Overview
A finite Heisenberg carrier, its non-trivial central character, and the associated Weil action are supplied model data rather than consequences of A1–A4 (Note 5). The thread $Q_8 \subset 2I \subset \mathrm{SU}(2)$ and its spin-$\tfrac{1}{2}$ sector are likewise programme inputs, not a uniquely selected physical carrier (Note 1). Do the structures of quantum mechanics already follow from this much?
Not currently. Q1's proved theorem is a representation-theoretic rigidity and no-go result, unrelated to phase coherence or the Born rule. Q2 proves only a finite Laplacian-eigenvalue coincidence between the spin-$\tfrac12$ and spin-$\tfrac32$ representations of $2I$. Q3 proves a theorem conditional on an explicit, undemonstrated hypothesis — that a bipartite state on a supplied carrier is invariant under the diagonal action of $2I$ — not derived from admissibility or from Born–Infeld indiscernibility. No result in this sub-programme derives phase coherence, a singlet correlator, a Tsirelson bound, the Born rule, or a physically preferred sector from the admissibility axioms alone.
Papers of the sub-programme
Q1 — Fourier-support rigidity (proved, unconditional).
- Q1 — an exact, unconditional identity in the finite Weil action associated with a supplied Heisenberg carrier: conjugate sector fingerprint vectors span identical subspaces on every complete BFS shell, with a precise no-go corollary for support-, rank-, and Gram-based diagnostics. Verified by 1,548 exact checks across six primes, zero failures. Establishes no phase coherence, singlet correlator, Tsirelson bound, or Born rule.
Q2 — an exact eigenvalue degeneracy on $2I$ (proved, unconditional).
- Q2 — on the binary icosahedral group $2I$, the spin-$\tfrac{1}{2}$ and spin-$\tfrac{3}{2}$ representations induce the same Cayley-graph Laplacian eigenvalue, $\lambda_{1/2} = \lambda_{3/2} = 18$, proved via Schur's lemma. Establishes no phase coherence, Casimir/isotropy normalisation, Born rule, Tsirelson-type bound, or $\mathrm{SU}(2)$ fixed-point/sector-selection argument.
Q3 — a conditional universal singlet.
- Q3 — given a supplied bipartite carrier and an explicit, undemonstrated hypothesis (H-inv) that the joint state is diagonally $2I$-invariant, Schur's lemma on $\mathrm{Hom}_{2I}(V_j^*,V_j)$ uniquely identifies the state as the $\mathrm{SU}(2)$ singlet $|\Omega_j\rangle$, with Casimir correlator $E(\hat{a},\hat{b}) = -\tfrac{j(j+1)}{3}(\hat{a}\cdot\hat{b})$, for all five admissible sectors of $2I$. Does not derive (H-inv) itself, phase coherence, or the Born rule.
Bell factorisation bridge (published claim refuted).
- Bell paper — claims that non-injectivity obstructs every Bell-factorisable representation. A finite model with non-injective fibres of size eight, deterministic local responses, measurement independence, and CHSH $=0$ disproves that implication. The paper's PR-box constraints have empty support. A post-publication procedure has been requested from Quantum Reports.
Inputs and outputs
Inputs. Of the results once cited here as upstream inputs — the admissible fibre and Heisenberg structure (Note 5), $\mathrm{Im}\,\mathbb{H} \cong \mathfrak{su}(2)$ and isotropic saturation (O23), the parity involution and unit-norm amplitudes (O18, O22), and BI indiscernibility of conjugate Weil blocks (BornInfeld) — none is actually used by the current, narrow content of Q1, Q2, or Q3. The only upstream input the current papers rely on is the Laplacian-eigenvalue data of the spectral admissibility programme (Note 1, SpectralAdmissibility), which fixes which representation sectors of $2I$ are admissible and is used by Q2 and Q3. Non-injectivity as a structural necessity ($\mathcal{S}_\Pi > 0$) does not supply a Bell non-factorisability result.
Outputs. No output of this sub-programme currently supplies phase coherence, a correlator, a Tsirelson bound, an $\mathrm{SU}(2)$ fixed-point identification, or the Born rule to any downstream branch. Q2's eigenvalue degeneracy fixes which five sectors are in scope for Q3. Bell non-factorisability remains open: the programme lacks a derived obstruction to a setting-independent global coupling.
Status
The sub-programme's original ambition — deriving the structures of quantum mechanics from admissibility alone, given the supplied complex $\mathrm{SU}(2)$ Weil carrier — is not established. Q1 proves an exact, unconditional Fourier-support rigidity theorem, unrelated to phase coherence or the Born rule. Q2 proves an exact, unconditional finite-group eigenvalue degeneracy on $2I$. Q3 proves a theorem conditional on an explicit, undemonstrated hypothesis (H-inv); neither (H-inv), phase coherence, nor the Born rule is derived by any constituent of this sub-programme. The published Bell paper's central implication is refuted by a finite Bell-local countermodel.
Open problems: deriving phase coherence of the supplied Weil carrier from the admissibility axioms; deriving Q3's invariance hypothesis (H-inv) rather than supplying it; deriving the Born rule, in the $\mathrm{SU}(2)$ sector or beyond it; and deriving a genuine cross-context obstruction sufficient for Bell non-factorisability.