Overview
The bridge condition as Q7 version 1.3 stated it required two asymptotic isotropy identifications: $A_Z = 2$ (asserted by Q8) and $A_H = 2$ (the subject of Q10). Q7 version 2.0 no longer reads either as settling its bridge conjecture. The second identification concerns the effective quadratic form on the measured admissible space $H_\mathrm{eff}$ of the Heisenberg–Weil representation.
Q10 proves that, under spectral universality [U] (itself proved by U1), the character-independence of the large-$q$ limit forces su(2)-isotropy of the effective quadratic form. The unique su(2)-isotropic quadratic form on the three-dimensional irreducible representation $\mathrm{Sym}^2(V_\rho)$ is the Casimir, which takes the value 2 there; on the two-dimensional spin-$\frac{1}{2}$ carrier it is $\frac{3}{4}$.
Together with Q8 ($A_Z = 2$) and U1 (proof of [U]), Q10 gives $A_H = A_Z = 2$. Q7 version 2.0 does not retain that as closing its bridge condition, and withdraws the numerical support these values rest on; this page is pending revision.
Core contributions
- Character-independence forces isotropy: the effective quadratic form on $H_\mathrm{eff}$ is character-independent in the large-$q$ limit, which under [U] forces su(2)-equivariance.
- Unique su(2)-isotropic form: on the three-dimensional irreducible su(2)-module $\mathrm{Sym}^2(V_\rho)$, the unique su(2)-invariant non-degenerate quadratic form is the Casimir $\langle \cdot, \cdot \rangle_{\mathrm{Cas}}$, with eigenvalue 2 there.
- Assertion $A_H = 2$: combining the su(2)-isotropy argument with the Casimir uniqueness reads the horizontal coefficient as $A_H = 2$, pending revision, matching the vertical coefficient $A_Z = 2$ from Q8.
- Relation to the Q7 bridge condition: with $A_H = A_Z = 2$, the asymptotic isotropy condition stated by Q7 version 1.3 would be satisfied. Q7 version 2.0 does not retain that reading: it records its bridge conjecture as neither confirmed nor refuted, and withdraws the numerical support these values rest on. This page is pending revision in the same correction cascade.
Role in the asymptotic isotropy chain
The Q7 bridge would connect admissibility data to emergent geometric coefficients, and would require as input the equality $A_H = A_Z$. Version 2.0 of Q7 states the requirements such a connection must meet and establishes none of it. The papers Q8 and Q10 supply this input from two directions presented as independent, which Q7 version 2.0 records as circular; the two arguments are:
- Q8: asserts $A_Z = 2$, pending revision, via the central admissibility constraint on the vertical (central) degree of freedom.
- Q10: asserts $A_H = 2$, pending revision, via su(2)-isotropy of the horizontal effective form under universality.
The agreement $A_H = A_Z = 2$ was presented as derived from independent structural arguments. Q7 version 2.0 records that support as circular rather than independent: the scalar was fixed by matching a value itself obtained by inserting the first.
Relation to the Cosmochrony programme
Q10 is a key node in the derivation chain leading from spectral admissibility to the identification of the effective Lorentzian co-metric. Its result feeds into Q11, which asserts the temporal coefficient $A_\tau = 2$ and reads the full Minkowski co-metric $g^{\mu\nu} \propto \eta^{\mu\nu}$ from it. That reading depends on the identification Q7 version 2.0 records as supplied by no source, and is pending revision.
The full chain through which the metric emerges reads: Q5a → Q5b → Q7 → Q8 → Q9 → Q10 → U1 → W1 → H2 → Q11.
References
Jérôme Beau. Asymptotic su(2)-Isotropy of the Effective Quadratic Form and the Identification A_H = 2, 2026. doi:10.5281/zenodo.19880900