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, resolved at a different stage of the sub-programme. The chain up to $\delta_{\mathrm{pair}}$ is unconditional with respect to the fibre structure of $\Pi$ (closed in O24 via the Verticality Lemma). 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. A further conditionality concerns the $SU(3)$ colour sector and the hypothesis [H-color].

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: unified closure of the three-generation mass spectrum through support exit for off-central ADE levels and Kesten–McKay saturation for the central mode.
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: theoretical resolution of the exact/proxy observable-class mismatch, deriving the corrected exact-block relation from $\delta$ to $\beta^*$, identifying block normalisation as the dominant measurable correction, and showing in pipeline mode that the remaining gap is structural rather than numerical.
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, $S_{\mathrm{BI}}[-\chi]=S_{\mathrm{BI}}[\chi]$ — an elementary, correct symmetry. Its further claim, that this evenness derives the fibre structure of the non-injective projection $\Pi$ and identifies it with the Weil conjugation $c\leftrightarrow q-c$, does not follow from the argument given; both remain open, 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: structural derivation of the threshold \(\Sigma_c(n_3)=3\) through quaternionic minimality, proving that the three-dimensional stable sector is not a geometric assumption or a spectral artefact, but the observable consequence of the minimal admissible associative non-commutative structure compatible with Born–Infeld fibre admissibility, namely \(\mathrm{Im}\,\mathbb{H}\).
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 \(\delta_{\mathrm{pair}}\), showing strong inter-pair concentration across all conjugate pairs, explaining the apparent drift of \(\delta_{\mathrm{pair}}(q)\) as a finite-size normalization effect controlled by the BFS window depth \(n_1(q)\), identifying \(n_1(q)/q\) rather than \(q\) itself as the correct asymptotic variable, and showing that the O14 normalization correction brings the corrected values back into the admissible window \([7.4,10.6]\).
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: establishes a representation-theoretic interpretation of the pair observable \(\sigma_{\mathrm{pair}}\) by identifying its pre-saturation growth with the Hilbert–Schmidt norm of an associated matrix trajectory, constructing a dictionary between conjugate Weil blocks and isotypic sectors, introducing a three-level identification hierarchy (growth equivalence, quotient identification, and conditional canonical identification), and proposing concrete falsifiability tests linking spectral admissibility to a binary-icosahedral \(SU(2)\)-type sector.
32. O27: proves that every admissible morphism \(\Phi_{q,\rho}\) necessarily factors through the quaternionic sector \(\mathfrak{su}(2)\), by formalizing admissibility-based naturality, establishing the universality of the admissible quotient, excluding non-quaternionic target structures, constructing the canonical factorization \(\Phi_{q,\rho} = \rho \circ \iota \circ \pi\), and thereby turning the \(Q_8 \subset 2I \subset SU(2)\) thread from a structural candidate into a rigidity theorem with a representation-theoretic interpretation of \(\beta^*\).
33. O28: 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 normalization analysis of the O-series capacity campaign is closed, and the measured depths are the transitional-regime data of the Critical Coverage theorem.
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: carrier identification via the Veronese collapse — the anti-linear Born–Infeld parity makes the target \(d_\rho^2 = 4\) inaccessible, and \(r_{\mathrm{eff}} = 3\) is the invariant of the irreducible adjoint (spin-\(1\)) carrier \(H_{\mathrm{eff}} \simeq \mathrm{Sym}^2(V_\rho)\); the spin-\(\tfrac{1}{2}\) space \(V_\rho\) (\(d_\rho = 2\)) is the Veronese square-root coordinate, confirmed across \(q \in \{61,101,151,211\}\).
37. O30: analytical derivation of the eigenvalue ratio \([1:\tfrac{1}{2}:\tfrac{1}{2}]\) in \(\mathrm{Sym}^2(V_\rho)\) — the Born–Infeld parity anti-linearity forces the equatorial constraint \(|\alpha_j|=|\beta_j|\), the covariance diagonalises under single-harmonic phase decorrelation, and the factor of~2 is the squared symmetric-tensor normalisation coefficient; correction \(O(q^{-1/2})\) from U1.
38. O31: structural framework for the SU(3) identification via colour triplet co-admissibility — defines the colour triplet \(\{c_1,c_2,c_3\}\) with \(c_1+c_2+c_3\equiv 0\pmod{q}\), proves the metaplectic automorphism forces \(\sigma_c(n)=\sigma_{\omega c}(n)\), and establishes \(G_{\mathrm{SM}} = \mathrm{SU}(3)\times\mathrm{SU}(2)\times\mathrm{U}(1)\) as a composite of three admissibility fixed points, unconditionally on the standard graph via the v1.5 single-frequency BI fingerprint mechanism (Proposition 4.23).
39. O32: numerical test of hypothesis [H-color] on \(q\in\{61,151,211,307\}\) — all test primes pass \(R\leq 3\), \(\delta_{\mathrm{tri}}\approx 3\delta_c\) at sub-percent level, and \(C_{\mathrm{color}}\) has numerical rank 1 in all cases; [H-color] pointwise now proved analytically via O31 v1.5, residual \(R_{\mathrm{var}}\) reinterpreted as finite-sample auxiliary-draw footprint.
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: phase coherence as a structural consequence of projective admissibility — proves that any admissible transition preserves Born–Infeld indiscernibility of conjugate Weil blocks, deriving the singlet correlator \(E(\hat{a},\hat{b})=-\hat{a}\cdot\hat{b}\) and the Tsirelson bound from admissibility alone, without postulating quantum mechanics.
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 \(\lambda=18\) (co-admissibility); Born rule, singlet correlator, and Tsirelson bound derived for spin-\(\tfrac{3}{2}\); \(\mathrm{SU}(2)\) established as the stable fixed point of the admissibility flow.
3. Q3: universal spin-\(j\) singlet from admissibility — Born–Infeld indiscernibility forces the bipartite proto-state to be the singlet \(|\Omega_j\rangle\) by Schur's lemma; universal correlator \(E(\hat{a},\hat{b}) = -j(j+1)/3\cdot(\hat{a}\cdot\hat{b})\) proved for all 5 admissible sectors \(j\in\{\tfrac{1}{2},1,\tfrac{3}{2},2,\tfrac{5}{2}\}\) of \(2I\).
5. Q5a: large-\(q\) Mosco convergence of the admissibility forms \(\mathcal{E}_q\) on \(\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})\) to the continuum limit \(L_\Pi = -A\partial_x^2\) on \(L^2(\mathbb{R})\); Q5b: BFS shell stratification of \(\mathrm{Heis}_3(\mathbb{R})\) producing a four-dimensional Lorentzian spacetime with temporal direction identified with BFS depth; [H-lift] proved by Q9.
6. Q6a: gauge structure from admissible non-injective projection — \(\mathrm{U}(1)\) and \(\mathrm{SU}(2)\) as admissibility fixed points, \(G_{\mathrm{SM}}=\mathrm{SU}(3)\times\mathrm{SU}(2)\times\mathrm{U}(1)\) as a composite (unconditional on the standard graph via O31 v1.5); Q6b: effective Lorentzian geometry from the admissibility filter \(\Pi_q\), inheriting Q5a hypotheses only.
7. Q7: structural bridge between the admissible three-dimensional sector \(H_{\mathrm{eff}} \simeq \mathbb{C}^3\) and the spatial three-dimensional sector of the emergent Lorentzian geometry, proving that any identification is uniquely constrained by representation theory (via \(H_{\mathrm{eff}} \simeq \mathrm{Sym}^2(V_\rho)\) and Schur rigidity) and reducing the problem to a single spectral criterion on \(\sigma_2(L_{\mathrm{eff}})\), with the absence of cross terms established analytically and the remaining condition \(A_H = A_z\) governing asymptotic isotropy.
8. Q8: sub-principal symbol of the effective operator forces \(A_Z = C_{\mathfrak{su}(2)} = 2\) unconditionally under Q5a via Casimir rigidity, independently of the spatial bridge problem.
9. Q9: proof of hypothesis [H-lift] — the modulation generator is suppressed at rate \(O(q^{-1/2})\), and \(A_H > 0\) follows by coercivity independently of the bridge; closes the main conditionality of Q5b.
10. 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 \(A_H\to 2\).
11. Q11: temporal Casimir rigidity closes Q5b open problem O3 — Schur rigidity forces \(A_\tau=2\), closing the co-metric \(g^{\mu\nu}=\mathrm{diag}(-2,2,2,2)\propto\eta^{\mu\nu}\); completes the full derivation chain Q5a\(\to\)Q5b\(\to\cdots\to\)Q11.
12. 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 \(a_4\); establishes that gravity (\(a_2\)) and gauge dynamics (\(a_4\)) are separated by geometric direction and spectral order rather than by a missing symmetry group.
13. Q13: gauge–gravity spectral synthesis — solves the joint variational problem \(\delta_{g,\mathcal{A}} S_\Pi = 0\) and derives the coupled Einstein–Yang–Mills system; computes the \(a_6\) cross-coupling \(R_{\mu\nu\rho\sigma}F^{\mu\nu}F^{\rho\sigma}\); constructs the Eddington–Born–Infeld joint completion; obtains the hierarchy ratio \(G_N g_{\mathrm{YM}}^2 \sim \ell_{\mathrm{sp}}^2/(\dim V)\cdot[\log(\Lambda/\mu)]^{-1}\) without fine-tuning as a structural consequence of spectral stratification.

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; reduces open Q5a hypotheses to [H1], [H-E1], [C].
• 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; closes all Q5a Mosco convergence hypotheses unconditionally.
• Heisenberg structure: non-factorisability and emergence of Heisenberg structure — admissibility constraints force a non-Abelian relational group structure uniquely identified as \(\mathrm{Heis}_3\) when the admissible dimension is three; structural explanation for \([X,P]=i\hbar\).

Inputs and outputs

Upstream inputs. The sub-programme takes from the foundational branch the fact that $\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})$ and its Weil representation are theorems of the axioms (not postulates), 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 admissibility thread $Q_8\subset 2I\subset SU(2)$ and the spin-$\tfrac12$ sector $V_\rho\cong\mathbb{C}^2$ (consumed by the quantum and geometric sub-programmes); the threshold $\Sigma_c(n_3)=3$ with $\mathrm{Im}\,\mathbb{H}\cong\mathfrak{su}(2)$; and the conditional $SU(3)$ co-admissible colour triplet.

Status

The sub-programme is analytically closed for the $SU(2)$ sector and numerically closed for $q\in\{29,61,101,151,211,307\}$. The $SU(3)$ extension is now unconditional on the standard graph: $[\text{H-color}]_{\mathrm{pointwise}}$ is proved analytically in O31 v1.5 (Proposition 4.23) via the single-frequency BI fingerprint structure, completing all four analytical levels (rank, averaged, effective, pointwise). 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. The remaining open deliverable on the positive side is the extension of the numerical campaign to $q=401$.