A registered result in its source context
Original ledger classification: Identity [I] (symbolic theorems) + Numerical [N] (branch census) + Physical [P] (residue)
Recorded statement: TORSION NORMAL FORM + THE DELTA = 1/2 THEOREM (executes the GATE.UWALL.05 'genericity future work'; the distinguished lepton transport value delta = 1/2 stops being a point observation): (1) FLATNESS = mu4 TORSION [I, symbolic generic M] -…
Statement, assumptions and sources
Paper context: Research contracts
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GATE.UWALL.06
Original ledger classification: Identity [I] (symbolic theorems) + Numerical [N] (branch census) + Physical [P] (residue)
Original claim text
TORSION NORMAL FORM + THE DELTA = 1/2 THEOREM (executes the GATE.UWALL.05 'genericity future work'; the distinguished lepton transport value delta = 1/2 stops being a point observation): (1) FLATNESS = mu4 TORSION [I, symbolic generic M] - prod_k U^k M U^{-k} = (MU)^4 exactly, so the Z4-family flatness constraint of the (U_wall) bundle IS the torsion statement 'T = MU is a fourth root of unity': the mu_4 atom appears as the ORDER of the twisted cusp generator. (2) DELTA = 1/2 THEOREM [I, both directions]: on the involutive branch (T^2 = 1) T is a Hermitian reflection 2vv^dag - 1 and the cusp trace condition 2v^dag U^{-1}v = 1 splits exactly into |v1|^2 = 1/2 (real part) + |v2| = |v3| (imaginary part), forcing diag M = (0, i/2, -i/2) with spec M = {1, w, w^2} AUTOMATIC (unitary + tr 0 + det 1 => char poly lam^3 - 1): delta = |M_22| = 1/2 EXACTLY AND CONSTANTLY on the whole 2-parameter branch - the value v20 used by hand and v40/v41 observed at the explicit point is a mu4-torsion identity. (3) REFLECTION LEMMA [I]: the D4 reflection forces diag M = (r, z, zbar) with r real and Re z = -r/2 - the D4 diagonal freedom is ONE real parameter, delta = 1/2 <=> r = 0. (4) BRANCH CENSUS [N, seeded scipy sampling, Python-only]: the full D4-fixed flat cusp locus realises only tr(MU) in {1, -1}; the involutive branch has diagonal CONSTANT (0, 1/2, 1/2) to 1e-15 across all samples; the {1, i, -i} branch has d1 varying over [0, 1] and the explicit v33/v40 point sits in its d1 = 0 slice - the same delta value. (5) RESIDUE [P] (recorded): whether the anchor splitting type O(-2)+O(-1)^2 (Mehta-Seshadri side) forces the d1 = 0 slice or selects the involutive branch is the remaining question; delta = 1/2 itself is no longer open
Original classification detail
Theorem: delta = 1/2 is a mu4-torsion identity on the involutive branch; flatness = (MU)^4 = 1 normal form; residue = bundle-side branch selection
Dependencies as recorded
GATE.UWALL.04; GATE.UWALL.05; FLAV.DELTA.01; v20; v30; v33
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- Research contracts
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- 2026-06-24 (Cross-surface consistency sweep: papers, website, gates, figures realigned)Research changelog; TFPT — Introduction and reading guide
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- Front matter / supportResearch contracts
- Research Contract 1 — [formula in source]tfpt research contracts
- 2026-06-11Research changelog; TFPT — Introduction and reading guide
- Front matter / supportTFPT — Introduction and reading guide; Verification reference catalog
- Research Contract 1 — [formula in source]tfpt research contracts
Registered verification modules
- v114_torsion_delta.pyRegistered verification module · registry
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TORSION NORMAL FORM + THE δ = 1/2 THEOREM (executes the v41 'genericity future work'; δ = 1/2 stops being an observation): FLATNESS = μ₄ TORSION [I, symbolic generic M] — ∏ₖ Uᵏ M U⁻ᵏ = (MU)⁴ exactly, so the ℤ₄-family flatness of the (U_wall) bundle IS the torsion statement 'T = MU is a fourth root of unity' (the μ₄ atom = the ORDER of the twisted cusp generator); δ THEOREM [I, both directions]: on the involutive branch (T² = 1) T is a Hermitian reflection 2vv†−1 and the cusp trace splits exactly into |v₁|² = ½ + |v₂| = |v₃| ⇒ diag M = (0, i/2, −i/2) with spec M = {1, ω, ω²} AUTOMATIC — δ = ½ exact and constant on the whole 2-parameter branch (the value v20 used by hand is a μ₄-torsion identity); REFLECTION LEMMA: D₄ forces diag M = (r, z, z̄), r real — ONE real parameter, δ = ½ ⟺ r = 0; BRANCH CENSUS [N, seeded]: only tr(MU) ∈ {1,−1} realised; involutive branch diagonal constant to 1e−15; the {1,i,−i} branch varies with the explicit v33/v40 point in its d₁ = 0 slice; residue [P]: bundle-side branch selection (anchor splitting), δ itself closed
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