Overview
This article continues the spectral admissibility programme after the proxy-level extraction achieved in O11. While O11 established that a representation-adapted Weil-block proxy could recover a stable decay regime on Heisenberg Cayley graphs, it left open the central question of whether the exact Weil projection, retaining the central coordinate $\gamma$, would yield the same exponent.
O12 answers that question by replacing the bidimensional proxy on the abelianised $(a,b)$ coordinates with the exact irreducible Weil representation. The resulting observable is the exact block-wise incremental projective capacity $\Sigma_n^{(c)} = \Delta r_n^{(c)} / |S_n|$, measured inside generic Weil blocks $H_c$ of dimension $q$.
The outcome is a clear exact/proxy regime shift. At the accessible primes $q \in \{29,53,61\}$, the exact exponent is consistently found in the range $\hat{\delta}_{\mathrm{exact}} \approx 4.4$--$4.8$, well above the O11 proxy value $\hat{\delta}_{\mathrm{cap}} \approx 3.39$. The paper identifies this as Case B: the exact Weil projection yields a steeper decay law than the proxy, and the difference is traced to the dynamically active central phase.
Core contributions
- Exact Weil-block observable: replacement of the O11 proxy by the full irreducible Weil projection, retaining the central coordinate $\gamma$ and its phase contribution $\psi_c(\gamma + bx)$.
- Exact incremental capacity definition: introduction of the observable $\Sigma_n^{(c)} = \Delta r_n^{(c)} / |S_n|$, measuring the genuinely new directions contributed by shell $S_n$ inside an exact Weil block $H_c$.
- First exact exponent extraction: fitted decay exponents $\hat{\delta}_{\mathrm{exact}} = 4.83 \pm 0.24,\ 4.40,\ 4.57$ for $q = 29,\ 53,\ 61$, with $R^2 > 0.98$ in all runs.
- Exact/proxy regime shift: systematic confirmation that $\hat{\delta}_{\mathrm{exact}} > \hat{\delta}_{\mathrm{cap}}$, establishing Case B of the exact/proxy comparison.
- Central-phase mechanism: introduction of the coherence observable $\ell_\gamma(n)$, showing that the central coordinate is dynamically active over the fitting window and contributes to the acceleration of exact-block saturation.
- Qualified universality: mean-level power-law decay is confirmed, while strict block-by-block universality is not, revealing a non-trivial exact-block heterogeneity rather than a failure of the model.
- Phenomenological tension: via $\beta^* = 1 / (\delta + \tfrac{1}{2})$, the exact exponent implies $\beta^* \approx 0.19$--$0.20$, outside the O3 target $\beta^* \in (0.09, 0.13)$.
Interpretation
O12 shows that the final ambiguity left by O11 is not a minor correction but a structural one. The exact and proxy observables do not lie in the same quantitative regime: once the central coordinate $\gamma$ is reinstated, the decay exponent shifts upward by about one unit.
- O8 identified geometric compression on LPS graphs.
- O9 restored an observable BFS window by moving to polynomial-growth Heisenberg geometry.
- O10 showed that dense fingerprints remain algorithmically misaligned with the dynamics.
- O11 restored observability at the proxy level via a bidimensional Weil-block construction on $(a,b)$.
- O12 closes the observable gap by measuring the decay law in the exact irreducible representation and identifying the metaplectic phase as the mechanism behind the exact/proxy discrepancy.
In this picture, the relevant signal is still a law of diminishing novelty, but its exact rate depends on the full non-commutative structure of the Heisenberg group. The central coordinate is not a passive correction to the proxy dynamics: it changes the observed decay class and therefore the inferred cascade exponent.
Relation to the Cosmochrony program
O12 follows directly from O11. Spectral admissibility, capacity, rigidity, and stratigraphy define the spectral backbone of the theory. O1 restores ordering through projective dynamics, O3 amplifies the hierarchy via valence growth, O4 constrains the cascade exponent from bounded relational flux, O5 localises the failure of vertex-level mechanisms, O6 proves the no-go for fixed finite-dimensional fingerprints, O7 reformulates the observable in terms of projective capacity, O8 identifies geometric compression on LPS graphs, O9 removes that compression by moving to polynomial-growth Heisenberg geometry, O10 isolates the dense-sketch bottleneck, and O11 restores observability at the proxy representation level.
The present paper moves from proxy to exact representation-level data. It confirms that the exact observable is measurable, that it differs structurally from the O11 proxy, and that the resulting exponent generates a real tension when propagated to the cascade exponent $\beta^*$. In that sense, O12 is the first paper in the series where the programme becomes quantitatively falsifiable at the exact representation level.
Current outcome and open problem
The exact measurement currently implies $\beta^* \approx 0.19$--$0.20$, outside the target range $\beta^* \in (0.09, 0.13)$ inferred earlier from charged lepton mass ratios. This incompatible outcome is reported explicitly and not removed by post hoc adjustment.
Three interpretations remain open: the accessible primes may still be too small to probe the asymptotic regime; the structural relation between $\delta$ and $\beta^*$ may require revision at the exact level; or the phenomenological target itself may need reconsideration. The next step of the programme is therefore a larger-prime exact computation (O12-O1), designed to decide between finite-size drift and genuine structural tension.
References
Jérôme Beau. Exact Weil-Block Projective Capacity on Heisenberg Graphs: Resolving the Final Obstruction to δ Extraction. Preprint.