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 (O18), shell alignment under projection locking (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 carriers (the Heisenberg–Schrödinger carrier, the Weil module associated with it, and the $\mathrm{SU}(2)$ spinor carrier assumed conditionally by O23, which is not identified with the Weil module) — 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 amplitudes on the supplied carriers (the Heisenberg–Schrödinger carrier, the Weil module associated with it, and the spinor carrier of O23, which is not identified with the Weil module) 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.