Follow a result back to its evidence.

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.

Linked claims

WOIT.OS.TWISTOR.01WOIT.THETA.FREE.01

Paper sections

Source section 1Source section 2Source section 3Source section 4Source section 5Source section 6Source section 7Source section 8Source section 9Source section 10Source section 11Source section 12Source section 13Source section 14Source section 15Source section 16Source section 17Source section 18Source section 19Source section 20Source section 21Source section 22Source section 23Source section 24Source section 25Source section 26Source section 27Source section 28Source section 29Source section 30Source section 31Source section 32Source section 33Source section 34Source section 35