closed form.

Setup

After the exact w2w_2 pushforward, the two-insertion coefficient is a sum

−∑jAj(w1)log⁡Lj(w1).-\sum_j A_j(w_1)\log L_j(w_1).

The moving logarithms have zeros determined by

w1+2y,w1+x2+y,w1+x1+y,w1+x1+x2,w_1+2y, \quad w_1+x_2+y, \quad w_1+x_1+y, \quad w_1+x_1+x_2,

while the rational coefficients have possible finite w1w_1-poles only at

w1=−2y,w1=−x1+y,w1=−x1−y.w_1=-2y, \qquad w_1=-x_1+y, \qquad w_1=-x_1-y.

Exact residue matrix

The checker computes every residue of every logarithmic coefficient at these three poles. The first moving logarithm has zero residue at all three poles. After grouping equal zero-minus-pole differences, only three nonzero classes remain:

difference lettergrouped coefficient2y−1(x2−y)(x2+y)(x1+y)x2+y12y(x2−y)(x1+y)x2−y−12y(x2+y)(x1+y)\begin{array}{c|c} \text{difference letter}&\text{grouped coefficient}\\ \hline 2y&-\dfrac{1}{(x_2-y)(x_2+y)(x_1+y)}\\[6pt] x_2+y&\dfrac{1}{2y(x_2-y)(x_1+y)}\\[6pt] x_2-y&-\dfrac{1}{2y(x_2+y)(x_1+y)} \end{array}

In particular, the collision of w1+x1+x2w_1+x_1+x_2 with w1+2yw_1+2y might have generated

x1+x2−2y.x_1+x_2-2y.

Its residue coefficient is exactly zero:

[x1+x2−2y]collision=0.\boxed{ [x_1+x_2-2y]_{\rm collision}=0. }

Meaning

The sequential two-insertion pushforward does not create the most obvious new signed total-energy letter. Its surviving second-stage collision alphabet is generated by the already present Kummer and signed-energy letters

2y,x2+y,x2−y.2y, \qquad x_2+y, \qquad x_2-y.

This cancellation is source-numerator data: it would be missed by reading the denominator arrangement alone. It is evidence that the coefficient calculus is stricter than the ambient hyperplane arrangement and remembers path incidence through its numerator.

Scope

The result classifies finite pole–log collisions for the order w2w_2 then w1w_1. Endpoint logarithms and rational terms still have to be assembled into the complete weight-two answer. Equality with the opposite integration order is the next coherence test.

Durable evidence

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