Overview
This article continues the spectral admissibility programme after O14. While O14 identified the observable-class mismatch between the scalar proxy capacity of O7 and the exact Weil-block capacity measured in O12 and O13, it still left open a sharper question: where exactly does the derivation of the map $\delta \mapsto \beta^*$ fail?
O15 answers that question by auditing the derivation chain from O3 to O7. The outcome is precise: the mass-ratio mechanism of O3 remains intact, the growth-law derivation of O6 remains valid in its scalar domain, but the passage from a scalar global observable to a mean over exact Weil blocks is not justified. The exact exponent $\hat{\delta}_{\mathrm{exact}}$ is therefore not automatically the dynamic exponent entering the O6 growth equation.
The paper also proves an aggregation no-go: no non-negative reweighting of the exact blocks can raise the dynamic exponent above the measured exact one. The remaining exact/proxy gap is thus no longer a numerical, finite-size, or averaging issue. It is a structural problem of the growth equation itself.
Core contributions
- Audit of the O3–O7 chain: O15 separates the derivation into its logical stages and shows that O3 remains valid, while O6 remains valid only in the scalar setting for which it was derived.
- Non-transferability theorem: the exponent $\hat{\delta}_{\mathrm{exact}}$ extracted from the exact block mean $\bar{\Sigma}_n = (q-1)^{-1}\sum_c \Sigma_n^{(c)}$ is not, in general, the dynamic exponent controlling the growth law for $p(n)$.
- Aggregation no-go: no choice of non-negative dynamic weights on the exact Weil blocks can produce $\alpha_{\mathrm{dyn}} > \hat{\delta}_{\mathrm{exact}}$. Better averaging or reweighting cannot recover the phenomenological target regime.
- Strict notation hierarchy: the paper distinguishes $\hat{\delta}_{\mathrm{exact}}$, $\alpha_{\mathrm{dyn}}$, $\delta_{\mathrm{eff}}$, and $\sigma(q)$, preventing the scalar and exact-block exponents from being conflated.
- Exact-block growth-law hypothesis: O15 formulates an effective growth law under exact-block normalisation, but explicitly marks its $1/q$ factor and associated exponent-level translation as conjectural rather than proved.
- Structural conclusion: the remaining exact/proxy gap is no longer an observable-level problem. It is localised at the level of the growth equation and the exact dynamic observable that should replace the O7 scalar quantity.
Interpretation
O15 does not reopen the finite-size question closed by O13, nor the observable-class correction established in O14. Instead, it asks whether the O7 derivation itself survives when the proxy scalar observable is replaced by exact Weil-block capacity.
- O12 introduced the exact Weil-block observable and found a higher decay exponent than the proxy one.
- O13 showed that this discrepancy persists asymptotically and is not a finite-size artefact.
- O14 corrected the observable-level mismatch but showed that the corrected exponent still fails to recover the phenomenological target.
- O15 proves that this failure cannot be repaired by block aggregation and must instead be addressed at the level of the dynamic observable entering the growth law.
The key conceptual shift is therefore from observable correction to dynamical correspondence. The problem is no longer ``which exact exponent should replace the proxy one?'' but ``which exact-block observable really enters the growth equation for $p(n)$?''
Relation to the Cosmochrony program
O15 follows directly from O14. 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, O11 restores observability at the proxy representation level, O12 implements the exact Weil-block observable, O13 closes the finite-size explanation, and O14 isolates the observable-class mismatch.
The present paper is the derivational bridge between exact measurement and the cascade law. It does not add new prime ranges. Its purpose is to determine whether the O7 map from decay exponent to cascade exponent survives in the exact-block regime. Its answer is negative in the naive scalar form.
Current outcome and open problem
The main negative result of O15 is structural: even before any detailed modelling of the exact-block growth law, the aggregation no-go implies $\alpha_{\mathrm{dyn}} \le \hat{\delta}_{\mathrm{exact}} < 5.0$ over the currently tested range. Since the phenomenological target requires a much larger effective decay exponent, no refinement based only on block reweighting can restore the O7 relation.
O15 therefore turns the remaining tension into a calculable programme. The next step is to extract the exact dynamic observable $R_n^{\mathrm{eff}}$ directly from the O12/O13 block data, test the conservative scenario in which the O7 functional form survives, and, if that scenario fails, derive the structural correction $\sigma(q)$ from Weil-block structure itself.
The remaining open problem is thus sharply localised: what replaces the scalar proxy observable of O7 in the exact Weil-block growth equation?
References
Jérôme Beau. Scalar-to-Block Breakdown of the δ→β* Map: Why the O7 Growth Law Does Not Transfer to Exact Weil Capacity on Heisenberg Graphs. Preprint.