Overview
The Weil representation of $\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})$ satisfies the exact conjugation identity $\rho_{q-c} = \overline{\rho_c}$, valid for every prime $q$ and every central character $c$, independent of any hypothesis about the non-injective projection $\Pi$. O16 examines whether conjugate blocks $\{c,\,q-c\}$ constitute the fibres of $\Pi$.
That identification rests on two independent premises: that $\Pi$ carries a parity involution $\Pi(x)=\Pi(-x)$, and that this involution is realised, in the Weil representation, as the character map $c \mapsto q-c$. Neither follows from the conjugation identity alone, and neither is derived from first principles within the Cosmochrony framework — both are open, structurally motivated hypotheses.
Under a further, separate hypothesis — that a per-block observable $\sigma_c(n)$ has a conjugate-pair ratio $r(c,q) = \sigma_{q-c}(n)/\sigma_c(n)$ independent of the shell index $n$ — the log-log slope of the pair-level product $\sigma_{\mathrm{pair}}(n) = \sigma_c(n)\,\sigma_{q-c}(n)$ is exactly double that of $\sigma_c(n)$: $\delta_{\mathrm{pair}} = 2\,\delta_c$, a direct algebraic consequence of differentiating a constant-ratio product. On the exact block observable of the real O12/O13 pipeline, the proved conjugate-block identity forces this ratio to be exactly $1$, so the pair-level product reduces definitionally to the single block squared: no separate doubling phenomenon to explain there.
Core contributions
- Two independent fibre hypotheses: that $\Pi$ carries a parity involution $\Pi(x)=\Pi(-x)$, and that this involution is realised in the Weil representation as $c\mapsto q-c$ — neither implied by the other or by the proved conjugation identity.
- Conjugate symmetry: $\rho_{q-c} = \overline{\rho_c}$, exact and unconditional representation theory, independent of any hypothesis about $\Pi$.
- Elementary bound: $\langle\sigma_c^2\rangle \ge \langle\sigma_c\rangle^2$ for any non-negative sequence, by Cauchy–Schwarz.
- Conjugate-block identity on the real pipeline: $\Sigma_n^{(c_1,c_2,c_3)} = \Sigma_n^{(-c_1,-c_2,-c_3)}$ holds exactly for the O12/O13 observable, forcing the conjugate-pair ratio to be identically $1$ there.
- Conditional exponent doubling: $\delta_{\mathrm{pair}} = 2\,\delta_c$, given an $n$-independent conjugate-pair ratio — an algebraic consequence, carrying no numerical content until a value of $\delta_c$ is supplied for a sourced observable.
Interpretation
O16 does not modify the growth law of O6/O7, and does not modify the underlying Weil-block pipeline of O12–O15. It examines a separate question: what is the correct unit of admissibility — the individual block, or the conjugate pair?
- O15: the scalar block-level growth law does not transfer to the exact Weil regime
- O16: conjugate pairs are the fibres of $\Pi$ only under two independent, open hypotheses
Three things are open, not one, and should not be blended together: whether $\Pi$ carries a parity involution at all; whether that involution is specifically realised as $c\mapsto q-c$ in the Weil representation; and, independently of both, whether any sourced, reproducible observable satisfies the doubling premise for a pair-level quantity distinct from the block squared.
Relation to the Cosmochrony program
O16 follows the block-level results of O12–O15 without modifying them. It proposes the conjugate pair as the fibre-level unit of admissibility and derives what follows conditionally from that proposal.
On the real O12/O13 pipeline, sampling the conjugate pair does not sample independent data: the exact conjugate-block identity means it computes the same block squared. The fixed Heisenberg Cayley graph also has frontier $N^{3/4}$, not the square-root frontier of O7, so no pair-capacity growth carrier connecting $\delta_{\mathrm{pair}}$ to $\beta^*$ is defined by the corpus.
Current outcome and open directions
The identification of conjugate pairs with fibres of $\Pi$ remains open, resting on the two independent hypotheses above. The exponent-doubling remark is conditional and algebraically valid, holding trivially (with ratio $1$) on the real pipeline's own block observable, and not yet tied to any other sourced observable.
A derivation, from first principles within Cosmochrony, of both fibre hypotheses would strengthen the fibre interpretation, but would not by itself close the separate capacity-to-rate bridge to $\beta^*$, nor supply a sourced observable satisfying the doubling premise for a pair-level quantity distinct from the block squared.
References
Jérôme Beau. Fibre Admissibility Hypotheses and Conditional Pair-Exponent Doubling.