The Spectral Admissibility Sub-Programme

The structural chain

$\underbrace{c_\chi}_{\text{BI saturation}} \;\Longrightarrow\; \underbrace{A_n^{\max} = c_\chi/\sqrt{\lambda_n}}_{\text{admissibility envelope}} \;\Longrightarrow\; \underbrace{\sigma^{\mathrm{can}}_{\mathrm{pair}}(n)}_{\text{pair capacity}} \;\Longrightarrow\; \underbrace{\delta_{\mathrm{pair}}}_{\text{capacity exponent}} \;\dashrightarrow\; \underbrace{\beta^* \approx 0.126}_{\text{cascade exponent (no native carrier)}}.$

Each solid arrow is a distinct analytical or numerical sub-problem studied at a different stage of the sub-programme. Weil conjugation is proved, but its identification with a Born–Infeld fibre remains open. The final arrow is dashed: the span-growth no-go (SGN) proves that the conversion $\beta^*\approx 1/(\delta_{\mathrm{pair}}+\tfrac12)$ has no derived carrier on the Heisenberg measurement substrate — both corpus-defined closures, the square-root frontier and the pair-growth carrier, fail, as does the exponent-coordinate identification. The numerical agreement with the mass-fitted window therefore stands as a phenomenological coincidence check, not a mass-hierarchy derivation. The former $SU(3)$ colour-sector claims are withdrawn by O31 version 2.0. Its rank and pointwise [H-color] levels rested on the superseded O31, whose current record neither asserts nor refutes them: they are unsupported and open. The block-averaged and effective levels proved in O32 stand. The colour problem is open.

Papers of the sub-programme

1. Spectral admissibility: bounds on the amplitude of individual spectral modes.
2. Spectral capacity: global aggregation of admissible amplitudes across representation sectors.
3. Spectral Gram rigidity: structural constraints on neutral generating sets in SU(2).
4. Spectral stratigraphy: discrete stabilisation levels of spectral modes along the relaxation cascade.
5. Spectral relaxation: dynamical behaviour of the projective threshold and its role in amplifying spectral hierarchies.
6. O1: variable-valence projective dynamics restoring the ordering of stabilisation events along the relaxation cascade.
7. O3: hierarchical amplification of ADE spectral separations into realistic mass ratios through growing relational valence.
8. O4: examines whether a structural upper bound on the cascade exponent follows from the bounded-flux constraint together with the Cheeger inequality, and finds that it does not follow from that argument.
9. O5: finite character-trace saturation, ruling out vertex-based class-function encodings as the source of the cascade exponent's mode-resolved novelty.
10. O6: representation-theoretic obstruction showing that no fixed finite-dimensional encoding can generate the cascade exponent, establishing bounded-depth saturation and the necessity of dynamical, depth-growing redundancy.
11. O7: coarse-grained projective capacity as the continuum limit of admissible redundancy, linking discrete saturation to emergent nonlinear dynamics.
12. O8: three-step path fingerprints escaping the O6 obstruction, revealing a new geometric compression mechanism where exponential shell growth collapses the observable capacity window.
13. O9: polynomial-growth Cayley graphs resolving the O8 geometric obstruction, showing that the observable capacity window regains polynomial BFS depth beyond expander compression.
14. O10: first large-q computation confirming polynomial ball growth, projective capacity decay, and the O7 state law, while identifying dense TensorSketch as the final algorithmic obstruction to extracting the capacity exponent.
15. O11: representation-adapted Weil-block proxy restoring a stable pre-saturation decay regime and enabling the first reliable extraction of the projective capacity exponent on Heisenberg graphs.
16. O12: exact Weil-block projective capacity $\Sigma_n^{(c)}$ replacing the O11 proxy, depending on the block only through a projective pair $(\alpha,\beta)$ and hence exactly independent of the central coordinate γ, with the intrinsic asymptotic exponent derived exactly, $\delta_{\mathrm{exact}}=3$, finite-$q$ fitted slopes identified as crossover statistics of the block saturation-depth distribution.
17. O13: extended exact Weil-block computation to $q\in\{101,151,211\}$, showing that the finite-$q$ exponent sequence decreases monotonically from $q=61$ onward and does not drift toward the phenomenological target, consistent with O12's exact value $\delta_{\mathrm{exact}}=3$.
18. O14: establishes why exact Heisenberg capacity does not determine the phenomenological cascade rate. A factor $q^{-\eta}$ changes the intercept of a fixed-$q$ regression and leaves its fitted slope invariant, the proposed central-phase correction vanishes identically, and the reciprocal prescription yields no $\beta^*$ diagnostic from these data.
19. O15: derivational audit of the O3–O7 chain proving that the scalar proxy growth law does not transfer to the exact Weil-block observable, and establishing a checkable, conditional aggregation bound $\alpha_{\mathrm{dyn}}\le\hat\delta_{\mathrm{exact}}$, verified at $q=151$ and $q=211$, open at $q\in\{29,61,101\}$.
20. O16: examines whether conjugate Weil blocks $\{c,q-c\}$ constitute the fibres of the non-injective projection Π, under two independent open hypotheses, deriving a conditional exponent-doubling result $\delta_{\mathrm{pair}}=2\delta_c$ that holds trivially, with ratio $1$, on the real O12/O13 pipeline.
21. O17: proves, within a scalar Weil-representation toy model, exact block-independence of the raw Gram–Schmidt redundancy count for any two blocks whatsoever, not only conjugate ones — too broad a fact to license a same-fibre criterion for conjugate pairs, and scoped to this toy model, not shown to extend to the real O12/O13 pipeline.
22. O18: proves the Born–Infeld action is even, $S_{\mathrm{BI}}[-\chi]=S_{\mathrm{BI}}[\chi]$, with parity acting covariantly on the response family (configuration and probe transformed together), and proves the no-go that evenness does not force fixed-probe indiscernibility (countermodel $R(x)=x^2$): the parity orbit is not forced into the fibres of $\Pi$. The fibre identification is a typed classification problem, of which (H-rank) is one clause; at the Weil level the conjugation $\rho_{q-c}=\overline{\rho_c}$ is an O17 theorem, while the identification of $\{c,q-c\}$ as a physical fibre remains an open bridge, as in O16.
23. O19: proves two generic Gram–Schmidt identities (phase invariance, span–rank equality). Its proposed canonical pair observable, meant to remove pipeline dependence from O16–O18, is not constructed for the real O12/O13 pipeline: the residual factor it sets out to normalise does not arise there, and the basis/instance tests needed to classify it were never performed.
24. O20: formulates a conditional threshold-crossing template — given a constant threshold and an exact power-law decay, crossing within a fixed shell window is algebraically equivalent to an exponent interval, a real, reusable result under those hypotheses. The threshold, the amplitude map it depends on, and the numerical window $\delta_{\mathrm{pair}}\in[7.4,10.6]$ itself are not derived.
25. O21: writes a two-point crossing rank $n_{3}^{\mathrm{obs}}$, valid as an extrapolated crossing rank under a fixed threshold and an exact power law. Its central claim of invariance under amplitude rescaling $\sigma\mapsto\lambda\sigma$ is algebraically false — the construction instead scales as $\lambda^{1/\delta_{\mathrm{local}}}$ — so it does not eliminate pipeline dependence. The fixed threshold $9/q^2$, the shell-alignment conjecture, and the cross-substrate transfer to $\beta^*$ remain open.
26. O22: proves that Born–Infeld projection locking forces the O21-defined $n_{3}^{\mathrm{obs}}$ onto a BFS shell within O22's stated model. Since that rank is not amplitude-invariant (O21), this locking result does not by itself close O21's intrinsic-definition problem.
27. O23: conditional adjoint-dimension theorem for the threshold \(\Sigma_c(n_3)=3\) — if the neutral sector is carried by an irreducible two-dimensional unitary representation \(V_\rho\cong\mathbb{C}^2\) (a supplied input), its traceless sector is \(\mathfrak{su}(2)\cong\mathrm{Im}\,\mathbb{H}\), of dimension 3; generator counting in \(Q_8\) cannot give 3. Carrier selection and the identification of \(\Sigma_c\) with that dimension remain open, so \(\Sigma_c(n_3)=3\) enters the pipeline as a supplied selection rule, not a derived constant.
28. O24: Verticality Lemma, showing that admissibility depends on observable rank rather than fibre cardinality and that non-injectivity may enlarge microscopic multiplicity without enlarging the admissible neutral traceless observable sector. This develops the transfer \(c_\chi \to \delta_{\mathrm{pair}}\) only under the stated fibre hypotheses; it supplies neither a first-principles identification of the fibre structure of \(\Pi\) (open since O16) nor a native derivation of \(\delta_{\mathrm{pair}} \to \beta^*\).
29. O25: first systematic pair-level campaign for \(\delta_{\mathrm{pair}}\), with exhaustive pair sampling through \(q=211\) and representative samples through \(q=601\). It finds fixed-\(q\) pair concentration, but does not identify the asymptotic variable or pin \(\delta_\infty\); the O14 correction overcorrects at large \(q\). The reciprocal image \(0.108\text{--}0.123\) is only a phenomenological comparison.
30. Temporal ordering: identification of the cumulative projected capacity \(I(n)=\sigma_{\mathrm{pair}}(0)-\sigma_{\mathrm{pair}}(n)\) as the correct monotone observable along the admissible cascade, establishing strict stepwise monotonicity across all conjugate pairs and tested primes, interpreting this behaviour as a one-way activation process in which projected spectral structure is never lost, defining a projective temporal ordering induced by non-injective projection rather than by substrate dynamics, and formulating the corresponding one-way activation conjecture \(\sigma_{\mathrm{pair}}(n)\leq\sigma_{\mathrm{pair}}(n-1)\) together with its spectral saturation interpretation.
31. O26 version 2.1: separates proved pre-saturation growth-exponent equivalence, conjectural quotient identification and canonical identification conditional on an admissible embedding. Under Hypothesis 4.4(a), the embedded products lie in \(\mathrm{Sym}^2(V_\rho)\), so their rank is at most three for \(d_\rho=2\); the rank-four target is excluded under that contract only. Tests 1–2 are implementation checks. The O28–O29 ranks measured in \(\mathrm{End}(H_{\mathrm{eff}})\) do not measure \(\mathrm{End}(V_\rho)\); the embedding remains open.
32. O27 version 2.0: quotient factorisation does not imply equivariance. Under explicit saturation, carrier, parity and equivariance inputs, the Schur-level counts are proved; a canonical equivariant vector lift and any interpretation of \(\beta^*\) remain open.
33. O28 version 2.0: asymptotic calibration of the BFS window and effective dimension of the admissible trajectory; out of sample (to \(q=601\)) the calibrated linear extrapolation fails (up to \(+88\%\)), the declared finite-size adjustment of the capacity exponent is a diagnostic rather than a derived normalisation correction, and the measured depths are the transitional-regime data of the Critical Coverage theorem. The rank-3 covariance value is finite-data and not an invariant, and the representation gap to \(V_\rho\) stays open.
34. Critical Coverage note: companion to O28 resolving the saturation-depth asymptotics by an exact interval theorem — every Weil–BFS fingerprint is a Fourier character, so \(r_c(n)=|T'_c+C\,I_{n+1}|\) with universal shellwise ceiling \(18\); the depth converges to the constant \(22\) and the critical coverage obeys \(x_1(q)\asymp q^{-2}\), with deterministic bounds for every prime \(q\ge 311\).
35. Span-Growth note: the exact native identity \(\Delta r=\sigma_c\,|S_n|\) gives an abstract real-weight three-branch theorem and a rigorous finite-window law for integer ranks. A three-legged no-go shows that the O4–O7 conversion \(1/(\delta_{\mathrm{pair}}+\tfrac12)\) has no native Heisenberg derivation for the corpus-defined objects: both valence–exploration closures fail, the frontier is \(N^{3/4}\) rather than \(p^{1/2}\) with no defined BI feedback carrier, and block capacity is neither the pair product nor expressed in the reduced-model exponent coordinates. The numerical \(\beta^*\) agreement remains a cross-substrate phenomenological comparison.
36. O29: reports a finite-checkpoint numerical \(6\to3\) collapse in the measured trajectory span \(W_{\mathrm{adm}}\subset H_{\mathrm{eff}}\), with rank three modal rather than invariant (32/50 pairs at \(q=101\), 103/105 at \(q=211\)). Its fitted commutators and commutant are compatible with a supplied adjoint carrier without identifying it. Its auxiliary two-dimensional PCA space \(U_c^{(2)}\) is not the fundamental doublet \(V_\rho\), does not construct a Veronese square root, and carries no canonical representation status.
37. O30: a conditional \(3\times3\) model on \(\mathrm{Sym}^2(V_\rho)\), where (H0)–(H3) give the equatorial form with its sign and the diagonal ratio \([1:\tfrac{1}{2}:\tfrac{1}{2}]\) requires two further weighted cancellations, neither derived nor established empirically. It is distinct from the \(9\times9\) covariance measured in O28 on \(\mathrm{End}(H_{\mathrm{eff}})\), and supplies no analytical account of its spectrum.
38. O31: version 2.0 is a withdrawal notice. It withdraws the claimed colour-profile co-admissibility, arithmetic and metaplectic steps, rank test, \(\mathrm{SU}(3)\) uniqueness and every complete gauge-group derivation, with explicit \(\mathrm{SO}(3)\), diagonal-torus and \(\{1,2,q-3\}\) counterexamples.
39. O32: retains the finite-\(q\) values measured for \(q\in\{61,151,211,307\}\), including the reported ratios, decay fits and numerical rank. Their former pointwise co-admissibility and \(\mathrm{SU}(3)\) interpretation is withdrawn.
40. O33: exact arithmetic of the O12 three-character Fourier filtration — closed cyclic sumset \(K_n\), spacing-parity porosity criterion, complete Weil stabiliser through the multiplicative group \(A_K\) (orbit-divisibility sieve, uniform exact \(C_4\) and \(C_6\) families, additive rigidity theorem \(A_K=\{\pm1\}\) outside a degenerate boundary regime, completeness open only there); internal Mackey–Hecke \(M_2(\mathbb{C})\) doublets in \(\mathrm{End}(V_\rho)\), universal Harper split \(1\pm q^{-1/2}\), boundary mixing controlled by an exact cyclotomic phase determinant, and parity protection of the odd Weyl-symbol sector.

Q-series:

1. Q1: exact conjugate-sector Fourier-support identity and rank/support no-go; it does not derive phase coherence, a singlet correlator, the Tsirelson bound, or the Born rule.
2. Q2: quantum structure beyond spin-\(\tfrac{1}{2}\) — on the binary icosahedral group \(2I\), spin-\(\tfrac{1}{2}\) and spin-\(\tfrac{3}{2}\) sectors share eigenvalue \(\lambda=18\). This finite-group degeneracy does not derive phase coherence, the Born rule, the Tsirelson bound, or an \(\mathrm{SU}(2)\) fixed point.
3. Q3: universal spin-\(j\) singlet from admissibility — under the explicit, unestablished diagonal-\(2I\)-invariance hypothesis, the bipartite proto-state is the singlet \(|\Omega_j\rangle\) by Schur's lemma; universal correlator \(E(\hat{a},\hat{b}) = -j(j+1)/3\cdot(\hat{a}\cdot\hat{b})\) follows for the five listed sectors of \(2I\); the invariance hypothesis, phase coherence, and the Born rule remain open.
4. Q5a version 3.2: the canonical Fourier-window filtration is exact, but the published form converges to the zero form at rate \(q^{-2}\), and no common scalar normalisation produces the proposed non-trivial toric differential operator under the stated inputs. The former Mosco derivation is withdrawn and the spatial limit is the open hypothesis [H-L]; Q5b: BFS shell stratification of \(\mathrm{Heis}_3(\mathbb{R})\) producing four-dimensional Carnot/BFS geometry. Its metric reading is conditional on [H-L], and [H-lift] remains open after Q9 2.0.
5. Q6a: gauge structure from admissible non-injective projection — \(\mathrm{U}(1)\) and \(\mathrm{SU}(2)\) as proposed admissibility fixed points; its former full-group derivation relied on the withdrawn O31 claims, so the colour factor and complete group are open; Q6b: effective Lorentzian geometry whose metric claims remain conditional on the independent spatial-limit hypothesis [H-L]; [H-F] does not close [H-lift] in Q9 2.0.
6. Q7: requirements for an equivariant bridge between the admissible three-dimensional sector \(H_{\mathrm{eff}} \simeq \mathbb{C}^3\) and the spatial sector of the proposed Lorentzian geometry. Under an explicit identification hypothesis \(H_{\mathrm{eff}} \simeq \mathrm{Sym}^2(V_\rho)\), which no source supplies, Schur rigidity makes such a bridge unique up to one scalar, and the test to be met is a criterion on \(\sigma_2(L_{\mathrm{eff}})\) that no paper of the corpus has evaluated. Version 2.0 withdraws the analytic no-cross-terms reading of the continuum symbol and the isotropy evidence: the measured numbers are Rayleigh quotients of the discrete Weil Laplacian on plane waves. The bridge is a missing identification, not a proved obstruction.
7. Q8: sub-principal symbol of the effective operator gives \(A_Z = C_{\mathfrak{su}(2)} = 2\) within the supplied Q5a carrier model via Casimir rigidity; its metric interpretation remains conditional on [H-L] and on the identification Q7 version 2.0 records as supplied by no source. Pending revision.
8. Q9: version 2.0 establishes a free kinetic Mosco limit under [K] and [R], with a scale-dependent modulation bound. It selects no central fibre and leaves [H-lift] open.
9. Q10: asymptotic \(\mathfrak{su}(2)\)-isotropy of the effective quadratic form — character-independence of the large-\(q\) limit forces isotropy under [U]; the unique isotropic form is the Casimir with value 2, giving \(A_H\to 2\). Reading that value as a coefficient depends on the same identification, and Q7 version 2.0 no longer reads it as settling its bridge condition. Pending revision.
10. Q11: under the frozen, unestablished [H-F], temporal Casimir rigidity gives \(A_\tau=2\) within the stated algebraic model. Reading \(g^{\mu\nu}=\mathrm{diag}(-2,2,2,2)\propto\eta^{\mu\nu}\) as an emergent metric remains conditional on the independent, unestablished [H-L] and on the identification Q7 version 2.0 records as supplied by no source.
11. Q12: Yang–Mills dynamics from the vertical variation of the projective spectral entropy \(S_\Pi[g,\mathcal{A}]\), deriving the Yang–Mills equations from the Seeley–DeWitt coefficient \(a_4\) at supplied fixed metric and compact gauge data; only its emergent-base reading consumes [H-L].
12. Q13 version 1.4: on supplied geometric and gauge data, the joint variation gives a conditional Einstein–Yang–Mills system. The \(a_6\) terms form a spanning list modulo integration by parts and Bianchi identities, with no unique cross-term or nonlinear completion asserted. Couplings are matching data and no hierarchy is predicted; only the emergent-base reading consumes [H-L].

Supporting papers:

• U1: uniform spectral universality for Weil fingerprint energies — proves hypothesis [U] with rate \(\varepsilon(q)=O(q^{-1/2})\) via Weil-generator Lipschitz bound and BFS–Carnot–Carathéodory convergence; used by Q10, W1, O30.
• W1: weight stabilisation of the admissible Dirichlet form — proves [H-w] (convergence \(a_q(s)\to A>0\)) as a structural consequence of fibre-invariance of admissible observables, made quantitative by U1. This closes a hypothesis of the former Q5a framework but does not establish the current spatial-limit hypothesis [H-L].
• H2: semiclassical consistency of the Weil representation — proves [H2] (strong convergence \(\hat{X}_q\to x\), \(\hat{P}_q\to -i\partial_x\)) via a quantitative Poisson aliasing lemma and discrete Sobolev identity. This closes [H2] in the former Q5a framework but does not establish the current spatial-limit hypothesis [H-L].
• Heisenberg structure: version 2.1 gives a finite obstruction to selecting a Heisenberg carrier from the published admissibility constraints. A non-trivial commutator does not force centrality or uniquely select \(\mathrm{Heis}_3\); the finite Heisenberg carrier remains a supplied realisation.

Inputs and outputs

Upstream inputs. The sub-programme takes a finite Heisenberg group and a non-trivial central character as supplied carrier data. Stone–von Neumann then selects the corresponding Heisenberg–Schrödinger representation; the associated Weil action requires separate symplectic data. It also takes the capacity axiom [A-cap] with its Born–Infeld saturation candidate and constant $c_\chi$, the structural necessity of non-injectivity, and the no-scale principle making $c_\chi$ the only dimensionful parameter. No result from the emergence branch is an input.

Outputs. The measured pair capacity exponent $\delta_{\mathrm{pair}}$, whose reciprocal $1/(\delta_{\mathrm{pair}}+\tfrac12)$ is compared with the lepton-hierarchy and charge–mass papers as a phenomenological check rather than transferred to them as a derived quantity. The thread $Q_8\subset 2I\subset SU(2)$, the spin-$\tfrac12$ carrier $V_\rho\cong\mathbb{C}^2$, and the selection rule $\Sigma_c(n_3)=3$ are supplied inputs, not outputs derived by this sub-programme. O31 supplies no colour-group output; the colour factor and complete gauge group are open.

Status

The sub-programme contains proved spectral and numerical results, but the carrier and gauge-group bridges remain conditional. O31 version 2.0 withdraws the proposed $SU(3)$ extension and the complete gauge-group derivation; O32 remains a numerical study without that interpretation. The downstream conversion to the cascade exponent $\beta^*$ is closed negatively: the span-growth no-go establishes that it has no native Heisenberg carrier, so the mass-hierarchy comparison is phenomenological and the derivation of the charged-lepton hierarchy remains open. A new colour theorem would require an independently defined carrier, a valid co-admissibility statement and a group-identification argument.