The UV seed is Fisher geometry on relational configuration space. The wavefunctional is
Ψ = R e^(iS/ħ), ρ = R²
and the Fisher metric is built from ρ, so the microscopic geometry is Riemannian, not Lorentzian.
Lemma 1 — UV positivity
The Fisher core is positive semidefinite because it is built from squared score functions:
G^core_IJ[R] = (∂_I R ∂_J R) / R²
So for any tangent vector v^I,
v^I G^core_IJ v^J ≥ 0.
With the regularizing mass term and admissibility conditions, the full metric is strictly positive on the allowed class:
G_IJ[R] ≻ 0.
So the UV theory starts from a genuine Riemannian geometry.
Lemma 2 — phase/current separation
The Fisher metric depends on R, not directly on S. The phase enters through the current sector,
J^I = ρ G^IJ ∂_J S
so the phase defines a directed flow, while the Fisher metric defines distinguishability. This is the first place where a preferred relational direction can appear, but it is still not yet a Lorentzian sign flip. The sign enters when the clock sector is constrained in Lemma 3.
Lemma 3 — clock reduction gives the minus sign
The relational constraint has the form
𝒞 = ½(h^ab P_a P_b + U − P_τ²) + O(ε) = 0
so the clock momentum P_τ already enters with opposite sign. That is the key point: the minus sign is in the constrained relational dynamics before any metric reduction happens.
The reduced effective metric then takes the form
g^(4)_μν(x) = −N²(x) ∂_μτ ∂_ντ + h_μν(x)
So the timelike direction comes from the clock/phase sector, not from Fisher distinguishability. The clock direction is unique because τ is scalar, so the negative direction is rank-1.
Lemma 4 — Euclidean branches are not stable fixed points
A purely positive-definite emergent metric is not a stable attractor of the coupled flow. Under the relational kernel, a Euclidean seed makes the clock-space coupling B collapse toward zero, so the Schur complement degenerates back to the positive spatial block. That means the iterate stays Riemannian rather than crossing into the Lorentzian basin.
So the Euclidean branch is not a competing stable fixed point. The Lorentzian branch is the stable infrared attractor.
The fixed-point packaging
All of this can be summarized by the composed map
T[g; Ψ] = ℛ_clock ∘ 𝒫^(G[R])_{Q₀[Ψ]}(g)
where:
𝒫^(G)_{Q₀} is Fisher-kernel coarse-graining,
ℛ_clock is clock-constraint reduction.
Then the theorem is:
T has a stable fixed point g*
g* is Lorentzian
the Lorentzian branch is universal and selected by the flow
Euclidean fixed points are not stable in the admissible class
If 𝒫^(G[R])₍Q₀₎ strictly contracts all Fisher modes orthogonal to ∇S (i.e. modes v satisfying Gᵢⱼ[R] vⁱ ∂ʲS = 0), and ℛ_clock maps the unique non-contracted monotone mode along ∇S into the timelike sector via the indefinite clock constraint, then 𝒯 = ℛ_clock ∘ 𝒫^(G[R])₍Q₀₎ has a Lorentzian infrared fixed point. Equivalently: spatial directions are the Fisher modes Gaussoherence contracts; the timelike direction is the unique mode it cannot contract.
The contraction hypothesis has a spectral formulation: ∇S defines the non-contracting direction of 𝒫^(G[R])₍Q₀₎, corresponding to a conserved-current zero-mode sector of the Fisher heat kernel under flows that preserve the continuity equation, while all transverse modes have eigenvalue strictly less than 1. The spectral gap between these sectors is computable from the heat kernel of Δ_G[R] and gives the contraction condition an explicit, checkable form.
In summary:
Fisher geometry gives the Riemannian UV seed. The phase/current sector gives the first distinguished flow direction. Clock reduction supplies the minus sign. The coupled map T makes Lorentzian signature the IR attractor. Lorentzian signature is selected by a dynamical, spectral mechanism.
