Toward Coefficient Isotropy from Character Uniformity: A Stability Lemma and Its Missing Geometric Inputs

An isotropic spatial geometry reconstructed from the admissible fibre of the Cosmochrony programme would need the horizontal and central coefficients of a positive three-dimensional spatial form to agree, and would need their common value. This paper isolates what uniformity of the O-series fingerprint observable across characters can contribute. The observable is a Gram–Schmidt rank increment per breadth-first shell; the hypothesis [U] asks that, on a specified block family and depth window, it stay within a relative distance $\varepsilon$ of a common positive profile. We prove a sharp lemma: two averages of such data with a common depth weighting have a ratio within $2\varepsilon/(1-\varepsilon)$ of one. This becomes a statement about geometric coefficients only through four inputs named and not supplied here: a sector assignment of blocks to spin weights, a linear response law, a positive rank-three target form, and an independent absolute scale. Under them the coefficient ratio obeys the same bound, and the absolute coefficient follows only from the supplied scale. The $\mathfrak{su}(2)$ Casimir acts as 2 on the spin-one module, but invariance fixes an invariant form only up to a positive factor, so the value 2 determines no coefficient. Exact conjugation parity holds for conjugately matched fingerprint data with an identical normalisation, which the published pair campaign, sampling its blocks independently, does not meet. Parity and concentrated pair exponents do not imply [U], and a three-dimensional neutral sector and rank stability constrain none of the amplitudes that [U] bounds. The available compression entries are Rayleigh-quotient identities of the discrete Weil Laplacian, not coefficients; no coefficient value is derived. Interpretive outlook: along this route, relative isotropy would be inherited from the uniformity of the admissible fibre across characters, while the absolute scale of emergent space would remain an independent datum rather than a consequence of representation theory.