This repository contains the source of the O17 Cosmochrony paper Exact Block-Independence of Raw Gram-Schmidt Redundancy in a Scalar Weil Model.

Within a scalar Weil-representation toy model (a representation on $\mathbb{C}^q$), the raw Gram-Schmidt redundancy count $\tilde\sigma_c(n;b_1,b_2)$ is exactly independent of the character $c$ and of both initial-condition parameters $(b_1,b_2)$: $\tilde\sigma_{c'}(n;,b_1',b_2') = \tilde\sigma_c(n;,b_1,b_2) \quad\text{exactly, for all } n\geq0,$ for any two blocks, conjugate or not. A verification script (reproducing a worked example between blocks with $b_1\neq b_1'$, plus an independent test between genuinely unrelated blocks past the point of BFS wraparound) is included in code/o17_block_independence_check.py.

This block-independence gives a clean, unconditional statement about the amplitude factor $r(c,q)$: since the raw count cannot carry any block-dependence, whatever block-dependence a normalised observable $\sigma_c := \tilde\sigma_c/D$ exhibits must come entirely from its normalisation $D$ — a direct proof. This conclusion is scoped strictly to this paper's own toy model.

Established, within this paper's own toy model:

• Exact block-independence of the raw redundancy count, for any two blocks whatsoever.
• Any normalisation-dependent observable's block-dependence lies entirely in its normalisation denominator — a direct logical consequence, not requiring separate numeric verification.
• The underlying Weil conjugation identity $\rho_{q-c}=\overline{\rho_c}$, used in the restricted theorem's proof, remains independently valid.

Open:

• Whether a corresponding block-independence result holds for the real O12/O13 pipeline. The toy model represents each block by a single position-basis vector; the real pipeline's exact fingerprints are single frequency-basis vectors indexed by a generic triple $(c_1,c_2,c_3)$, not this paper's scalar $c$. No argument connects the two.
• The fibre structure of the non-injective projection $\Pi$: the block-independence result holds too broadly (for any two blocks, not only conjugate ones) to serve, by itself, as a criterion for grouping specifically conjugate blocks into fibres.
• The normalisation $D$ this paper leaves unspecified.

O16's fibre-level proposal and this paper's own results were, along with each other, cited downstream in O18–O21. Their correctness does not follow automatically from either paper and must be reassessed on each paper's own text.

• v1.0 — initial release. Opened from O16's original (since withdrawn) finite-window baseline and amplitude-factor claims, and asserted that this paper's toy model identifies conjugate blocks with the fibres of $\Pi$ "as measured by the real pipeline," and that O14–O15 were correct-but-incomplete block-level analyses. The proof of the arbitrary-blocks corollary asserted, without arguing, that Gram-Schmidt dynamics is independent of $e_{b_1}$'s labelling.
• v1.1 — corrective revision. Withdrew the pipeline identification (no argument connects the toy model's position-basis fingerprints to the real pipeline's frequency-basis ones) and the O14–O15 requalification. Replaced the asserted labelling-independence step with a direct combinatorial proof establishing something stronger: exact block-independence for any two blocks, not only conjugate ones — proved without appeal to the conjugation identity at all. This stronger fact undercuts, rather than supports, the withdrawn fibre claim: applied uniformly, "identical dynamics ⟹ same fibre" would place every block in one fibre.
• v1.2 — the PDF body was rewritten to state only the current, established content natively, with no narrative of what was withdrawn or corrected; that history is preserved here and in this paper's Zenodo record notes instead. The title's "A Corrective Revision" suffix was dropped. No mathematical content changed between v1.1 and v1.2.

paper/
├── out/ # Compiled O17 PDF
├── code/ # Verification script for the block-independence result
├── tex/ # LaTeX sources
└── README.md

If you reference this work, please cite:

J. Beau, Exact Block-Independence of Raw Gram-Schmidt Redundancy in a Scalar Weil Model, Zenodo, 2026.

Portions of the derivations, conceptual synthesis, numerical strategy, and editorial refinement benefited from iterative interactions with large language models used as analytical assistants. All theoretical results, computations, and interpretations remain the sole responsibility of the author.

This repository is intended as a research reference.

Critical feedback, independent verification, and further analysis of:

• whether a corresponding block-independence result holds for the real O12/O13 pipeline
• the fibre structure of the non-injective projection $\Pi$
• the normalisation $D$ this paper leaves unspecified

are welcome.

Please open an issue to discuss conceptual points, technical details, or possible extensions.