We prove an exact, unconditional identity in the finite Weil (oscillator) representation of the Heisenberg group $\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})$, $q$ an odd prime: for every complete (inversion-symmetric) BFS shell of the Cayley graph $\mathrm{Cay}(\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z}), S_q)$, the orbit vectors generated by conjugate Weil sectors $c$ and $q-c$ span identical subspaces of $\mathbb{C}^q$, for the special character block and generic character triples alike. The proof reduces every orbit vector to a single pure Fourier mode and shows the reachable frequency set is closed under negation whenever the generating shell is inversion-symmetric — unconditional, coherence-independent, verified independently by $1{,}548$ exact integer-arithmetic checks across six primes with zero failures, plus a negative control on a deliberately truncated shell. We characterise precisely which diagnostics this forecloses: not only Fourier support, rank, projectors, dimensions, and principal angles, but any quantity invariant under independent rephasing and reindexing of the frame — including normalised Bargmann invariants, proved trivial by an explicit frame-orbit argument. A bibliography search confirms $\rho_{q-c}=\overline{\rho_c}$ is standard; no matching result was found for the subspace-equality identity or its no-go corollaries. This paper corrects an earlier version (v1.1.2): independent review found its central-theorem proof unsound and its separate Born-rule proof independently defective; both are withdrawn, and the singlet correlator and Tsirelson bound survive only as a standard, conditional remark. What remains is the rigidity result above and its precisely scoped no-go corollaries. Read as a methodological instance for the wider programme (interpretive, not a further result): a numerical near-zero-residual observation can be forced by a construction's own structural degeneracy rather than signal an intended physical mechanism.
We prove an exact, unconditional identity in the finite Weil (oscillator) representation of the Heisenberg group Heis3(Z/qZ), q an odd prime: for every complete (inversion-symmetric) BFS shell of the Cayley graph Cay(Heis3(Z/qZ),Sq), the orbit vectors generated by conjugate Weil sectors c and q−c span identical subspaces of Cq, for the special character block and generic character triples alike. The proof reduces every orbit vector to a single pure Fourier mode and shows the reachable frequency set is closed under negation whenever the generating shell is inversion-symmetric — unconditional, coherence-independent, verified independently by 1,548 exact integer-arithmetic checks across six primes with zero failures, plus a negative control on a deliberately truncated shell. We characterise precisely which diagnostics this forecloses: not only Fourier support, rank, projectors, dimensions, and principal angles, but any quantity invariant under independent rephasing and reindexing of the frame — including normalised Bargmann invariants, proved trivial by an explicit frame-orbit argument. A bibliography search confirms ρq−c=ρc is standard; no matching result was found for the subspace-equality identity or its no-go corollaries. This paper corrects an earlier version (v1.1.2): independent review found its central-theorem proof unsound and its separate Born-rule proof independently defective; both are withdrawn, and the singlet correlator and Tsirelson bound survive only as a standard, conditional remark. What remains is the rigidity result above and its precisely scoped no-go corollaries. Read as a methodological instance for the wider programme (interpretive, not a further result): a numerical near-zero-residual observation can be forced by a construction's own structural degeneracy rather than signal an intended physical mechanism.
The quantum structure sub-programme comprises three companion papers on the binary icosahedral group 2I, plus one independently published result. Q1 proves an exact, unconditional Fourier-support rigidity theorem in the finite Weil representation of the Heisenberg group, with a precise no-go corollary for support-, rank-, and Gram-based diagnostics; it establishes no phase coherence, singlet correlator, Tsirelson bound, or Born rule. Q2 proves an exact finite-group fact: the spin-21 and spin-23 representations of 2I induce the same Cayley-graph Laplacian eigenvalue; it establishes no Casimir/isotropy normalisation, Born rule, Tsirelson-type bound, or SU(2) fixed-point/sector-selection argument. Q3 proves a theorem conditional on an explicit, undemonstrated hypothesis: given that a bipartite state on a supplied composition is invariant under the diagonal action of 2I — not derived from admissibility or from Born–Infeld indiscernibility — the state is uniquely the SU(2) singlet, with an explicit Casimir correlator, for all five admissible sectors of 2I; it does not derive that invariance hypothesis, phase coherence, or the Born rule. No chain currently connects these three results to each other or to the structures of quantum mechanics. The Bell paper, published independently in Quantum Reports, establishes the structural non-applicability of Bell-type factorizability in non-injective effective descriptions; the proof of that central theorem uses only non-injectivity and informational completeness. This note records the current, narrow scope of each constituent and the open problems that remain.
The companion entanglement note and the orthogonality paper E1 identify two distinct entanglement entropies of the conjugate Weil pair {c,q−c}, which must not be conflated. The first is the entropy of the minimal admissible pair sector: the rank-3 admissible projection Πq is spectrally atomic, equal to span{e0,eξc,eq−ξc}; removing the zero mode leaves the two-dimensional Born–Infeld parity pair {eξc,eq−ξc}, identified with the spin-21 representation. The proto-state on this sector is the singlet, and its reduced state is maximally mixed, so Sentadm=log2 for general canonical blocks, unconditionally relative to the cited upstream theorems. This is the sector relevant to the singlet correlator and the Tsirelson bound. The second is the full residual fibre-level rank rpair(n)=R∞−R(n) defined on the full Gram–Schmidt basis (rank q at saturation, not the rank-3 admissible projection). E1 proves Sent(n)=log(rpair(n)) unconditionally for matched single-character blocks; for independently sampled canonical blocks it remains conditional on the residual admissibility indiscernibility hypothesis [Hres]. The metaplectic dilation ϕω is excluded as a bridge for [Hres] because it acts between blocks, not within the residual support of one block.
The companion entanglement paper established numerically that the normalized Gram–Schmidt admission residual satisfies wj=1 for every admitted Weil fingerprint direction, across q∈{29,61,101,151} and multiple generic blocks. The equality was left as an open analytic strengthening. We close it here by explicit computation. The k=3 Weil fingerprint of Heis3(Z/qZ) for any block (c1,c2,c3) is, up to a global phase, the discrete Fourier mode u^A=(e2πiAt/q)t=0q−1 at frequency A=c1b1+c2b2+c3b3modq, where (b1,b2,b3) are the b-components of the three path elements. Since distinct Fourier modes are orthogonal, any newly admitted fingerprint is already orthogonal to the current span, giving wj=∥wjGS∥/∥vj∥=1 exactly. For matched single-character blocks, the conjugate pair produces sign-reflected frequency sets, so the paired Fourier supports are canonically identified; conditional on the companion paper's supplied purity of the joint state, this fixes the diagonal Schmidt-partner pairing. For general canonical blocks, the equal-count identity is recovered, while the literal Schmidt pairing remains conditional on a canonical block identification. Combined, these results prove analytically that, conditional on the companion paper's supplied purity of the joint state, the Schmidt spectrum of the conjugate pair is flat on the residual support for matched blocks, and the reduced entropy is Sent(n)=logrpair(n), where rpair(n)=R∞−R(n) is the residual rank.
A compact abelian group acts unitarily on a finite-dimensional space. This paper proves that the overlap of two states, one shifted by a group element, is a finite Fourier series whose coefficients are positive semi-definite kernels indexed by the isotypic components of the conjugation representation; the coefficient of the trivial character is the inner product of the twirled states, and the sum of the squared moduli of the other coefficients is the variance of the overlap over the group. For the circle the series is a trigonometric polynomial whose degree is bounded by the total charge, so it is determined exactly by its values at finitely many equally spaced shifts. The same decomposition is applied to a reset channel, a fixed unitary that absorbs encoded inputs one at a time into a hidden register and discards the input between steps. Its readout under a common shift of the inputs is a finite Fourier series whose zero mode is the readout of the jointly twirled input; this need not equal the readout of inputs twirled one at a time, and an explicit Clifford channel separates the two. The degree of the readout is at most the charge bound times the number of steps, and for every charge bound an explicit channel attains it. Every coefficient can be estimated from overlaps measured at finitely many shifts, with a number of runs quadratic in the inverse error and logarithmic in the number of coefficients, and a charge sector of an encoded state can be read at the amplitude level by exact phase estimation of the charge followed by a Hadamard test. Sector kernels of product encodings and Fourier sectors of Clifford channels with Pauli readout are shown to be classically computable in polynomial time. The paper claims no quantum advantage. A final section carries the Fourier argument to the signature of a piecewise-linear path, read as a vector in a Fock space, and decomposes the signature kernel under rotation of the increments into harmonics that are themselves kernels.
Paper O28 established that the per-pair covariance Cc in End(Heff) has rank reff=3 with invariant eigenvalue structure [1:21:21]. The present paper gives the structural explanation of this result. The value reff=3 is real and stable across primes; the conjugation identity ρq−c=ρˉc (an O17 theorem, recalled in O18 Theorem 3.1 ) acts anti-linearly, so the conjugate-pair outer products Mj=πc(vj)⊗πq−c(vj)∗ are (up to a small, q-decreasing defect) complex symmetric; and the O26 target reff=dρ2=4 is structurally inaccessible from conjugate-pair data. A direct characterisation on the O25/Q5a checkpoints shows that the trajectory {wj=πc(vj)} spans all three complex dimensions of Heff (singular ratios σ3/σ1≃0.7–1.0), and that reff=3 is a hard 6→3 collapse of the Veronese (squaring) image: the symmetric squares wj⊗wj satisfy exactly three independent quadratic relations. The three constraint forms are traceless, and their commutators span exactly so(3); the Schur commutant of the action on Heff is one-dimensional, so Heff=C3 is irreducible. This identifies reff=3 as the invariant of the adjoint action Heff≃su(2) of O23/O27 — the spin-1 (vector) representation. Since the adjoint is irreducible, Heff contains no two-dimensional invariant subspace: the spin-21 space Vρ is the Schur target of the morphism Φˉ:su(2)→Vρ (O27), not a subspace of Heff. Test 4 measures the adjoint carrier Heff≃Sym2(Vρ), not Vρ directly: the spin-21 coordinate Vρ is recovered only through the Veronese square-root structure, not as the rank measured by reff. Consequently Test 4 confirms the carrier (spin-1 on Heff) and the inaccessibility of dρ2=4, but does not by itself identify the spin-21 sector; a genuine spin-21 test requires an independent observable on Vρ (the locking-broken protocol of Section sec:results reaches the full dρ2=4 and motivates it).
We compute the entanglement entropy of a closed conjugate Weil pair {c,q−c} along the spectral admissibility cascade of the Cosmochrony programme, where entanglement is the infra-projectable invariant of a non-decomposable joint fibre of the non-injective projection. The bipartite state lives on the residual fibre-level support of the Gram–Schmidt span of Weil fingerprints on the Heisenberg Cayley graph. We show that the normalized admissibility residual of each newly admitted independent direction equals unity: the admitted Weil fingerprints are mutually orthogonal (proved analytically in the companion paper ENT-E1), so the reduced Schmidt spectrum is flat and the reduced entropy is exactly Sent(n)=logrpair(n), the logarithm of the residual Gram–Schmidt rank. No Born–Infeld spectral correction survives: the admissibility-weight deficit is identically zero. Because the cascade only adjoins independent directions, rpair(n) is non-increasing structurally, so the residual conjugate-pair entanglement is maximal at the earliest admissible stage and decreases monotonically as correlated capacity is stabilized into projected structure. The capacity-level statement (non-negativity of the stabilization rate) is a conjecture of the projective temporal ordering analysis, supported numerically only within a relative tolerance of 10−6. The entropy computed here is the full residual-rank observable on the complete Gram–Schmidt basis; its relation to the minimal admissible pair sector, whose entropy is log2 conditional on two explicit hypotheses (H-carrier) and (H-inv), is established separately in the companion paper ENT-E2. The remaining finite-q task is to measure the contraction law of rpair(n) and compare its exponent with the pair-capacity exponent δpair.