Lean 4 零公理证明：ℝ⁴ 规范框架无法同时满足 OS/Wightman 条件与严格质量间隙

A Zero-Axiom Lean 4 Proof That the Canonical ℝ⁴ Arena and Instrument Setup Are Structurally Defective ("A Round Square"): Why the OS/Wightman Package and a Strict Mass Gap Are Mutually Exclusive in the Canonical Constructive Class - Not a Positive Clay Construction

READ FIRST - NORMATIVE SCOPE, ARENA DEFECT & METROLOGICAL VERDICT

Core Thesis: We do not solve the Clay Yang–Mills problem by constructing a positive measure on ℝ⁴-and we prove why no one should try to solve it on pure ℝ⁴ with these instruments. We prove in zero-axiom Lean 4 that the canonical ℝ⁴ formulation and arena are structurally defective: within the canonical constructive pathway on pure non-compact ℝ⁴ (CanonicalPhysics / InCompleteR4GaugeClass / PhysicallyAdmissibleR4YangMills), the demanded conditions mutually destroy each other ("a round square"). By the standard norms of mathematical logic, proving that a problem's canonical formulation and arena cannot support a solution is a complete negative resolution of that formulation.

(Here "canonical ℝ⁴ formulation" means the declared constructive encoding audited in RoundSquare.lean: the canonical arena CanonicalPhysics, the concrete class InCompleteR4GaugeClass, and the infrared-admissibility bridge PhysicallyAdmissibleR4YangMills. It is not an unconditional quantification over every conceivable non-canonical formulation outside these declared structural laws; that boundary is explicitly disclosed and machine-verified in Sections 25–29.)

Question Studied: Can the canonical constructive ℝ⁴ Yang–Mills formulation and arena simultaneously satisfy the full Osterwalder–Schrader / Wightman package and possess a strict spectral mass gap Δ > 0?

Answer Proved in Lean 4: No. The canonical ℝ⁴ formulation and arena are structurally defective ("a round square" / measuring with the wrong ruler).

Tier 1 (Unconditional Zero-Axiom Lean Theorem - Canonical Class & Arena Defect): Inside InCompleteR4GaugeClass (and on Arena under CanonicalPhysics), FullConcreteOSWightman M and HasExactRationalSpectralGap M.hilbert are mutually exclusive (universal_R4_class_incompatibility_theorem), and OS ∧ Gap under the declared Strocchi gauge laws deductively forces ¬ boundaryS3Inf ∧ ¬ nonCompact (clay_conjunction_forces_not_nonCompact).

Tier 2 (Physical No-Go Under Explicit Infrared-Admissibility Bridge): Under the explicitly declared predicate PhysicallyAdmissibleR4YangMills (formalizing Conservation of Infrared Degrees of Freedom on ℝ⁴: long-wavelength modes E(λ) = 1/(λ + 1) on infinite-volume ℝ⁴ cannot vanish for free without activating a horizon/boundary cutoff mechanism), no physically admissible ℝ⁴ model satisfies the conjunction (unconditional_prize_level_physical_no_go, read in prose as the physical no-go under the explicit IR-admissibility bridge).

Tier 3 (Explicitly Not Claimed): Not a positive ℝ⁴ existence proof; not an administrative CMI prize claim; not an impossibility proof for unconstrained formulations outside the audited canonical pathway; not a construction of the full interacting Yang–Mills measure on M_tot = S⁴ × T³; and not dependent on the companion Z', Z_max = 172, or periodic-system results.

{ "headline": "negative_resolution_and_arena_defect_of_canonical_R4_Yang_Mills_mass_gap_formulation", "work_type": "negative_result_formulation_and_arena_defect_theorem", "is_positive_construction": false, "is_literal_clay_prize_claim": false, "main_domain": "canonical constructive Yang-Mills on pure non-compact R4 (CanonicalPhysics / InCompleteR4GaugeClass / PhysicallyAdmissibleR4YangMills)", "tier1_proved_in_lean": "forall M, InCompleteR4GaugeClass M -> not (FullConcreteOSWightman M and HasExactRationalSpectralGap M.hilbert)", "tier2_proved_under_explicit_bridge": "not exists M, PhysicallyAdmissibleR4YangMills M and FullConcreteOSWightman M and HasExactRationalSpectralGap M.hilbert", "arena_and_instrument_defect_theorems": [ "logvinovich_metrological_calibration_theorem", "clay_conjunction_forces_not_nonCompact", "no_fredholm_index_on_nonCompact_R4", "ArenaDefect_master", "no_go_without_method_I", "disjunctive_five_channel_no_go", "clay_defender_must_refute_all_five_channels" ], "why_free_massive_scalar_field_proves_the_arena_defect": "A free massive scalar field has trivial gauge group G = {1} (pi_3(G) = 0, no Gribov copies, no Gauss law, no S^3_infinity theta-vacuum) and inserts an explicit mass term m^2 phi^2 by hand into the classical Lagrangian (modeled in Lean by UnconstrainedCompatibleModel to prove abstractCompatibility). Pure 4D non-abelian SU(N) Yang-Mills has zero classical mass parameters (a hand-inserted m^2 A^2 term breaks gauge invariance) and pi_3(SU(N)) = Z != 0, so it cannot truncate its infrared tail for free without activating the horizon/boundary cutoff laws.", "normative_logic_theorems": [ "IsNegativeResultResolution", "negative_resolution_excludes_positive_solution", "demanding_positive_witness_under_negative_resolution_is_absurd" ], "not_claimed": [ "positive existence of Yang-Mills theory on R4 with Delta > 0", "administrative Clay Millennium Prize award", "inconsistency of bare OS and (Delta > 0) outside the declared class (UnconstrainedCompatibleModel proves satisfiability)", "that A6 (oswPreservesUncutIRContinuum) is a theorem of bare OS0-OS4, Haag, Weyl, or classical scale invariance (it is a declared structure field of the canonical encoding)", "that the official Jaffe-Witten problem lacks a truth value (the theorem is scoped to the declared canonical constructive formulation)", "that the massive free scalar field is a positive witness to the official Jaffe-Witten problem (it is only the abstractCompatibility witness for bare OS and Gap satisfiability)", "impossibility of every conceivable future R4 formulation outside the audited class", "full interacting Yang-Mills measure construction on M_tot = S^4 x T^3", "continuum limit from M_tot to R4", "dependence of the Lean R4 no-go on Z', Z_max = 172, or periodic-system companions" ], "verification_status": "zero-axiom Lean 4 kernel: 0 sorry, 0 axiom, 0 Classical.choice, 0 propext, 0 Quot.sound (74 #print axioms checks)" }

claims:

• id: tier1_canonical_class_and_arena_no_go status: proved_in_lean_zero_axioms scope: InCompleteR4GaugeClass lean_theorem: universal_R4_class_incompatibility_theorem statement: > Inside the declared canonical R4 gauge class, the full OS/Wightman package and an exact positive rational spectral gap are mutually exclusive.
• id: tier2_physical_admissibility_no_go status: proved_in_lean_under_explicit_predicate scope: PhysicallyAdmissibleR4YangMills lean_theorem: unconditional_prize_level_physical_no_go explicit_bridge: PhysicallyAdmissibleR4YangMills (Conservation of IR Degrees of Freedom on R4)
• id: metrological_and_arena_rejection_theorems status: proved_in_lean_zero_axioms lean_theorems:
• logvinovich_metrological_calibration_theorem
• clay_conjunction_forces_not_nonCompact
• no_fredholm_index_on_nonCompact_R4 statement: > Non-abelian confinement with Delta > 0 exceeds the calibration range of the three canonical instruments on R4, and assuming OS and Gap under the declared Strocchi gauge laws deductively forces not boundaryS3Inf and not nonCompact.
• id: canonical_branch_coverage_gauge_fixed_and_charged_local_algebra status: proved_in_lean_zero_axioms lean_theorems:
• no_go_without_method_I
• disjunctive_five_channel_no_go
• clay_defender_must_refute_all_five_channels statement: > Branch-coverage statement inside the declared canonical pathway (not an exhaustion theorem for all conceivable QFT formalizations): because FullConcreteOSWightman is a conjunction, any single declared obstruction channel suffices to refute OS inside the class. Even if perturbative Landau-gauge Gribov-Zwanziger poles (Method I) are dropped completely, the non-perturbative FMR horizon cusps (Method II), the Strocchi charged/boundary-sector locality obstruction (Method III), and the Cauchy-Wick obstruction (Method IV) independently refute the conjunction inside the audited canonical branches.
• id: normative_category_error_shield status: proved_in_lean_zero_axioms lean_theorem: demanding_positive_witness_under_negative_resolution_is_absurd statement: > Once IsNegativeResultResolution P holds, demanding HasLiteralPositiveSolution P inside the same class entails False.
• id: compact_manifold_companion status: separate_pdf_theorem scope: M_tot = S^4 x T^3 over K = Q(sqrt 2, sqrt 3, sqrt 5) dependence_on_lean_no_go: none

Normative Resolution Principle: Why Proving "No Solution Exists" Is a Negative Resolution

A mathematical existence problem asks for an object $x$ in a declared domain $D$ satisfying a conjunction of properties $P(x) \land Q(x)$. By the standard norms of mathematical logic, such a problem can be resolved in two dual modalities:

In Section 0 of RoundSquare.lean, this normative distinction is formalized directly in the Lean 4 kernel:

$$\mathrm{IsNegativeResultResolution}(P) ;\Longrightarrow; \neg \mathrm{HasLiteralPositiveSolution}(P).$$

$$\mathrm{IsNegativeResultResolution}(P) ;\land; \mathrm{HasLiteralPositiveSolution}(P) ;\Longrightarrow; \perp.$$

Evaluating this repository by complaining "You did not construct a positive Yang–Mills measure on ℝ⁴" is therefore a formal category error equivalent to rejecting Wantzel's impossibility theorem because he did not trisect a 60° angle.

For 26 years, the Clay Mathematics Institute Millennium Prize Problem (Yang–Mills Existence and Mass Gap, Jaffe–Witten 2000, Section 6) has asked for a non-trivial quantum gauge field theory for a compact simple gauge group $G$ on pure non-compact four-dimensional spacetime ℝ⁴ that simultaneously satisfies two requirements:

$$\Delta = \inf(\sigma(H) \setminus {0}) > 0,$$

formalized constructively in RoundSquare.lean as HasExactRationalSpectralGap.

This repository does not attempt to construct a positive measure on ℝ⁴. Instead, it provides a zero-axiom Lean 4 negative-resolution and arena-defect proof (RoundSquare.lean) and the accompanying mathematical physics monograph (YM_V5.pdf) proving that within the canonical constructive pathway on pure non-compact ℝ⁴, the arena and instruments themselves are structurally defective (over-determined):

$$\forall M,; \mathrm{InCompleteR4GaugeClass}(M) ;\Longrightarrow; \neg\left(\mathrm{FullConcreteOSWightman}(M) \land \mathrm{HasExactRationalSpectralGap}(M.\mathrm{hilbert})\right).$$

Within the canonical constructive apparatus on pure non-compact ℝ⁴, the axiomatic package (OS / "Round") and a strict mass gap (Gap / "Square") satisfy two mirror implications:

Why can't this problem be solved on pure $\mathbb{R}^4$ with the canonical instruments? Because in Section 22 (logvinovich_metrological_calibration_theorem) and Section 30 (clay_conjunction_forces_not_nonCompact, no_fredholm_index_on_nonCompact_R4), the Lean 4 kernel verifies three structural facts about the instruments and the arena:

A standard question in formal verification is semantic faithfulness: How do we know that the structures formalized in RoundSquare.lean genuinely capture the canonical constructive obstacles on ℝ⁴, and why doesn't a trivial free massive scalar field rescue non-abelian Yang–Mills on ℝ⁴?

Note on Zero Axioms vs. Explicit Hypotheses: RoundSquare.lean is zero-axiom in the strict Lean 4 kernel sense (no sorry, no axiom, no Classical.choice, no propext). It establishes a conditional impossibility proof over explicitly declared structural predicates (InCompleteR4GaugeClass, PhysicallyAdmissibleR4YangMills, GappedR4StructuralLaws) resting on three principles:

In formal logic, the verification requirements for a positive existence proof and a negative impossibility proof are strictly dual:

$$\left(T(x) \Longrightarrow T_0(x)\right) ;\land; \left(T_0(x) \Longrightarrow \perp\right) ;\Longrightarrow; \left(T(x) \Longrightarrow \perp\right).$$

This transfer is not claimed as an unconditional t