"""v445 -- SEAM.RIGIDITY.FORCING.01: the rigidity FORCING theorem -- it upgrades v442's "block-diagonal PERMITS commuting" to "commuting with the order-4 clock FORCES block-diagonal", isolating the single remaining physical premise. v442 (SEAM.RIGIDITY.UNIFORM.01) showed, uniform in N, that a mu4-character-block-diagonal K commutes with the clock rho=diag(i^n) -- one direction (block => commute), i.e. block structure is PERMITTED. The open residual of v398/v442 was that RP+gap+holomorphy FORCE Lambda_Sigma block-diagonal (vs merely permit). This module proves the CONVERSE, load-bearing direction as an EXACT linear-algebra fact: the commutant of the order-4 clock is EXACTLY the mu4-character block algebra, with an exact dimension and zero off-block slack, and the clock ORDER (4 = |mu4| = h(A3)) is what fixes that structure -- an order-2 clock has a strictly larger commutant. So commuting FORCES block-diagonality; the only thing left to assume is that the physical transfer commutes with the clock (clock- invariance of the seam state), the cited rigidity/BW step. [E] 1. FORCING (commute <=> block-diagonal, EXACT). for rho=diag(i^n), [rho,K]=0 iff K_ij=0 whenever i!=j mod 4 -- proved ENTRYWISE for every (i,j) and swept N=4..128. The converse of v442: commuting FORCES mu4-block-diagonality, not merely permits. [E] 2. EXACT COMMUTANT DIMENSION. dim{K:[rho,K]=0} = sum_s n_s^2 (n_s the four sector sizes), verified as the ad_rho nullspace; it is a PROPER subspace of the N^2 full algebra (sum n_s^2 < N^2) -- the forcing is a genuine, quantitative restriction. [E] 3. OFF-BLOCK <=> NON-COMMUTING (sharpness). every off-block matrix unit E_ij (i!=j mod 4) has ||[rho,E_ij]|| = |i^i - i^j| >= sqrt(2) > 0 -- no off-block direction survives; the forcing leaves zero slack. [E] 4. THE ORDER IS THE DISCRIMINATOR. the order-4 clock gives 4 sectors and commutant dim sum n_s^2(4); the order-2 clock rho2=diag((-1)^n) gives only 2 sectors and a STRICTLY LARGER commutant -- so the FOUR Gauss-Bonnet marks (order 4) force strictly more block structure than two; only the order-4/index-4 extension is (E8)_1 (det 1; v281/v154/v351), the index-2 would be SO(16) (det 4). [E] 5. NONDEGENERACY. rho has EXACTLY 4 distinct eigenvalues {1,i,-1,-i} for all N>=4, so the four sectors never merge -- a clock with a repeated eigenvalue would collapse sectors; distinctness is what makes the block decomposition unique. [C] 6. SYNTHESIS (force, not permit). commuting with the order-4 mu4 clock is EQUIVALENT to mu4-block-diagonality (an iff, exact, uniform in N), so RP- definability (the transfer fixed by RP+reflection, v54/v398) PLUS clock-invariance of the state FORCE the block-diagonal Lambda_Sigma -- v442's "permit" upgraded to "force" for the linear-algebra core. [C]/[O] 7. VERDICT. the forcing is a proved iff; the SINGLE remaining premise is that the physical transfer Lambda_Sigma commutes with the clock on the full L^2 (clock-invariance = QGEO.SYM.01, downstream of SEAM.EQUIV.01 + BW; v323/v335). SEAM.RIGIDITY.01/SEAM.EQUIV.01 stays [O] -- but the gap is now precisely "transfer in commutant", not "does block-diagonal permit". Python-only (numpy; exact integer/commutant linear algebra; the order-4 => index-4 => E8 det facts it leans on are Wolfram-mirrored via v89/v281/v422). """ import numpy as np from tfpt_constants import check, summary, reset, g_car, N_fam def _sector_sizes(N, m): return [sum(1 for k in range(N) if k % m == s) for s in range(m)] def _commutant_dim_via_nullspace(N): """dim ker(ad_rho) for rho=diag(i^n), computed as an N^2 x N^2 SVD nullspace.""" rho = np.diag([(1j) ** n for n in range(N)]) I = np.eye(N) ad = np.kron(I, rho) - np.kron(rho.T, I) # vec(rho K - K rho) s = np.linalg.svd(ad, compute_uv=False) return int(np.sum(s < 1e-9)) def run(): reset() print("v445 SEAM.RIGIDITY.FORCING: commuting with the order-4 clock FORCES " "mu4-block-diagonality (the converse of v442)") # ---- 1. forcing: commute <=> block-diagonal, entrywise, swept N ---- iff_ok = True for N in (4, 8, 12, 16, 32, 64, 128): rd = np.array([(1j) ** n for n in range(N)]) for i in range(N): for j in range(N): commutes0 = abs(rd[i] - rd[j]) < 1e-12 # [rho,E_ij]=0 same_sector = (i % 4) == (j % 4) if commutes0 != same_sector: iff_ok = False check("FORCING (commute <=> block-diagonal, EXACT) [E]: for rho=diag(i^n), [rho,K]=0 " "iff K_ij=0 whenever i!=j mod 4 -- verified ENTRYWISE for every (i,j), swept " "N=4..128. The converse of v442: commuting FORCES mu4-block-diagonality, not " "merely permits it", iff_ok) # ---- 2. exact commutant dimension = sum n_s^2, proper subspace ---- dim_ok = True proper_ok = True for N in (4, 8, 12, 16): nd = _commutant_dim_via_nullspace(N) formula = sum(n * n for n in _sector_sizes(N, 4)) if nd != formula: dim_ok = False if not (formula < N * N): proper_ok = False check("EXACT COMMUTANT DIMENSION [E]: dim{K:[rho,K]=0} = sum_s n_s^2 (the four sector " "sizes), verified as the ad_rho nullspace for N=4..16 (e.g. N=16: 64=4*4^2 < " "256=N^2); a PROPER subspace of the full algebra -- the forcing is a genuine, " "quantitative restriction", dim_ok and proper_ok) # ---- 3. off-block <=> non-commuting (sharpness) ---- N = 16 rd = np.array([(1j) ** n for n in range(N)]) min_offblock = min(abs(rd[i] - rd[j]) for i in range(N) for j in range(N) if (i % 4) != (j % 4)) check("OFF-BLOCK <=> NON-COMMUTING (sharpness) [E]: every off-block matrix unit E_ij " "(i!=j mod 4) has ||[rho,E_ij]||=|i^i - i^j|=%.4f >= sqrt(2) > 0 -- no off-block " "direction survives; the forcing leaves zero slack" % min_offblock, min_offblock >= np.sqrt(2) - 1e-9) # ---- 4. the order is the discriminator (4 marks force more than 2) ---- order_strict = True for N in (4, 8, 16, 32, 64): d4 = sum(n * n for n in _sector_sizes(N, 4)) d2 = sum(n * n for n in _sector_sizes(N, 2)) if not (d4 < d2): order_strict = False check("THE ORDER IS THE DISCRIMINATOR [E]: the order-4 clock gives 4 sectors and " "commutant dim sum n_s^2(4); the order-2 clock diag((-1)^n) gives 2 sectors and " "a STRICTLY LARGER commutant (e.g. N=16: 64 vs 128) -- the FOUR Gauss-Bonnet " "marks (order 4=|mu4|=h(A3)) force strictly more block structure than two; only " "the order-4/index-4 extension is (E8)_1 (det 1; v281/v154/v351)", order_strict) # ---- 5. nondegeneracy: exactly 4 distinct eigenvalues ---- nondeg = True for N in (4, 7, 16, 33, 128): rd = np.array([(1j) ** n for n in range(N)]) if len({np.round(x, 9) for x in rd}) != 4: nondeg = False check("NONDEGENERACY [E]: rho has EXACTLY 4 distinct eigenvalues {1,i,-1,-i} for all " "N>=4 (N=4,7,16,33,128 checked), so the four sectors never merge -- a clock with " "a repeated eigenvalue would collapse sectors; distinctness makes the block " "decomposition unique", nondeg) # ---- 6. synthesis (force, not permit), typed [C] ---- forcing = iff_ok and dim_ok and proper_ok and order_strict and nondeg check("SYNTHESIS (force, not permit) [C]: commuting with the order-4 mu4 clock is " "EQUIVALENT to mu4-block-diagonality (an iff, exact, uniform in N, |mu4|=%d " "marks => 4 sectors), so RP-definability (transfer fixed by RP+reflection, " "v54/v398) PLUS clock-invariance of the state FORCE the block-diagonal " "Lambda_Sigma -- v442's 'permit' upgraded to 'force' for the linear-algebra core" % (g_car - 1), forcing and (g_car - 1) == 4) # ---- 7. verdict, typed [C]/[O] ---- check("VERDICT [C]/[O]: the forcing is a proved iff; the SINGLE remaining premise is " "that the physical transfer Lambda_Sigma commutes with the clock on the full " "L^2 (clock-invariance = QGEO.SYM.01, downstream of SEAM.EQUIV.01 + BW; " "v323/v335). SEAM.RIGIDITY.01/SEAM.EQUIV.01 stays [O] -- but the gap is now " "precisely 'transfer in commutant', not 'does block-diagonal permit'", forcing and g_car == 5 and N_fam == 3) return summary("v445 SEAM.RIGIDITY.FORCING: commuting with the order-4 mu4 clock FORCES " "mu4-block-diagonality -- the commutant is EXACTLY the 4-sector block " "algebra (entrywise iff swept N=4..128; exact dim sum n_s^2 < N^2; " "off-block slack zero; order-2 commutant strictly larger; 4 distinct " "eigenvalues). Upgrades v442's 'permit' to 'force'; the single residual " "is the transfer lying in the commutant (clock-invariance), SEAM.EQUIV.01 " "stays [O]") if __name__ == "__main__": raise SystemExit(1 if run() else 0)