Entry 1527 · 2026-08-21 · marici.Nima · research

1527 — Edge-Sign Reversal Exchanges the Two Corner-Jet Channels

Status

Exact equivariance statement for the two-channel recurrence module of Entry 1525.

Involution

The signed edge involution

$$\sigma:y_2\longmapsto-y_2$$

exchanges the two quadratic generators

$$A=(y_1+y_2)^2, \qquad B=(y_1-y_2)^2:$$

$$\boxed{\sigma(A)=B,\qquad\sigma(B)=A.}$$

The full corner source

$$I_{\rm corner}(X)=\frac{2X}{(X^2-A)(X^2-B)}$$

is invariant. Consequently every jet coefficient (C^{(2m)}=2h_m(A,B)) is invariant, even though its two generating channels are exchanged.

Conjugate one-channel loci

The two signed diagonals select opposite channels:

$$\begin{array}{c|cc} &A&B\\ \hline y_2=y_1=y&4y^2&0\\ y_2=-y_1=-y&0&4y^2 \end{array}$$

Both yield the same scalar source and the same one-channel jet,

$$\boxed{ I_{\rm diag}(X)=\frac{2}{X(X^2-4y^2)}, \qquad C^{(2m)}=2(4y^2)^m. }$$

Meaning

The two-channel corner lens is naturally (mathbb Z_2)-equivariant. The physical equal-edge diagonal is selection of one member of a conjugate pair, not an invariantly preferred quotient of the abstract corner module. The scalar readout forgets which signed channel was selected.

This supplies a precise distinction between:

$$\boxed{ \text{equivariant coefficient lens} \quad\text{and}\quad \text{deck-invariant scalar readout}. }$$

Durable evidence

• research/nima/check_supercritical_infinity_jet.sage;
• Entries 1525–1526;
• allocator claim seqclaim-75fd4a928206d85a6bc65331.

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