AperiodicFugue Thanks for focusing the discussion. Here are direct answers to your four open questions:
1) Probability Conservation (Born rule).
We place states in a right-quaternionic Hilbert space. A consistent probabilistic interpretation is obtained by a time-independent, positive-definite metric G fixed by PT symmetry, with evolution
i ∂_t ψ = H ψ, H† G = G H .
Then the inner product ⟨ψ|φ⟩_G = ψ† G φ gives a positive density j^0 = ψ† G ψ and
d/dt ∫ j^0 d^3x = 0 .
With the PT constraint, G is unique up to unitaries that commute with H; when the extra geometry is trivial (ε = 0), the construction reduces to standard complex QM.
2) Outer disks and local conservation (explicit object + balances + no preferred frame).
The kinematic surplus is encoded as the divergence of an antisymmetric stress built from a local axial 2-form. Concretely,
F_{μν} = ε · ℓ^2 · ε_{μναβ} u^α ω^β ,
ω^β = (1/2) ε^{βγδσ} u_γ ∇_δ u_σ ,
whose spatial dual defines a weak magnetic-like field B_ε^i = (1/2) ε^{ijk} F_{jk} along the disk spin axis. The momentum balance (thin disk, weak field) is
∂_t(ρ v^i) + ∇_j Π^{ij} = ρ a_0^i + ∇_j S^{ij} ,
S^{ij} = ε · ρ · ε^{ijk} B_{ε,k} , S^{ij} = - S^{ji} .
Because the force density is a divergence of an antisymmetric stress, local conservation holds once field stress–energy is included. F_{μν} is built only from g_{μν}, u^μ, and ε_{μναβ}, so no global preferred frame is selected. For a rotating thin disk this yields the linear tail
v^2(r) ≃ v_bar^2(r) + (ε · c · H0) · r ,
with a0 = ε · c · H0. Using H0 ≈ 2.2×10^-18 s^-1 and c ≈ 3×10^8 m/s gives c H0 ≈ 6.6×10^-10 m/s^2; e.g. ε = 0.2 ⇒ a0 ≈ 1.3×10^-10 m/s^2.
3) The single dial: why labs see almost nothing.
A single dimensionless parameter ε (and a coupling λ) controls phase accumulation via a path–curvature functional,
Δφ ∝ λ ∬_Σ R_eff dS .
Labs: short loops and tiny curvature ⇒ Δφ is negligible, evading orientation-odd bounds.
Galaxies: enormous paths over gentle curvature ⇒ the same tiny integrand accumulates to an observable effect. The same dial sets a0 = ε · c · H0 above.
4) The discriminator (conservative choice).
A clean “smoking gun” is frequency-squared dispersion of GW speed across PTA/LISA/LVK:
c_T^2(f) = 1 + ξ (2π f / Λ)^2 + O(f^4) .
A joint, multi-band fit that drives this f^2 slope below the minimally viable range (given ξ, Λ, ε) rules out the framework. This respects c_T(0) = c from multi-messenger events.
This is my complete response on the technical core. Thanks again for the precise questions.
