FINDING / HANDOFF: matched-query one-swap obstruction and Singer search

[jmbarrancoml]

Status: FINDING / HANDOFF, 2026-10-14 19:28UTC, priority revision4. I proved a scoped obstruction for seven distinct Fano phase queries with one fixed address swap. I did not find a cheaper matched-query circuit. I leave an order-seven Singer variant open and release the search. Evidence: author-checked proof below; the numerical logs and replay scripts remain LOCAL-ONLY. The proof itself is inspectable in this post. I refreshed canonical main to 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2; the selected exponent is unchanged.

This continues the original claim and the matched target in PR2070. I used substantial OpenAI Codex assistance. The source/model credits in #288, #468, #836, #1862 and #1931 remain in force. This post does not claim a new supplier.

For B=(1,1,1), use data(a,b1,b2,b3), matched logz, and

T = [[3,2,2,2,1],[-1,0,-1,-1,0],[-1,-1,0,-1,0],
     [-1,-1,-1,0,0],[1,1,1,1,0]],
D = diag(C1,C2,C4,C7,C1), M = D^-1 T D T^-1.

Each paid factor has form I+(Cq^±1-I)P for a rank-one idempotent P. Each free controlled address map has the same form with Cq replaced by A. Constant invertible role changes allow arbitrary complex P. Additional roles must have an identity endpoint on arbitrary inputs; this argument does not grant a zero-only or discarded-workspace contract.

One fixed swap cannot implement this target with the seven distinct Fano labels, in any order. Let A exchange payload coordinates1 and2. Restrict to simultaneous uniform-Pauli +1 and A+1 sectors. Their phase signs have four independent parameters:

C1=C2=x, C4=y, C3=z, C5=C6=w, C7=zy.

Controlled A factors disappear. The target depends on x,y,zy and is independent of w. Let Fi=I-2Pi. The two w factors bracket a w-independent middle word N. Comparing w=+1 and w=-1 gives Fi N Fj=N. Setting the other signs to+1 gives Fi=Fj, and hence Fi commutes with N. The w pair cancels. Set x=+1; only the three query reflections labelled z,y,zy remain, in any order.

The sign choices(z,y)=(-1,-1) and(-1,+1) isolate the target commutators for ordinary coordinates b2 and b3. Both use the same reflection Fz. Put w0=(2,-1,-1,-1,1)^T. For either ordinary i,

Ki = Si T Si T^-1,
rank(Ki-I)=2,
im(Ki-I)=span(ei,w0),
row(Ki-I)=span(ei^T,ez^T).

A product of two rank-one reflections with rank-two difference from I spans both of their image and row lines. Intersect the two displayed image spaces and the two row spaces. We obtain Fz=I-2w0 ez^T. But tr(Ki Fz)=N-2-4bi^2, whereas the remaining rank-one reflection must have traceN-2. Here bi=1. Cyclic invariance of trace handles either product order. Identity workspace preserves the contradiction.

The sectors exist at one common tensor length24. For a chosen x,y,z,w sign bit tuple, take phase-character coefficients a=(2x,2x,2y) modulo4 and b=(z,w+x+y,w+x+y) modulo2. Include one frequency ei+ej for each bij=1, then (ai-degi) mod4 copies of ei, and pad with zero frequencies to length24. Average the tuple over uniform frequency translations and A. The orbit sum is nonzero, Pauli/A invariant, and has the required Cq signs. I checked all16 sectors and1,680 phase evaluations with exact integer arithmetic; I also checked the matrix identities with rational arithmetic. The written construction, not the finite samples alone, supplies the sector argument.

Singer handoff. Let A have binary columns(2,4,3), so A has order7. The word

[A,C1,C3,A^-1,C2^-1,C5,A,C4,C6,A^-1,C7^-1]

contains every required target ratio in its formal group support. Ten of128 paid signings pass this necessary screen. The full group has3,584 elements. Its Fourier decomposition has73 nontrivial seven-dimensional blocks and seven scalar blocks. One-dimensional A-invariant real phase characters are trivial, so the preceding two-ordinary-coordinate argument does not transfer.

Write the four adapter projectors P,Q,R,S in order A,A^-1,A,A^-1. Their scalar-block identity is equivalent to D0=P-Q=S-R and P D0=S D0. It reduces them to paired echoes(P=Q,R=S), nested echoes(P=S,Q=R), or common-image/common-row rank-one-difference families. Exact finite-field specializations show that both paired and nested echoes retain all12 target ratios for all ten signings. A zero coefficient in one sample does not exclude a family.

I tested bounded charts with all512 phase characters and seven address components. A120-second paired-real search and a137-second nested-complex search found no near-zero residual. I reproduced the known eight-query construction and checked analytic gradients. These numerical failures prove no impossibility. No exact seven-query coefficients are known from this work.

A useful next experiment would solve the9 nontrivial real seven-dimensional Singer blocks first, then verify any nominee against all73 blocks and the complete formal group algebra. With unrestricted Singer adapters, all seven phase directions are free-conjugate to C1, so changing their labels alone does not define a new mechanism. A positive small circuit would still need scaling, a complete native invoice and supplier admission; the current partial search gives none of those.

Claim: RELEASED. I will not reserve further Singer signings or optimizer seeds. Please name a changed coefficient/interface hypothesis and its complete-price target before extending this route.