BEKENSTEIN PHASE-0 SPECTROMETER (exact GF(2) linear algebra) reference constants: 1/pi = 0.31831 2pi = 6.28319 phi^5/pi = 3.53011 fibers are constant BY LINEARITY (kernel cosets); Lean verified this by enumeration at N=2,3 (SharedCutMarginal.lean). All values exact. ============================================================================== STRIP 1xN (open, 2x(N+1) vertices). Subregion = k-face prefix. ============================================================================== -- closure = GLOBAL -- N dimC S/face(tot) marg(1) marg(k=N-1) perface(k=N-1) defect d(N-1) 2 5 5/2 4 4 4 0 3 7 7/3 4 6 3 1 4 9 9/4 4 8 8/3 4/3 5 11 11/5 4 10 5/2 3/2 6 13 13/6 4 12 12/5 8/5 7 15 15/7 4 14 7/3 5/3 8 17 17/8 4 16 16/7 12/7 9 19 19/9 4 18 9/4 7/4 10 21 21/10 4 20 20/9 16/9 11 23 23/11 4 22 11/5 9/5 12 25 25/12 4 24 24/11 20/11 gluing law check (N = 12 host): D(m+n) - D(m) - D(n) = seam D(1+1) - D(1) - D(1) = 2 D(1+2) - D(1) - D(2) = 2 D(1+3) - D(1) - D(3) = 2 D(1+4) - D(1) - D(4) = 2 D(2+1) - D(2) - D(1) = 2 D(2+2) - D(2) - D(2) = 2 D(2+3) - D(2) - D(3) = 2 D(2+4) - D(2) - D(4) = 2 D(3+1) - D(3) - D(1) = 2 D(3+2) - D(3) - D(2) = 2 D(3+3) - D(3) - D(3) = 2 D(3+4) - D(3) - D(4) = 2 D(4+1) - D(4) - D(1) = 2 D(4+2) - D(4) - D(2) = 2 D(4+3) - D(4) - D(3) = 2 D(4+4) - D(4) - D(4) = 2 asymptotic defect/face -> 1.8182 [nearest 2 (|err| = 0.1818)] -- closure = LOCAL -- N dimC S/face(tot) marg(1) marg(k=N-1) perface(k=N-1) defect d(N-1) 2 4 2 3 3 3 1 3 5 5/3 3 4 2 2 4 6 3/2 3 5 5/3 7/3 5 7 7/5 3 6 3/2 5/2 6 8 4/3 3 7 7/5 13/5 7 9 9/7 3 8 4/3 8/3 8 10 5/4 3 9 9/7 19/7 9 11 11/9 3 10 5/4 11/4 10 12 6/5 3 11 11/9 25/9 11 13 13/11 3 12 6/5 14/5 12 14 7/6 3 13 13/11 31/11 gluing law check (N = 12 host): D(m+n) - D(m) - D(n) = seam D(1+1) - D(1) - D(1) = 2 D(1+2) - D(1) - D(2) = 2 D(1+3) - D(1) - D(3) = 2 D(1+4) - D(1) - D(4) = 2 D(2+1) - D(2) - D(1) = 2 D(2+2) - D(2) - D(2) = 2 D(2+3) - D(2) - D(3) = 2 D(2+4) - D(2) - D(4) = 2 D(3+1) - D(3) - D(1) = 2 D(3+2) - D(3) - D(2) = 2 D(3+3) - D(3) - D(3) = 2 D(3+4) - D(3) - D(4) = 2 D(4+1) - D(4) - D(1) = 2 D(4+2) - D(4) - D(2) = 2 D(4+3) - D(4) - D(3) = 2 D(4+4) - D(4) - D(4) = 2 asymptotic defect/face -> 2.8182 [nearest 3 (|err| = 0.1818)] ============================================================================== 2D PATCH n x n faces (open). Subregion = a x a sub-patch (corner). ============================================================================== -- closure = GLOBAL -- n faces dimC S/face(tot) marg(1) marg(axa) perface defect 2 4 8 2 4 4 4 0 3 9 15 5/3 4 9 9/4 7/4 4 16 24 3/2 4 16 16/9 20/9 5 25 35 7/5 4 25 25/16 39/16 6 36 48 4/3 4 36 36/25 64/25 7 49 63 9/7 4 49 49/36 95/36 8 64 80 5/4 4 64 64/49 132/49 (perface column is the JOINT marginal per face of the (n-1)x(n-1) corner sub-patch) -- closure = LOCAL -- n faces dimC S/face(tot) marg(1) marg(axa) perface defect 2 4 5 5/4 3 3 3 1 3 9 7 7/9 3 5 5/4 11/4 4 16 9 9/16 3 7 7/9 29/9 5 25 11 11/25 3 9 9/16 55/16 6 36 13 13/36 3 11 11/25 89/25 7 49 15 15/49 3 13 13/36 131/36 8 64 17 17/64 3 15 15/49 181/49 (perface column is the JOINT marginal per face of the (n-1)x(n-1) corner sub-patch) ============================================================================== TORUS n x n faces (closed surface, the horizon topology class). Subregion = a x a patch on the torus (a = n-1: proper, all but a frame). ============================================================================== -- closure = GLOBAL -- n faces dimC S/face(tot) marg(1) marg(axa) perface defect 3 9 8 8/9 4 8 2 2 4 16 15 15/16 4 15 5/3 7/3 5 25 24 24/25 4 24 3/2 5/2 6 36 35 35/36 4 35 7/5 13/5 7 49 48 48/49 4 48 4/3 8/3 8 64 63 63/64 4 63 9/7 19/7 9 81 80 80/81 4 80 5/4 11/4 10 100 99 99/100 4 99 11/9 25/9 -- closure = LOCAL -- n faces dimC S/face(tot) marg(1) marg(axa) perface defect 3 9 5 5/9 3 5 5/4 11/4 4 16 7 7/16 3 7 7/9 29/9 5 25 9 9/25 3 9 9/16 55/16 6 36 11 11/36 3 11 11/25 89/25 7 49 13 13/49 3 13 13/36 131/36 8 64 15 15/64 3 15 15/49 181/49 9 81 17 17/81 3 17 17/64 239/64 10 100 19 19/100 3 19 19/81 305/81 ============================================================================== PER-PIXEL SUM vs JOINT (the PerPixelRecordAdditivity fork), 2D open patch. sum_i marg(face_i) vs marg(all faces jointly), whole n x n patch as the region inside a larger (n+2) x (n+2) host so the region is proper. ============================================================================== -- closure = GLOBAL -- n sum marg perface joint perface kappa(sum) kappa(joint) 2 16 4 9 9/4 4.000 2.250 3 36 4 16 16/9 4.000 1.778 4 64 4 25 25/16 4.000 1.562 5 100 4 36 36/25 4.000 1.440 6 144 4 49 49/36 4.000 1.361 7 196 4 64 64/49 4.000 1.306 8 256 4 81 81/64 4.000 1.266 -- closure = LOCAL -- n sum marg perface joint perface kappa(sum) kappa(joint) 2 12 3 5 5/4 3.000 1.250 3 27 3 7 7/9 3.000 0.778 4 48 3 9 9/16 3.000 0.562 5 75 3 11 11/25 3.000 0.440 6 108 3 13 13/36 3.000 0.361 7 147 3 15 15/49 3.000 0.306 8 192 3 17 17/64 3.000 0.266 ============================================================================== READING GUIDE ============================================================================== kappa candidates on the table (bits/pixel): 4 = per-pixel traced marginal, GLOBAL closure (LEG-A kappa=4) 3 = per-pixel traced marginal, LOCAL closure (the nullity reading) ~1-2 = joint marginal per face (kappa -> 1 fork) ->0 = joint marginal per face, LOCAL 2D (perimeter law) defect d = 4 - kappa. The judge's Live Bet 1 (defect calibrates 1/pi in G) requires the defect sequence to approach a transcendental; a clean integer asymptote with an integer seam gluing law is the kill condition ("structureless rational") UNLESS the gluing seam itself carries the structure.