# command: /Users/christopheryamba/.pyenv/versions/3.11.9/bin/python3 -I /Users/christopheryamba/YambaCode/Math/parikh-permutation-statistics/checks/check_rigidity.py --full
# working directory: /Users/christopheryamba/YambaCode/Math/parikh-permutation-statistics/checks
# python: 3.11.9 (macOS-27.0.1-arm64-arm-64bit)
# started: 2026-10-21T09:05:01+02:00
== check_rigidity: Theorem A (rigidity) and Lemma 3.2, notes/article2-design.md section 3 and notes/proof-rigidity-local.md  [--full]
   A-i(1) profile level: 1 <= a,c <= 7, 1 <= b <= 8: 174440 cells, 187709371 profile pairs, 37373 rigid cells
      (E) sub-case z=0: 13720
      (E) sub-case z=0+ay: 392
      (E) sub-case z=0+cx: 392
      (E) sub-case z=ay: 6664
      (E) sub-case z=ay+cx: 392
      (E) sub-case z=cx: 6664
      rigid by Ch: 3444
      rigid by Ch+G: 644
      rigid by E: 18228
      rigid by E+Ch: 2352
      rigid by E+Ch+G: 1552
      rigid by E+G: 2172
      rigid by E+N: 2352
      rigid by E+N+Ch+G: 16
      rigid by E+N+G: 1552
      rigid by G: 945
      rigid by N: 3444
      rigid by N+Ch+G: 28
      rigid by N+G: 644
   A-i(2) formula level: 1 <= a,b,c <= 20: 128339000 cells, 4641628 rigid; Prop 3.4 criterion <=> not rigid in every cell (literal per-h form on a, c <= 4)
   A-i(3) word level: positive (a,b,c), a+b+c <= 12: 772626 words, 22099 cells, 11035 rigid
   P3.4: Lemma 3.3/3.3' on all profiles with 1 <= a, b <= 10 (A side; C side with c = a); Prop 3.4 with its witnesses, Cor 3.5 and the layer i+j = b+1 (756 cells) on the A-i(1) range; (3.1) on 919853 profile pairs (a, c <= 5, b <= 6)
   A-ii k=3: totals <= 13 (state-merged search; word by word <= 12): 287 contents, 37856 fibres, 16818 single-class, 123312 classes
   A-ii k=4: totals <= 13 (state-merged search; word by word <= 11): 715 contents, 313742 fibres, 128284 single-class, 1123197 classes
   A-ii k=5: totals <= 13 (state-merged search; word by word <= 9): 1287 contents, 1251245 fibres, 573460 single-class, 3334411 classes
   A-ii k=6: totals <= 13 (state-merged search; word by word <= 9): 1716 contents, 2848959 fibres, 1569076 single-class, 5552639 classes
   A-ii k=7: 1 extra contents, 2025 fibres
   A-ii extra contents (state-merged): (4, 4, 4), (5, 5, 5), (3, 3, 3, 3), (2, 3, 2, 3, 2), (1, 2, 2, 1, 2, 2, 1); on every fibre also per-triple constancy, Cor 4.1, Remark 4.2 and the transfer count per content
   A-iii k=3: 287 contents, psi_2 determines the matrix on 76
   A-iii k=4: 715 contents, psi_2 determines the matrix on 145
   A-iii k=5: 1287 contents, psi_2 determines the matrix on 265
   A-iii k=6: 1716 contents, psi_2 determines the matrix on 393
   A-iv (2^k) rigid vectors, transfer matrix: k=3: 20, k=4: 86, k=5: 364, k=6: 1548, k=7: 6576, k=8: 27944, k=9: 118736, k=10: 504528, k=11: 2143808, k=12: 9109344
   A-iv brute force over vectors agrees for k = 3..10; word level (single-class fibres = rigid vectors) for k = 3..6; section 6 of the note (M, recurrence, both series, roots, M^2 > 0, a_k/a_(k-1) for 12 <= k <= 40) exact; a_40/a_39 = 4.2491405381
   A-iv(GF) (2,)-periodic: N/Q certified (bound B = 8 <= 40 terms); counts k=3..8: [20, 86, 364, 1548, 6576, 27944]
   A-iv(GF) (3,)-periodic: N/Q certified (bound B = 14 <= 40 terms); counts k=3..8: [44, 284, 1520, 8900, 49896, 285652]
   A-iv(GF) (4,)-periodic: N/Q certified (bound B = 22 <= 40 terms); counts k=3..8: [78, 736, 4492, 35262, 238212, 1757228]
   A-iv(GF) (5,)-periodic: N/Q certified (bound B = 32 <= 40 terms); counts k=3..8: [122, 1614, 10646, 109602, 838680, 7784930]
   A-iv(GF) (1, 2)-periodic: N/Q certified (bound B = 7 <= 40 terms); counts k=3..8: [9, 27, 81, 243, 729, 2187]
   A-iv(GF) (2, 3)-periodic: N/Q certified (bound B = 20 <= 40 terms); counts k=3..8: [32, 154, 800, 3644, 19264, 87080]
   A-iv(GF) (1, 2, 2)-periodic: N/Q certified (bound B = 12 <= 40 terms); counts k=3..8: [14, 40, 120, 560, 1600, 4800]
   A-iv(GF) word-level single-class counts compared on 16 (content, claim) pairs
   L3.2: k=3 totals <= 12: 772626 words, k=4 totals <= 11: 4554048 words, k=5 totals <= 9: 978840 words; 28777218 (word, j<m<l) instances, 13923681 with D>0, 13673448 avoiding (m+1)m(m-1); 1180089 with D=0 although w contains (m+1)m(m-1), of which 0 at (j,l)=(m-1,m+1) (so the converse of the vanishing claim fails only for wider j..l), 536472 with j < m-1 = l-2 and 536472 with l > m+1 = j+2
   L3.2(d) by chains: k=3: 772626 words, k=4: 4554048 words, k=5: 978840 words; by brute force over position triples: totals <= 8
   L3.2(f): decreasing word, k <= 7, n_i <= 3: 79137 wide (content, j, m, l) instances, all D = 0 with (m+1)m(m-1) present
   Cor 2.1 on every L3.2 word: 1966200 (N') and 1966200 (Ch') instances
   L1.1: projections, (1.1), the full product and the insertion construction on 182 contents (totals <= 8); conjugation on every A-i(3) word and at profile level for b, c <= 8
   WIT: 477 contents ((k, max n_i, max total) in [(4, 4, 12), (5, 3, 11), (6, 2, 12)]), 73401 non-rigid vectors, 102104 (vector, non-rigid triple) witness pairs built and checked
   forward = backward interval counts on every L3.2 word; occ() cross-check, minor determinant and degenerate cases: totals <= 8; state-merged = word-by-word on 760 contents
   (parallel part 255.3 s with 2 workers)
   ok   A-i(1) profile level: 419055 checks, 0 failures
   ok   A-i(2) formula level: 133510786 checks, 0 failures
   ok   A-i(3) word level: 835517 checks, 0 failures
   ok   A-ii Cor 4.1, per-triple constancy, Remark 4.2: 35125847 checks, 0 failures
   ok   A-ii psi_2-fibres: 4453827 checks, 0 failures
   ok   A-iii completeness: 4007 checks, 0 failures
   ok   A-iv (2^k) rigid vectors: 47 checks, 0 failures
   ok   A-iv transfer count = single-class fibres: 4006 checks, 0 failures
   ok   A-iv(GF) generating-function certificates: 57 checks, 0 failures
   ok   EXA examples of the note (word level): 7 checks, 0 failures
   ok   L1.1 projections, (1.1), conjugation: 3314975 checks, 0 failures
   ok   L3.2 Cor 2.1 (N'), (Ch'), (2.1): 3932400 checks, 0 failures
   ok   L3.2 minor identity: 90293768 checks, 0 failures
   ok   L3.2(d) exact criterion: 30436944 checks, 0 failures
   ok   P3.4 Lemma 3.3 and 3.3': 157373 checks, 0 failures
   ok   P3.4 Prop 3.4 criterion <=> not rigid (formula range): 128643120 checks, 0 failures
   ok   P3.4 Prop 3.4, Cor 3.5, layer, (3.1) (profile level): 1505914 checks, 0 failures
   ok   WIT A(ii) necessity witnesses: 306315 checks, 0 failures
   ok   cross-checks: 9586154 checks, 0 failures
check_rigidity: 442530119 checks, 0 failures, 260.9 s
# exit status: 0
# wall time: 261.0 s
