Spectral Admissibility Programme

Structural constraints on relational spectral systems under bounded flux.

Overview

The spectral admissibility programme studies how relational systems constrained by bounded flux select admissible spectral configurations. It combines representation theory, spectral graph theory, and relational constraints to identify structurally stable regimes.

Within the Cosmochrony framework, this programme clarifies which relational structures remain compatible with bounded relaxation dynamics and how those constraints shape the admissible hierarchy of relational graphs.

Programme structure

  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 \(c_\chi \to \delta_{\mathrm{pair}} \to \beta^*\).