A registered result in its source context
Original ledger classification: Formal
Recorded statement: LEAN: THE mu4 MARK-LOCALITY COMMUTATION AS REAL LINEAR ALGEBRA (2026-06-22). Lean 4 (TfptCarrier/Mu4Commutation.lean) turns the v201/v309/v323 finite-model fact (the mu4 clock rho=diag(i^n) commutes with a DtN sub-principal coupling IFF…
Statement, assumptions and sources
Paper context: Origin Theory
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FORM.MU4COMM.01
Original ledger classification: Formal
Original claim text
LEAN: THE mu4 MARK-LOCALITY COMMUTATION AS REAL LINEAR ALGEBRA (2026-06-22). Lean 4 (TfptCarrier/Mu4Commutation.lean) turns the v201/v309/v323 finite-model fact (the mu4 clock rho=diag(i^n) commutes with a DtN sub-principal coupling IFF that coupling is mark-local, i.e. couples only Fourier modes differing by a multiple of 4) into an EXACT, kernel-checked statement over the Gaussian integers Z[i]: [E] i^4=1 (an offset-4 mode shift is invisible to the mu4 clock) and i^2=-1 (an offset-2 shift flips the sign, so rho sees it), by decide; [E] mu4_commutes_offset4: rho * M4 = M4 * rho for a concrete offset-4 coupling on 6 modes (the Z4-invariant/mark-local case), by decide; [E] mu4_not_commutes_offset2: rho * M2 != M2 * rho for an offset-2 coupling (the non-mark-local negative control), by decide. Built end-to-end (lake build) on Lean v4.29.1 + Mathlib. The VALUE is grounding the v201 block-diagonality criterion (mark-locality <=> omega o rho = omega) in exact, kernel-decidable linear algebra over Z[i], no native_decide.
Original classification detail
[E] Lean-built: i^4=1, i^2=-1, and the mu4 commutation (offset-4 commutes, offset-2 does not) on concrete 6x6 matrices over Z[i], all kernel decide (no native_decide). Grounds the v201/v323 mark-locality criterion in exact linear algebra.
Dependencies as recorded
QGEO.SUBPRIN.01; QGEO.BW.01; QGEO.NECESS.01; v201; v309; v323; v331
Open the original ledger source, row 434Sections that explicitly cite this claim
A literal claim identifier is present in these paper sections. The section and its premises determine what the citation supports.
- Honest statusinput 4 origin theory
- Honest statusorigin theory
- 2026-06-22 (endgame round: the necessity of [formula in source] reduced to the order-4 CM selection, the QG metric sector characterized as the conformal-mode problem, [formula in source] and the [formula in source] commutation grounded in Lean: v331, v332, FORM.MU4COMM.01)Research changelog; TFPT — Introduction and reading guide
- Honest statusOrigin Theory
- Honest statusinput 4 origin theory
Navigation through a shared verification script
These sections cite a script associated with the claim. This is a source navigation join, rather than an explicit citation of the claim or a theorem equivalence.
- The unique attractor: a gapped transport selects ONE fixed pointorigin theory
- The unique attractor: a gapped transport selects ONE fixed pointinput 4 origin theory
- 2026-06-22 (can the one open arrow be fully solved? the closure-mode classification: the flat [formula in source] spherical base locus, four modes, no free theorem – only rigidity or axiom [formula in source] : v347)Research changelog; TFPT — Introduction and reading guide
- The unique attractor: a gapped transport selects ONE fixed pointinput 4 origin theory
- Front matter / supportTFPT — Introduction and reading guide; Verification reference catalog
- The unique attractor: a gapped transport selects ONE fixed pointOrigin Theory
Registered verification modules
- v331_necessity_of_H.pyRegistered verification module · frontier
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The NECESSITY of H reduced: rotation-invariance of the seam DtN is EQUIVALENT to the four marks at the square (tau=i), so the one open lemma is exactly the order-4 CM selection (the necessity companion to v329's sufficiency). [E] SQUARE => [rho,Lambda]=0: marks at {0,pi/2,pi,3pi/2} make the curvature Z4-symmetric (Fourier support 0 mod 4), M_f couples only modes differing by 4Z (the v201/v210 sufficiency as a DtN |k|+M_f operator). [E] NECESSITY: a mark off the square breaks Z4 and gives [rho,Lambda]!=0 -- equivalence, not just implication. [E] ORDER-4 CM: cross-ratio(1,i,-1,-i)=2 => j=1728, the unique 4-config with an order-4 automorphism (v214/v267); H <=> square <=> mu4. [E] NEG: a Z2 mark set commutes with rho^2 but not rho. [O] residual: the dynamical SELECTION of the square (the 4 marks forced by v216, the square the unique order-4 CM) is the irreducible postulate. Does NOT close the bedrock.
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