What question did this study set out to answer?

The aim is to formalize various physics concepts into a structured format using Lean 4 software.

June 21, 2026Open Access

UGP Physics Lean — Physics application layer (Lean 4 formalization)

Key Points

The aim is to formalize various physics concepts into a structured format using Lean 4 software.
Developed 382 modules within the UGP Physics framework in Lean 4.
Utilized machine verification to ensure correctness of derived results in gauge theory and cosmology.
Performed multiple rounds of graduations on various theorems and constructs up to June 2026.
Successfully formalized gauge theory and related concepts into 382 machine-verifiable modules.
Derived specific predictions such as the pion mass at 139.57 MeV and closed all three PMNS mixing angles with given parameters.
Generated numerous additional theorems contributing to theoretical physics, including structural and connectivity insights.

Abstract

Lean 4 software archive for the UGP Physics program: physics application modules formalizing gauge theory, EW structure, color dynamics, cosmology, and related results derived from the Universal Generative Principle. 382 modules total (ugp-lean) — all CatAL results independently machine-verifiable; formalization paper 142 pages. June 2026 completions (Rounds 81–83D): FKTT etaB CatAL (6. 109e-10, +0. 15 sigma) ; pion mass mₚi = 139. 57 MeV CatAL (GOR+condensate, -0. 001% from PDG) ; all three PMNS mixing angles closed CatAD (sin² theta₁2=4/13, sin² theta₂3=19/42, sin theta₁3=11/73) ; mₜau structural prediction vH/ (98*sqrt (2) ) disclosed. Additional CatAL theorems: phimdlₚotentialₛu2lᵢnvariant (SU (2) L weak force, zero named axioms — all four forces now CatAL) ; mdlₜowerbundle (MDL non-circularity at three nested levels) ; particlescomputationₛpacetimeₜrinity (PCT Trinity CatAL) ; gteₚolynomialfiveᵣolesₖₑxtraᵦero (Single-Source Principle named theorem) ; pscₑnumerationforcesₙgen₃ (34, 560-universe PSC Layer I scan, nativedecide — all 12 survivors Ngen=3) ; incompletenessᵢmpliesₙonzeroₒmegaₗambda (PSC→OmegaLambda > 0). Z7InvariantSubsets graduation (10 zero-sorry theorems certifying invariant subset classification for the GTE polynomial over Z₇, Rule 110 uniqueness as maximal proper invariant sub-CA, and QNR Binary Floor). Round 98 graduation (2026-06-09, 8719 build jobs, zero sorry): GTECausalTree. lean (8 theorems — perfectTreeₙumNodes, gteᵣulegteₜengenerations certifying 1023 events at depth 10, perfectTreeₕortonᵣatio, gtecausalₜreeₛummary) ; PolyExplorations. lean (35 theorems — polyₚᵤniformgsᵣoots, period₄75ᵢsₘinimal, fourₒbjectGTEₚairwisedistinct, Tbₒutputₙotdeterminedbyₘod7, polyₚᵥacuumbasincardₑq₅2, and others) ; NgenThreeMechanismsUnified. lean; CUP3DUniqueness additions (gen2gen3ₜopologydegenerate, CatAL). Round 100 graduation (2026-06-09, zero sorry): MDLThreeLevelUnification. lean — P51 OP3 closed: three-level MDL Tower non-circularity machine-certified. Wave-2 graduation (2026-06-11, commit d989f57, 8728 build jobs, zero sorry): TriangleLiftTheorem. lean + TriangleLiftStructural. lean (Direct-Interpolation Lift, sparsity floor, chirality census — ugpₒrbitᵢnterpolationₗift + orbitchiralitycensus, CatAL) ; SpinSevenWallSpectroscopy. lean (14 theorems, exact wall/bump spectroscopy) ; CMCAPhysicalPoint. lean (8 theorems — tape spacing a=hbar*c/LambdaGTE derived, xiₖink=|Z₇|=7, Boundary Equivalence Theorem, criterion proven minimal) ; tapeₛaturationₜheorem (CatAD | CSC) ; ParityProjectionBattery + ParityProjectionForcing (777-form + 16807-recoding censuses) ; GTECompilation trajectory containment theorems. Total: 334 modules. Wave-3 graduation (2026-06-13, commit ef22e9d, 8764 build jobs, zero sorry): PSL (2, 7) algebraic unification suite (f21ᵢsborelₚsl27, pgl27generatedbyₛingerₐndborel, psl27ᵢsₐutfano, f21ᵣegularₒnfanoflags) ; A4/Eisenstein suite (eisensteinₐ4fromᵢnert₂, gteₘanifestflavorᵢsₛ3ᵢnₐ4) ; Klein quartic corollary (kleinquarticgenusₑqₙgen, genus = Ngen = 3, CatAL) ; V-A coupling (vaₒp6bundle, gA/gV = -1 exactly, CatAL) ; C2 selector closure (d1d5forcesgibbsfamily, c2ₛelectorₘemberᵢndependent, CatAL) ; golden-fiber taxonomy (goldenᵤniversalityclassₜheorem, CatAL) ; cosmological constant (lambdaₐllₒrderᵦeroₚureₚhi, Lambda=0 all-order in pure-phi sector, CatAL) ; spectral convergence (spin7ₛoftₛpectrumconvergesₖink, CatAL) ; Landauer heat floor (landauerₕeatfloorₜransputation, CatAL) ; and others. Total: 361 modules. Companion to the ugp-physics paper series (P00–P53). Wave-4 graduation (2026-06-14, zero sorry): VacuumGHConvergence. lean (OQ-QG-1a CLOSED CatAL — vacuumcmcaghconvergesₜoflatₛpace: FinGrid L GH-converges to UnitCube at rate 1/L) ; MatterGHPrecompactness. lean (singleₖinkghconvergesₜoflat CatAL) ; SU2LDoubletHilbert. lean (su2ldoubletₕilbertcertified CatAL — SU (2) L doublet Hilbert space certified with orthonormal doublet partners and SU (2) L generators). Total: 364 modules. EPIC₀91 graduation (2026-06-15, zero sorry): GTE arithmetic fully derived from Z₇×Z₃ — ridge constant 16=2^ (1+ord₇ (2) ) from CRT (gt001gt009ᵣidgeclosed) ; quotient gap 13=F₇ from field structure (gt012gapfromᵦ7ᵦ3) ; Fibonacci lift uniquely forced (gt011ᵣidgeₒffsetₑqₚisano₇) ; a-values 1, 9, 5 as orbit projections (semanticfloorₐconditionfromcascade) ; even-step c-prime uniquely selected (mersenneₗadderᵤniquenessₚroved) ; Nc=3 → b₁=73 (gt013ₚscₙcbundle) ; circle closure Chain A + Chain B (ugpₒrbitᵢnterpolationₗift). Five new modules: GTE/RidgeDerivation. lean, GTE/FibonacciDerivation. lean, GTE/SemanticFloorDerivation. lean, GTE/MersenneLadder. lean, GTE/PSCColorRankBundle. lean. Total: 368 modules. Round 7 graduation (2026-06-15, commits ea84a74 + dc6cf79, zero sorry): 8 new modules — ClassicalLambda (classicalₗambdaᵦero CatAL: vacuum energy = 0, zero sorry), EWRhoCorrectedMZ (mZᵣhocorrectedᵢnterval CatAD, pole mass 91. 2±0. 1 GeV), AlphaEMArchivalIdentities, Phi7UnificationTheorem, MobiusTripleClassification, PolynomialLipschitz, KahlerStateManifold, LorentzianCausalityNecessity, IMTStructuralPair. Wolfenberg → Weinberg theorem rename complete. 20 CatAL certifications graduated from exp to canonical. Total: 382 modules. Formalization paper: 142 pages, 131 theorems.

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