Datasets:

PureOne
/

Nuyori-VII-Sharp-Nonlinear-Laminate-Design

#!/usr/bin/env python3
"""Certified all-film bounds for declared nonlinear, non-Lie-closed inputs.
Synthetic mathematical examples only. Floating-point shooting proposes rational
flux brackets. Integer outward matrix-exponential arithmetic verifies them; the
global ceiling follows from affine coefficient comparison and the analytical
minimax theorem, not from an unverified numerical search.
"""
from __future__ import annotations
from dataclasses import dataclass
from fractions import Fraction as F
from pathlib import Path
import csv,json,math
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

ROOT=Path(__file__).resolve().parent
BITS=76;Q=1<<BITS
def ceildiv(a,b):return -((-a)//b)
@dataclass(frozen=True)
class I:
    lo:int
    hi:int
    @staticmethod
    def f(x):
        x=F(x);return I(x.numerator*Q//x.denominator,ceildiv(x.numerator*Q,x.denominator))
    def __add__(self,o):
        if not isinstance(o,I):o=I.f(o)
        return I(self.lo+o.lo,self.hi+o.hi)
    __radd__=__add__
    def __neg__(self):return I(-self.hi,-self.lo)
    def __sub__(self,o):return self+-o if isinstance(o,I) else self+-I.f(o)
    def __rsub__(self,o):return I.f(o)+-self
    def __mul__(self,o):
        if not isinstance(o,I):o=I.f(o)
        p=[self.lo*o.lo,self.lo*o.hi,self.hi*o.lo,self.hi*o.hi]
        return I(min(p)//Q,ceildiv(max(p),Q))
    __rmul__=__mul__
    def __truediv__(self,o):
        if not isinstance(o,I):o=I.f(o)
        assert not o.lo<=0<=o.hi
        lo=[];hi=[]
        for x in [self.lo,self.hi]:
            for y in [o.lo,o.hi]:lo.append(x*Q//y);hi.append(ceildiv(x*Q,y))
        return I(min(lo),max(hi))
    def divint(self,k):return I(self.lo//k,ceildiv(self.hi,k))
    def fractions(self):return [str(F(self.lo,Q)),str(F(self.hi,Q))]

def madd(a,b):return [[a[i][j]+b[i][j] for j in range(2)]for i in range(2)]
def mmul(a,b):return [[sum((a[i][k]*b[k][j] for k in range(2)),I.f(0))for j in range(2)]for i in range(2)]
def mnorm(a):return max(sum(max(abs(v.lo),abs(v.hi))for v in row)for row in a)
def mexp(a):
    """Outward Taylor/scaling/squaring; induced-infinity-norm tail <=2^-BITS."""
    nscale=0
    while 2*mnorm(a)>Q:
        a=[[v.divint(2)for v in row]for row in a];nscale+=1
    n=32
    assert 2*F(1,2)**(n+1)/math.factorial(n+1)<F(1,Q)
    unit=[[I.f(1 if i==j else 0)for j in range(2)]for i in range(2)]
    term=unit;ans=unit
    for k in range(1,n+1):
        term=[[v.divint(k)for v in row]for row in mmul(term,a)]
        ans=madd(ans,term)
    ans=[[I(v.lo-1,v.hi+1)for v in row]for row in ans]
    for _ in range(nscale):ans=mmul(ans,ans)
    return ans

L=F(1);HA=F(4,5);HB=F(1,5);SKIN=F(1,20);U=F(3,2);C=F(12,5)
SPLIT=[('A',SKIN),('B',HB/2),('A',HA-2*SKIN),('B',HB/2),('A',SKIN)]
CENTRAL=[('A',HA/2),('B',HB),('A',HA/2)]

def endpoint_float(j,layers,rho):
    a,b,c=map(float,rho);u=0.
    for phase,h in layers:
        x=j*float(h)
        if phase=='A':u+=x;continue
        d=b*b/4-a*c
        if d>1e-14:
            s=math.sinh(math.sqrt(d)*x)/math.sqrt(d);co=math.cosh(math.sqrt(d)*x)
        elif d< -1e-14:
            s=math.sin(math.sqrt(-d)*x)/math.sqrt(-d);co=math.cos(math.sqrt(-d)*x)
        else:s=x;co=1.
        u=((co+s*b/2)*u+s*a)/(-s*c*u+co-s*b/2)
    return u

def endpoint_interval(j,layers,rho):
    a,b,c=rho;u=I.f(0);jj=I.f(j);checks=[]
    for phase,h in layers:
        if phase=='A':u=u+jj*h;continue
        x=jj*h
        matrix=[[x*(b/2),x*a],[-x*c,-x*(b/2)]]
        # For EVERY time within the layer, ||exp(tM)-I|| <=2d if d<=1/2.
        # Thus q(t)>=1-2d max(1,u_start)>0. This rules out hidden Riccati poles.
        d=F(mnorm(matrix),Q);umax=max(F(1),F(u.hi,Q))
        assert 2*d<=1 and 2*d*umax<1 and u.lo>=0
        e=mexp(matrix)
        denominator=e[1][0]*u+e[1][1]
        assert denominator.lo>0
        u=(e[0][0]*u+e[0][1])/denominator
        assert u.lo>=0
        checks.append({'layer':str(h),'matrix_norm_upper':str(d),'pole_margin_lower':str(1-2*d*umax)})
    return u,checks

def certified_flux(layers,rho):
    a,b,c=rho
    # All six declared models and their affine envelopes are increasing on [0,U].
    assert a>0 and b>=0 and c>=0
    rmin=min(F(1),a);rmax=max(F(1),a+b*U+c*U*U)
    jmin=U/(L*rmax);jmax=U/(L*rmin)
    lo=float(jmin);hi=float(jmax)
    for _ in range(80):
        mid=(lo+hi)/2
        if endpoint_float(mid,layers,rho)<float(U):lo=mid
        else:hi=mid
    root=(lo+hi)/2
    center=F(round(root*(1<<52)),1<<52);radius=F(1,1<<45)
    for _ in range(15):
        jl=center-radius;jh=center+radius
        assert jl>0
        el,pl=endpoint_interval(jl,layers,rho);eh,ph=endpoint_interval(jh,layers,rho)
        if el.hi<F(U)*Q<eh.lo:break
        radius*=2
    else:raise AssertionError('A floating root proposal failed exact validation')
    return {'J_lower':str(jl),'J_upper':str(jh),'endpoint_at_lower':el.fractions(),
            'endpoint_at_upper':eh.fractions(),'pole_checks_lower':pl,'pole_checks_upper':ph}

def ri(rec,lam):
    ji=I(I.f(F(rec['J_lower'])).lo,I.f(F(rec['J_upper'])).hi)
    return I.f(C)+I.f(lam)/ji

def main():
    records=[];rows=[]
    for eps in [F(1,50),F(1,10),F(1,4)]:
        rho=[F(1),F(1,2),eps]
        # Exact global interval identities:
        # chord-rho=eps*u(U-u); rho-tangent=eps*(u-U/2)^2.
        lower=[1-eps*U*U/4,F(1,2)+eps*U,F(0)]
        upper=[F(1),F(1,2)+eps*U,F(0)]
        assert lower[0]>0
        rho_max=rho[0]+rho[1]*U+rho[2]*U*U
        qA_min=F(3,4);qB_min=4-F(5,4)*rho_max
        assert qB_min>0
        for sign in [1,-1]:
            lam=sign*F(7,4)
            layers=CENTRAL if sign>0 else SPLIT
            envelope_layers=layers
            actual=certified_flux(layers,rho);actual_r=ri(actual,lam)
            chosen=upper if sign>0 else lower
            envelope=certified_flux(layers,chosen);envelope_r=ri(envelope,lam)
            opposite=SPLIT if sign>0 else CENTRAL
            alternative=certified_flux(opposite,rho);alternative_r=ri(alternative,lam)
            # A nonlinear law need not retain the affine architecture ranking.
            # Select a concrete feasible lower witness after evaluating both.
            if alternative_r.lo>actual_r.lo:
                layers,opposite=opposite,layers
                actual,alternative=alternative,actual
                actual_r,alternative_r=alternative_r,actual_r
            # The upper proof ALWAYS uses the affine envelope's own optimizer,
            # which may be a different physical stack from the actual candidate.
            upper_bound=F(envelope_r.hi,Q);lower_bound=F(actual_r.lo,Q)
            assert lower_bound>0 and upper_bound>=F(actual_r.hi,Q)
            name='eps_'+str(eps).replace('/','_')+('_positive'if sign>0 else'_negative')
            rec={'name':name,'eps':str(eps),'potential_sign':sign,'Lambda':str(lam),
                 'rho':list(map(str,rho)),'rho_lower':list(map(str,lower)),'rho_upper':list(map(str,upper)),
                 'positive_qA_lower':str(qA_min),'positive_qB_lower':str(qB_min),
                 'candidate_architecture':'central'if layers==CENTRAL else'protected_split',
                 'affine_envelope_architecture':'central'if sign>0 else'protected_split',
                 'candidate_differs_from_affine_architecture':layers!=envelope_layers,
                 'candidate_stack':[[p,str(h)]for p,h in layers],
                 'actual_candidate_flux':actual,'actual_candidate_resistance':actual_r.fractions(),
                 'global_envelope':'upper_chord'if sign>0 else'lower_midpoint_tangent',
                 'global_envelope_stack':[[p,str(h)]for p,h in envelope_layers],
                 'global_envelope_flux':envelope,'global_envelope_resistance':envelope_r.fractions(),
                 'all_film_bracket':[str(lower_bound),str(upper_bound)],
                 'exact_relative_gap':str((upper_bound-lower_bound)/lower_bound),
                 'alternative_stack':[[p,str(h)]for p,h in opposite],
                 'alternative_flux':alternative,'alternative_resistance':alternative_r.fractions()}
            records.append(rec)
            rows.append({'name':name,'eps':float(eps),'sign':sign,
                         'candidate':rec['candidate_architecture'],
                         'affine_envelope_architecture':rec['affine_envelope_architecture'],
                         'ranking_reversal_certified':actual_r.lo>alternative_r.hi and layers!=envelope_layers,
                         'candidate_lower':float(lower_bound),'candidate_upper':actual_r.hi/Q,
                         'global_upper':float(upper_bound),'relative_global_gap':float((upper_bound-lower_bound)/lower_bound),
                         'alternative_resistance':(alternative_r.lo+alternative_r.hi)/(2*Q),
                         'candidate_flux_width':float(F(actual['J_upper'])-F(actual['J_lower']))})
    family_trials=[]
    for arch,st in [('central',CENTRAL),('protected_split',SPLIT)]:
        scenario=[]
        for rec in records:
            field='actual_candidate_resistance'if rec['candidate_architecture']==arch else'alternative_resistance'
            scenario.append({'name':rec['name'],'resistance':rec[field]})
        family_trials.append({'architecture':arch,'stack':[[p,str(h)]for p,h in st],
                              'scenario_responses':scenario,
                              'worst_interval':[str(min(F(s['resistance'][0])for s in scenario)),
                                                str(min(F(s['resistance'][1])for s in scenario))]})
    family_choice=max(family_trials,key=lambda c:F(c['worst_interval'][0]))
    family_lower=F(family_choice['worst_interval'][0])
    family_upper=min(F(rec['all_film_bracket'][1])for rec in records)
    assert family_lower>0 and family_upper>=F(family_choice['worst_interval'][1])
    family={'operating_conditions':len(records),'humidity_orientations':2*len(records),
            'candidate_trials':family_trials,'chosen_architecture':family_choice['architecture'],
            'chosen_stack':family_choice['stack'],
            'all_film_robust_bracket':[str(family_lower),str(family_upper)],
            'exact_relative_gap':str((family_upper-family_lower)/family_lower),
            'upper_bound_logic':'sup_F min_s R_s(F) <= min_s sup_F R_s(F); each upper is already all-film'}
    cert={'format':'Nuyori-VI nonlinear affine-envelope certificate v1','synthetic':True,
          'fixed_point_bits':BITS,'inputs':{'L':str(L),'H_A':str(HA),'H_B':str(HB),'skin':str(SKIN),
            'a_L':'0','a_H':'1','K_A':'1+a','G':'a+a^2/2','U':str(U),
            'K_B':'K_A/rho(G)','phi_base':'1+2*u/3-4*u^2/9',
            'q_A':'2+phi(G)','q_B':'4+rho(G)*phi(G)','C':str(C)},'cases':records,
          'common_film_family':family}
    (ROOT/'certificate.json').write_text(json.dumps(cert,indent=2)+'\n')
    result={'status':'PASS: validated synthetic nonlinear all-film bounds','cases':len(rows),
            'actual_water_not_Lie_closed':True,'both_oxygen_laws_nonconstant':True,
            'global_affine_bounds_exact_algebra':True,'rows':rows,
            'maximum_relative_gap':max(r['relative_global_gap']for r in rows),
            'primary_eps_1_10_gaps':[r['relative_global_gap']for r in rows if r['eps']==.1],
            'certified_departures_from_affine_candidate':[r['name']for r in rows if r['ranking_reversal_certified']],
            'common_film_family':{'operating_conditions':len(records),'humidity_orientations':2*len(records),
                'architecture':family_choice['architecture'],'worst_lower':float(family_lower),
                'global_upper':float(family_upper),'relative_global_gap':float((family_upper-family_lower)/family_lower)},
            'limitations':['No measured material properties or superiority are established.',
              'The true quadratic water law does not have the exact Lie-closed architecture theorem; affine comparison yields a certified gap, not exact optimality.',
              'The oxygen potential identity is imposed in these inputs; residual envelopes are needed for other measured oxygen laws.',
              'The midpoint tangent minimizes uniform paired envelope width, not necessarily the final flux or objective gap.',
              'The feasible candidate is selected from central and protected-split stacks; it need not equal the affine envelope optimizer or the unknown actual global optimizer.',
              'No interfacial cost, minimum layer thickness, swelling, lateral defects or transient transport is covered.']}
    (ROOT/'results.json').write_text(json.dumps(result,indent=2)+'\n')
    (ROOT/'verification_summary.txt').write_text(json.dumps(result,indent=2)+'\n')
    with (ROOT/'cases.csv').open('w',newline='')as f:
        w=csv.DictWriter(f,fieldnames=list(rows[0]));w.writeheader();w.writerows(rows)
    fig,ax=plt.subplots(1,2,figsize=(10.4,3.7),layout='constrained')
    x=[1.5*j/200 for j in range(201)];eps=.1
    ax[0].plot(x,[1+.5*v+eps*v*v for v in x],color='#117b85',label='Actual nonlinear ratio')
    ax[0].plot(x,[1+(.5+eps*1.5)*v for v in x],color='#b28a42',linestyle='--',label='Affine upper chord')
    ax[0].plot(x,[1-eps*1.5**2/4+(.5+eps*1.5)*v for v in x],color='#667d88',linestyle=':',label='Affine lower tangent')
    ax[0].set_xlabel('Water primitive u=G(a)');ax[0].set_ylabel('Water reciprocal ratio rho');ax[0].legend(fontsize=7)
    ax[0].set_title('Global coefficient enclosure, eps=0.1')
    for sign,col,label in [(1,'#117b85','Positive potential integral'),(-1,'#b28a42','Negative potential integral')]:
        rr=[r for r in rows if r['sign']==sign]
        ax[1].plot([r['eps']for r in rr],[100*r['relative_global_gap']for r in rr],'o-',color=col,label=label)
    ax[1].set_xlabel('Quadratic coefficient eps');ax[1].set_ylabel('Certified all-film relative gap (%)')
    ax[1].set_title('Actual finite film versus a global ceiling');ax[1].legend(fontsize=7)
    fig.suptitle('Synthetic smooth laws; exact oxygen identity and validated nonlinear fluxes',fontsize=10)
    fig.savefig(ROOT/'nonlinear_certificate.png',dpi=190);plt.close(fig)
    print(json.dumps({k:v for k,v in result.items()if k not in ['rows','limitations']},indent=2))
if __name__=='__main__':main()

	#!/usr/bin/env python3
	"""Certified all-film bounds for declared nonlinear, non-Lie-closed inputs.

	Synthetic mathematical examples only. Floating-point shooting proposes rational
	flux brackets. Integer outward matrix-exponential arithmetic verifies them; the
	global ceiling follows from affine coefficient comparison and the analytical
	minimax theorem, not from an unverified numerical search.
	"""
	from __future__ import annotations
	from dataclasses import dataclass
	from fractions import Fraction as F
	from pathlib import Path
	import csv,json,math
	import matplotlib
	matplotlib.use('Agg')
	import matplotlib.pyplot as plt

	ROOT=Path(__file__).resolve().parent
	BITS=76;Q=1<<BITS
	def ceildiv(a,b):return -((-a)//b)
	@dataclass(frozen=True)
	class I:
	lo:int
	hi:int
	@staticmethod
	def f(x):
	x=F(x);return I(x.numeratorQ//x.denominator,ceildiv(x.numeratorQ,x.denominator))
	def __add__(self,o):
	if not isinstance(o,I):o=I.f(o)
	return I(self.lo+o.lo,self.hi+o.hi)
	__radd__=__add__
	def __neg__(self):return I(-self.hi,-self.lo)
	def __sub__(self,o):return self+-o if isinstance(o,I) else self+-I.f(o)
	def __rsub__(self,o):return I.f(o)+-self
	def __mul__(self,o):
	if not isinstance(o,I):o=I.f(o)
	p=[self.loo.lo,self.loo.hi,self.hio.lo,self.hio.hi]
	return I(min(p)//Q,ceildiv(max(p),Q))
	__rmul__=__mul__
	def __truediv__(self,o):
	if not isinstance(o,I):o=I.f(o)
	assert not o.lo<=0<=o.hi
	lo=[];hi=[]
	for x in [self.lo,self.hi]:
	for y in [o.lo,o.hi]:lo.append(xQ//y);hi.append(ceildiv(xQ,y))
	return I(min(lo),max(hi))
	def divint(self,k):return I(self.lo//k,ceildiv(self.hi,k))
	def fractions(self):return [str(F(self.lo,Q)),str(F(self.hi,Q))]

	def madd(a,b):return [[a[i][j]+b[i][j] for j in range(2)]for i in range(2)]
	def mmul(a,b):return [[sum((a[i][k]*b[k][j] for k in range(2)),I.f(0))for j in range(2)]for i in range(2)]
	def mnorm(a):return max(sum(max(abs(v.lo),abs(v.hi))for v in row)for row in a)
	def mexp(a):
	"""Outward Taylor/scaling/squaring; induced-infinity-norm tail <=2^-BITS."""
	nscale=0
	while 2*mnorm(a)>Q:
	a=[[v.divint(2)for v in row]for row in a];nscale+=1
	n=32
	assert 2F(1,2)*(n+1)/math.factorial(n+1)<F(1,Q)
	unit=[[I.f(1 if i==j else 0)for j in range(2)]for i in range(2)]
	term=unit;ans=unit
	for k in range(1,n+1):
	term=[[v.divint(k)for v in row]for row in mmul(term,a)]
	ans=madd(ans,term)
	ans=[[I(v.lo-1,v.hi+1)for v in row]for row in ans]
	for _ in range(nscale):ans=mmul(ans,ans)
	return ans

	L=F(1);HA=F(4,5);HB=F(1,5);SKIN=F(1,20);U=F(3,2);C=F(12,5)
	SPLIT=[('A',SKIN),('B',HB/2),('A',HA-2*SKIN),('B',HB/2),('A',SKIN)]
	CENTRAL=[('A',HA/2),('B',HB),('A',HA/2)]

	def endpoint_float(j,layers,rho):
	a,b,c=map(float,rho);u=0.
	for phase,h in layers:
	x=j*float(h)
	if phase=='A':u+=x;continue
	d=bb/4-ac
	if d>1e-14:
	s=math.sinh(math.sqrt(d)x)/math.sqrt(d);co=math.cosh(math.sqrt(d)x)
	elif d< -1e-14:
	s=math.sin(math.sqrt(-d)x)/math.sqrt(-d);co=math.cos(math.sqrt(-d)x)
	else:s=x;co=1.
	u=((co+sb/2)u+sa)/(-scu+co-sb/2)
	return u

	def endpoint_interval(j,layers,rho):
	a,b,c=rho;u=I.f(0);jj=I.f(j);checks=[]
	for phase,h in layers:
	if phase=='A':u=u+jj*h;continue
	x=jj*h
	matrix=[[x(b/2),xa],[-xc,-x(b/2)]]
	# For EVERY time within the layer, \|\|exp(tM)-I\|\| <=2d if d<=1/2.
	# Thus q(t)>=1-2d max(1,u_start)>0. This rules out hidden Riccati poles.
	d=F(mnorm(matrix),Q);umax=max(F(1),F(u.hi,Q))
	assert 2d<=1 and 2d*umax<1 and u.lo>=0
	e=mexp(matrix)
	denominator=e[1][0]*u+e[1][1]
	assert denominator.lo>0
	u=(e[0][0]*u+e[0][1])/denominator
	assert u.lo>=0
	checks.append({'layer':str(h),'matrix_norm_upper':str(d),'pole_margin_lower':str(1-2dumax)})
	return u,checks

	def certified_flux(layers,rho):
	a,b,c=rho
	# All six declared models and their affine envelopes are increasing on [0,U].
	assert a>0 and b>=0 and c>=0
	rmin=min(F(1),a);rmax=max(F(1),a+bU+cU*U)
	jmin=U/(Lrmax);jmax=U/(Lrmin)
	lo=float(jmin);hi=float(jmax)
	for _ in range(80):
	mid=(lo+hi)/2
	if endpoint_float(mid,layers,rho)<float(U):lo=mid
	else:hi=mid
	root=(lo+hi)/2
	center=F(round(root*(1<<52)),1<<52);radius=F(1,1<<45)
	for _ in range(15):
	jl=center-radius;jh=center+radius
	assert jl>0
	el,pl=endpoint_interval(jl,layers,rho);eh,ph=endpoint_interval(jh,layers,rho)
	if el.hi<F(U)*Q<eh.lo:break
	radius*=2
	else:raise AssertionError('A floating root proposal failed exact validation')
	return {'J_lower':str(jl),'J_upper':str(jh),'endpoint_at_lower':el.fractions(),
	'endpoint_at_upper':eh.fractions(),'pole_checks_lower':pl,'pole_checks_upper':ph}

	def ri(rec,lam):
	ji=I(I.f(F(rec['J_lower'])).lo,I.f(F(rec['J_upper'])).hi)
	return I.f(C)+I.f(lam)/ji

	def main():
	records=[];rows=[]
	for eps in [F(1,50),F(1,10),F(1,4)]:
	rho=[F(1),F(1,2),eps]
	# Exact global interval identities:
	# chord-rho=epsu(U-u); rho-tangent=eps(u-U/2)^2.
	lower=[1-epsUU/4,F(1,2)+eps*U,F(0)]
	upper=[F(1),F(1,2)+eps*U,F(0)]
	assert lower[0]>0
	rho_max=rho[0]+rho[1]U+rho[2]U*U
	qA_min=F(3,4);qB_min=4-F(5,4)*rho_max
	assert qB_min>0
	for sign in [1,-1]:
	lam=sign*F(7,4)
	layers=CENTRAL if sign>0 else SPLIT
	envelope_layers=layers
	actual=certified_flux(layers,rho);actual_r=ri(actual,lam)
	chosen=upper if sign>0 else lower
	envelope=certified_flux(layers,chosen);envelope_r=ri(envelope,lam)
	opposite=SPLIT if sign>0 else CENTRAL
	alternative=certified_flux(opposite,rho);alternative_r=ri(alternative,lam)
	# A nonlinear law need not retain the affine architecture ranking.
	# Select a concrete feasible lower witness after evaluating both.
	if alternative_r.lo>actual_r.lo:
	layers,opposite=opposite,layers
	actual,alternative=alternative,actual
	actual_r,alternative_r=alternative_r,actual_r
	# The upper proof ALWAYS uses the affine envelope's own optimizer,
	# which may be a different physical stack from the actual candidate.
	upper_bound=F(envelope_r.hi,Q);lower_bound=F(actual_r.lo,Q)
	assert lower_bound>0 and upper_bound>=F(actual_r.hi,Q)
	name='eps_'+str(eps).replace('/','_')+('_positive'if sign>0 else'_negative')
	rec={'name':name,'eps':str(eps),'potential_sign':sign,'Lambda':str(lam),
	'rho':list(map(str,rho)),'rho_lower':list(map(str,lower)),'rho_upper':list(map(str,upper)),
	'positive_qA_lower':str(qA_min),'positive_qB_lower':str(qB_min),
	'candidate_architecture':'central'if layers==CENTRAL else'protected_split',
	'affine_envelope_architecture':'central'if sign>0 else'protected_split',
	'candidate_differs_from_affine_architecture':layers!=envelope_layers,
	'candidate_stack':[[p,str(h)]for p,h in layers],
	'actual_candidate_flux':actual,'actual_candidate_resistance':actual_r.fractions(),
	'global_envelope':'upper_chord'if sign>0 else'lower_midpoint_tangent',
	'global_envelope_stack':[[p,str(h)]for p,h in envelope_layers],
	'global_envelope_flux':envelope,'global_envelope_resistance':envelope_r.fractions(),
	'all_film_bracket':[str(lower_bound),str(upper_bound)],
	'exact_relative_gap':str((upper_bound-lower_bound)/lower_bound),
	'alternative_stack':[[p,str(h)]for p,h in opposite],
	'alternative_flux':alternative,'alternative_resistance':alternative_r.fractions()}
	records.append(rec)
	rows.append({'name':name,'eps':float(eps),'sign':sign,
	'candidate':rec['candidate_architecture'],
	'affine_envelope_architecture':rec['affine_envelope_architecture'],
	'ranking_reversal_certified':actual_r.lo>alternative_r.hi and layers!=envelope_layers,
	'candidate_lower':float(lower_bound),'candidate_upper':actual_r.hi/Q,
	'global_upper':float(upper_bound),'relative_global_gap':float((upper_bound-lower_bound)/lower_bound),
	'alternative_resistance':(alternative_r.lo+alternative_r.hi)/(2*Q),
	'candidate_flux_width':float(F(actual['J_upper'])-F(actual['J_lower']))})
	family_trials=[]
	for arch,st in [('central',CENTRAL),('protected_split',SPLIT)]:
	scenario=[]
	for rec in records:
	field='actual_candidate_resistance'if rec['candidate_architecture']==arch else'alternative_resistance'
	scenario.append({'name':rec['name'],'resistance':rec[field]})
	family_trials.append({'architecture':arch,'stack':[[p,str(h)]for p,h in st],
	'scenario_responses':scenario,
	'worst_interval':[str(min(F(s['resistance'][0])for s in scenario)),
	str(min(F(s['resistance'][1])for s in scenario))]})
	family_choice=max(family_trials,key=lambda c:F(c['worst_interval'][0]))
	family_lower=F(family_choice['worst_interval'][0])
	family_upper=min(F(rec['all_film_bracket'][1])for rec in records)
	assert family_lower>0 and family_upper>=F(family_choice['worst_interval'][1])
	family={'operating_conditions':len(records),'humidity_orientations':2*len(records),
	'candidate_trials':family_trials,'chosen_architecture':family_choice['architecture'],
	'chosen_stack':family_choice['stack'],
	'all_film_robust_bracket':[str(family_lower),str(family_upper)],
	'exact_relative_gap':str((family_upper-family_lower)/family_lower),
	'upper_bound_logic':'sup_F min_s R_s(F) <= min_s sup_F R_s(F); each upper is already all-film'}
	cert={'format':'Nuyori-VI nonlinear affine-envelope certificate v1','synthetic':True,
	'fixed_point_bits':BITS,'inputs':{'L':str(L),'H_A':str(HA),'H_B':str(HB),'skin':str(SKIN),
	'a_L':'0','a_H':'1','K_A':'1+a','G':'a+a^2/2','U':str(U),
	'K_B':'K_A/rho(G)','phi_base':'1+2u/3-4u^2/9',
	'q_A':'2+phi(G)','q_B':'4+rho(G)*phi(G)','C':str(C)},'cases':records,
	'common_film_family':family}
	(ROOT/'certificate.json').write_text(json.dumps(cert,indent=2)+'\n')
	result={'status':'PASS: validated synthetic nonlinear all-film bounds','cases':len(rows),
	'actual_water_not_Lie_closed':True,'both_oxygen_laws_nonconstant':True,
	'global_affine_bounds_exact_algebra':True,'rows':rows,
	'maximum_relative_gap':max(r['relative_global_gap']for r in rows),
	'primary_eps_1_10_gaps':[r['relative_global_gap']for r in rows if r['eps']==.1],
	'certified_departures_from_affine_candidate':[r['name']for r in rows if r['ranking_reversal_certified']],
	'common_film_family':{'operating_conditions':len(records),'humidity_orientations':2*len(records),
	'architecture':family_choice['architecture'],'worst_lower':float(family_lower),
	'global_upper':float(family_upper),'relative_global_gap':float((family_upper-family_lower)/family_lower)},
	'limitations':['No measured material properties or superiority are established.',
	'The true quadratic water law does not have the exact Lie-closed architecture theorem; affine comparison yields a certified gap, not exact optimality.',
	'The oxygen potential identity is imposed in these inputs; residual envelopes are needed for other measured oxygen laws.',
	'The midpoint tangent minimizes uniform paired envelope width, not necessarily the final flux or objective gap.',
	'The feasible candidate is selected from central and protected-split stacks; it need not equal the affine envelope optimizer or the unknown actual global optimizer.',
	'No interfacial cost, minimum layer thickness, swelling, lateral defects or transient transport is covered.']}
	(ROOT/'results.json').write_text(json.dumps(result,indent=2)+'\n')
	(ROOT/'verification_summary.txt').write_text(json.dumps(result,indent=2)+'\n')
	with (ROOT/'cases.csv').open('w',newline='')as f:
	w=csv.DictWriter(f,fieldnames=list(rows[0]));w.writeheader();w.writerows(rows)
	fig,ax=plt.subplots(1,2,figsize=(10.4,3.7),layout='constrained')
	x=[1.5*j/200 for j in range(201)];eps=.1
	ax[0].plot(x,[1+.5v+epsv*v for v in x],color='#117b85',label='Actual nonlinear ratio')
	ax[0].plot(x,[1+(.5+eps1.5)v for v in x],color='#b28a42',linestyle='--',label='Affine upper chord')
	ax[0].plot(x,[1-eps1.52/4+(.5+eps1.5)*v for v in x],color='#667d88',linestyle=':',label='Affine lower tangent')
	ax[0].set_xlabel('Water primitive u=G(a)');ax[0].set_ylabel('Water reciprocal ratio rho');ax[0].legend(fontsize=7)
	ax[0].set_title('Global coefficient enclosure, eps=0.1')
	for sign,col,label in [(1,'#117b85','Positive potential integral'),(-1,'#b28a42','Negative potential integral')]:
	rr=[r for r in rows if r['sign']==sign]
	ax[1].plot([r['eps']for r in rr],[100*r['relative_global_gap']for r in rr],'o-',color=col,label=label)
	ax[1].set_xlabel('Quadratic coefficient eps');ax[1].set_ylabel('Certified all-film relative gap (%)')
	ax[1].set_title('Actual finite film versus a global ceiling');ax[1].legend(fontsize=7)
	fig.suptitle('Synthetic smooth laws; exact oxygen identity and validated nonlinear fluxes',fontsize=10)
	fig.savefig(ROOT/'nonlinear_certificate.png',dpi=190);plt.close(fig)
	print(json.dumps({k:v for k,v in result.items()if k not in ['rows','limitations']},indent=2))
	if __name__=='__main__':main()