============================================================================================
A. the 1-D parity maps: J^2 = I exactly, masses EVEN, derivative ODD
============================================================================================
 Nx   M  |Pt^2-I|  |Pb^2-I|   Mtt EVEN   Mbb EVEN    dbt ODD  dbt EVEN(wrong)
  2   4   0.0e+00   0.0e+00   8.40e-17   8.40e-17   7.95e-17         2.00e+00
  2   5   0.0e+00   0.0e+00   8.11e-17   8.11e-17   1.39e-16         2.00e+00
  2   6   0.0e+00   0.0e+00   1.68e-16   1.68e-16   6.59e-17         2.00e+00
  2   7   0.0e+00   0.0e+00   8.40e-17   8.40e-17   1.11e-16         2.00e+00
  2   8   0.0e+00   0.0e+00   8.40e-17   8.40e-17   1.03e-16         2.00e+00
  3   4   0.0e+00   0.0e+00   6.30e-17   6.30e-17   7.95e-17         2.00e+00
  3   5   0.0e+00   0.0e+00   8.11e-17   8.11e-17   1.39e-16         2.00e+00
  3   6   0.0e+00   0.0e+00   1.89e-16   1.89e-16   6.59e-17         2.00e+00
  3   7   0.0e+00   0.0e+00   6.30e-17   6.30e-17   1.11e-16         2.00e+00
  3   8   0.0e+00   0.0e+00   6.30e-17   6.30e-17   1.03e-16         2.00e+00
  4   4   0.0e+00   0.0e+00   8.40e-17   8.40e-17   7.95e-17         2.00e+00
  4   5   0.0e+00   0.0e+00   8.11e-17   8.11e-17   1.39e-16         2.00e+00
  4   6   0.0e+00   0.0e+00   1.68e-16   1.68e-16   6.59e-17         2.00e+00
  4   7   0.0e+00   0.0e+00   8.40e-17   8.40e-17   1.11e-16         2.00e+00
  4   8   0.0e+00   0.0e+00   8.40e-17   8.40e-17   1.03e-16         2.00e+00
  5   4   0.0e+00   0.0e+00   5.25e-17   5.25e-17   7.95e-17         2.00e+00
  5   5   0.0e+00   0.0e+00   8.11e-17   8.11e-17   1.39e-16         2.00e+00
  5   6   0.0e+00   0.0e+00   1.58e-16   1.58e-16   6.59e-17         2.00e+00
  5   7   0.0e+00   0.0e+00   1.05e-16   1.05e-16   1.11e-16         2.00e+00
  5   8   0.0e+00   0.0e+00   5.25e-17   5.25e-17   1.03e-16         2.00e+00

J^2 = I exactly on 20/20 (Nx, M) combinations
mass EVEN envelope   5.25e-17 .. 1.89e-16
deriv ODD envelope   6.59e-17 .. 1.39e-16
deriv EVEN (the WRONG sign) 2.00e+00 .. 2.00e+00  <- must be O(1)

============================================================================================
B. the 2-D blocks: nine eps masses EVEN, four derivative blocks ODD
============================================================================================

centro pair generic (2,2), M=6
  block    test trial  expect   |P M P - s M|/|M|   |P M P + s M|/|M|(wrong)
  Ggram1   V1   V1         +1           2.109e-16                  2.000e+00
  Ggram2   V2   V2         +1           2.109e-16                  2.000e+00
  Gw       Vw   Vw         +1           2.109e-16                  2.000e+00
  CwE1     Vw   V1         -1           1.989e-16                  2.000e+00
  CwE2     Vw   V2         -1           2.198e-16                  2.000e+00
  P13      V1   V3         -1           2.198e-16                  2.000e+00
  P23      V2   V3         -1           1.989e-16                  2.000e+00
  A11      V1   V1         +1           1.757e-16                  2.000e+00
  A12      V1   V2         +1           2.267e-16                  2.000e+00
  A13      V1   V3         +1           1.598e-16                  2.000e+00
  A21      V2   V1         +1           2.043e-16                  2.000e+00
  A22      V2   V2         +1           2.008e-16                  2.000e+00
  A23      V2   V3         +1           2.018e-16                  2.000e+00
  A31      V3   V1         +1           1.851e-16                  2.000e+00
  A32      V3   V2         +1           2.028e-16                  2.000e+00
  A33      V3   V3         +1           1.814e-16                  2.000e+00

(3,3) ring lc + interior lossy, M=5
  block    test trial  expect   |P M P - s M|/|M|   |P M P + s M|/|M|(wrong)
  Ggram1   V1   V1         +1           8.112e-17                  2.000e+00
  Ggram2   V2   V2         +1           8.112e-17                  2.000e+00
  Gw       Vw   Vw         +1           8.112e-17                  2.000e+00
  CwE1     Vw   V1         -1           1.388e-16                  2.000e+00
  CwE2     Vw   V2         -1           1.388e-16                  2.000e+00
  P13      V1   V3         -1           1.388e-16                  2.000e+00
  P23      V2   V3         -1           1.388e-16                  2.000e+00
  A11      V1   V1         +1           8.112e-17                  2.000e+00
  A12      V1   V2         +1           8.112e-17                  2.000e+00
  A13      V1   V3         +1           8.112e-17                  2.000e+00
  A21      V2   V1         +1           8.112e-17                  2.000e+00
  A22      V2   V2         +1           8.112e-17                  2.000e+00
  A23      V2   V3         +1           8.112e-17                  2.000e+00
  A31      V3   V1         +1           8.112e-17                  2.000e+00
  A32      V3   V2         +1           8.112e-17                  2.000e+00
  A33      V3   V3         +1           8.112e-17                  2.000e+00

centro pair NON-RECIPROCAL (2,2), M=7
  block    test trial  expect   |P M P - s M|/|M|   |P M P + s M|/|M|(wrong)
  Ggram1   V1   V1         +1           1.406e-16                  2.000e+00
  Ggram2   V2   V2         +1           1.406e-16                  2.000e+00
  Gw       Vw   Vw         +1           1.406e-16                  2.000e+00
  CwE1     Vw   V1         -1           1.989e-16                  2.000e+00
  CwE2     Vw   V2         -1           1.099e-16                  2.000e+00
  P13      V1   V3         -1           1.099e-16                  2.000e+00
  P23      V2   V3         -1           1.989e-16                  2.000e+00
  A11      V1   V1         +1           1.656e-16                  2.000e+00
  A12      V1   V2         +1           1.629e-16                  2.000e+00
  A13      V1   V3         +1           1.234e-16                  2.000e+00
  A21      V2   V1         +1           1.629e-16                  2.000e+00
  A22      V2   V2         +1           1.228e-16                  2.000e+00
  A23      V2   V3         +1           8.690e-17                  2.000e+00
  A31      V3   V1         +1           1.234e-16                  2.000e+00
  A32      V3   V2         +1           8.690e-17                  2.000e+00
  A33      V3   V3         +1           1.031e-16                  2.000e+00

EVEN blocks (36): residual 8.11e-17 .. 2.27e-16; the ODD reading of the same blocks 2.00e+00 .. 2.00e+00
ODD blocks  (12): residual 1.10e-16 .. 2.20e-16; the EVEN reading 2.00e+00 .. 2.00e+00

============================================================================================
B2. the two ELIMINATIONS carry the derived signs (E3 EVEN, G3 ODD)
============================================================================================
centro pair generic (2,2)          M=6  E3 -> +P3 E3 : 7.061e-14   (-P3 reading 2.000e+00)
                                        G3 -> -Pw G3 : 1.373e-13   (+Pw reading 2.000e+00)
(3,3) ring lc + interior lossy     M=5  E3 -> +P3 E3 : 3.473e-14   (-P3 reading 2.000e+00)
                                        G3 -> -Pw G3 : 2.692e-14   (+Pw reading 2.000e+00)

============================================================================================
C. WRONG patterns.  dA = max|R A R + A| / max|A| ; dB likewise.
============================================================================================

SHIPPED  blkdiag(P1,P2,P2,P1)
   signs                     dA          dB  engages(1e-10)
   (1, 1, 1, 1)       2.000e+00   8.112e-17  False         
   (1, 1, 1, -1)      1.592e+00   8.112e-17  False         
   (1, 1, -1, 1)      2.000e+00   8.112e-17  False         
   (1, 1, -1, -1)     2.329e-15   8.112e-17  True            <== PASSES
   (1, -1, 1, 1)      2.000e+00   8.112e-17  False         
   (1, -1, 1, -1)     2.420e-01   8.112e-17  False         
   (1, -1, -1, 1)     2.000e+00   8.112e-17  False         
   (1, -1, -1, -1)    1.592e+00   8.112e-17  False         
   (-1, 1, 1, 1)      1.592e+00   8.112e-17  False         
   (-1, 1, 1, -1)     2.000e+00   8.112e-17  False         
   (-1, 1, -1, 1)     2.420e-01   8.112e-17  False         
   (-1, 1, -1, -1)    2.000e+00   8.112e-17  False         
   (-1, -1, 1, 1)     2.329e-15   8.112e-17  True            <== PASSES
   (-1, -1, 1, -1)    2.000e+00   8.112e-17  False         
   (-1, -1, -1, 1)    1.592e+00   8.112e-17  False         
   (-1, -1, -1, -1)   2.000e+00   8.112e-17  False         

WRONG-perm blkdiag(P1,P2,P1,P2)
   signs                     dA          dB  engages(1e-10)
   (1, 1, 1, 1)       1.509e+00   9.280e-01  False         
   (1, 1, 1, -1)      1.509e+00   9.280e-01  False         
   (1, 1, -1, 1)      1.225e+00   9.280e-01  False         
   (1, 1, -1, -1)     1.079e+00   9.280e-01  False         
   (1, -1, 1, 1)      1.225e+00   9.280e-01  False         
   (1, -1, 1, -1)     1.079e+00   9.280e-01  False         
   (1, -1, -1, 1)     1.509e+00   9.280e-01  False         
   (1, -1, -1, -1)    1.509e+00   9.280e-01  False         
   (-1, 1, 1, 1)      1.509e+00   9.280e-01  False         
   (-1, 1, 1, -1)     1.509e+00   9.280e-01  False         
   (-1, 1, -1, 1)     1.079e+00   9.280e-01  False         
   (-1, 1, -1, -1)    1.225e+00   9.280e-01  False         
   (-1, -1, 1, 1)     1.079e+00   9.280e-01  False         
   (-1, -1, 1, -1)    1.225e+00   9.280e-01  False         
   (-1, -1, -1, 1)    1.509e+00   9.280e-01  False         
   (-1, -1, -1, -1)   1.509e+00   9.280e-01  False         

WRONG-perm blkdiag(P1,P1,P2,P2)
   signs                     dA          dB  engages(1e-10)
   (1, 1, 1, 1)       1.509e+00   9.280e-01  False         
   (1, 1, 1, -1)      1.509e+00   9.280e-01  False         
   (1, 1, -1, 1)      1.225e+00   9.280e-01  False         
   (1, 1, -1, -1)     1.079e+00   9.280e-01  False         
   (1, -1, 1, 1)      1.225e+00   9.280e-01  False         
   (1, -1, 1, -1)     1.079e+00   9.280e-01  False         
   (1, -1, -1, 1)     1.509e+00   9.280e-01  False         
   (1, -1, -1, -1)    1.509e+00   9.280e-01  False         
   (-1, 1, 1, 1)      1.509e+00   9.280e-01  False         
   (-1, 1, 1, -1)     1.509e+00   9.280e-01  False         
   (-1, 1, -1, 1)     1.079e+00   9.280e-01  False         
   (-1, 1, -1, -1)    1.225e+00   9.280e-01  False         
   (-1, -1, 1, 1)     1.079e+00   9.280e-01  False         
   (-1, -1, 1, -1)    1.225e+00   9.280e-01  False         
   (-1, -1, -1, 1)    1.509e+00   9.280e-01  False         
   (-1, -1, -1, -1)   1.509e+00   9.280e-01  False         

WRONG-perm all-V3
   signs                     dA          dB  engages(1e-10)
   (1, 1, 1, 1)       2.000e+00   7.222e-01  False         
   (1, 1, 1, -1)      1.592e+00   7.222e-01  False         
   (1, 1, -1, 1)      2.000e+00   7.222e-01  False         
   (1, 1, -1, -1)     7.660e-01   7.222e-01  False         
   (1, -1, 1, 1)      2.000e+00   7.222e-01  False         
   (1, -1, 1, -1)     7.660e-01   7.222e-01  False         
   (1, -1, -1, 1)     2.000e+00   7.222e-01  False         
   (1, -1, -1, -1)    1.592e+00   7.222e-01  False         
   (-1, 1, 1, 1)      1.592e+00   7.222e-01  False         
   (-1, 1, 1, -1)     2.000e+00   7.222e-01  False         
   (-1, 1, -1, 1)     7.660e-01   7.222e-01  False         
   (-1, 1, -1, -1)    2.000e+00   7.222e-01  False         
   (-1, -1, 1, 1)     7.660e-01   7.222e-01  False         
   (-1, -1, 1, -1)    2.000e+00   7.222e-01  False         
   (-1, -1, -1, 1)    1.592e+00   7.222e-01  False         
   (-1, -1, -1, -1)   2.000e+00   7.222e-01  False         

2 of 64 (pattern, sign) combinations satisfy the structure: [('SHIPPED  blkdiag(P1,P2,P2,P1)', [1, 1, -1, -1]), ('SHIPPED  blkdiag(P1,P2,P2,P1)', [-1, -1, 1, 1])]
smallest FAILING dA 2.420e-01 -- 9.4 decades above the 1e-10 bar

C2.  corruptions of _stag_parity_1d (the map, not the block signs)
   variant                                   dA          dB   |R^2-I| engages 
   shipped                            2.329e-15   8.112e-17   0.0e+00 True    
   bubble sign -> +1                  3.309e-01   2.778e-01   0.0e+00 False   
   bubble sign flipped                9.888e-01   8.333e-01   0.0e+00 False   
   half-hats do NOT swap              4.186e-01   2.778e-01   0.0e+00 False   
   identity (no parity)               6.954e-02   0.000e+00   0.0e+00 False   
