The Quantum Structure Sub-Programme

Abstract

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-$\tfrac12$ and spin-$\tfrac32$ representations of ${2I}$ induce the same Cayley-graph Laplacian eigenvalue; it establishes no Casimir/isotropy normalisation, Born rule, Tsirelson-type bound, or ${\mathrm{SU}}(2)$ 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 ${\mathrm{SU}}(2)$ 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.