This repository contains the source of the O15 Cosmochrony paper
Observable-Class Derivation Mismatch and the Heisenberg Transfer No-Go: From Proxy Capacity to Block-Level Span Dynamics.

Version 1.16.1. Adds the first finite verifications of open direction O15-O1: at $q=151$ and $q=211$, the exact identity $R_n^{\mathrm{eff}}=\bar\Sigma_n(1+V_n)$ is confirmed to machine precision (7ドル.1\times10^{-15}$ and 3ドル.6\times10^{-15}$ respectively), and the checkable OLS-slope condition of Corollary "Aggregation bound" holds at both (slopes 0ドル.2842>0$ and 0ドル.0419>0$), giving $\alpha_{\mathrm{dyn}}\leq\hat\delta_{\mathrm{exact}}$ at both primes (3ドル.9824\leq4.2666$; 3ドル.5466\leq3.5885$). The pointwise-monotone special case does not hold on either window, making both a genuine test of the weaker, actually-required OLS condition, not the stronger sufficient one. O15-O1 is now partially completed (verified at $q=151$ and $q=211$; still open at $q\in{29,61,101}$), with reproduction code added at code/.

Version 1.16. The non-equivalence theorem is unchanged. The aggregation result is corrected: it now states an exact identity for the natural dynamical weights, $R_n^{\mathrm{eff}}=\bar\Sigma_n\cdot(1+V_n)$, and bounds the aggregation exponent conditionally on a checkable monotonicity criterion, rather than for arbitrary weights. Provenance errors in the numeric table and the inter-block variance attribution are fixed, and block notation now matches the actual O12/O13 pipeline (generic triples, sampled finite sets) throughout. The native Heisenberg identity remains $\Delta r_n=|S_n|\Sigma_n$, whereas the Born--Infeld square-root factor belongs to the LPS model. The legacy C1/C2 formulas remain phenomenological diagnostics, not native growth laws.

This work extends the spectral admissibility sub-programme by auditing the derivation chain O3–O7 in light of the exact Weil-block results of O12–O14.

O13 established, and O12's independent corrective revision confirms analytically, that the exact observable does not satisfy the proxy-level relation between the capacity exponent and the cascade exponent as a finite-size artefact. This still left open a crucial question: where exactly does the derivation fail?

The present work answers that question.

It shows that the mismatch is not due to:

• finite-size effects
• numerical instability
• central-phase bias
• or an insufficiently refined statistical aggregation of blocks

Instead, the failure occurs at the level of the growth equation itself: the scalar observable entering the O6/O7 derivation is not equivalent to the exact Weil-block observable measured in O12–O14.

The paper establishes that the structural relation $\beta^* = \frac{1}{\delta + \tfrac12}$ derived in O6–O7 is conditional on the changing-degree LPS model and does not transfer to the exact-block setting as constituted in the corpus.

The main logical result is a non-transferability theorem:

• the exponent $\hat\delta_{\mathrm{exact}}$ extracted from the block mean $\bar\Sigma_n = \frac{1}{|\mathcal{C}{\mathrm{gen}}|}\sum{c\in\mathcal{C}_{\mathrm{gen}}} \Sigma_n^{(c)}$ is not, in general, the exponent controlling the dynamic growth law of (p(n))

Thus the O7 chain $\bar\Sigma_n \sim n^{-\delta} \quad \Longrightarrow \quad \beta^* = \frac{1}{\delta + \tfrac12}$ fails in the exact Weil regime.

The native Heisenberg statement is instead

$\Delta r_n=|S_n|\Sigma_n\asymp n^{D-1-\delta}$

on a finite pre-saturation window. It contains neither the changing valence $p(n)$ nor a native $c_{\mathrm{BI}}\sqrt p$ factor.

The paper proves three central points.

The derivation is decomposed into its logical steps:

• O3: mass hierarchy from exit ranks
• O6: growth-law derivation
• O7: capacity reformulation

The result is:

• O3 remains valid
• O6 remains valid in its scalar domain
• the failure is localised at the point where a scalar global observable is implicitly substituted by a mean over exact Weil blocks

So the issue is not a mathematical error in O3–O7, but a domain-of-validity failure.

The paper proves that the exact observable $\bar\Sigma_n = \frac{1}{|\mathcal{C}{\mathrm{gen}}|}\sum{c\in\mathcal{C}_{\mathrm{gen}}} \Sigma_n^{(c)}$ cannot be identified with the scalar redundancy observable entering the O6 growth law.

This establishes that:

• (\hat\delta_{\mathrm{exact}}) is a measured block-mean exponent
• it is not a scalar exponent that may be inserted into the O6 growth equation

The natural block-weighted redundancy $R_n^{\mathrm{eff}}$ (Definition, weights proportional to $\Delta r_n^{(c)}$) satisfies an exact identity: $R_n^{\mathrm{eff}} = \bar\Sigma_n\cdot(1+V_n)$, where $V_n$ is exactly the inter-block variance ratio already reported by O12/O13.

From this identity, $\alpha_{\mathrm{dyn}} \le \hat\delta_{\mathrm{exact}}$ follows whenever $\log(1+V_n)$ has non-negative OLS slope against $\log n$ over the fitting window (in particular whenever 1ドル+V_n$ is non-decreasing there) — a condition checkable from existing pipeline data, now verified at two of the five tested primes: at $q=151$ and $q=211$ the OLS condition holds (slopes 0ドル.2842$ and 0ドル.0419$) even though the stronger pointwise-monotone special case does not at either, confirming the two are genuinely distinct (open direction O15-O1, partially completed; $q\in{29,61,101}$ remain open). Where it holds, since $\hat\delta_{\mathrm{exact}} < 5.0 \ll 7.4$, this specific, physically motivated reweighting cannot reach the target range $\delta \in [7.4, 10.6]$.

This is narrower than an earlier version of this result, which claimed the bound for any non-negative block weights. That stronger claim is false: weights that concentrate mass asymmetrically across the fitting window can produce $\alpha_{\mathrm{dyn}} > \hat\delta_{\mathrm{exact}}$. The corrected, checkable statement covers the one weighting scheme the growth equation actually motivates — "concentrate weight on the highest-rank blocks" — not every conceivable reweighting.

To avoid mixing distinct objects, the paper introduces a strict hierarchy of symbols:

• (\hat\delta_{\mathrm{exact}}):
  measured exponent from O12/O13 exact Weil-block capacity

• (\alpha_{\mathrm{dyn}}):
  exponent of the block-weighted aggregation observable

• (\delta_{\mathrm{eff}}):
  legacy endpoint diagnostic in C1

• (\sigma(q)):
  additional denominator displacement in the legacy C2 diagnostic

This hierarchy is central to the paper: the whole point of O15 is precisely that these quantities must not be conflated.

The paper proves the exact identity $\Delta r_n=|S_n|\Sigma_n$ and audits the attempted transplantation of the LPS growth equation.

What is rigorously established:

• the scalar O6 law does not transfer
• no native pair-capacity growth carrier is defined
• the natural dynamical reweighting satisfies an exact identity, $R_n^{\mathrm{eff}}=\bar\Sigma_n\cdot(1+V_n)$, bounding its exponent by $\hat\delta_{\mathrm{exact}}$ whenever $\log(1+V_n)$ has non-negative OLS slope against $\log n$ over the fitting window (in particular whenever 1ドル+V_n$ is non-decreasing there)

What remains conjectural or unverified:

• a new native growth carrier, or an explicit two-substrate hypothesis
• any inter-$q$ estimator and its normalisation
• the coupling of a block observable to relational growth
• whether the OLS-slope condition above holds at the three remaining tested primes (open direction O15-O1, partially completed at $q=151$ and $q=211$) — until checked at each prime, "aggregation alone cannot save the target range" is conditional there, not unconditional

This distinction is made explicit throughout the paper.

O15 is a clarification paper, but a decisive one.

It converts the S2 tension from:

• an unexplained discrepancy
• or a possible numerical artefact
• or a vague structural problem

into a well-posed mathematical programme.

After O15, the remaining issue is no longer:

"Why does the exact exponent disagree with the target?"

but rather:

"Which native Heisenberg growth process, if any, can carry the block-level pair observable?"

That is a much sharper and more calculable question.

O15 follows the chain:

• O12: exact Weil-block extraction
• O13: asymptotic elimination of the finite-size hypothesis
• O14: observable and estimator layers separated
• O15: derivation-level failure localised at the growth equation

So O15 is the paper that establishes:

• the issue is not observational
• the issue is not numerical
• the issue is not statistical
• the issue is dynamic and structural

O15 introduces several key advances:

• a full audit of the O3–O7 derivation chain
• a precise theorem on non-transferability of the scalar proxy derivation
• an exact identity for the natural block-weighted redundancy and a conditional aggregation bound for it
• a strict hierarchy of exponents and correction terms
• an explicit separation between:
  • proved statements
  • structural hypotheses
  • open derivational tasks
• a reformulation of the remaining problem as a calculable next step

The central conceptual outcome is:

• the exact-block mismatch is not due to bad measurement
• it is not due to finite size
• it is not due to central-phase bias
• it is not due to poor averaging across blocks

Instead, the O7 scalar growth law has no established native carrier in the fixed-degree Heisenberg cascade.

This means that the remaining gap must be addressed at the level of:

• the native growth observable
• an equation derived on the same substrate
• and possibly the representation-dependent structure of Weil blocks

O15 does not resolve the S2 tension completely.

What it does resolve is the location and nature of the problem.

The paper establishes that:

• the scalar-to-block passage is the actual failure point
• the block aggregation route cannot restore the target range
• the next step must derive a native carrier on the Heisenberg substrate itself

So O15 transforms S2 from a broad tension into a concrete programme.

The remaining open problems are now sharply separated:

• measure $R_n^{\mathrm{eff}}$ to quantify aggregation
• construct a native pair-capacity growth carrier, or state a new two-substrate hypothesis

Two historical diagnostics are retained for traceability:

The legacy endpoint bookkeeping defines $\delta_{\mathrm{eff}},$ and inserts it into $\beta^* = \frac{1}{\delta_{\mathrm{eff}} + \tfrac12}$ as a phenomenological comparison.

The second diagnostic introduces $\sigma(q),$ chosen to quantify the displacement needed to hit the reference window.

Neither diagnostic is a native Heisenberg growth law.

1. Extraction of (R_n^{\mathrm{eff}}) (O15-O1) — partially completed Quantify the aggregation observable from the existing O12/O13 block data. Done at $q=151$ and $q=211$; still open at $q\in{29,61,101}$.

2. Native growth carrier Construct a Heisenberg process carrying the pair observable

3. Inter-$q$ estimator Define its target before assigning a $q$-normalisation exponent

4. Representation-level growth law
   Clarify how Born–Infeld boundedness translates from relational variables to block-internal representation degrees of freedom

5. Extension beyond the lepton sector
   Investigate whether the same scalar-to-block breakdown appears in quark or neutrino-related observables

The programme is now:

• free of algebraic obstruction (O6)
• free of geometric obstruction (O8–O9)
• free of proxy-level representation obstruction (O10–O11)
• exact at the Weil-block level (O12)
• asymptotically stabilised (O13)
• observable-level mismatch identified (O14)
• growth-equation mismatch localised (O15)

It does not assume:

• that the O7 scalar map extends to exact Weil blocks
• that block averaging can recover the target
• that (\hat\delta_{\mathrm{exact}}) is an exponent entering a native growth equation
• that the remaining tension is merely numerical

O15 is primarily a derivation and audit paper. Every numerical value in Table~1 (the fitted exponents $\hat\delta_{\mathrm{exact}}(q)$) and the inter-block variance ratios $V_n^{\max}(q)$ is drawn directly, unmodified, from O12's and O13's own published, independently reproducible pipelines.

One measurement O15 motivates — extracting the aggregation exponent $\alpha_{\mathrm{dyn}}$ from $R_n^{\mathrm{eff}}$ (open direction O15-O1) — has since been performed at two of the five sampled primes:

• code/o15_o1_nogo_verification.py 151 — reproduces the exact identity check and the OLS-slope condition at $q=151$ (params: m_block=10, n_max=37, bfs_frac=0.29, seed RNG_BASE+151, deterministic). Takes approximately 35 minutes on a standard machine (BFS truncation at this prime is expensive).
• code/o15_o1_nogo_verification.py 211 — same check at $q=211$ (params: m_block=10, n_max=50, bfs_frac=0.11, seed RNG_BASE+211, deterministic). Takes approximately two hours on a standard machine — BFS truncation at this prime is markedly more expensive than at $q=151$, and per-block cost is highly variable (some blocks finish in seconds, others take tens of minutes).
• code/results/o15_o1_q151.json, code/results/o15_o1_q211.json — the exact numeric output of each run, committed for provenance.
• code/spectral_O12.py — the shared O12/O13 pipeline, vendored into this repository so the scripts above are self-contained (this repository is separate from O12's and O13's; a script that only works from a sibling checkout is not reproducible from a fresh clone of this repo alone).

The other three sampled primes ($q\in{29,61,101}$) remain explicit future work, not results claimed by this paper. The script accepts any of the five as an argument; running it at those primes would close the remaining part of O15-O1, but that has not been done here.

paper/
├── out/ # Compiled O15 PDF
├── code/ # O15-O1 verification script (q=151, q=211) and vendored pipeline
├── tex/ # LaTeX sources
└── README.md

If you reference this work, please cite:

J. Beau, Observable-Class Derivation Mismatch and the Heisenberg Transfer No-Go: From Proxy Capacity to Block-Level Span Dynamics, 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, alternative derivations of a native block-level growth carrier, and direct extraction methods for $R_n^{\mathrm{eff}}$ are welcome.

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