Overview
O28 reported an effective covariance rank $r_{\mathrm{eff}} = 3$ in the supplied three-coordinate space $H_{\mathrm{eff}}$. O29 asks what that finite-data result actually identifies.
In an extended recomputation, rank three remains modal but is not invariant: it occurs in $32/50$ stored samples at $q = 101$ and $103/105$ at $q = 211$.
The audit turns on separating three spaces that had been read as one. The measured trajectory span lies in $H_{\mathrm{eff}}$; its leading two-dimensional PCA truncation $U_c^{(2)}$ is data-dependent; and the spin-½ module $V_\rho$ is supplied elsewhere. No calculation in this paper identifies these objects with one another.
Core contributions
Three results stand, each carrying its own epistemic label.
- Numerical. At the declared tolerance and checkpoints, the Veronese evaluation matrix has numerical rank three, the fitted constraint commutators have numerical rank three in the chosen $\mathfrak{so}(3)$ basis, and the computed commutant is one-dimensional. Combined with the carrier supplied conditionally by O23 and the factorisation statement of O27, these observations are compatible with the adjoint spin-1 carrier $H_{\mathrm{eff}} \simeq \mathfrak{su}(2)$; they do not identify it from the data.
- Mixed evidence. Under the stated parity identity the conjugate-pair outer products are symmetric. The tested finite samples do not reach rank 4, and the locking-broken control recovers four operator directions only inside the auxiliary space $U_c^{(2)}$.
- Conditional. No subspace of $H_{\mathrm{eff}}$ is identified here with the spin-½ target $V_\rho$. Under the supplied actions it is not reachable equivariantly from the adjoint, since $\operatorname{Hom}_{\mathfrak{su}(2)}(\mathbf{3}, \mathbf{2}) = 0$. Its selection rests on the O26 minimality argument, pending an independent observable.
Interpretation
The collapse diagnoses geometry within measured coordinates. It does not select a physical representation. A genuine spin-½ test still requires an independent observable and a typed bridge to $V_\rho$.
The distinction matters beyond this paper because it is the programme's recurrent failure mode: not an incorrect calculation, but an unproved identification between a measured quantity and a representation chosen elsewhere. Here the calculation stands and the identification is withdrawn.
Relation to the Cosmochrony program
O25 supplies the checkpoint trajectories, O28 the covariance measurement, O26 the minimality criterion, O23 the conditional spinor carrier, and O27 the separation between factorisation through the admissibility quotient and equivariance, together with the open vector lift.
O29 revises the O26 criterion. That criterion assumed the conjugate-pair blocks were independently distributed in $V_\rho$; for conjugate-pair data the blocks are anti-linearly related, so the target-space prediction is replaced by a carrier test on $H_{\mathrm{eff}}$. The identification chain from measured covariance to a fundamental doublet therefore remains open, and closing it is a task for an independent observable rather than for more block statistics.
Reproduction
The analysis code, its declared seeds and thresholds, and the SHA-256 sums of the checkpoint archives are committed in the paper's repository. The raw archives themselves are too large to vendor and do not yet have a public download location, so a reader who does not already hold them can verify the code, seeds and hashes but cannot regenerate the numbers from a fresh clone. Publishing that data record is a separate open task of the sub-programme.