>_PPC/PADIC_FUNCTIONAL_DECOUPLING

Paper → Proof → Code

Formalization of Rowan Brad Quni-Gudzinas' p-adic Hamiltonian framework. The Lean lane adds the missing formal bridge: p-adic depth truncation is implemented as a real lattice-theoretic nucleus — not an analogy — and slots directly into the existing Hossenfelder boundary-gap and Miranda periodicity infrastructure.

Core construction: DepthState p = (Set ℤ_[p])ᵒᵈ — the order-dual state lattice where meet/join dynamics operate on observable sets of p-adic integers.

12

Lean Modules

28

Public Theorems

0

Sorry Count

453

Lines

8

C Files

8

Rust Modules

>_VERIFICATION.SEAL

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 VERIFIED12 MODULES0 SORRY

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

Key Mathematics

\|x+y\|_p \le \max(\|x\|_p,\|y\|_p)

Ultrametric inequality — every triangle is isosceles.

j_k : \mathrm{Core.Nucleus}\;\bigl((\mathrm{Set}\;\mathbb{Z}_p)^{od}\bigr)

Depth-k restriction promoted via Prenucleus.toCoreNucleus.

\forall k,\;\exists S,\; \operatorname{boundaryGap}(N_k, S) \ne \varnothing

Hossenfelder bridge: finite depth always leaves hidden collateral.

Downloads

Lean proofs (tar.gz)Verified C (tar.gz)Safe Rust (tar.gz)

Research project page →GitHub repo →

Modules

Click to expand. Switch between Mathematics and Lean 4 views.

p-adic norm inequalities from Mathlib and the truncation witness used throughout the project.

Theorems

padic_norm_sum_le_max

\|q + r\|_p \le \max(\|q\|_p, \|r\|_p)

Ultrametric inequality (nonarchimedean property).

appr_lt_pow

x.\mathrm{appr}(k) < p^k

Mathlib truncation witness: approximation is bounded.

Lean artifact inventory

Valuation.lean Ultrametric.lean UltrametricLightCone.lean RandomWalk.lean Tripartite.lean EnergyConservation.lean PadicRounding.lean FixedPoints.lean GapAtDepth.lean HossenfelderConnection.lean MirandaConnection.lean

C artifacts

padic_valuation.c— p-adic norm + ultrametric inequality ultrametric_lightcone.c— Causal balls + isosceles random_walk.c— Bounded walk position tracking tripartite_hamiltonian.c— Three-body energy struct + conservation energy_conservation.c— TimeStep + step_conserves_energy padic_rounding.c— DepthState + depthRestrict + nucleus fixed_points.c— Fixed-point enumeration at depth k gap_at_depth.c— Witness + boundary gap computation

Rust artifacts

Cargo.toml— Crate manifest src/lib.rs— Module re-exports (8 modules)src/valuation.rs— p-adic norm + ultrametric src/ultrametric_lightcone.rs— Causal ball + isosceles src/random_walk.rs— Walk position + bound check src/tripartite.rs— TripartiteHamiltonian struct src/energy_conservation.rs— TimeStep + conservation src/padic_rounding.rs— DepthState + nucleus src/fixed_points.rs— Fixed-point iterator src/gap_at_depth.rs— Gap witness + monotonicity