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Original ledger classification: Identity/Lattice (exact sympy + exact Cl(16); 40-digit gap-certified RP inertia certificates)

Recorded statement: THE WOIT-ALPHA MILESTONE (2026-07-22): the real structure exists, and free reflection positivity picks the SAME family -- the alpha stage of WOIT.OS.TWISTOR.01 executed as one module (Part W = Theta, Part R = free RP). [E] THETA…

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WOIT.THETA.FREE.01

Original ledger classification: Identity/Lattice (exact sympy + exact Cl(16); 40-digit gap-certified RP inertia certificates)

Original claim text

THE WOIT-ALPHA MILESTONE (2026-07-22): the real structure exists, and free reflection positivity picks the SAME family -- the alpha stage of WOIT.OS.TWISTOR.01 executed as one module (Part W = Theta, Part R = free RP). [E] THETA CLASSIFICATION COMPLETE: exactly TWO families of anti-linear structures on C^2 normalise the clock rho = diag(i,1) -- family D (z -> mu conj(z), |mu| = 1): Theta rho Theta = rho^-1 EXACTLY, Theta^2 = +1 (-1 IMPOSSIBLE: M conj(M) = diag(|mu|^2,1), Kramers-free), inverts the deck, fixed set = the great circle through the deck poles; family A (z -> mu/conj(z)): centralises the clock projectively (NEVER inverts), Theta^2 = +-1 per mu, fixed set |z| = 1 resp. empty; ROLE SEPARATION: family A defines the euclidean section (Woit's rho_tw side: rho_tw^2 = -1, no real points -- replicated exactly on C^4), family D reflects it (the OS conjugation, sigma_std = conjugation with real points RP^3). [E] MARK PINNING: mu in mu4 iff the marks are preserved -- a 4-element mu4-torsor (2 mark axes + 2 silver-midpoint axes), clock conjugation rho Theta_mu rho^-1 = Theta_{-mu} => 2 clock orbits. [E] SPIN/FOCK: Z8 plane without phase leaks (conj(zeta8^k) = zeta8^-k); Cl(16): Theta_Fock = U_r o K with Theta_Fock^2 = 2^7 > 0 (normalised +1), inverts the clock tower (V -> 4096 V^-1) and normalises the deck; KRAMERS DICHOTOMY: the deck-induced candidate Theta_t = Utilde o K has Theta_t^2 = (-1)^F -- the v510 split/nonsplit dichotomy IS the Theta^2 = +1 vs (-1)^F dichotomy; the deck does NOT furnish Theta, the seam-circle REFLECTION does. [E] KILL TEST 1 (free level): does NOT fire -- Theta exists with Theta^2 = +1 AND Theta rho Theta = rho^-1 and satisfies marks + deck normalisation + spin/Fock simultaneously. [E] FREE RP: NS vacuum C(d) = (2/N)/sin(pi d/N), C^2 = -1 exact; preconditions exact (RCR^T = -C, SCS^T = C, RSR^-1 = S^-1; the deck fails anti-invariance); BOND CUT: one-particle Gram (8,0,0) PD (min EV 1.888e-3, 40 digits), even deg <= 2 (29,0,0) PD, N = 8 full half-sided algebra PD (even + odd each (8,0,0)) -- RP without degree truncation; twist eta = +i forced (eta = 1 non-Hermitian); CONTROLS: site cut fails exactly (det = 0, inertia (3,3,1); continuum Cauchy-Stieltjes control strictly PD -- a lattice-placement artifact, RP selects the site placement relative to the marks); the clock-centralising family-A structure fails RP structurally ((4,4,0)) -- RP and Theta rho Theta = rho^-1 select the SAME family; bonus: anti-chiral state -> odd sector (0,8,0) ND (the free shadow of kill test 3). [C]/[O] NOT shown: the interacting algebra A_hol (kill test 1 stays live there), gauge-fixed RP (kill test 2), OS reconstruction, chirality without mirrors (kill tests 3/6), mu4 incidence (kill test 7) -- the WOIT-beta/gamma roadmap. NO marker moves; WOIT.OS.TWISTOR.01 stays Open.

Original classification detail

[E] the free/equivariant alpha stage (Theta classification, exact relations, kill test 1 discharged at the free level, free RP on the bond cut selecting the same family); [C]/[O] everything interacting -- kill test 1 stays live on A_hol; no marker moves

Dependencies as recorded

WOIT.OS.TWISTOR.01; SEAM.BIT.FREEDOM.01; SEAM.CLOCK.RIGIDITY.01; SEAM.BIT.ORIGIN.01; CELEST.SEAM.01

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Registered verification modules

  • v519_woit_theta_rp_free.pyRegistered verification module · frontier
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    WOIT.THETA.FREE.01: the WOIT-alpha milestone of the OS twistor bridge WOIT.OS.TWISTOR.01 -- 'the real structure exists, and free reflection positivity picks the SAME family' (Part W = the anti-linear Theta: classification, exact relations, kill test 1 on the free/equivariant level; Part R = reflection positivity of the free 16-Majorana NS seam system under the pinned Theta; NO marker moves -- WOIT.OS.TWISTOR.01 stays [O]: kill test 1 is discharged on the FREE level only and stays formally live on the interacting algebra). Exact sympy + exact Cl(16) Fractions; the RP definiteness certificates are 40-digit spectra of exact Hermitian matrices (tol 1e-25). [E] THETA CLASSIFICATION COMPLETE: exactly TWO families of anti-linear structures on C^2 normalise the clock rho = diag(i,1) -- family D (z -> mu conj(z), |mu| = 1): Theta rho Theta = rho^-1 EXACTLY, Theta^2 = +1 (-1 IMPOSSIBLE: M conj(M) = diag(|mu|^2, 1) -- Kramers-free), inverts the deck, fixed set = the great circle through the deck poles; family A (z -> mu/conj(z)): centralises the clock projectively (NEVER inverts), Theta^2 = +-1 per mu, fixed set |z| = 1 resp. empty. ROLE SEPARATION: family A defines the euclidean section (Woit's rho_tw side: rho_tw^2 = -1, no real points -- replicated exactly on C^4), family D reflects it (the OS conjugation; sigma_std = the standard conjugation with real points RP^3). [E] MARK PINNING: z -> mu conj(z) preserves the mu4 marks iff mu in mu4 -- a 4-element mu4-TORSOR (2 mark axes + 2 silver-midpoint axes); clock conjugation rho Theta_mu rho^-1 = Theta_{-mu} => 2 clock orbits. [E] SPIN/FOCK: Z8 plane without phase leaks (conj(zeta8^k) = zeta8^-k); Cl(16): Theta_Fock = U_r o K with Theta_Fock^2 = 2^7 > 0 (normalised +1), inverts the clock tower (V -> 4096 V^-1, exact scalar via the implementer route) and normalises the deck (256 Utilde^-1); KRAMERS DICHOTOMY: the deck-induced candidate Theta_t = Utilde o K has Theta_t^2 = 256 gamma_1..gamma_16 = (-1)^F -- the v510 split/nonsplit dichotomy IS the Theta^2 = +1 vs (-1)^F dichotomy; the deck does NOT furnish Theta, the seam-circle REFLECTION does. [E] KILL TEST 1 (free level): does NOT fire -- Theta exists with Theta^2 = +1 AND Theta rho Theta = rho^-1 and satisfies all three side conditions simultaneously (marks, deck normalisation, spin/Fock). [E] FREE RP: NS vacuum C(d) = (2/N)/sin(pi d/N), C^2 = -1 exact; preconditions exact (R C R^T = -C, S C S^T = C, R S R^-1 = S^-1; the deck FAILS anti-invariance); BOND CUT: one-particle Gram (8,0,0) PD (min eigenvalue 1.888e-3 at 40 digits), even deg <= 2 sector (29,0,0) PD, N = 8 complete half-sided algebra PD (even + odd each (8,0,0)) -- RP with NO degree truncation; twist eta = +i FORCED (eta = 1 non-Hermitian); CONTROLS: the site cut fails exactly (det = 0, inertia (3,3,1); continuum control: Cauchy-Stieltjes positive -- a lattice-PLACEMENT artifact, RP selects the site placement relative to the marks); the clock-CENTRALISING family-A structure fails RP structurally ((4,4,0)) -- RP and Theta rho Theta = rho^-1 select the SAME family; bonus: the anti-chiral state flips the odd sector to (0,8,0) ND (the free shadow of kill test 3). [C]/[O] NOT shown: the interacting algebra A_hol (kill test 1 stays live there), gauge-fixed RP (kill test 2), OS reconstruction, chirality without mirrors (kill tests 3/6 -- only the free odd-sector shadow), mu4 incidence (kill test 7); WOIT-beta/gamma roadmap: (beta-1) Theta on the gauge-invariant subalgebra + gauge-fixed RP on the equivariant SDYM(E8) sector; (beta-2) the OS quotient of the free system explicit; (beta-3) the PT <-> PT* duality vs sigma_std typed; (gamma) the chirality theorem + the mark incidence. Two contract PRECISIONS (prose, no marker): (i) 'Theta induced by the seam reflection' means the seam-circle REFLECTION, not the deck (the deck furnishes the (-1)^F Kramers class -- the v510 bridge); (ii) the marks sit RP-side on the BOND MIDPOINTS of the 16-Majorana circle.

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