>_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.

Lean Modules

Public Theorems

Sorry Count

453

Lines

C Files

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

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>_MENTAT.JOIN

“Once men turned their thinking over to machines in the hope that this would set them free. But that only permitted other men with machines to enslave them.”
Frank Herbert, Dune

A janitor who proves a theorem outranks a tenured professor who publishes noise.

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ONTOLOGICAL ENGINEER8 designations

IDEA

A valid, original framing or conjecture

THEORY

Formal argument with paper-level rigor

APPLICATION

Connecting theory to observable outcomes

CODE

Working software the project depends on

EXPERIMENT

Reproducible research with methodology and data

PROOF

Machine-verified claim checked by a proof assistant

KERNEL

Foundational, load-bearing implementation

BRIDGE

Connecting subsystems or knowledge domains end-to-end

NOETIC ENGINEER8 designations

VISIONARY

Strategic direction & roadmaps

NARRATOR

Writing, documentation & papers

DESIGNER

Visual, UX & information design

EDUCATOR

Teaching, tutorials & workshops

CULTIVATOR

Community, outreach & events

DIPLOMAT

Partnerships, governance & policy

INTERPRETER

Translation, media & accessibility

SENTINEL

Ethics, review & quality assurance

Every accepted contribution receives a MENTAT Contribution Record — cryptographically signed, IPFS-pinned, permanently yours. No committee decides your worth. The type checker does.

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