Distinctions = Curvature
In sPaceNPilottime (sPNP), all physics, quantum and gravitational, arises from one unified geometry: the curvature of the universe’s configuration‑space. Every distinction injects curvature, and particles simply follow its geodesics.
- Configuration‑Space & the Projection Kernel
Shape space (dim 3N–6) holds all relational configurations X. Naturally accomodating a Jacobi geodesic, work on the reduced 3N–6 relational manifold, obtained by quotienting out global translations and rotations, though Fisher integrals can run over the full configuration density ρ(X). Push forward the 3N–6 measure into the full 3N density via symmetry reduction Jacobians. Jacobi time arises naturally via the energy constraint E = T + V fixing the parameterization along geodesics in configuration space.
The projection kernel 𝐾(𝐗,𝐱) is the covariant, RG-stable point-spread function that maps configuration-space distinctions into local spacetime fields. To preserve functional Lorentz covariance, I define 𝐾 from invariant clock and spatial landmarks constructed within configuration space, enforce causal support via a light-cone–peaked transverse Gaussian (equivalently a Euclidean heat kernel analytically continued), and select an embedding 𝐗 ↦ 𝐱 by the expectation/variational pullback. With the quotient Jacobian included, smearing the Fisher–Rao curvature by 𝐾 reproduces an emergent metric g_μν, whose semiclassical Einstein equations follow from the same covariant action that defines the configuration-space geodesic flow; conservation follows from diffeomorphism invariance. The specific Einstein form follows because the emergent spacetime dynamics are obtained by varying a single covariant action that couples the configuration-space geometry (sourced via the smeared Fisher curvature) to the spacetime metric; diffeomorphism invariance of this action yields the Bianchi identity and the symmetric divergence-free structure of the left-hand side, producing the Einstein tensor structure upon identification κ = 8πG_N. The unique minimal covariant action consistent with the Fisher metric and diffeomorphism invariance is the Einstein–Hilbert form, so higher-curvature terms are suppressed as Q₀² corrections.
Distinction: any small shift δX changes ρ(X) ⇒ informational slope.
Projection kernel (K(X,x)) glues distinctions in shape‑space into emergent spacetime:
K(X,x) ∝ exp[ – ( (x⁰ − t(X))² − |x − X|² )⊥ / (4 Q₀²) ] · Θ((x⁰ − t(X))² − |x − X|²) · [1 + O(Q₀² R_F)]
– D^2 = Minkowski distance
– Q0 sets coarse‑graining: bigger Q0 ⇒ smoother distinctions
To enforce causal support adopt a light-cone-peaked transverse Gaussian: K(X,x) is peaked on the causal cone (x0−t(X))2≥∣x−X∣2 and damps transverse separations with a Gaussian of width Q₀. This avoids the Minkowski-Gaussian growth issue while preserving Lorentz covariance of the kernel’s argument.
D^2 = (x^0 - t(X))^2 - |x - X|^2 is still used, but the Gaussian only damps the transverse directions to this null surface.
Origins of 𝑄0
Renormalization‑Group Origin: Integrating out high‑frequency (short‑wavelength) modes in the path integral naturally induces a Gaussian regulator of width Q_0 ~ Λ⁻¹ where Λ is the UV cutoff. Both routes converge to the same Gaussian form with width Q₀, showing it is not an arbitrary parameter but the natural coarse-graining scale.
Coherent‑State Derivation: Inserting a resolution of the identity in overcomplete coherent states yields exactly K(X,x) ∝ exp[– D²(X,x) / (4 Q₀²)] as the smearing kernel, with 𝑄0 set by the coherent‑state width.
Where X→(t(X),xᵢ(X)) carries the emergent Jacobi time coordinate. Here t(X) is the Jacobi time from the energy constraint E = T + V, and xᵢ(X) are relational landmarks. t(X) denotes an internally constructed scalar clock functional on configuration space. For example, the phase of a heavy semiclassical subsystem chosen so that t(X) transforms covariantly under the emergent Lorentz transformations and does not introduce an external foliation.
g_IJ = ∫ dX ρ(X) ∂_I ln ρ(X) ∂_J ln ρ(X)
After smearing with Q₀, one can equivalently rewrite it in a “local” form plus controlled corrections O(Q₀²).
- The Full Fisher–Rao Metric: Encoding Distinctions
Here we appeal to the full, non‑degenerate Fisher–Rao metric (not its rank‑1 shorthand), ensuring that all subsequent connections, Riemann tensors, and quantum potentials are well‑defined.*
The Fisher metric measures how sharply the probability density ρ changes when you move in configuration space. The core of the metric is sensitive to the gradient of the wavefunction’s amplitude: g_IJ(X) ∝ (∂_I R)(∂_J R) / R²
Distinction = nonzero derivative ∂_I ln ρ
No distinction (∂ ln ρ = 0) ⇒ metric flat in that direction
*The full, canonical sPNP metric is a three‑component object that also includes a classical mass term and a Hessian term (∂_I ∂_J R / R), which is crucial for phenomena like quantum tunneling. See sPNP appendix for the complete definition.
- From Metric Variation to Curvature
Christoffel symbols Γ^K_IJ arise from variations in g_IJ.
Riemann tensor R^I_{ JKL} arises from variations in Γ.
Distinctions ↔ ∂ ln ρ ↔ ∂ g ↔ curvature
No distinctions ⇒ g constant ⇒ Γ=0 ⇒ R=0. Since ρ(X) is never perfectly uniform in a real universe, distinctions always exist. The Fisher metric g_IJ(X) is therefore generically curved: even small deviations in R(X) introduce nonzero ∂₍I₎ ln ρ, making g non-flat.
Any non‑uniformity in ρ injects nonzero curvature.
- Wavefunction, Distinctions & Quantum Potential
Write the global wavefunctional:
Ψ(X) = R(X) · exp[i S(X)/ħ]
Amplitude distinctions (R(X)) build g_IJ via ∂_I R.
Phase distinctions (S(X)) define Bohmian flow v^I = (1/m) ∂^I S.
Quantum potential is the Laplace–Beltrami on R:
Δ_g R = (1/√det g)·∂_I[√det g · g^IJ · ∂_J R] Q(X) = –(ħ^2/2m) · [Δ_g R / R]
In the regime Q₀→0 the Laplace–Beltrami reduces to the flat ∇², recovering the standard Q; finite‑Q₀ corrections are higher‑curvature terms. Provided g_IJ→δ_IJ in that limit, which holds for sufficiently smooth R(X).
Distinction: any curvature in g (from R) is the quantum force.
- Quantum Clumping of Distinctions: Micro & Macroscopic Gravity
Microscopic “Gravity”: Spin‑Distinctions as Curvature
Spin flip of one electron: two nearby states, spin‑up vs. spin‑tilted by dφ, define a tiny “hill” in information‑space.
Fisher distance for spin: ds^2 = dφ^2 (Bloch sphere radius 1 ⇒ constant curvature)
That nonzero curvature at the spin‑level acts like a micro‑gravity tug on the electron’s trajectory, guiding even single‑particle behavior (interference, tunneling).
Microscopic: each spin flip, scattering, energy jump is a local distinction δρ → tiny curvature δR (“micro‑gravity”). In simple spin‑½ interferometry one can compute that this curvature reproduces the standard fringe spacing phase condition ΔS = 2π n ħ which governs observable interference patterns.
Spin‑Distinctions on the Bloch Sphere: The full spin‑½ state manifold is a 2‑sphere with metric ds² = dθ² + sin²θ dφ² so that small rotations at fixed θ have ds ≈ sinθ dφ which for θ = π/2 (equator) reduces to ds = dφ.
The Bloch sphere curvature is a concrete illustration of the general rule: distinctions between states naturally live on curved statistical manifolds, and sPNP takes this as the universal rule, not an isolated feature of spin.
Macroscopic: trillions of distinctions (atoms clumped in a planet) integrate into large curvature (“macro‑gravity”).
Emergence: smear high‑D curvature into 4D with K(X,x):
g_μν(x) ≈ η_μν
- κ · ∫ dμ(X) · ρ(X) · K(X,x) · F_IJ(X) · [E^I_μ · E^J_ν] → G_μν = 8π T_μν (fix κ=8π G_N)
Conservation of ρ under Hamiltonian/Jacobi flow ensures ∂τρ+…=0 pushes forward to ∇^μ T{μν}=0, matching the Bianchi identity ∇^μ G_{μν}=0.
When integrating over reduced shape space, include the symmetry-reduction Jacobian Δ₍gauge₎(X); explicitly the convolution becomes: g_μν(x) = η_μν + κ ∫ dμ(X) Δ₍gauge₎(X) ρ(X) K(X,x) F_{IJ}(X) E^{I}_μ E^{J}_ν.
Select the embedding via the expectation pullback
XI(x) = <X_I>_{K,ρ}, EI_μ(x) = ∂_μ XI^eff(x)
(alternative: a dynamical variational embedding is possible and yields the same leading-order pushforward).
Distinctions at the quantum‑level become spacetime curvature at the classical level.
Summary
Distinctions = variations ∂ ln ρ(X) in configuration‑space
Fisher metric g_IJ encodes those distinctions
Curvature R^I_{ JKL} is nonzero whenever distinctions exist
Laplace–Beltrami on R(X) yields the quantum potential
Projection kernel K projects shape‑space curvature to 4D Einstein curvature
Distinctions = Curvature is the geometric heart of sPNP, where every possible configuration distinction, from single‑spin flips (micro‑gravity) to planetary clumps (macro‑gravity), sculpts the curvature that guides the one actual trajectory. No extra worlds, just a landscape of distinctions. In sPNP, gravity is not added to quantum theory, it is already in the distinctions.
