### Floating-point LP explorations (HiGHS via scipy; NOT part of any proof). Date (UTC): 2026-10-10T06:03:33Z

## 1. Shift-invariant-marginal LP on the k x k window, word A B^(L-1)   (scripts/marg_lp.py k L)
k=2 L=2: LP value 3.0 time 0.001405954360961914
k=3 L=3: LP value 2.0 time 0.0028810501098632812
k=4 L=4: LP value 1.6 time 0.12206721305847168

## 2. Same LP for all binary words of length <= 4, window len(w) x len(w)   (scripts/marg_lp_word.py)
AB: LP value 3.000000000   known C_2 = 3
ABB: LP value 2.000000000   known C_2 = 2
ABA: LP value 3.000000000   known C_2 = 3
AABB: LP value 1.500000000   known C_2 = 1.5
ABAB: LP value 3.000000000   known C_2 = 3
ABBA: LP value 2.000000000   known C_2 = 2
AABA: LP value 2.000000000   known C_2 = 2
ABBB: LP value 1.600000000   known C_2 = 1.6

## 3. Halberstam--Schildkraut local LP (their Prop. 5.2 / Section 5.3), word ABBB   (scripts/hs_lp.py SHAPE 4)
box5 (5x5 window; reproduces 59526/35459 = 1.6787275...): RESULT box5 L 4 bound 1.6787275444879997
hs24 (24-cell window used by H-S for BABBB): RESULT hs24 L 4 bound 1.7009674582233947
box6 (6x6 window, run of this session, 319 s): RESULT box6 L 4 bound 1.6474464579901151
box7 (7x7 window): stopped after 800 s without convergence; last master-LP value 1.609395230 (a lower bound for the final value of that LP)

## 4. ABBB, 4x4 window: bound from D4-symmetrised monomial potentials by degree / bounding box   (scripts/poly_dual2.py)
deg 2 bbox 3x4: orbits 23 classes 9 features 14 constraints 8294 bound 1.786121673
deg 3 bbox 2x2: orbits 17 classes 4 features 13 constraints 6208 bound 1.795918367
deg 3 bbox 2x3: orbits 39 classes 10 features 29 constraints 8138 bound 1.701547432
deg 3 bbox 3x3: orbits 55 classes 16 features 39 constraints 8294 bound 1.649587412
deg 3 bbox 2x4: orbits 57 classes 17 features 40 constraints 8532 bound 1.677145217
deg 3 bbox 3x4: orbits 91 classes 33 features 58 constraints 8534 bound 1.600000000
deg 4 bbox 3x3: orbits 118 classes 36 features 82 constraints 8295 bound 1.625000000
deg 4 bbox 2x4: orbits 115 classes 36 features 79 constraints 8535 bound 1.666666667
deg 9 bbox 3x3: orbits 259 classes 85 features 174 constraints 8295 bound 1.625000000
deg 8 bbox 2x4: orbits 191 classes 62 features 129 constraints 8535 bound 1.666666667

## 5. ABBB, 4x4 window: forced-tight pattern orbits and best uniform slack   (scripts/class_slack.py deg)
deg 3: orbits 91 classes 33 features 58 pattern orbits 8548
forced tight: 2 partial: 0
status 0 max-min slack on the rest: 0.0013179133445226981
deg 4: orbits 279 classes 116 features 163 pattern orbits 8548
forced tight: 2 partial: 0
status 0 max-min slack on the rest: 0.04285714285714296

## 6. ABBB: LP_4 value of concentration + kappa * density(A)   (scripts/kappa_lp.py)
kappa -8.0000: LP value -0.000000000   8/5+kappa/5 = 0.000000000   excess 0.000e+00
kappa -4.0000: LP value 0.800000000   8/5+kappa/5 = 0.800000000   excess 0.000e+00
kappa +0.0000: LP value 1.600000000   8/5+kappa/5 = 1.600000000   excess 0.000e+00
kappa +0.5000: LP value 1.700000000   8/5+kappa/5 = 1.700000000   excess -1.388e-16
kappa +0.5714: LP value 1.714285714   8/5+kappa/5 = 1.714285714   excess 4.163e-17
kappa +0.5800: LP value 1.717500000   8/5+kappa/5 = 1.716000000   excess 1.500e-03
kappa +1.0000: LP value 1.875000000   8/5+kappa/5 = 1.800000000   excess 7.500e-02
kappa +1.6000: LP value 2.100000000   8/5+kappa/5 = 1.920000000   excess 1.800e-01
kappa +2.0000: LP value 2.250000000   8/5+kappa/5 = 2.000000000   excess 2.500e-01

## 7. ABB, dimension 3, window 3x3x3: polynomial potentials (cutting planes over all 2^27 patterns)
degree 2 (scripts/abb3d_quad.py): FINAL bound 6.496170397172602 (target 6)
degree 3 (scripts/abb3d_poly.py): FINAL bound 6.047478657434043 (target 6)
degree 4 (scripts/abb3d_poly4.py): master LP value reached 6.0000000 at iteration 116 (log line: it 116 cuts 2614 LP M 6.0000000 true max 7.0148144 t 246);
   feasibility phase (scripts/abb3d_phase2.py): FEASIBLE: bound 6 holds in floating point; margin phase: MARGIN PHASE DONE: every pattern within the margin is in the pool
   exact rounding (scripts/abb3d_exact.py q=10080):
   float max 6.000000953674316
   patterns within 0.01 of the bound: 39
   tight patterns (float): 39  largest non-tight value: None
   distinct tight rows: 6
   rank of the tight system: 3
   all tight equalities hold exactly: True  max |theta_exact - theta_float| = 0.00014500246603654787
   common denominator Qden = 1451520  max |Z| = 5362624
   (a) class sums are zero
   (b) exact: max over 2^27 patterns of 9*Qden*(Psi0+Q) = 78382080;  54*Qden = 78382080;  valid = True;  patterns with equality = 39
       largest value below the bound: 6 - 1079/90720 = 5.988106261022928
   time 8.0
