Overview
An isotropic spatial geometry reconstructed from the admissible fibre would need the horizontal coefficient $A_H$ and the central coefficient $A_Z$ of a positive three-dimensional spatial form to agree, and would need their common value. Q10 isolates what uniformity of the O-series fingerprint observable across characters can contribute to that question.
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. [U] is open.
Main results
- Ratio stability: if data $x_{i,n}$ satisfy $|x_{i,n} - s(n)| \le \varepsilon\,s(n)$ with $s > 0$ and $\varepsilon < 1$, any two averages with a common depth weighting have a ratio within $2\varepsilon/(1-\varepsilon)$ of one. The bound is sharp, and the common depth weighting is essential.
- Conditional isotropy: under [U], a sector assignment of blocks to spin weights [W], a linear response law identifying the coefficient ratio with a ratio of fingerprint averages [R], and a supplied positive rank-three target [T], $A_H/A_Z$ obeys the same bound. An absolute value needs an independent scale [S].
- Invariance fixes no coefficient: invariant Hermitian forms on the spin-one module form one ray; the Casimir eigenvalue $2$ singles out a normalisation, not a coefficient.
- Exact parity, with its conditions: conjugation parity of the observable holds for conjugately matched fingerprint data with an identical normalisation. The O25 campaign samples its paired blocks independently and does not select them to meet this condition.
- No implication: exact parity and identical pair exponents are compatible with the failure of [U] for every $\varepsilon < 1/3$; a three-dimensional neutral sector and rank stability constrain none of the amplitudes that [U] bounds.
- Numbers: the available compression entries $2 + 4\sin^2(\pi k/q)$ are Rayleigh-quotient identities of the discrete Weil Laplacian, not metric coefficients.
Relation to the Cosmochrony programme
Q10 separates two questions. Relative isotropy could be inherited from the uniformity of the admissible fibre across characters, if geometric coefficients respond linearly to fingerprint data. The absolute scale is of a different kind: representation theory leaves it free, and Q8 shows that isotropy between horizontal and central directions selects a length scale. Neither the response law nor the extended spatial operator it presupposes is constructed yet. Q5b records [H-lift] and the coefficient values as open; Q11 version 2.0 shows that spatial $\mathrm{SU}(2)$ invariance leaves the temporal coefficient free as well.
References
Jérôme Beau. Toward Coefficient Isotropy from Character Uniformity: A Stability Lemma and Its Missing Geometric Inputs, 2026. doi:10.5281/zenodo.19880900