Datasets:

PureOne
/

Iyeumaring-AION-LATCH-VI

"""Certificate-guided exact two-circuit search; finite NumPy prototype.
Standard quadratic/Cauchy tools are used; no worldwide-priority claim.
Bounds are exact-arithmetic theorems; implementation adds numerical margins,
but is not an outward-rounded interval arithmetic verifier.
"""
from __future__ import annotations
import argparse,csv,hashlib,json,time
from pathlib import Path
import numpy as np
from scipy.linalg import qr

ROOT=Path(__file__).resolve().parent

def instance(seed,n,q,r,regime):
    rng=np.random.default_rng(np.random.SeedSequence([seed,n,q,r,{'low_rank':1,'full_rank':2,'suppressor':3}[regime]]));p=q+r
    B=rng.normal(size=(r,p));raw=rng.normal(size=(n,p))
    if regime=='low_rank':
        latent=rng.normal(size=(n,8));raw=latent@rng.normal(size=(8,p))/np.sqrt(8)+.0003*raw
    if regime=='suppressor':
        # The informative difference is weak as two individual directions.
        common=rng.normal(size=n);signal=rng.normal(size=n)
        raw[:,r]=common+.03*signal;raw[:,r+1]=common-.03*signal
    start=time.perf_counter_ns()
    piv=qr(B,pivoting=True,mode='economic')[2][:r] if regime!='suppressor' else np.arange(r)
    non=np.array([j for j in range(p) if j not in set(piv)],int)
    T=np.linalg.solve(B[:,piv],B[:,non]);U=raw[:,non]-raw[:,piv]@T
    Q=np.eye(q)+T.T@T
    setup=(time.perf_counter_ns()-start)/1e9
    if regime=='suppressor':
        # Make the innovation pair retain the designed cancellation. Raw pivot
        # features are zero only in the new-task domain, not in anchors.
        raw[:,piv]=0.;U=raw[:,non].copy();y=signal+.02*rng.normal(size=n)
    else:
        coef=rng.normal(size=q);coef/=np.linalg.norm(coef);y=U@coef+.05*rng.normal(size=n)
    return dict(U=U,y=y,Q=Q,setup_seconds=setup,p=p,q=q,r=r,n=n,regime=regime,seed=seed)

def exact_scores(K,h):
    a=np.diag(K);det=a[:,None]*a[None,:]-K*K
    numer=a[None,:]*h[:,None]**2+a[:,None]*h[None,:]**2-2*K*h[:,None]*h[None,:]
    with np.errstate(divide='ignore',invalid='ignore'):G=.5*numer/det
    G=np.where(np.triu(np.ones(K.shape,dtype=bool),1),G,-np.inf)
    return G

def pair_value(cross,i,j,diag,h):
    det=diag[i]*diag[j]-cross*cross
    if det<=0:return float('inf')
    return float(.5*(diag[j]*h[i]**2+diag[i]*h[j]**2-2*cross*h[i]*h[j])/det)

def choose(values):
    # Consistent finite-precision tie policy for the exact output.
    m=np.max(values);inds=np.flatnonzero(values>=m-1e-10*(1+abs(m)))
    return np.unravel_index(inds[0],values.shape),float(m)

def dense(task,reg=1e-4):
    U,y,Q=task['U'],task['y'],task['Q'];start=time.perf_counter_ns();n,q=U.shape
    h=U.T@y/n;K=U.T@U/n+reg*Q;G=exact_scores(K,h);pair,gain=choose(G)
    seconds=(time.perf_counter_ns()-start)/1e9
    return dict(method='dense_all_pairs',seconds=seconds,end_to_end_seconds=seconds+task['setup_seconds'],pair=list(map(int,pair)),gain=gain,exact_cross_queries=q*(q-1)//2,landmarks=0,bound_rounds=0,certified=True,fallback=False)

def screened(task,rule='gain',reg=1e-4,max_landmarks=16,queries_per_round=2):
    U,y,Q=task['U'],task['y'],task['Q'];start=time.perf_counter_ns();n,q=U.shape
    h=U.T@y/n;d=np.einsum('ij,ij->j',U,U)/n;diag=d+reg*np.diag(Q);yy=float(y@y/n)
    # Include the response as a landmark. A response vector is an observed
    # training input, never a validation/test outcome used to tune this rule.
    v=y/max(np.linalg.norm(y),1e-300);orth=[v]
    L=[v@U/np.sqrt(n)];known=np.zeros((q,q),bool);observed=np.full((q,q),-np.inf)
    best=-np.inf;bestpair=None;queries=0;rounds=0;certified=False;fallback=False
    floor_upper=exact_scores(reg*Q,h)
    mask=np.triu(np.ones((q,q),bool),1)
    for step in range(max_landmarks+1):
        LP=np.array(L);center=LP.T@LP;remaining=np.maximum(d-np.sum(LP*LP,axis=0),0)
        # Inflate the Cauchy envelope instead of treating numerical projections
        # as exact. This is a tested margin, not a rigorous rounding proof.
        padding=2e-10*(1+np.max(d));remaining+=padding
        radius=np.sqrt(remaining[:,None]*remaining[None,:]);lo=reg*Q+center-radius;hi=reg*Q+center+radius
        def upper_at(x):
            det=diag[:,None]*diag[None,:]-x*x
            num=diag[None,:]*h[:,None]**2+diag[:,None]*h[None,:]**2-2*x*h[:,None]*h[None,:]
            with np.errstate(divide='ignore',invalid='ignore'):g=.5*num/det
            return np.where(det>1e-24,g,np.inf)
        ub=np.minimum(np.maximum(upper_at(lo),upper_at(hi)),floor_upper)
        ub=np.minimum(ub,yy/2)
        ub=np.where(mask,ub,0.);ub+=1e-9*(1+np.abs(ub));ub=np.where(mask&~known,ub,-np.inf);rounds+=1
        for _ in range(queries_per_round):
            if np.max(ub)<=best-1e-10*(1+abs(best)):
                certified=True;break
            ij=np.unravel_index(np.argmax(ub),ub.shape);i,j=map(int,ij)
            if not np.isfinite(ub[i,j]):break
            cross=float(U[:,i]@U[:,j]/n+reg*Q[i,j]);g=pair_value(cross,i,j,diag,h)
            known[i,j]=True;observed[i,j]=g;ub[i,j]=-np.inf;queries+=1
            if g>best+1e-10*(1+abs(best)) or bestpair is None or (abs(g-best)<=1e-10*(1+abs(best)) and (i,j)<bestpair):best=g;bestpair=(i,j)
        if certified:break
        if np.max(ub)<=best-1e-10*(1+abs(best)):
            certified=True;break
        if step==max_landmarks:break
        if rule=='gain':
            i,j=np.unravel_index(np.argmax(ub),ub.shape);candidates=[int(i),int(j)];idx=max(candidates,key=lambda k:remaining[k])
        elif rule=='variance':idx=int(np.argmax(remaining))
        else:
            rng=np.random.default_rng(task['seed']+step+438);idx=int(rng.choice(np.flatnonzero(remaining>padding*2))) if np.any(remaining>padding*2) else int(np.argmax(remaining))
        z=U[:,idx].copy()
        # Reorthogonalize to reduce cancellation in nearly rank-deficient data.
        for _ in range(2):
            for vv in orth:z-=vv*(vv@z)
        norm=np.linalg.norm(z)
        if norm<1e-12*np.sqrt(max(d[idx],1.)*n):break
        z/=norm;orth.append(z);L.append(z@U/np.sqrt(n))
    if not certified:
        # An explicit fallback bounds how many scalar queries are issued before
        # using the strongest dense BLAS path. Its cost remains fully counted.
        K=U.T@U/n+reg*Q;G=exact_scores(K,h);pair,best=choose(G);bestpair=tuple(map(int,pair));queries+=q*(q-1)//2;fallback=True
    seconds=(time.perf_counter_ns()-start)/1e9
    return dict(method='screen_'+rule,seconds=seconds,end_to_end_seconds=seconds+task['setup_seconds'],pair=list(bestpair),gain=best,exact_cross_queries=queries,landmarks=len(L),bound_rounds=rounds,certified=certified,fallback=fallback)

def batched(task,adaptive=True,reg=1e-4):
    """Observable pilot chooses a dense or batched projection backend.
    It is a repair of repeated scalar bookkeeping, not a new nullspace solver.
    A final dense fallback preserves the candidate family and best gain.
    """
    U,y,Q=task['U'],task['y'],task['Q'];start=time.perf_counter_ns();n,q=U.shape
    h=U.T@y/n;d=np.einsum('ij,ij->j',U,U)/n;diag=d+reg*np.diag(Q);yy=float(y@y/n)
    pilot=U[np.linspace(0,n-1,min(48,n),dtype=int)]
    _,R,piv=qr(pilot,pivoting=True,mode='economic');sv=np.abs(np.diag(R));rank=int(np.sum(sv>max(sv[0],1e-12)*.003))
    fallback=False;queries=0;landmarks=0;rounds=0;certified=False
    if not adaptive or rank<=16:
        k=min(max(rank,2),16) if adaptive else min(8,q)
        # One matrix-matrix product replaces many scalar projection passes.
        Z=np.column_stack([y,U[:,piv[:k]]]);V=np.linalg.qr(Z,mode='reduced')[0]
        L=V.T@U/np.sqrt(n);landmarks=L.shape[0];center=L.T@L
        rem=np.maximum(d-np.einsum('ij,ij->j',L,L),0)+2e-10*(1+np.max(d));rad=np.sqrt(rem[:,None]*rem[None,:])
        def at(cross):
            det=diag[:,None]*diag[None,:]-cross*cross
            num=diag[None,:]*h[:,None]**2+diag[:,None]*h[None,:]**2-2*cross*h[:,None]*h[None,:]
            with np.errstate(divide='ignore',invalid='ignore'):v=.5*num/det
            return np.where(det>1e-24,v,np.inf)
        ub=np.maximum(at(reg*Q+center-rad),at(reg*Q+center+rad));ub=np.minimum(ub,exact_scores(reg*Q,h));ub=np.minimum(ub,yy/2)
        mask=np.triu(np.ones((q,q),bool),1);ub=np.where(mask,ub,0.);ub+=1e-9*(1+abs(ub));ub=np.where(mask,ub,-np.inf);rounds=1
        best=-np.inf;bestpair=None
        # Only a small number of expensive exact dots before strong fallback.
        for _ in range(16):
            if np.max(ub)<=best-1e-10*(1+abs(best)):
                certified=True;break
            i,j=map(int,np.unravel_index(np.argmax(ub),ub.shape));cross=float(U[:,i]@U[:,j]/n+reg*Q[i,j]);gain=pair_value(cross,i,j,diag,h);ub[i,j]=-np.inf;queries+=1
            if bestpair is None or gain>best+1e-10*(1+abs(best)):best=gain;bestpair=(i,j)
        if np.max(ub)<=best-1e-10*(1+abs(best)):certified=True
    if not certified:
        K=U.T@U/n+reg*Q;G=exact_scores(K,h);bestpair,best=choose(G);fallback=True;queries+=q*(q-1)//2
    sec=(time.perf_counter_ns()-start)/1e9
    return dict(method='pilot_batch' if adaptive else 'fixed_batch8',seconds=sec,end_to_end_seconds=sec+task['setup_seconds'],pair=list(map(int,bestpair)),gain=float(best),exact_cross_queries=queries,landmarks=landmarks,bound_rounds=rounds,certified=certified,fallback=fallback)

def test_bounds():
    rng=np.random.default_rng(611);checks=0;excess=0.
    for _ in range(60):
        U=rng.normal(size=(30,9));y=rng.normal(size=30);W=rng.normal(size=(2,9));Q=np.eye(9)+W.T@W
        reg=.03;h=U.T@y/len(y);d=np.mean(U*U,axis=0);diag=d+reg*np.diag(Q)
        V=np.linalg.qr(np.column_stack([y,U[:,:3]]))[0];L=V.T@U/np.sqrt(len(y));cen=L.T@L;rem=np.maximum(d-np.sum(L*L,axis=0),0)
        K=U.T@U/len(y)+reg*Q;actual=exact_scores(K,h)
        for i in range(9):
            for j in range(i+1,9):
                radius=np.sqrt(rem[i]*rem[j]);bounds=[pair_value(reg*Q[i,j]+cen[i,j]+s*radius,i,j,diag,h) for s in [-1,1]]
                gap=actual[i,j]-max(bounds);excess=max(excess,gap);assert gap<=1e-9;checks+=1
    return dict(pair_bound_checks=checks,max_bound_excess=excess)

def run(out,seeds=8,phase='explore'):
    out.mkdir(parents=True,exist_ok=True)
    configs=[(1024,128,2,'low_rank'),(1024,128,2,'full_rank'),(2048,512,4,'low_rank'),(2048,512,4,'full_rank'),(2048,512,4,'suppressor'),(8192,512,4,'low_rank')]
    rows=[];rng=np.random.default_rng(99101)
    for seed in range(seeds):
        for n,q,r,regime in configs:
            task=instance(seed+(100000 if phase=='confirm' else 0),n,q,r,regime)
            results=[]
            for method in rng.permutation(['dense','gain','variance','random','pilot_batch','fixed_batch8']):
                a=dense(task) if method=='dense' else batched(task,method=='pilot_batch') if method in ('pilot_batch','fixed_batch8') else screened(task,method)
                a.update(seed=seed,n=n,q=q,r=r,regime=regime,phase=phase);results.append(a)
            reference=next(r for r in results if r['method']=='dense_all_pairs')
            for a in results:
                a['gain_error']=abs(a['gain']-reference['gain']);a['same_pair']=a['pair']==reference['pair'];a['gain_match']=a['gain_error']<=1e-8*(1+reference['gain']);rows.append(a)
    with (out/'rows.csv').open('w') as f:
        writer=csv.DictWriter(f,fieldnames=list(rows[0]));writer.writeheader();writer.writerows(rows)
    summ=[]
    for n,q,r,regime in configs:
        for method in ['dense_all_pairs','screen_gain','screen_variance','screen_random','pilot_batch','fixed_batch8']:
            rs=[x for x in rows if x['n']==n and x['q']==q and x['regime']==regime and x['method']==method]
            summ.append(dict(n=n,q=q,r=r,regime=regime,method=method,seeds=len(rs),mean_seconds=float(np.mean([x['seconds'] for x in rs])),mean_end_to_end_seconds=float(np.mean([x['end_to_end_seconds'] for x in rs])),mean_queries=float(np.mean([x['exact_cross_queries'] for x in rs])),mean_landmarks=float(np.mean([x['landmarks'] for x in rs])),certified_fraction=float(np.mean([x['certified'] for x in rs])),fallback_fraction=float(np.mean([x['fallback'] for x in rs])),gain_mismatches=sum(not x['gain_match'] for x in rs)))
    report=dict(phase=phase,seeds=seeds,summary=summ,checks=test_bounds(),scope='Exact two-circuit winner discovery conditional on finite arithmetic margins; uses standard Cauchy/convexity tools. Complete compound-packet space restricted to pairs of supplied fundamental circuits. Strong dense and variance-landmark controls included; feature/circuit setup charged separately.')
    (out/'summary.json').write_text(json.dumps(report,indent=2));print(json.dumps(report,indent=2))

if __name__=='__main__':
    p=argparse.ArgumentParser();p.add_argument('--out',type=Path,default=ROOT/'exploratory');p.add_argument('--seeds',type=int,default=4);p.add_argument('--phase',default='explore');a=p.parse_args();run(a.out,a.seeds,a.phase)

	"""Certificate-guided exact two-circuit search; finite NumPy prototype.

	Standard quadratic/Cauchy tools are used; no worldwide-priority claim.
	Bounds are exact-arithmetic theorems; implementation adds numerical margins,
	but is not an outward-rounded interval arithmetic verifier.
	"""
	from __future__ import annotations
	import argparse,csv,hashlib,json,time
	from pathlib import Path
	import numpy as np
	from scipy.linalg import qr

	ROOT=Path(__file__).resolve().parent

	def instance(seed,n,q,r,regime):
	rng=np.random.default_rng(np.random.SeedSequence([seed,n,q,r,{'low_rank':1,'full_rank':2,'suppressor':3}[regime]]));p=q+r
	B=rng.normal(size=(r,p));raw=rng.normal(size=(n,p))
	if regime=='low_rank':
	latent=rng.normal(size=(n,8));raw=latent@rng.normal(size=(8,p))/np.sqrt(8)+.0003*raw
	if regime=='suppressor':
	# The informative difference is weak as two individual directions.
	common=rng.normal(size=n);signal=rng.normal(size=n)
	raw[:,r]=common+.03signal;raw[:,r+1]=common-.03signal
	start=time.perf_counter_ns()
	piv=qr(B,pivoting=True,mode='economic')[2][:r] if regime!='suppressor' else np.arange(r)
	non=np.array([j for j in range(p) if j not in set(piv)],int)
	T=np.linalg.solve(B[:,piv],B[:,non]);U=raw[:,non]-raw[:,piv]@T
	Q=np.eye(q)+T.T@T
	setup=(time.perf_counter_ns()-start)/1e9
	if regime=='suppressor':
	# Make the innovation pair retain the designed cancellation. Raw pivot
	# features are zero only in the new-task domain, not in anchors.
	raw[:,piv]=0.;U=raw[:,non].copy();y=signal+.02*rng.normal(size=n)
	else:
	coef=rng.normal(size=q);coef/=np.linalg.norm(coef);y=U@coef+.05*rng.normal(size=n)
	return dict(U=U,y=y,Q=Q,setup_seconds=setup,p=p,q=q,r=r,n=n,regime=regime,seed=seed)

	def exact_scores(K,h):
	a=np.diag(K);det=a[:,None]a[None,:]-KK
	numer=a[None,:]h[:,None]2+a[:,None]h[None,:]*2-2Kh[:,None]h[None,:]
	with np.errstate(divide='ignore',invalid='ignore'):G=.5*numer/det
	G=np.where(np.triu(np.ones(K.shape,dtype=bool),1),G,-np.inf)
	return G

	def pair_value(cross,i,j,diag,h):
	det=diag[i]diag[j]-crosscross
	if det<=0:return float('inf')
	return float(.5(diag[j]h[i]*2+diag[i]h[j]*2-2crossh[i]h[j])/det)

	def choose(values):
	# Consistent finite-precision tie policy for the exact output.
	m=np.max(values);inds=np.flatnonzero(values>=m-1e-10*(1+abs(m)))
	return np.unravel_index(inds[0],values.shape),float(m)

	def dense(task,reg=1e-4):
	U,y,Q=task['U'],task['y'],task['Q'];start=time.perf_counter_ns();n,q=U.shape
	h=U.T@y/n;K=U.T@U/n+reg*Q;G=exact_scores(K,h);pair,gain=choose(G)
	seconds=(time.perf_counter_ns()-start)/1e9
	return dict(method='dense_all_pairs',seconds=seconds,end_to_end_seconds=seconds+task['setup_seconds'],pair=list(map(int,pair)),gain=gain,exact_cross_queries=q*(q-1)//2,landmarks=0,bound_rounds=0,certified=True,fallback=False)

	def screened(task,rule='gain',reg=1e-4,max_landmarks=16,queries_per_round=2):
	U,y,Q=task['U'],task['y'],task['Q'];start=time.perf_counter_ns();n,q=U.shape
	h=U.T@y/n;d=np.einsum('ij,ij->j',U,U)/n;diag=d+reg*np.diag(Q);yy=float(y@y/n)
	# Include the response as a landmark. A response vector is an observed
	# training input, never a validation/test outcome used to tune this rule.
	v=y/max(np.linalg.norm(y),1e-300);orth=[v]
	L=[v@U/np.sqrt(n)];known=np.zeros((q,q),bool);observed=np.full((q,q),-np.inf)
	best=-np.inf;bestpair=None;queries=0;rounds=0;certified=False;fallback=False
	floor_upper=exact_scores(reg*Q,h)
	mask=np.triu(np.ones((q,q),bool),1)
	for step in range(max_landmarks+1):
	LP=np.array(L);center=LP.T@LP;remaining=np.maximum(d-np.sum(LP*LP,axis=0),0)
	# Inflate the Cauchy envelope instead of treating numerical projections
	# as exact. This is a tested margin, not a rigorous rounding proof.
	padding=2e-10*(1+np.max(d));remaining+=padding
	radius=np.sqrt(remaining[:,None]remaining[None,:]);lo=regQ+center-radius;hi=reg*Q+center+radius
	def upper_at(x):
	det=diag[:,None]diag[None,:]-xx
	num=diag[None,:]h[:,None]2+diag[:,None]h[None,:]*2-2xh[:,None]h[None,:]
	with np.errstate(divide='ignore',invalid='ignore'):g=.5*num/det
	return np.where(det>1e-24,g,np.inf)
	ub=np.minimum(np.maximum(upper_at(lo),upper_at(hi)),floor_upper)
	ub=np.minimum(ub,yy/2)
	ub=np.where(mask,ub,0.);ub+=1e-9*(1+np.abs(ub));ub=np.where(mask&~known,ub,-np.inf);rounds+=1
	for _ in range(queries_per_round):
	if np.max(ub)<=best-1e-10*(1+abs(best)):
	certified=True;break
	ij=np.unravel_index(np.argmax(ub),ub.shape);i,j=map(int,ij)
	if not np.isfinite(ub[i,j]):break
	cross=float(U[:,i]@U[:,j]/n+reg*Q[i,j]);g=pair_value(cross,i,j,diag,h)
	known[i,j]=True;observed[i,j]=g;ub[i,j]=-np.inf;queries+=1
	if g>best+1e-10(1+abs(best)) or bestpair is None or (abs(g-best)<=1e-10(1+abs(best)) and (i,j)<bestpair):best=g;bestpair=(i,j)
	if certified:break
	if np.max(ub)<=best-1e-10*(1+abs(best)):
	certified=True;break
	if step==max_landmarks:break
	if rule=='gain':
	i,j=np.unravel_index(np.argmax(ub),ub.shape);candidates=[int(i),int(j)];idx=max(candidates,key=lambda k:remaining[k])
	elif rule=='variance':idx=int(np.argmax(remaining))
	else:
	rng=np.random.default_rng(task['seed']+step+438);idx=int(rng.choice(np.flatnonzero(remaining>padding2))) if np.any(remaining>padding2) else int(np.argmax(remaining))
	z=U[:,idx].copy()
	# Reorthogonalize to reduce cancellation in nearly rank-deficient data.
	for _ in range(2):
	for vv in orth:z-=vv*(vv@z)
	norm=np.linalg.norm(z)
	if norm<1e-12np.sqrt(max(d[idx],1.)n):break
	z/=norm;orth.append(z);L.append(z@U/np.sqrt(n))
	if not certified:
	# An explicit fallback bounds how many scalar queries are issued before
	# using the strongest dense BLAS path. Its cost remains fully counted.
	K=U.T@U/n+regQ;G=exact_scores(K,h);pair,best=choose(G);bestpair=tuple(map(int,pair));queries+=q(q-1)//2;fallback=True
	seconds=(time.perf_counter_ns()-start)/1e9
	return dict(method='screen_'+rule,seconds=seconds,end_to_end_seconds=seconds+task['setup_seconds'],pair=list(bestpair),gain=best,exact_cross_queries=queries,landmarks=len(L),bound_rounds=rounds,certified=certified,fallback=fallback)

	def batched(task,adaptive=True,reg=1e-4):
	"""Observable pilot chooses a dense or batched projection backend.

	It is a repair of repeated scalar bookkeeping, not a new nullspace solver.
	A final dense fallback preserves the candidate family and best gain.
	"""
	U,y,Q=task['U'],task['y'],task['Q'];start=time.perf_counter_ns();n,q=U.shape
	h=U.T@y/n;d=np.einsum('ij,ij->j',U,U)/n;diag=d+reg*np.diag(Q);yy=float(y@y/n)
	pilot=U[np.linspace(0,n-1,min(48,n),dtype=int)]
	_,R,piv=qr(pilot,pivoting=True,mode='economic');sv=np.abs(np.diag(R));rank=int(np.sum(sv>max(sv[0],1e-12)*.003))
	fallback=False;queries=0;landmarks=0;rounds=0;certified=False
	if not adaptive or rank<=16:
	k=min(max(rank,2),16) if adaptive else min(8,q)
	# One matrix-matrix product replaces many scalar projection passes.
	Z=np.column_stack([y,U[:,piv[:k]]]);V=np.linalg.qr(Z,mode='reduced')[0]
	L=V.T@U/np.sqrt(n);landmarks=L.shape[0];center=L.T@L
	rem=np.maximum(d-np.einsum('ij,ij->j',L,L),0)+2e-10(1+np.max(d));rad=np.sqrt(rem[:,None]rem[None,:])
	def at(cross):
	det=diag[:,None]diag[None,:]-crosscross
	num=diag[None,:]h[:,None]2+diag[:,None]h[None,:]*2-2crossh[:,None]h[None,:]
	with np.errstate(divide='ignore',invalid='ignore'):v=.5*num/det
	return np.where(det>1e-24,v,np.inf)
	ub=np.maximum(at(regQ+center-rad),at(regQ+center+rad));ub=np.minimum(ub,exact_scores(reg*Q,h));ub=np.minimum(ub,yy/2)
	mask=np.triu(np.ones((q,q),bool),1);ub=np.where(mask,ub,0.);ub+=1e-9*(1+abs(ub));ub=np.where(mask,ub,-np.inf);rounds=1
	best=-np.inf;bestpair=None
	# Only a small number of expensive exact dots before strong fallback.
	for _ in range(16):
	if np.max(ub)<=best-1e-10*(1+abs(best)):
	certified=True;break
	i,j=map(int,np.unravel_index(np.argmax(ub),ub.shape));cross=float(U[:,i]@U[:,j]/n+reg*Q[i,j]);gain=pair_value(cross,i,j,diag,h);ub[i,j]=-np.inf;queries+=1
	if bestpair is None or gain>best+1e-10*(1+abs(best)):best=gain;bestpair=(i,j)
	if np.max(ub)<=best-1e-10*(1+abs(best)):certified=True
	if not certified:
	K=U.T@U/n+regQ;G=exact_scores(K,h);bestpair,best=choose(G);fallback=True;queries+=q(q-1)//2
	sec=(time.perf_counter_ns()-start)/1e9
	return dict(method='pilot_batch' if adaptive else 'fixed_batch8',seconds=sec,end_to_end_seconds=sec+task['setup_seconds'],pair=list(map(int,bestpair)),gain=float(best),exact_cross_queries=queries,landmarks=landmarks,bound_rounds=rounds,certified=certified,fallback=fallback)

	def test_bounds():
	rng=np.random.default_rng(611);checks=0;excess=0.
	for _ in range(60):
	U=rng.normal(size=(30,9));y=rng.normal(size=30);W=rng.normal(size=(2,9));Q=np.eye(9)+W.T@W
	reg=.03;h=U.T@y/len(y);d=np.mean(UU,axis=0);diag=d+regnp.diag(Q)
	V=np.linalg.qr(np.column_stack([y,U[:,:3]]))[0];L=V.T@U/np.sqrt(len(y));cen=L.T@L;rem=np.maximum(d-np.sum(L*L,axis=0),0)
	K=U.T@U/len(y)+reg*Q;actual=exact_scores(K,h)
	for i in range(9):
	for j in range(i+1,9):
	radius=np.sqrt(rem[i]rem[j]);bounds=[pair_value(regQ[i,j]+cen[i,j]+s*radius,i,j,diag,h) for s in [-1,1]]
	gap=actual[i,j]-max(bounds);excess=max(excess,gap);assert gap<=1e-9;checks+=1
	return dict(pair_bound_checks=checks,max_bound_excess=excess)

	def run(out,seeds=8,phase='explore'):
	out.mkdir(parents=True,exist_ok=True)
	configs=[(1024,128,2,'low_rank'),(1024,128,2,'full_rank'),(2048,512,4,'low_rank'),(2048,512,4,'full_rank'),(2048,512,4,'suppressor'),(8192,512,4,'low_rank')]
	rows=[];rng=np.random.default_rng(99101)
	for seed in range(seeds):
	for n,q,r,regime in configs:
	task=instance(seed+(100000 if phase=='confirm' else 0),n,q,r,regime)
	results=[]
	for method in rng.permutation(['dense','gain','variance','random','pilot_batch','fixed_batch8']):
	a=dense(task) if method=='dense' else batched(task,method=='pilot_batch') if method in ('pilot_batch','fixed_batch8') else screened(task,method)
	a.update(seed=seed,n=n,q=q,r=r,regime=regime,phase=phase);results.append(a)
	reference=next(r for r in results if r['method']=='dense_all_pairs')
	for a in results:
	a['gain_error']=abs(a['gain']-reference['gain']);a['same_pair']=a['pair']==reference['pair'];a['gain_match']=a['gain_error']<=1e-8*(1+reference['gain']);rows.append(a)
	with (out/'rows.csv').open('w') as f:
	writer=csv.DictWriter(f,fieldnames=list(rows[0]));writer.writeheader();writer.writerows(rows)
	summ=[]
	for n,q,r,regime in configs:
	for method in ['dense_all_pairs','screen_gain','screen_variance','screen_random','pilot_batch','fixed_batch8']:
	rs=[x for x in rows if x['n']==n and x['q']==q and x['regime']==regime and x['method']==method]
	summ.append(dict(n=n,q=q,r=r,regime=regime,method=method,seeds=len(rs),mean_seconds=float(np.mean([x['seconds'] for x in rs])),mean_end_to_end_seconds=float(np.mean([x['end_to_end_seconds'] for x in rs])),mean_queries=float(np.mean([x['exact_cross_queries'] for x in rs])),mean_landmarks=float(np.mean([x['landmarks'] for x in rs])),certified_fraction=float(np.mean([x['certified'] for x in rs])),fallback_fraction=float(np.mean([x['fallback'] for x in rs])),gain_mismatches=sum(not x['gain_match'] for x in rs)))
	report=dict(phase=phase,seeds=seeds,summary=summ,checks=test_bounds(),scope='Exact two-circuit winner discovery conditional on finite arithmetic margins; uses standard Cauchy/convexity tools. Complete compound-packet space restricted to pairs of supplied fundamental circuits. Strong dense and variance-landmark controls included; feature/circuit setup charged separately.')
	(out/'summary.json').write_text(json.dumps(report,indent=2));print(json.dumps(report,indent=2))

	if __name__=='__main__':
	p=argparse.ArgumentParser();p.add_argument('--out',type=Path,default=ROOT/'exploratory');p.add_argument('--seeds',type=int,default=4);p.add_argument('--phase',default='explore');a=p.parse_args();run(a.out,a.seeds,a.phase)