Overview
This article continues the spectral admissibility programme after the proxy-level extraction achieved in O11. O12 replaces the bidimensional proxy on the abelianised $(a,b)$ coordinates with the exact irreducible Weil representation, retaining the central coordinate $\gamma$. 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 $q$.
An exact algebraic identity shows that $\Sigma_n^{(c)}$ depends on the block only through a projective pair $(\alpha,\beta)$: it is therefore exactly independent of the central coordinate $\gamma$ and of the translational coordinate $a$. Finite-$q$ fitted slopes (found at $q \in \{29,53,61\}$ in the range $\hat\delta_{\mathrm{exact}} \approx 4.3$–$4.8$) are, by this closed form, crossover statistics of the block saturation-depth distribution — not estimates of an intrinsic asymptotic exponent.
Using that closed form, O12 derives the intrinsic unfolded exponent exactly, holding block-by-block rather than only in the mean: \[ \delta_{\mathrm{exact}} = 3 \qquad (q\to\infty,\ n\to\infty,\ n = o(\sqrt q)). \]
Core contributions
- Exact Weil-block observable: replacement of the O11 proxy by the full irreducible Weil projection, retaining the central coordinate $\gamma$.
- Exact incremental capacity definition: 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 $H_c$.
- Closed-form projective-pair reduction: $\Sigma_n^{(c)}$ depends on the block only through a projective pair $(\alpha,\beta)$ — an algebraic identity, exactly independent of $\gamma$ and of the translational coordinate $a$.
- Finite-$q$ slopes reclassified: the fitted decay exponents $\hat\delta_{\mathrm{exact}} \approx 4.3$–$4.8$ at $q\in\{29,53,61\}$ ($R^2 > 0.98$) are crossover statistics of the block saturation-depth distribution, independently reproduced by pure combinatorial set arithmetic.
- Intrinsic exponent derived exactly: $\delta_{\mathrm{exact}} = 3$ as $q\to\infty$, $n=o(\sqrt q)$, holding block-by-block from Heisenberg shell growth alone.
- Central coordinate provably inert: $\gamma$ has no effect on this observable — an algebraic consequence of the projective-pair reduction, not a numerical coincidence.
Interpretation
O12 completes the extraction chain initiated in O8, and is the first paper where the exponent is measured in the exact irreducible representation.
- 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 measures the exact irreducible observable and derives its intrinsic asymptotic exponent in closed form.
The gap between the finite-$q$ fitted slopes and the O11 proxy value $\hat\delta_{\mathrm{cap}} \approx 3.39$ is real and reproducible, but is a crossover phenomenon of the finite-$q$ regime rather than a shift in the asymptotic exponent: the block capacity's exact dependence on $(\alpha,\beta)$ alone rules out any central-coordinate or coherence mechanism as its source.
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 and delivers the programme's first exact, closed-form asymptotic capacity exponent. Its extension to a wider range of primes is carried out in O13; the separate question of how this block-level exponent relates to the cascade exponent $\beta^*$ is addressed downstream, beginning with O15.
Current outcome
The intrinsic asymptotic exponent of the exact block-wise projective capacity is now established in closed form, $\delta_{\mathrm{exact}} = 3$, independent of the central coordinate $\gamma$. The finite-$q$ fitted slopes reported here and extended in O13 are understood as crossover statistics of the block saturation-depth distribution, not as direct estimates of that intrinsic exponent.
The remaining question for the programme is no longer the observability or asymptotic convergence of this exponent — both are now settled — but which regime of the block-level observable legitimately enters the O6/O7 derivation of the cascade exponent $\beta^*$, a question taken up starting with O15.
References
Jérôme Beau. Exact Weil-Block Projective Capacity on Heisenberg Graphs: Resolving the Final Obstruction to δ Extraction. Preprint.