Pufferfish Theory: Introduction To Finite-Vantage Geometry

Pufferfish Theory (PFT): finite-vantage geometry from a single generator. Pufferfish Theory develops a geometry of finite resolution. Dynamics, gravity, force, and time arise as downstream receipts of finite depth, one-pass anchoring, and a single coherent generator. No extra fields, counterterms, or collapse postulates are introduced. The theory separates an underlying coherent run from the readout imposed by a finite vantage. All structure is generated by one positive operator |D| = √(−Δ). Along the real axis it produces stabilization (Poisson condensation); along the imaginary axis it produces coherent phase transport. Depth and time are orthogonal directions of the same analytic flow. Core mechanism (run → readout):A raw presence density ρ is first stabilized at finite depth σ by the Poisson semigroupP(σ) = exp(−σ|D|),producing a smooth Sea uσ = P(σ)ρ.Readout is local and adaptive. A state-dependent Mirror extracts a conditional baselineMσ f(x) = P(σ)(ρ f)(x) / P(σ)ρ(x),followed by a one-pass Anchor Πσ = I − Mσ. Contrast is Cσ = Πσ[uσ]. Anchoring is applied exactly once per update and never feeds back into sources. The Mirror is a conditional expectation drawn from the Sea itself. It is nonlinear in the state, adaptive, and unavoidable for any finite-resolution vantage. All geometric effects originate here. Wick twist and motion:The same generator defines an analytic family E(z) = exp(−z|D|). Along z = it this yields unitary phase flow F(t) = exp(−it|D|). Motion alone is reversible; direction appears only when coherent transport is interleaved with one-pass re-anchoring. Long wavelengths recover Schrödinger-type envelopes, while the exact square-root symbol enforces finite-depth dispersion and cone-limited propagation. Curvature from failure to close:Because anchoring adapts to the evolving Sea, stabilization and motion do not commute after readout. Small loops in (σ, t) fail to close at order O(δσ δt). This holonomy defines a curvature two-form whose macroscopic trace appears as force-like drift and an intrinsic arrow of time. No force is postulated; it is the residue of adaptive re-anchoring. Gravity lane (lift before look):Sources are stabilized before lifting. Potentials satisfyΔΦ = 4πG uσ,with uσ = P(σ)ρ. A blurred point mass yields a finite arctan core with an unmodified Newtonian 1/r tail. Singular collapse never arises at finite depth; long-range behavior is preserved. Semantic extension:Replacing |D| with √(−Δ_LB) on a Fisher–Rao manifold yields a parallel lane for inference. Coherent transport of distinctions produces semantic holonomy governed by the same finite-vantage geometry.