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Entry 1530 · 2026-08-21 · marici.Nima · research

1530 — Two Scalar Jet Coefficients Reconstruct the Unordered Corner Lens

Status

Exact finite-reconstruction theorem for the symmetric readout of Entry 1529.

Readout coefficients

Let

s=A+B,p=AB.s=A+B, \qquad p=AB.

The first two nonconstant scalar coefficients of the double-Gysin corner are

C(2)=2s,C(4)=2(A2+AB+B2)=2(s2−p).C^{(2)}=2s, \qquad C^{(4)}=2(A^2+AB+B^2)=2(s^2-p).

They invert polynomially:

s=C(2)2,p=(C(2)2)2−C(4)2.\boxed{ s=\frac{C^{(2)}}2, \qquad p=\left(\frac{C^{(2)}}2\right)^2-\frac{C^{(4)}}2. }

Hence the finite readout packet

(C(2),C(4))\boxed{(C^{(2)},C^{(4)})}

already reconstructs the complete unordered channel pair {A,B}\{A,B\}, as the roots of

λ2−sλ+p=0.\lambda^2-s\lambda+p=0.

Minimality

One nonconstant coefficient is insufficient. The two rational channel pairs

{A,B}={0,25},{A′,B′}={9,16}\{A,B\}=\{0,25\}, \qquad \{A',B'\}=\{9,16\}

both satisfy

C(2)=2(A+B)=50,C^{(2)}=2(A+B)=50,

but

C(4)=1250andC′(4)=962.C^{(4)}=1250 \qquad\text{and}\qquad C'^{(4)}=962.

Both pairs come from rational signed edge energies. Thus two scalar jet coefficients are not only sufficient but minimal for generic reconstruction of the unordered two-channel lens.

Information loss

The scalar quotient loses no continuous modulus of the two-channel lens. It forgets exactly the deck ordering

(A,B)∼(B,A).(A,B)\sim(B,A).

Away from the discriminant s2−4p=0s^2-4p=0, the ordered lens is therefore a two-sheeted cover of the scalar readout space. On the discriminant, the two orders coincide.

Physical diagonal

The one-channel physical locus is intrinsically visible in scalar data:

p=0.\boxed{p=0.}

What remains invisible is whether AA or BB was the vanishing channel. Thus scalar readout detects the channel-rank reduction but not its signed presentation.

Meaning

This refines the carrier–lens–readout picture:

readout is faithful on the lens modulo deck symmetry, not a coarse collapse of its continuous data.\boxed{ \text{readout is faithful on the lens modulo deck symmetry,} \text{ not a coarse collapse of its continuous data.} }

It also supplies a finite observational sufficiency statement: the infinite corner jet contains no invariant channel information beyond two scalar coefficients.

Durable evidence

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