Toward Three Spatial Directions from a Supplied Spinor Carrier: Requirements for an Equivariant Bridge

Abstract

Two sub-programmes of the Cosmochrony framework produce the integer three as a structural output, and this paper states what an equivariant bridge between them would require. On the geometric side, paper Q5b derives an effective Lorentzian co-metric on $\mathbb{R}_\tau \times {\mathrm{Heis}_3}(\mathbb{R})$ whose spatial sector has three directions, the two horizontal generators $\tilde X, \tilde Y$ and the central element $\tilde Z = [\tilde X, \tilde Y]$; its extraction is conditional on the unestablished spatial limit hypothesis [H-L], its central slot is degenerate at principal-symbol order, and the Lorentzian signature carries a further hypothesis. On the representation side, paper O23 proves that the neutral sector of a supplied spinor carrier $V_\rho \cong \mathbb{C}^2$ is ${\mathfrak{su}}(2) \simeq {\mathrm{Im}\,\mathbb{H}}$, of real dimension three, and paper O28 works with an admissible projection space ${H_{\mathrm{eff}}} = \mathbb{C}^3$, supplied as a parameter consistent with that selection rule, whose per-pair covariance has rank three as a finite-data value of its protocol and not as an invariant. The identification of that measured space with a spin-$1$ module of ${\mathfrak{su}}(2)$ is supplied, not derived: O29 identifies no such structure, and $d_\rho = 2$ is O26's minimality selection. Three requirements are proved here. First, the bridge cannot be a Lie algebra isomorphism: ${\mathfrak{su}}(2)$ is semisimple, ${\mathfrak{heis}_3}$ is nilpotent. Second, under the identification hypothesis, ${\mathrm{Sym}}^2({V_\rho})$ is, up to isomorphism, the unique irreducible three-dimensional complex module of ${\mathfrak{su}}(2)$, so any non-zero equivariant map to the complexified adjoint module is an isomorphism, unique up to one non-zero complex scalar; choosing it real and positive additionally requires a phase convention. Third, the ${\mathfrak{su}}(2)$ Casimir acts on ${\mathrm{Sym}}^2({V_\rho})$ as $2 \cdot \mathrm{Id}$, and a non-zero equivariant bridge pulls the Hermitian spatial form back to a positive-definite form of rank three whenever the two coefficients are positive; invariance under the Cartan generator $J_3$ separates the spin-weight $\pm 1$ sector from the spin-weight $0$ sector and gives horizontal isotropy when compatible with the real rotation structure, but does not force $A_H = A_z$. Version 2.0 re-types the numerical section. The quantities previously reported as symbol coefficients are compressions of the discrete Weil Laplacian $L_{\mathrm{Weil}} = 4I - W_a - W_a^\dagger - W_b - W_b^\dagger$ onto the three stored basis rows, which are pure Fourier modes: the central value is the flat-mode Rayleigh quotient $4 - 2 - 0 = 2$, exactly and for every tested pair, and the horizontal value is $4 - 2\cos(2\pi k/q)$ for the mode index $k$ selected by the pipeline. Consequently the identification of the central value with the Casimir eigenvalue and the fitted convergence of the isotropy gap are no longer asserted: for these modes $A_H - 2 = 2(1 - \cos(2\pi k/q))$, so vanishing of the gap is equivalent to $\mathrm{dist}(k/q, \mathbb{Z}) \to 0$, a property of the mode-selection rule that remains to be established. The broader bridge to an extended target remains a missing construction. The existing Q5b spatial principal symbol has rank two and cannot equal the rank-three Casimir form; the revised conjecture requires a new target. Separately, under Q5a's depth and weight hypotheses, no common scalar normalisation of its published admissibility form yields a toric differential limit.