Fundamental Fisher: EPI supports sPNP

sPaceNPilottime (sPNP) takes a lot of motivation from Roy Frieden’s EPI. In EPI, Fisher Information can produce physics. MFI (J- I = max) and the EPI Variational Principle could explain some fundamental aspects of the universe. I have had difficulty turning EPI into a Unified Theory and I think an explanation for phase S could help a lot. sPNP is very similar to EPI, but sPNP shows how a Unified Theory can be derived from the Jacobi-Fisher metric being reflexive with the wavefunction. The Fisher is naturally extremal in EPI, but below I also derived a variational principle of the Jacobi, which supports the phase and accounts for the dynamics, from EPI. The Jacobi-Fisher metric is Jacobi dynamics and Fisher Curvature.

The EPI variational principle can be seen as an explanatory bridge: it shows why a wavefunction of the form ψ = R e^{iS/ℏ} arises from a purely information-theoretic extremum, with Fisher curvature encoding fluctuations and the Jacobi term restoring transport through phase. In that sense, EPI motivates the existence of the Jacobi–Fisher metric and the dual structure (ρ,S). But once this structure is recognized, sPaceNPilottime no longer requires EPI as scaffolding. Just as string theory or loop quantum gravity posit their fundamental geometry directly, sPNP can take the Jacobi–Fisher geometry as given and derive full dynamics from it; EPI then functions less as a necessity and more as a philosophical justification for why such a structure is natural. We will explore this turtle here by building the variational principle for sPNP with phase S included.

EPI builds dynamics solely from amplitude curvature, so it captures internal fluctuations but not coherent transport. In contrast, sPNP treats (ρ,S), or (R = amplitude, S) as a coupled pair: the Fisher curvature of ρ yields the quantum potential (fluctuations and distinctions), while the gradient of S supplies the guiding momentum (transport). Ruggeri’s reasoning supports this dual structure in sPNP: the amplitude statistics alone look entropy-free for free particles, but the phase restores the full physics. sPNP therefore resolves the gap in EPI by unifying Fisher information with phase dynamics in a single geometric action. If I want to add phase here are some options.

Option A — Add S as a conjugate field Start with the EPI functional: Φ[p]=J[p]−αI[p]. Introduce a phase field S and append the continuity/action constraint: Φup[p,S]=J[p]−αI[p]+∫dt∫dxp(∂tS+2m(∇S)2+V). Extremize Φup over both p and S. With α=ℏ2/(8m) (or chosen to match units), this yields the Madelung system: amplitude dynamics with Fisher-curvature quantum potential + S-driven phase evolution.

Option B — Enrich the source functional J Split J into amplitude and phase parts: J=JR+JS, where JS encodes background information about momenta/velocities (i.e. phase). Then the variational principle δ(J−I)[p,S]=0 on the enlarged space (p,S) yields both amplitude and phase equations without imposing an extra conjugate term.

• Option A: Surgical fix — keep Frieden’s I and add an explicit phase/kinetic term.

• Option B: Conceptual fix — admit that J must already encode both amplitude and phase information.

I took some direction from this paper: Frieden, Roy. Spontaneous Formation of Universes from Vacuum via Information-induced Holograms September 2019 DOI:10.48550/arXiv.1909.11435

Hybrid variational derivation (EPI + Jacobi) — compact version

Notation & setup (single particle, nonrelativistic). Let ρ(x,t) ≥ 0 be the probability density and R = √ρ (so ρ = R^2). Let S(x,t) be Hamilton’s principal function (phase/action). Define the wavefunction ψ = R e^{iS/ℏ}. Fisher information (in amplitude form) I[ρ] = ∫ d^3x (∇ρ)^2 / ρ = 4 ∫ d^3x (∇R)^2. We define the hybrid functional (EPI + Jacobi piece + optional background J): Φ[ρ,S] = J[ρ,S] − α I[ρ] + ∫ dt ∫ d^3x ρ( ∂_t S + (∇S)^2/(2m) + V ). Here α>0 is chosen to recover the standard quantum potential; later we set α = ℏ^2/(8m). (The sign “−α I” is chosen so variation produces the standard form of the quantum potential.) J[ρ,S] is Frieden’s background/source functional (can impose normalization, global constraints, etc.). In what follows we treat J as inert (or weakly ρ-dependent) so the variational derivatives come from the Fisher piece and the Jacobi/action piece.

1. Variation w.r.t. S → continuity equation Only the last integral depends explicitly on S. Varying S gives δ_S Φ = ∫ dt ∫ d^3x δS ( ∂_t ρ + ∇·( ρ ∇S / m ) ). Stationarity for arbitrary δS yields the continuity (mass conservation) equation ∂_t ρ + ∇·( ρ (∇S)/m ) = 0. (C) So v = ∇S/m is the velocity field for the probability flow.

2. Variation w.r.t. R (or ρ) → quantum Hamilton–Jacobi Write the action/Jacobi term as A[ρ,S] = ∫ dt ∫ d^3x ρ( ∂_t S + (∇S)^2/(2m) + V ). Varying A with respect to R (using ρ = R^2) yields δ_R A = ∫ dt ∫ d^3x 2 R ( ∂_t S + (∇S)^2/(2m) + V ) δR. Now vary the Fisher piece. With I = 4 ∫ (∇R)^2 we have δ I = 4 δ ∫ (∇R)^2 = 4 ⋅ ( −2 ) ∫ δR ∇^2 R = −8 ∫ δR ∇^2 R. Therefore δ( − α I ) = −α ⋅ ( −8 ) ∫ δR ∇^2 R = 8 α ∫ δR ∇^2 R. Combine variations (ignoring δR J for simplicity): δ_R Φ = ∫ dt ∫ d^3x [ 2 R ( ∂_t S + (∇S)^2/(2m) + V ) + 8 α ∇^2 R ] δR = 0. Divide by 2R to get the quantum Hamilton–Jacobi-like equation: ∂_t S + (∇S)^2/(2m) + V + (4 α / R) ∇^2 R = 0. (HJ)

3. Fix α to recover the standard quantum potential The usual Madelung quantum potential is Q = − (ℏ^2 / 2m) ⋅ ( ∇^2 R / R ). Comparing (HJ) we need (4 α / R) ∇^2 R = Q term, i.e.: 4 α ∇^2 R / R = − (ℏ^2 / 2m) ∇^2 R / R. Hence choose α = ℏ^2 / (8 m). (With this choice and the sign convention Φ contains “− α I”, (HJ) becomes the standard quantum Hamilton–Jacobi equation with Q = −(ℏ^2/2m)(∇^2 R/R).) So with α = ℏ^2/(8m) the Euler–Lagrange equations from Φ give: • Continuity (C): ∂_t ρ + ∇·(ρ∇S/m) = 0. • Quantum Hamilton–Jacobi (HJ): ∂_t S + (∇S)^2/(2m) + V + Q = 0, with Q = −(ℏ^2/2m)(∇^2 R / R).

4. Reconstruct the Schrödinger equation Define ψ = R e^{iS/ℏ}. One checks that (C) and (HJ) are algebraically equivalent to the linear Schrödinger equation i ℏ ∂_t ψ = − (ℏ^2 / 2m) ∇^2 ψ + V ψ. Thus the hybrid variational principle Φ (EPI + Jacobi term) yields the full Madelung pair and hence Schrödinger dynamics.

5. Where the Jacobi–Fisher geometry appears • The Jacobi term ∫ ρ (∇S)^2/(2m) d^3x is the kinetic/Hamilton–Jacobi piece: classically, Jacobi’s principle rewrites dynamics as geodesic motion with the Jacobi metric ds_J^2 ∝ 2m(E − V) g_{ij}. Hamilton’s principal function S generates geodesic flow (momentum = ∇S). • The Fisher information term I[ρ] = 4∫(∇R)^2 gives a statistical metric (the Fisher metric) on the space of distributions. For distribution families parameterized by θ^i, • g_{ij}^{(F)}(θ) = ∫ dx (1/ρ(x;θ)) ∂{i}ρ(x;θ) ∂{j}ρ(x;θ). • The hybrid functional couples the two: the Jacobi conformal factor (E − V) multiplies the local inner product supplied by the Fisher metric, giving a Jacobi–Fisher metric on configuration-space parameter manifold: • g_{ij}^{(JF)}(θ) ∼ (E − V(θ)) ⋅ g_{ij}^{(F)}(θ). Intuitively: the Jacobi piece enforces action-driven geodesic flow; the Fisher piece endows the space with curvature via amplitude curvature (which becomes the quantum potential). Together they pick a geometry whose geodesics are guided by S and whose curvature is controlled by Fisher information, i.e. the Jacobi–Fisher geometry used in sPNP.

6. Short interpretation / role of J J[ρ,S] (Frieden’s “eternal J”) acts as global/background constraint: normalization, priors, conserved background FI, boundary data, or scale setting. Demanding stationarity of the combined J − I + Jacobi functional selects the physical solution sector (the extremal wavefunction) with Jacobi–Fisher geometry.

7. Summary Start with the hybrid functional Φ = J − α I + ∫ ρ(∂_t S + (∇S)^2/(2m) + V). Varying S yields continuity; varying R yields the Hamilton–Jacobi equation with an amplitude-derived term proportional to ∇^2 R/R. Choosing α = ℏ^2/(8m) produces the standard quantum potential Q = −(ℏ^2/2m) ∇^2 R/R. The pair (continuity, quantum HJ) is algebraically equivalent to the Schrödinger equation for ψ = R e^{iS/ℏ}. Geometrically, the Jacobi kinetic piece enforces geodesic/action motion while the Fisher term provides curvature, together defining a Jacobi–Fisher geometry on configuration space, which is the desired structure for sPNP.

This Turtle shows for natural sPNP is in the lens of EPI. It shows a compelling explanation for the fundamental nature of Fisher Information, but that does not mean this is how the universe is created. The turtles could continue to go up and down or there may even be diagonal turtles. I’ve also explored a possible turtle from Likelihood Operators and the direct link to Fisher Information: LO curvature → Fisher curvature a clean projection (the symmetric expectation of the LO two-form → QFIM; imaginary part → Uhlmann curvature). Precise conditions (pure states; faithful ρ; Gaussian families). LO is noncommutative, QFI is commuting, Riemannian. arxiv.org/pdf/2502.20055