Overview
This article continues the spectral admissibility programme by auditing the O3–O7 derivation chain in light of the exact Weil-block results of O12–O14. 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 exponent $\hat\delta_{\mathrm{exact}}$ is therefore not automatically the dynamic exponent entering the O6 growth equation.
The native Heisenberg identity is $\Delta r_n = |S_n|\,\Sigma_n$. From it, O15 derives an exact identity for the natural block-weighted redundancy, $R_n^{\mathrm{eff}} = \bar\Sigma_n \cdot (1+V_n)$, where $V_n$ is the inter-block variance ratio already reported by O12/O13. This identity bounds the aggregation exponent, $\alpha_{\mathrm{dyn}} \le \hat\delta_{\mathrm{exact}}$, whenever $\log(1+V_n)$ has non-negative OLS slope against $\log n$ over the fitting window — a condition checkable from existing pipeline data.
That checkable condition is now verified at two of the five tested primes: at $q=151$ and $q=211$ the OLS slope is positive (0.2842 and 0.0419 respectively), giving $\alpha_{\mathrm{dyn}} \le \hat\delta_{\mathrm{exact}}$ at both ($3.9824 \le 4.2666$; $3.5466 \le 3.5885$), while the stronger, pointwise-monotone special case does not hold on either window — a genuine test of the weaker, actually-required condition. It remains open at $q\in\{29,61,101\}$.
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 $p(n)$.
- Exact aggregation identity: the natural block-weighted redundancy satisfies $R_n^{\mathrm{eff}} = \bar\Sigma_n \cdot (1+V_n)$ exactly, where $V_n$ is the inter-block variance ratio.
- Conditional aggregation bound: $\alpha_{\mathrm{dyn}} \le \hat\delta_{\mathrm{exact}}$ holds whenever $\log(1+V_n)$ has non-negative OLS slope against $\log n$ — a checkable condition, not an assumption on arbitrary weights.
- Finite-prime verification: the checkable condition holds at $q=151$ and $q=211$ (positive OLS slopes, pointwise-monotone special case failing at both), giving a genuine test of the weaker, actually-required condition; open at $q\in\{29,61,101\}$.
- Structural conclusion: the remaining exact/proxy gap is localised at the level of the growth equation and the exact dynamic observable that should replace the O7 scalar quantity, not at the level of block aggregation.
Interpretation
O15 does not reopen the finite-size question closed by O13. 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 derived its intrinsic asymptotic exponent exactly.
- O13 confirmed that value over an extended prime range.
- O15 proves that the O7 growth law does not transfer to the exact-block setting in its naive scalar form, and gives a checkable, conditional bound on how far block aggregation alone can push the dynamic exponent.
The key conceptual shift is from observable correction to dynamical correspondence: the question is no longer ``which exact exponent should replace the proxy one?'' but ``which exact-block observable really enters the growth equation for $p(n)$, and under what condition does aggregation bound it?''
Relation to the Cosmochrony program
O15 audits the O3–O7 chain in light of O12–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 confirms its asymptotic value, and O14 diagnoses the observable-class mismatch.
The present paper is the derivational bridge between exact measurement and the cascade law. Its answer is negative for the naive scalar form of the O7 relation, but it supplies a checkable route — the aggregation bound — that downstream papers, beginning with O16, build on.
Current outcome
The aggregation bound $\alpha_{\mathrm{dyn}} \le \hat\delta_{\mathrm{exact}}$ is verified, via its checkable OLS-slope condition, at $q=151$ and $q=211$; it remains open at $q\in\{29,61,101\}$. At both verified primes, the bound holds even though the stronger, pointwise-monotone special case fails — a decisive illustration that the checkable condition, not the stronger sufficient one, is what the aggregation identity actually requires.
The remaining open problem is what replaces the scalar proxy observable of O7 in the exact Weil-block growth equation, and whether the aggregation bound's checkable condition extends to the three primes where it has not yet been tested.
References
Jérôme Beau. Observable-Class Derivation Mismatch and the Heisenberg Transfer No-Go: From Proxy Capacity to Block-Level Span Dynamics. Preprint.