Entry 1529 · 2026-08-21 · marici.Nima · research

1529 — The Scalar Corner Readout Is the Symmetric Quotient of the Two-Channel Lens

Status

Closed invariant-theoretic description of the lens/readout separation in Entries 1525–1527.

Equivariant lens

The supported corner lens is generated by

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

with edge-sign reversal exchanging (A) and (B). Its scalar infinity jet depends only on the symmetric combinations

$$\boxed{ s=A+B=2(y_1^2+y_2^2), \qquad p=AB=(y_1^2-y_2^2)^2. }$$

Accordingly the scalar readout factors through

$$\boxed{ \mathbb Q[A,B] \longrightarrow \mathbb Q[A,B]^{\mathbb Z_2} =\mathbb Q[s,p]. }$$

The even-jet recurrence is precisely

$$E_m=sE_{m-1}-pE_{m-2}.$$

Branch locus

The discriminant of the unordered channel pair is

$$\boxed{ s^2-4p=(A-B)^2=16y_1^2y_2^2. }$$

Thus the quotient is branched where either edge energy vanishes, not on the generic physical equal-edge locus.

Physical diagonal

For (y_2=\pm y_1=\pm y), both conjugate signed diagonals have the same invariant image

$$\boxed{(s,p)=(4y^2,0).}$$

Away from (y=0), this point is not on the quotient branch locus:

$$s^2-4p=16y^4\ne0.$$

Therefore the scalar readout identifies the two channel selections because it forgets their ordering, not because the two channels collide.

Meaning

This is an exact realization of the program's three-layer distinction:

$$\boxed{ \text{supported carrier corner} \to \text{equivariant two-channel coefficient lens} \to \text{symmetric scalar readout}. }$$

The readout loses a discrete channel label even at a smooth quotient point. Hence physically indistinguishable scalar data need not imply identical coefficient presentations or a singular carrier degeneration.

Durable evidence

• research/nima/check_supercritical_infinity_jet.sage;
• Entries 1525–1527;
• allocator claim seqclaim-10a52d82843ecb22f0762df9.

View source on GitHub