Apoth3osis
>_PADIC.DECOUPLING

P-adic Functional Decoupling

Formalization of Rowan Brad Quni-Gudzinas' p-adic Hamiltonian framework inside HeytingLean. The key extension is not just the physics: it is the proof that finite-depth p-adic visibility behaves like a genuine nucleus and therefore plugs directly into the existing Hossenfelder boundary-gap machinery.

>_VERIFICATION.SEAL
FORMALLY VERIFIED • LEAN 4 • MACHINE-CHECKED • APOTH3OSIS∀∃⊢→∧∨¬≡QED28 theorems • 0 sorry11 modules0 SORRY

Formal Verification Certificate

Every theorem in this project has been machine-checked by the Lean 4 kernel. No axiom is assumed without proof. No gap exists in the verification chain.

28 THEOREMS VERIFIED11 MODULES0 SORRY

P-adic Functional Decoupling • Lean 4 + Mathlib • Apoth3osis Labs

Core mathematical chain

Htotal=HW+HS+HintH_{\mathrm{total}} = H_W + H_S + H_{\mathrm{int}}
causalBall⁡(c,r)={x∣∥x−c∥p≤r}\operatorname{causalBall}(c,r) = \{x \mid \|x-c\|_p \le r\}
jk(S)=S∩roundedSkeleton⁡(p,k)j_k(S) = S \cap \operatorname{roundedSkeleton}(p,k)

The observational nucleus lives on sets of p-adic integers, not on an invented placeholder lattice. This keeps the paper-level intuition tied to actual Mathlib p-adic data and existing Heyting nucleus infrastructure.

Artifact surface

Paper → Proof → Code pageC artifactRust artifactLean nucleus artifact

Phase 1–3: Ultrametric Mechanics

  • • Mathlib p-adic norm imported directly; no fake arithmetic layer
  • • Causal balls and isosceles law formalized over the nonarchimedean norm
  • • Random walk boundedness proved by induction through the ultrametric inequality

Phase 4: Depth Truncation Nucleus

  • • Observable state space chosen as OrderDual (Set Z_[p])
  • • Depth restriction promoted through Prenucleus.toCoreNucleus
  • • Finite fixed points and nonzero finite-depth gap proved on the visible skeleton

Phase 5–6: Boundary + Runtime Bridge

  • • BoundaryNucleus and boundaryGap reused from the existing Hossenfelder formalization
  • • Miranda bridge states the shared nucleus/fixed-point pattern without inventing extra geometry
  • • PPC route exposes Lean, C, and Rust artifacts as downloadable runtime surfaces

Lean modules

Open PPC page
Padic.Valuation
Padic.Ultrametric
Padic.UltrametricLightCone
Padic.RandomWalk
Hamiltonian.Tripartite
Hamiltonian.EnergyConservation
Nucleus.PadicRounding
Nucleus.FixedPoints
Nucleus.GapAtDepth
Bridge.HossenfelderConnection
Bridge.MirandaConnection