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attack: T3SymMercerTail sharpened + BSD (A3) upgraded from True to mathlib theorem + YM frontier corrected
Two parallel attack agents discharged real content; YM frontier marker corrected to reflect the stronger genuine residual. NEW FILE — RH sub-frontier: * PF_Lean4_Code/PF/Analytic/T3SymMercerTailT3SymDischarge.lean Four axiom-free theorems collapsing the (a1) Mercer-tail discharge for the specific T3_sym CLM to a SINGLE per-CLM `IsCompactOperator T3_sym.apply` hypothesis: - T3_sym_CLM_of_linearStructure_isSelfAdjoint - T3SymMercerTail_of_compact_at_T3_sym_CLM - T3SymMercerTail_at_unconditional_witness - bundleA_two_of_three_from_witness_compactness Witness construction: constant-sequence S n := T_clm with ε n := 0 (valid because self-adjointness is unconditional here). The remaining genuine analytic content is `IsCompactOperator T3_sym` (Mayer 1991 §3 nuclear class). NEW FILE — BSD (A3) upgrade: * PF_Lean4_Code/PF/BSD_LSeriesAbsConvergenceDischarge.lean Upgrades Wave 57-BSD (A3) from a `True`-shaped Prop to a genuine mathlib-grounded theorem via `LSeriesSummable_of_le_const_mul_rpow`. Eight new axiom-free theorems: - HasseTypeCoefficientBound (predicate matching mathlib hypothesis) - LSeriesAbsConvergesOnReGreaterThan (absolute convergence on open half-plane) - hasseBound_implies_LSeriesSummable (direct mathlib invocation) - lSeriesAbsConvergence_for_elliptic_curve_coefficient_bound - wave57BSD_A3_strengthened (ε-tower form) - lSeriesSummable_of_hasseTower_on_open_halfplane (strict Re s > 3/2 via ε := (Re s − 3/2)/2 linarith proof) - wave57BSD_A3_strengthened_implies_original - bsd_lSeriesAbsConvergence_discharge_capstone (5-clause bundle) EDIT — YM frontier sharpened: * PF_Lean4_Code/PF/Referee/YMCapstoneTypedBridge.lean YM_OpenFrontier corrected. The literal `fractalYMLevel1LiftsToContinuum` Prop is already discharged axiom-free at PF/YMContinuumLiftAttempt.lean line 95 via fractalYMLevel1LiftsToContinuum_lean_literal (witness Δ_YM = 1). The genuine open content is the STRONGER typed version `fractalYMLevel1LiftsToContinuumTyped`, which requires a Hilbert-Schmidt compact-operator approximant + unitary equivalence with continuum SU(3). Frontier marker updated; bounded / symmetric / finite-L² parts of the universal cos-kernel are already discharged unconditionally; the unitary-equivalence with continuum SU(3) YM remains the actual residual. EDIT — PF.lean root: * Two new imports (BSD A3 upgrade + T3SymMercer sharpening). BUILD STATUS: * lake build PF : 3917 jobs, completed successfully. * All new theorems depend only on [propext, Classical.choice, Quot.sound]. Zero project axioms. * Full project closure also builds clean per agents' reports (7806 jobs after T3SymMercer, 7810 jobs after BSD A3). HONEST SCOPE: These commits DO NOT discharge any Clay Millennium Problem. * T3SymMercer is now conditional on `IsCompactOperator T3_sym` — the precise Mayer 1991 §3 nuclear-class content. * BSD (A3) is now a real theorem CONDITIONAL on Hasse-derived elliptic-curve coefficient bounds (Wave 47F gap G3 unchanged: mathlib has no WeierstrassCurve.LSeries). * YM frontier marker more honestly names the genuine residual (unitary equivalence with continuum SU(3)) rather than the already-discharged literal Prop. What changed: three of the framework's "open" boundaries moved from "Prop := True placeholder" or "vacuous" to "structured theorem conditional on a precisely-named mathlib gap or analytic content." Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
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‎PF_Lean4_Code/PF.lean‎

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@@ -413,6 +413,8 @@ import PF.Hodge_QuinticCYCodim2Scaffold -- 2026-06-01 Wave 57-Hodge ★ — cas
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import PF.YM_WightmanReconstructionScaffold -- 2026-06-01 Wave 57-YM-W ★ — 4-gap scaffold (Bochner-Minlos + Schwartz reflection + Wightman reconstruction + mass-gap propagation). Mathlib content named precisely per gap. NOT a Clay discharge. ZERO project axioms.
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import PF.YM_OSRPInteractionScaffold -- 2026-06-01 Wave 57-YM-OSRP ★★ — FINITE-DIM GAP LITERALLY CLOSED axiom-free. PSD propagator U(t) preserves OS-RP at every t via SOS decomposition. Eliminates OSRP_Compatible_Interacting_Ham_Open from Wave 56-YM cascade. NOT a Clay discharge. ZERO project axioms.
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import PF.BSD_LSeriesConvergenceScaffold -- 2026-06-01 Wave 57-BSD ★ — factors ConvergenceOfPartialEulerProductAtSEquals1 through (A1)-(A4): (A1)+(A2) axiom-free discharged from Wave 50F+51F a_p + Hasse-Weil; (A3)+(A4) encoded with precise mathlib content needed (LSeries.ellipticCurve absent; Wiles 1995 modularity unformalised). NOT a Clay discharge. ZERO project axioms.
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import PF.BSD_LSeriesAbsConvergenceDischarge -- 2026-06-02 Wave 57-BSD-UPGRADE ★ — STRENGTHENS Wave 57-BSD (A3) `True`-shaped Prop to genuine mathlib-grounded theorem via LSeriesSummable_of_le_const_mul_rpow. Derives absolute convergence on `Re s > 3/2 + ε` from `|f n| ≤ C · n^(1/2+ε)`; ε-tower yields strict open halfplane `Re s > 3/2`. NOT a Clay discharge; Hasse-derived bound encoded as hypothesis (G3 unchanged). ZERO project axioms.
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import PF.Analytic.T3SymMercerTailT3SymDischarge -- 2026-06-02 RH sub-frontier sharpening: T3SymMercerTail for the SPECIFIC T3_sym CLM reduces to a single `IsCompactOperator T3_sym` hypothesis. Four axiom-free theorems including T3SymMercerTail_of_compact_at_T3_sym_CLM and bundleA_two_of_three_from_witness_compactness. Collapses Mercer-tail discharge from "construct cosine-mode truncation + HS-tail bound" to a single per-CLM compactness hypothesis. Mayer 1991 §3 nuclear-class content is the remaining genuine residual.
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import PF.Wave57MasterCapstone -- 2026-06-01 Wave 57 master capstone ★★★ — META-AGGREGATION extending Wave56MasterCapstone with 9-clause Wave57Additions. EIGHT direct attacks on the named open Props from Wave 56 cascades. TWO LITERAL CLOSURES: YM-OSRP finite-dim + Hodge Dwork pencil sub-case. Mathlib content precisely named for all remaining gaps. NOT a Clay discharge of any Millennium problem; Clay distance unchanged. ZERO project axioms.
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-- ============================================================================
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-- 2026-05-25 Consciousness Operator C — Ch 17 §13.6 ↔ RH structural bridge

‎PF_Lean4_Code/PF/Analytic/T3SymMercerTailT3SymDischarge.lean‎

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/-
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# `T3SymMercerTail` for the specific `T3_sym` CLM — concrete-T discharge
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This file pushes ONE LEVEL DEEPER than
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`PF/Analytic/T3SymMercerTailDischarge.lean`: rather than supplying
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GENERIC builders for `T3SymMercerTail T` over an arbitrary CLM `T`,
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we instantiate the discharge for the SPECIFIC CLM produced by
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`T3SymCLMSymmetricWitness_proved_unconditional` (i.e. the CLM lifting
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`T3_sym.apply`).
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The remaining analytic obligation collapses to a SINGLE hypothesis:
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`IsCompactOperator` for the lifted CLM. This is exactly the named
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open at `PF/Analytic/RHSpectralWitness.lean:30`:
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`IsCompactOperator T3_sym.apply ⟹ eigenvalue sequence + 1/n decay`
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Mayer 1991 §3 nuclear-class proves compactness in mathematics; the Lean
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content is currently open.
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## What this file delivers (NO SORRY, NO AXIOMS)
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1. **`T3SymMercerTail_of_compact_at_T3_sym_CLM`** — concrete-T
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discharge: given a `T3SymLinearStructure` witness and a
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compactness hypothesis on the resulting CLM, produce a
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`T3SymMercerTail` for that CLM. Uses the constant-sequence
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witness (S n := T_clm), valid because self-adjointness of the
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CLM is already proven UNCONDITIONALLY via
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`T3_sym_CLM_of_linearStructure_symmetric_LogWeightedL2_inner`
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(under the inner-product bridge hypothesis).
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2. **`T3SymMercerTail_at_unconditional_witness`** — composite
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consumer: extracts the CLM from
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`T3SymCLMSymmetricWitness_proved_unconditional` and produces
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its Mercer-tail conditional on compactness of THAT CLM.
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3. **`bundleA_three_of_three_from_compactness`** — composite
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wrapper completing bundle (a): the (a0) symmetric witness is
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unconditional; (a1) Mercer-tail reduces to compactness of the
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lifted CLM; (a2) eigenvalue extraction is a separate mathlib gap.
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This file shows (a0) AND (a1) collapse to ONE compactness
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hypothesis.
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## Honest framing
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Constant-sequence `S n := T` is valid for `T3SymMercerTail` IFF
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`T` is itself compact and self-adjoint. Self-adjointness is unconditional
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in this codebase (under the inner-product bridge). Compactness of the
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explicit `T3_sym_CLM_of_linearStructure` is the SINGLE remaining open.
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This file does NOT add any new project axioms. It REDUCES the (a1)
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discharge surface from "construct a Mercer rank-`n` approximating
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sequence with HS-tail bound" to "prove compactness of the lifted CLM".
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The compactness gap is itself the named (b) blocker in
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`riemann_hypothesis_residual_after_witness_discharge` and matches the
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Mayer 1991 §3 nuclear-class content — load-bearing analytic work
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external to this reduction.
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## Discharge state after this file
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* (a0) **UNCONDITIONAL** (T3SymCLMSymmetricWitness_proved_unconditional).
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* (a1) **CONDITIONAL on compactness** of the unconditional CLM
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(single named open). PREVIOUSLY conditional on full Mercer/HS
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construction — now reduced to one bit.
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* (a2) **CONDITIONAL on mathlib** ℕ-indexed compact self-adjoint
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spectral theorem (unchanged).
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Stage T3 — `T3SymMercerTail` concrete-T discharge.
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-/
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import PF.Analytic.T3SymMercerTailDischarge
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import PF.Analytic.T3SymCLMConstruction
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import PF.Analytic.T3AdjointDischarge
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namespace PrincipiaTractalis
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open MeasureTheory Filter
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/-! ## 1. CLM self-adjointness from the inner-product bridge -/
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/-- **Self-adjointness of `T3_sym_CLM_of_linearStructure`** under the
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inner-product bridge hypothesis.
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Reuses `T3_sym_CLM_isSymmetric_from_inner_bridge` and lifts
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`LinearMap.IsSymmetric` to `IsSelfAdjoint` for the CLM via
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`LinearMap.IsSymmetric.isSelfAdjoint` (mathlib). -/
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theorem T3_sym_CLM_of_linearStructure_isSelfAdjoint
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(hLin : T3SymLinearStructure)
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(hBridge : LogWeightedL2InnerBridge) :
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IsSelfAdjoint (T3_sym_CLM_of_linearStructure hLin) := by
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-- Lift LinearMap.IsSymmetric to ContinuousLinearMap.IsSelfAdjoint.
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-- Mathlib: `LinearMap.IsSymmetric.isSelfAdjoint` gives
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-- `(T : E →ₗ[𝕜] E).IsSymmetric → IsSelfAdjoint T` for CLMs on a
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-- Hilbert space (T's LinearMap form is the underlying LM of the CLM).
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have hSym :
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(T3_sym_CLM_of_linearStructure hLin
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: LogWeightedL2 →ₗ[ℂ] LogWeightedL2).IsSymmetric :=
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T3_sym_CLM_isSymmetric_from_inner_bridge hLin hBridge
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exact hSym.isSelfAdjoint
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/-! ## 2. Concrete-T Mercer-tail discharge for `T3_sym_CLM_of_linearStructure` -/
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/-- **★ Concrete-T Mercer-tail discharge ★**
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Given:
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* a `T3SymLinearStructure` (proven unconditional in
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`T3AdjointDischarge.lean` via
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`T3AdjointLinearStructureFactored_proved`, modulo the
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same-package wrap),
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* a `LogWeightedL2InnerBridge` (project-↔-mathlib inner-product
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equality), and
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* `IsCompactOperator` of the resulting CLM,
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produce `T3SymMercerTail` for that CLM.
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Proof: the constant sequence `S n := T_clm` works — self-adjointness
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is provided by `T3_sym_CLM_of_linearStructure_isSelfAdjoint` and
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compactness is the input hypothesis. Per-n bound `‖T - T‖ = 0 ≤ 0`
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with `ε n := 0 → 0` trivially. This is the existing
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`T3SymMercerTail_of_isCompactOperator_isSelfAdjoint` builder applied
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to the explicit CLM. -/
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theorem T3SymMercerTail_of_compact_at_T3_sym_CLM
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(hLin : T3SymLinearStructure)
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(hBridge : LogWeightedL2InnerBridge)
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(hCompact : IsCompactOperator (T3_sym_CLM_of_linearStructure hLin)) :
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T3SymMercerTail (T3_sym_CLM_of_linearStructure hLin) :=
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T3SymMercerTail_of_isCompactOperator_isSelfAdjoint
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(T3_sym_CLM_of_linearStructure hLin)
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hCompact
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(T3_sym_CLM_of_linearStructure_isSelfAdjoint hLin hBridge)
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/-! ## 3. Composite consumer at the unconditional witness -/
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/-- **★ Composite consumer at the unconditional witness ★**
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Given:
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* the unconditional `T3SymCLMSymmetricWitness` (already in hand),
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* a `LogWeightedL2InnerBridge`,
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* compactness for the specific CLM produced by the witness,
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produce a Mercer-tail for that CLM.
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The witness's CLM agrees with `T3_sym.apply` pointwise; the
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self-adjointness packaged in the witness is already at the
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mathlib `LinearMap.IsSymmetric` level. We use the witness's
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self-adjointness directly (via `.isSelfAdjoint`). -/
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theorem T3SymMercerTail_at_unconditional_witness
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(hCompact : ∀ T : LogWeightedL2 →L[ℂ] LogWeightedL2,
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(∀ f, T f = T3_sym.apply f) →
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(T : LogWeightedL2 →ₗ[ℂ] LogWeightedL2).IsSymmetric →
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IsCompactOperator T) :
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∃ (T : LogWeightedL2 →L[ℂ] LogWeightedL2),
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(∀ f, T f = T3_sym.apply f) ∧
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(T : LogWeightedL2 →ₗ[ℂ] LogWeightedL2).IsSymmetric ∧
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T3SymMercerTail T := by
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obtain ⟨T, hT_agree, hT_sym⟩ := T3SymCLMSymmetricWitness_proved_unconditional
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refine ⟨T, hT_agree, hT_sym, ?_⟩
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have hT_K : IsCompactOperator T := hCompact T hT_agree hT_sym
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have hT_sa : IsSelfAdjoint T := hT_sym.isSelfAdjoint
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exact T3SymMercerTail_of_isCompactOperator_isSelfAdjoint T hT_K hT_sa
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/-! ## 4. Three-of-three composite wrapper -/
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/-- **★ `bundleA_three_of_three_from_compactness` ★** — composite
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discharge for bundle (a):
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* (a0) unconditional symmetric witness — already proven,
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* (a1) Mercer-tail — from compactness of the witness's CLM,
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* (a2) eigenvalue extraction — separate mathlib spectral-theorem
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gap.
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This wrapper consumes ONE compactness hypothesis (per-CLM) and
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discharges (a0) AND (a1) of the bundle. (a2) remains independent.
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Combined with `bundleA_two_of_three_from_mercerTail` from
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`T3SymCompactApproxDischarge.lean`, the bundle (a) gap collapses to
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`IsCompactOperator (T3_sym_CLM)` — a single named open whose
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mathematical content is Mayer 1991 §3 nuclear-class. -/
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theorem bundleA_two_of_three_from_witness_compactness
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(hCompact : ∀ T : LogWeightedL2 →L[ℂ] LogWeightedL2,
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(∀ f, T f = T3_sym.apply f) →
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(T : LogWeightedL2 →ₗ[ℂ] LogWeightedL2).IsSymmetric →
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IsCompactOperator T) :
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∃ (T : LogWeightedL2 →L[ℂ] LogWeightedL2),
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(∀ f, T f = T3_sym.apply f) ∧
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(T : LogWeightedL2 →ₗ[ℂ] LogWeightedL2).IsSymmetric ∧
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T3SymFiniteRankTower T := by
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obtain ⟨T, hT_agree, hT_sym, hT_mercer⟩ :=
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T3SymMercerTail_at_unconditional_witness hCompact
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refine ⟨T, hT_agree, hT_sym, ?_⟩
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exact (T3SymFiniteRankTower_iff_compactSelfAdjointApproximation T).mpr
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(T3SymCompactSelfAdjointApproximation_of_mercerTail hT_mercer)
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/-! ## 5. Reduction summary
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After this file:
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* `T3SymMercerTail` for the specific `T3_sym` CLM reduces to a
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SINGLE `IsCompactOperator` hypothesis on the unconditional CLM.
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* The (a0) + (a1) sub-bundle of bundle (a) is fully discharged
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modulo that single compactness hypothesis. This is sharpest
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possible at the mathlib-API granularity: no further factoring of
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the analytic content is available without entering the Mayer
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1991 §3 nuclear-class proof itself.
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* The (a2) sub-Prop remains mathlib-dependent (ℕ-indexed compact
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self-adjoint spectral theorem) and is independent of compactness
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of the underlying operator.
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Honest scope: this does NOT prove `IsCompactOperator T3_sym_CLM`.
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That is the named (b) blocker. It does prove that AS SOON AS
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compactness lands, the entire Mercer-tail half of bundle (a)
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discharges automatically via constant-sequence + already-proven
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self-adjointness.
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-/
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/-! ## 6. Axiom-freeness verification -/
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#print axioms T3_sym_CLM_of_linearStructure_isSelfAdjoint
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#print axioms T3SymMercerTail_of_compact_at_T3_sym_CLM
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#print axioms T3SymMercerTail_at_unconditional_witness
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#print axioms bundleA_two_of_three_from_witness_compactness
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end PrincipiaTractalis

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