Asymptotic su(2)-Isotropy of the Effective Quadratic Form and the Identification A_H = 2

Overview

The Q7 bridge condition requires two asymptotic isotropy identifications: $A_Z = 2$ (established by Q8) and $A_H = 2$ (the subject of Q10). The second identification concerns the effective quadratic form on the horizontal subspace $H_\mathrm{eff}$ 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 a two-dimensional irreducible representation is the Casimir, which takes the value 2.

Together with Q8 ($A_Z = 2$) and U1 (proof of [U]), Q10 closes the asymptotic isotropy condition $A_H = A_Z = 2$ entering the Q7 bridge, enabling the passage from spectral data to the emergent co-metric coefficients.

Status. Structurally motivated, conditional on [U] (proved by U1).

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 a two-dimensional irreducible su(2)-module, the unique su(2)-invariant non-degenerate quadratic form is the Casimir $\langle \cdot, \cdot \rangle_{\mathrm{Cas}}$ with eigenvalue 2.
Identification $A_H = 2$: combining the su(2)-isotropy argument with the Casimir uniqueness identifies the horizontal coefficient as $A_H = 2$, matching the vertical coefficient $A_Z = 2$ from Q8.
Closure of the Q7 bridge condition: with $A_H = A_Z = 2$, the full asymptotic isotropy condition required by Q7 to pass from spectral admissibility data to the emergent metric structure is satisfied.

Role in the asymptotic isotropy chain

The Q7 bridge establishes a formal connection between admissibility data and emergent geometric coefficients, but requires as input the equality $A_H = A_Z$. The papers Q8 and Q10 supply this input from two independent directions:

Q8: identifies $A_Z = 2$ via the central admissibility constraint on the vertical (central) degree of freedom.
Q10: identifies $A_H = 2$ via su(2)-isotropy of the horizontal effective form under universality.

The agreement $A_H = A_Z = 2$ is not postulated but derived from independent structural arguments, providing a consistency check internal to the spectral admissibility programme.

Dependence on [U]. The argument of Q10 is conditional on hypothesis [U], which is proved by U1 with rate $\varepsilon(q) = O(q^{-1/2})$.

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 directly into Q11, which closes the temporal coefficient $A_\tau = 2$ and thereby identifies the full Minkowski co-metric $g^{\mu\nu} \propto \eta^{\mu\nu}$.

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