Overview
O16 proposes a pair-level observable $\sigma_{\mathrm{pair}}(n) = \sigma_c(n)\,\sigma_{q-c}(n)$, built from the exact conjugation identity $\rho_{q-c}=\overline{\rho_c}$, as a candidate fibre-level unit of admissibility, conditional on two independent open hypotheses about the non-injective projection $\Pi$. Independently of those hypotheses, O17 asks: why is the raw Gram–Schmidt dynamics identical for $\rho_c$ and $\rho_{q-c}$, and what is the structural origin of the amplitude factor $r(c,q)$?
O17 answers both questions within an explicitly defined scalar toy model. Within that model, the raw redundancy count $\tilde\sigma_c(n;b_1,b_2)$ is exactly independent of the central character $c$ and of both initial-condition parameters $(b_1,b_2)$ — not only equal for conjugate characters $c$ and $q-c$, but identical for any two blocks whatsoever. The proof is purely combinatorial (translation-invariance of shell-label collisions on the underlying Cayley graph) and needs no conjugation identity.
A direct consequence: for any normalised observable $\sigma_c := \tilde\sigma_c/D$ defined within this model, all of its block-dependence resides entirely in the normalisation $D$, not in the raw count itself.
Core contributions
- Identity of raw dynamics, restricted form: for blocks sharing the same initial support, $\tilde\sigma_{q-c}=\tilde\sigma_c$ for all shells, via the conjugation identity $\rho_{q-c}=\overline{\rho_c}$.
- Block-independence, general form: $\tilde\sigma_{c'}(n;b_1',b_2') = \tilde\sigma_c(n;b_1,b_2)$ exactly, for any two blocks — conjugate or not — proved by a purely combinatorial argument needing no conjugation identity.
- Normalisation artefact: any block-dependence of a normalised observable $\sigma_c := \tilde\sigma_c/D$ comes entirely from $D$, a direct consequence of block-independence.
- Scope limit on the fibre question: because block-independence holds for any two blocks, not only conjugate ones, it cannot by itself license a same-fibre conclusion for conjugate pairs specifically — applied uniformly, it would place every block in one fibre.
Interpretation
O17 does not introduce a new growth law, and does not establish that conjugate pairs are the fibres of $\Pi$. It answers a narrower, purely mathematical question about this paper's own toy model.
- O16: proposes conjugate pairs as fibres, conditional on two open hypotheses
- O17: proves the underlying raw dynamics is block-independent in a scalar toy model — too broadly to select conjugate pairs specifically
A genuinely selective criterion, distinguishing conjugate pairs from other block pairings, is not supplied here. Whether an analogous block-independence result holds for the real pipeline, and what its normalisation actually is, is open.
Relation to the Cosmochrony program
O17 follows O16 without modifying its proposal. It resolves two narrow, purely mathematical questions about the raw Weil-block dynamics underlying that proposal, entirely within its own toy model, and reports what that resolution can and cannot support for the fibre question.
The toy model's fingerprints are single position-basis vectors; the real O12/O13 pipeline's fingerprints are frequency-basis vectors indexed by a generic triple. No argument here connects the two, so none of O17's results is asserted for that pipeline.
Current outcome and open directions
Within its own toy model, O17 establishes exact block-independence of the raw redundancy count for any two blocks, and shows that this fact is too broad to support a fibre criterion specific to conjugate pairs.
Two directions remain open: whether a corresponding block-independence result holds for the real O12/O13 pipeline's raw redundancy count, and what its normalisation actually is; and, independently, what genuinely selective criterion — if any — identifies conjugate pairs specifically as the fibres of $\Pi$.
References
Jérôme Beau. Exact Block-Independence of Raw Gram-Schmidt Redundancy in a Scalar Weil Model.