The Spectral Admissibility Sub-Programme

The structural chain

$\underbrace{c_\chi}_{\text{BI saturation}} \;\Longrightarrow\; \underbrace{A_n^{\max} = c_\chi/\sqrt{\lambda_n}}_{\text{admissibility envelope}} \;\Longrightarrow\; \underbrace{\sigma^{\mathrm{can}}_{\mathrm{pair}}(n)}_{\text{pair capacity}} \;\Longrightarrow\; \underbrace{\delta_{\mathrm{pair}}}_{\text{capacity exponent}} \;\Longrightarrow\; \underbrace{\beta^* \approx 0.126}_{\text{cascade exponent}}.$

Each arrow is a distinct analytical or numerical sub-problem, resolved at a different stage of the sub-programme. The chain is unconditional with respect to the fibre structure of $\Pi$ (closed in O24 via the Verticality Lemma): no free parameter is adjusted to produce $\beta^*$, via the structural relation $\beta^*\approx 1/(\delta_{\mathrm{pair}}+\tfrac12)$. The sole remaining 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: admissible frontier dynamics and identification of the structural origin of the hierarchy exponent through matrix-level redundancy.
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 projection replacing the O11 proxy, revealing a regime shift in the decay exponent, identifying the central coordinate γ as dynamically active, and providing the first exact representation-level extraction of the projective capacity exponent on Heisenberg graphs.
17. O13: extended exact Weil-block computation at larger primes ruling out the strong finite-size explanation of the δ–β* tension, showing a monotone decrease of the exact decay exponent from q = 61 onward, and reclassifying the remaining mismatch as structural rather than numerical.
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, establishing an aggregation no-go, and localising the remaining $\delta$–$\beta^*$ gap at the level of the growth equation rather than measurement or averaging.
20. O16: identification of the correct fibre-level observable in the exact Weil regime via conjugate block pairs, proving exponent doubling $\delta \to 2\delta$, resolving the $\delta$-deficit without modifying the O6–O7 growth law, and showing that admissibility is defined on fibres rather than individual blocks.
21. O17: structural derivation of the fibre-level observable from conjugate Weil dynamics, proving the exact equality $\delta_{q-c}=\delta_c$, establishing that conjugate blocks carry identical dynamical information, and showing that the factor $r(c,q)$ is a normalisation artefact rather than a structural invariant.
22. O18: derivation of the minimal fibre structure of the non-injective projection from Born–Infeld indiscernability, proving that every fibre contains the parity involution $\chi\mapsto-\chi$, establishing $c\leftrightarrow q-c$ as its Weil-level realisation, and closing the foundational gap underlying the fibre-level observable of O17.
23. O19: explicit construction of the Gram–Schmidt pipeline normalisation, classification of the residual amplitude factor $r(c,q)$ into basis, instance, or mixed components, and definition of a canonical pair observable independent of pipeline conventions, completing the amplitude-level normalisation of the fibre observable introduced in O16–O18.
24. O20: formulation of a persistence-based admissibility criterion selecting the physical exponent window $\delta_{\mathrm{pair}}\in[7.4,10.6]$, introducing a threshold-crossing condition for the canonical pair observable $\sigma_{\mathrm{pair}}^{\mathrm{can}}(n)$, and identifying the remaining dependence on an external saturation scale and a fit parameter as the last non-intrinsic element of the programme.
25. O21: intrinsic derivation of the persistence criterion through the construction of a canonical crossing rank $n_{3}^{\mathrm{obs}}$ defined directly from $\sigma_{\mathrm{pair}}^{\mathrm{can}}(n)$, eliminating the external threshold $\sigma_{\mathrm{BI}}$ and the fit parameter $C$, establishing a fully scale-invariant and parameter-free formulation of admissibility, and restoring a structural determination of $\delta_{\mathrm{pair}}$ and its relation to $\beta^*$.
26. O22: theorem-level derivation of shell-alignment through projection locking, proving that admissible saturation can occur only where the continuous Born–Infeld saturation locus meets the discrete projectively admissible shell support, eliminating shell-alignment as an independent structural hypothesis, and establishing it as the observable consequence of non-injective admissible realisation under \(\Pi\).
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: proof that admissibility depends on observable rank rather than fibre cardinality, establishing the verticality of all Born–Infeld-admissible symmetries with respect to \(\Pi\), showing that non-injectivity may enlarge microscopic multiplicity without enlarging the admissible neutral traceless observable sector, and thereby closing the unconditional transfer \(c_\chi \to \delta_{\mathrm{pair}} \to \beta^*\).
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, establishing the correct asymptotic variable \(n_1(q)/q\) and closing the normalization analysis of the O-series capacity campaign.
34. O29: spin-\(\tfrac{1}{2}\) sector identification via the symmetric rank formula — the Born–Infeld parity constraint forces every outer product \(M_j\) into the complex symmetric subspace \(\mathrm{Sym}(V_\rho,\mathbb{C})\), yielding covariance rank \(r_{\mathrm{eff}} = d_\rho(d_\rho+1)/2\); the unique positive inversion at \(r_{\mathrm{eff}} = 3\) gives \(d_\rho = 2\), confirming spin-\(\tfrac{1}{2}\) universally across \(q \in \{61,101,151,211\}\).
35. 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.
36. 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).
37. 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.

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 Born–Infeld saturation functional with 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 cascade exponent $\beta^*\approx 0.126$ (consumed by the lepton-hierarchy and charge–mass papers); 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 sole remaining open deliverable is the extension of the numerical campaign to $q=401$.