LarsenClose/witness-asymmetry: Witness Extraction Asymmetry Across Logic, Complexity, and Fixed-Point Mathematics

Machine-verified formalization in Lean 4 / Mathlib v4.28.0.

The reflexive object D ≅ [D,D] has two equations:

fold ∘ unfold = id        (certification is free)
unfold ∘ fold = selfApp   (extraction generates computation)

Under resource bounds, certification stays free and extraction becomes costly. The gap between them, parameterized by the resource regime, generates three known stratifications:

Logical    : ∅ / Markov / EM
Polynomial : P / NP (polynomial Markov bridge)
Categorical: bounded / unbounded "!" contraction cost

These are not three analogies. They are three instances of a single abstract WitnessSystem, with the collapse predicate being EM, P = NP, and ¬HasGrowthGap respectively. The collapse correspondences are mediated by explicit bridge theorems, not by definitional renaming.

The key bridge theorem:

PolyMarkov ↔ P = NP

Markov's principle restricted to polynomial predicates — irrefutable bounded existence implies polynomial findability — is equivalent to P = NP through axiomatized search-to-decision and finder-to-decider reductions.

The computability side proves a three-gap decomposition of self-referential computation:

Naming gap:       CLOSED (selfApp-epi gives internal naming, no Gödel numbering)
Construction gap: CLOSED (fixed-point construction cost ≤ linear)
Depth gap:        FORCED (finite-table representability forces tower growth gap)

Naming + Construction + Growth gap = False   (anti-compression at [])

The growth gap is proved for Nat.Partrec.Code via Mathlib's computability library. The barrier diagnostic confirms the growth gap relativizes (Baker-Gill-Solovay blocks direct P ≠ NP separation), while identifying a category mismatch with natural proofs (the gap is global slice-counting, not truth-table-local).

The ISP (invariant subspace problem) side proves:

Chain classification:  extremal chains produce only trivial fixed points
Deflation theorem:     pre-fixed + non-collapse → nontrivial fixed point
Operator bridge:       compact → DCC path, normal → spectral path, general → orbit chain
Independence:          both ISP success and failure are satisfiable in abstract lattice models

The deflation theorem recovers the order-theoretic template of Lomonosov's 1973 compact-operator result as a special case of the deflation mechanism.

The axiom profile methodology measures which logical principles each theorem requires:

Layer 0 (∅):     Y combinator, Gödel I & II, halting, Kleene recursion, Myhill iso
Layer 1 (Markov): Rice's theorem (exact boundary)
Layer 2 (EM):    Post backward ≡ EM, full hierarchy properness

Applied to algebra: Bezout, CRT, Lagrange, Sylow compile at Layer 0. Maximal ideals and infinite-dimensional bases require Layer 2 (Choice/Zorn). The Markov layer is empty in algebra — specific to computation.

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All paper-critical theorems verified at [] or [propext, Quot.sound] via Lean.collectAxioms.
Classical.choice enters only through Mathlib bridge files and deliberate countermodel constructions.
Full axiom verification in AXIOM_VERIFICATION.md (3179 declarations checked).

Key companion documents:

• THEOREM_MAP.md — every theorem with axiom profile and scope
• AXIOM_VERIFICATION.md — 3179 declarations verified via Lean.collectAxioms

Supports three papers:

1. The Axiom Profile of Computation
2. Reflexive Compression Boundaries in Graded Categories
3. Chain Obstructions and Deflation in the Invariant Subspace Problem