This repository contains the source of the O9 Cosmochrony paper
Projective Capacity Beyond Expanders: Polynomial-Growth Relaxation Graphs and the Capacity Exponent.

This work extends the spectral admissibility sub-programme by resolving the geometric obstruction identified in O8, which prevented the extraction of the capacity exponent on LPS expander graphs.

While O8 demonstrated that the $k=3$ permutation-path fingerprint escapes the fixed finite-dimensional obstruction of O6, it also showed that exponential shell growth on LPS graphs compresses the entire pre-saturation window into only $O(\log q)$ BFS steps, making exponent extraction impossible.

The present work removes this obstruction by replacing LPS graphs with a family of Cayley graphs exhibiting polynomial shell growth, allowing the observable window to unfold over a polynomial number of BFS steps.

The central object remains the three-step permutation-path fingerprint of O8, now evaluated on a different geometric background.

Core Result

The paper establishes that the obstruction identified in O8 is purely geometric and not intrinsic to the fingerprint construction.

Starting from:

• the $k=3$ permutation-path fingerprint
• the O7 capacity observable $\Sigma_n$
• the state-law prediction $R_n^{(3)}\approx\Phi(\eta_n)$
• a replacement of LPS graphs by Cayley graphs of the discrete Heisenberg group

the analysis shows that:

• polynomial-growth Cayley graphs satisfy $|B_n| \sim n^D$ with homogeneous dimension $D=4$
• a vertex window of size $O(q^2)$ now spans at least $\Omega(q^{1/2})$ BFS steps (Window-Depth Theorem)
• in practice, the available BFS depth scales as $\Theta(q)$
• the observable window contains polynomially many data points, unlike the $O(\log q)$ window on LPS graphs
• numerical BFS confirms:
  • diameter $\Theta(q)$
  • growth exponent $\hat{D} \to 4$
  • window-depth ratios up to $5.4\times$ compared to LPS

Thus:

• the O6 algebraic obstruction is bypassed (O8)
• the O8 geometric obstruction is resolved (O9)
• the remaining limitation is no longer structural but purely finite-size

Structural Role of O9

O9 does not yet extract the phenomenological value of $\delta$.

Instead, it proves that the failure observed in O8 was not due to:

• insufficient fingerprint complexity
• or an intrinsic limitation of projective capacity dynamics

but solely to:

• the exponential shell growth of the LPS graph family

The logical chain is now:

• O6: fixed finite-dimensional fingerprints fail algebraically
• O7: the correct observable is projective capacity
• O8: exponential shell growth geometrically compresses the window
• O9: polynomial-growth graphs remove this compression

The obstruction is therefore fully localised and resolved.

What O9 Adds

O9 introduces several decisive structural results:

• a formal proof of LPS window compression ($O(\log q)$ BFS depth)
• a replacement principle: exponent extraction requires $|B_n| \sim n^D$
• introduction of polynomial-growth Cayley graphs (discrete Heisenberg group)
• the Window-Depth Theorem: a vertex window $O(q^2)$ yields $\Omega(q^{1/2})$ BFS depth
• identification of the true scaling: $n_{\max} = \Theta(q)$
• numerical confirmation of:
  • polynomial shell growth
  • convergence toward homogeneous dimension $D=4$
  • polynomial expansion of the observable window

Interpretation of the Result

The main conceptual outcome is the separation between:

• graph geometry, which controls the observable depth
• projective dynamics, which determines the exponent $\delta$

In particular:

• the fingerprint and state law are geometry-independent
• but their observability requires a geometry with sufficiently slow growth

Thus, $\delta$ is not a property of a specific graph family, but a property of the underlying projective dynamics, provided that the graph does not obstruct its measurement.

Relation to Previous Steps

O9 preserves all previous structural results:

• spectral admissibility from Step 1
• binary-polyhedral maximality from Step 2
• three-level ADE stratigraphy from Step 3
• projective dynamics and support contraction from O1
• hierarchical amplification via growing valence from O3
• structural upper bound on the cascade exponent from O4
• admissible-frontier saturation from O5
• fixed finite-dimensional no-go from O6
• capacity reformulation and state law from O7
• growing fingerprint and geometric obstruction from O8

It does not modify the target exponent. Instead, it establishes the correct geometric setting in which it becomes observable.

Conceptual Structure

O9 advances the structural chain as follows:

1. Spectral admissibility → mode selection
2. Spectral capacity → binary-polyhedral maximality
3. Spectral stratigraphy → discrete ADE levels
4. O1 → ordering via support contraction
5. O3 → amplification via valence growth
6. O4 → structural upper bound on $\beta$
7. O5 → admissible-frontier saturation
8. O6 → fixed finite-dimensional no-go
9. O7 → projective capacity formulation
10. O8 → growing fingerprint + geometric obstruction
11. O9 → resolution via polynomial-growth geometry

The programme now excludes both:

• fixed finite-dimensional encodings
• exponential-growth graph geometries

as viable settings for exponent extraction.

What O9 Resolves

O9 provides:

• a proof that the O8 obstruction is purely geometric
• a construction of a graph family where this obstruction disappears
• a theorem guaranteeing polynomially large observable windows
• a validation that the fingerprint framework is structurally sound
• a complete isolation of the remaining open problem

Residual Open Problem

What remains is no longer a question of geometry or fingerprint construction.

The only remaining task is:

• extracting $\delta$ in the polynomial-growth setting
• and deriving it from first principles

This requires:

• larger values of $q$ (to access the $n^4$ regime cleanly)
• refined numerical or analytical tools

Open Directions

1. Extraction of $\delta$ at large $q$
   Perform measurements on Heisenberg graphs for $q \gg 100$

2. First-principles derivation of $\delta$
   Derive the exponent from ADE representation theory and projective capacity

3. Universality across polynomial-growth families
   Test higher Heisenberg groups and abelian lattices

4. Intermediate-growth graphs
   Determine whether polynomial growth is necessary or only sufficient

5. Continuum limit
   Connect discrete measurements to the continuous capacity framework of O7

Status

This framework is now:

• free of algebraic obstruction (O6)
• free of geometric obstruction (O8 → O9)
• expressed in capacity variables (O7)
• ready for exponent extraction (O10)

It does not assume:

• that a growing fingerprint is sufficient by itself
• that expander geometries are suitable measurement environments
• that finite-size data can directly reveal the asymptotic exponent

Repository Structure

paper/
├── out/ # Compiled O9 PDF
├── tex/ # LaTeX sources
└── README.md

Citation

If you reference this work, please cite:

J. Beau, Projective Capacity Beyond Expanders: Polynomial-Growth Relaxation Graphs and the Capacity Exponent, Zenodo, 2026.

Acknowledgements

Portions of the derivations, conceptual synthesis, and editorial refinement benefited from iterative interactions with large language models used as analytical assistants. All theoretical results and interpretations remain the sole responsibility of the author.

Contributions

This repository is intended as a research reference.

Critical feedback, independent verification, and alternative constructions of polynomial-growth graph families are welcome.

Please open an issue to discuss conceptual points, technical details, or possible extensions.