Datasets:

PureOne
/

Nuyori-VII-Sharp-Nonlinear-Laminate-Design

#!/usr/bin/env python3
"""Validated exact interface frontier for a synthetic nonseparable film model.
The continuum upper certificate is analytical. All numerical root signs use
outward integer Taylor enclosures. No measured polymer properties are used.
"""
from __future__ import annotations
from dataclasses import dataclass
from fractions import Fraction as F
from pathlib import Path
from collections import Counter
import json,math,csv
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

ROOT=Path(__file__).resolve().parent
BITS=160;Q=1<<BITS
def ceildiv(a,b):return -((-a)//b)
@dataclass(frozen=True)
class I:
    lo:int;hi:int
    @staticmethod
    def f(v):
        v=F(v);return I(v.numerator*Q//v.denominator,ceildiv(v.numerator*Q,v.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 -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)
        z=[self.lo*o.lo,self.lo*o.hi,self.hi*o.lo,self.hi*o.hi]
        return I(min(z)//Q,ceildiv(max(z),Q))
    __rmul__=__mul__
    def divint(self,n):return I(self.lo//n,ceildiv(self.hi,n))

def expi(x):
    steps=0
    while max(abs(x.lo),abs(x.hi))*2>Q:x=x.divint(2);steps+=1
    n=48
    assert 2*F(1,2)**(n+1)/math.factorial(n+1)<F(1,Q)
    term=I.f(1);ans=term
    for k in range(1,n+1):term=(term*x).divint(k);ans=ans+term
    ans=I(ans.lo-1,ans.hi+1)
    for _ in range(steps):ans=ans*ans
    return ans
def sinhi(x):return (expi(x)-expi(-x)).divint(2)
def coshi(x):return (expi(x)+expi(-x)).divint(2)
def asfr(x):return [str(F(x.lo,Q)),str(F(x.hi,Q))]

L=F(1);H=F(1,20);K=F(4,5);DELTA=F(1);LAM=F(-1);C=F(3)
M=L-4*H
NS=[1,2,4,5,8,16,41,128]

def layer_stack(n):
    if F(n)<=L/(4*H):
        b=L/(4*n);out=[('A',b)]
        for j in range(n):
            out.append(('B',2*b))
            out.append(('A',b if j==n-1 else 2*b))
        return out,'low',b
    a=M/(4*(n-1));out=[('A',H),('B',H+a)]
    for _ in range(n-2):out.extend([('A',2*a),('B',2*a)])
    out.extend([('A',2*a),('B',H+a),('A',H)])
    return out,'high',a

def F_interval(j,n=None):
    j=I.f(j)
    if n is None:return (4/K)*sinhi(K*H*j)+M*j
    if F(n)<=L/(4*H):return (4*n/K)*sinhi(K*(L/(4*n))*j)
    a=M/(4*(n-1));return (4/K)*(sinhi(K*H*j)+(n-1)*sinhi(K*a*j))

def bracket(n=None):
    lo=F(0);hi=DELTA/L
    assert F_interval(lo,n).hi<DELTA*Q<F_interval(hi,n).lo
    for _ in range(96):
        mid=(lo+hi)/2;v=F_interval(mid,n)
        if v.hi<DELTA*Q:lo=mid
        elif v.lo>DELTA*Q:hi=mid
        else:
            # F' >= L for every branch, so ambiguity already gives a tiny
            # exact enclosure rather than requiring an equality decision.
            e=max(DELTA-F(v.lo,Q),F(v.hi,Q)-DELTA)/L
            lo=max(lo,mid-e);hi=min(hi,mid+e);break
    assert lo>0 and F_interval(lo,n).hi<DELTA*Q<F_interval(hi,n).lo
    return lo,hi

def resistance(br):
    lo,hi=br
    assert LAM<0
    return C+LAM*DELTA/lo,C+LAM*DELTA/hi

def occupation(stack):
    q=F(0);coef=Counter();qpoints=[q]
    for phase,h in stack:
        qq=q+(K if phase=='B' else -K)*h
        coef[max(q,qq)]+=1;coef[min(q,qq)]-=1
        q=qq;qpoints.append(q)
    assert q==0
    return sorted((x,c) for x,c in coef.items() if c),qpoints

def main():
    infj=bracket();infr=resistance(infj)
    ji=I(I.f(infj[0]).lo,I.f(infj[1]).hi)
    fp=4*H*coshi(K*H*ji)+M
    # Interval for the exact asymptotic leading constant.
    num=(-LAM*DELTA)*K*K*M*M*M
    astar_lo=num*infj[0]/(96*F(fp.hi,Q));astar_hi=num*infj[1]/(96*F(fp.lo,Q))
    cases=[];rows=[]
    for n in NS:
        st,branch,scale=layer_stack(n)
        assert len(st)==2*n+1
        assert sum((h for _,h in st),F(0))==L
        assert all(sum((h for p,h in st if p==phase),F(0))==L/2 for phase in ['A','B'])
        assert st==st[::-1] and st[0][1]>=H
        occ,qp=occupation(st)
        br=bracket(n);rr=resistance(br)
        gap=(infr[0]-rr[1],infr[1]-rr[0]);assert gap[0]>0
        scaled=((n-1)**2*gap[0],(n-1)**2*gap[1])
        case={'n_B_layers':n,'interfaces':2*n,'layers':2*n+1,'branch':branch,'scale':str(scale),
              'stack':[[p,str(h)] for p,h in st],'occupation_coefficients':[[str(q),v] for q,v in occ],
              'Q_extrema':[str(x) for x in qp],'flux_interval':list(map(str,br)),
              'resistance_interval':list(map(str,rr)),'unrestricted_regret_interval':list(map(str,gap)),
              'scaled_regret_interval':list(map(str,scaled))}
        cases.append(case)
        rows.append({'n_B_layers':n,'interfaces':2*n,'layers':2*n+1,'branch':branch,
          'flux_lower':float(br[0]),'flux_upper':float(br[1]),'resistance_lower':float(rr[0]),
          'resistance_upper':float(rr[1]),'regret_upper':float(gap[1]),'scaled_regret':float(sum(scaled)/2)})
    floor=F(1,100);nmax=math.floor(1+M/(2*floor));assert nmax==41
    st,_,_=layer_stack(nmax);assert min(h for _,h in st)>=floor
    cert={'format':'Nuyori VI exact finite-interface frontier v1','synthetic':True,'fixed_point_bits':BITS,
      'inputs':{'L':str(L),'H_A':str(L/2),'H_B':str(L/2),'skin':str(H),'kappa':str(K),
        'a_L':'0','a_H':str(DELTA),'alpha_A':'3','alpha_B':'3','psi':'-1','Lambda':str(LAM*DELTA),
        'r_A':'1-kappa*a','r_B':'1+kappa*a','q_A':'2+kappa*a','q_B':'2-kappa*a'},
      'unrestricted':{'flux_interval':list(map(str,infj)),'resistance_interval':list(map(str,infr)),
          'attained_by_pure_film':False,'asymptotic_constant_interval':[str(astar_lo),str(astar_hi)]},
      'cases':cases,'minimum_thickness':{'floor':str(floor),'n_max':nmax,'layers':2*nmax+1,'interfaces':2*nmax,
          'minimum_actual_layer':str(min(h for _,h in st))}}
    (ROOT/'certificate.json').write_text(json.dumps(cert,indent=2)+'\n')
    with (ROOT/'minimum_thickness_layers.csv').open('w',newline='') as f:
        w=csv.writer(f);w.writerow(['layer','phase','start_depth','end_depth','thickness','relative_thickness'])
        y=F(0)
        for idx,(phase,thick) in enumerate(st,1):
            w.writerow([idx,phase,str(y),str(y+thick),str(thick),str(thick/L)]);y+=thick
    result={'status':'PASS: exact analytic global frontier with validated synthetic root intervals',
      'synthetic':True,'finite_interface_cases':len(cases),'nonattainment_for_any_pure_measurable_film':True,
      'unrestricted_flux_interval':[float(x) for x in infj],'unrestricted_resistance_interval':[float(x) for x in infr],
      'asymptotic_constant_interval':[float(astar_lo),float(astar_hi)],'cases':rows,
      'minimum_thickness':{'floor':float(floor),'n_max':nmax,'layers':2*nmax+1,'interfaces':2*nmax,
        'resistance_interval':[float(x) for x in map(F,cases[NS.index(nmax)]['resistance_interval'])]},
      'limitations':['Synthetic ideal 1D model; no manufactured polymer claim.',
       'Exact frontier applies to equal inventories, opposing affine reciprocal water slopes, potential-compatible oxygen laws and the stated A skins.',
       'Unrestricted optimum is a supremum and requires a relaxed mixing limit, not a finite pure film.',
       'The numerical minimum-layer example has d<=h; the general theorem uses h_eff=max(h,d).']}
    (ROOT/'results.json').write_text(json.dumps(result,indent=2)+'\n')
    with (ROOT/'frontier.csv').open('w',newline='') as f:
        w=csv.DictWriter(f,fieldnames=list(rows[0]));w.writeheader();w.writerows(rows)
    fig,axs=plt.subplots(1,3,figsize=(12.3,3.7),layout='constrained')
    xs=[r['interfaces'] for r in rows];gaps=[r['regret_upper'] for r in rows]
    axs[0].loglog(xs,gaps,'o-',color='#117b85',label='Exact global finite-interface gap')
    axs[0].set_xlabel('Maximum physical interfaces');axs[0].set_ylabel('Gap to unrestricted supremum')
    axs[0].set_title('Manufacturing frontier');axs[0].grid(alpha=.2)
    sel=[r for r in rows if r['n_B_layers']>1]
    axs[1].semilogx([r['n_B_layers']-1 for r in sel],[r['scaled_regret'] for r in sel],'o-',color='#117b85')
    axs[1].axhline(float((astar_lo+astar_hi)/2),color='#b18b31',linestyle='--',label='Proved leading constant')
    axs[1].set_xlabel('n - 1');axs[1].set_ylabel('(n - 1)^2 × gap');axs[1].set_title('Sharp inverse-square law');axs[1].legend(fontsize=7);axs[1].grid(alpha=.2)
    y=0.
    for phase,h in st:
        axs[2].barh(0,float(h),left=y,height=.5,color='#173c49' if phase=='A' else '#d4b35b');y+=float(h)
    axs[2].set_xlim(0,float(L));axs[2].set_ylim(-.6,.6);axs[2].set_yticks([]);axs[2].set_xlabel('Physical depth / total thickness')
    axs[2].set_title('1% minimum layer: 83-layer optimum')
    fig.suptitle('Synthetic exact model: interface count and layer thickness are explicit constraints',fontsize=10)
    fig.savefig(ROOT/'frontier.png',dpi=190);plt.close(fig)
    (ROOT/'verification_summary.txt').write_text(json.dumps(result,indent=2)+'\n')
    print(json.dumps({k:v for k,v in result.items() if k not in ['cases','limitations']},indent=2))

if __name__=='__main__':main()

	#!/usr/bin/env python3
	"""Validated exact interface frontier for a synthetic nonseparable film model.

	The continuum upper certificate is analytical. All numerical root signs use
	outward integer Taylor enclosures. No measured polymer properties are used.
	"""
	from __future__ import annotations
	from dataclasses import dataclass
	from fractions import Fraction as F
	from pathlib import Path
	from collections import Counter
	import json,math,csv
	import matplotlib
	matplotlib.use('Agg')
	import matplotlib.pyplot as plt

	ROOT=Path(__file__).resolve().parent
	BITS=160;Q=1<<BITS
	def ceildiv(a,b):return -((-a)//b)
	@dataclass(frozen=True)
	class I:
	lo:int;hi:int
	@staticmethod
	def f(v):
	v=F(v);return I(v.numeratorQ//v.denominator,ceildiv(v.numeratorQ,v.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 -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)
	z=[self.loo.lo,self.loo.hi,self.hio.lo,self.hio.hi]
	return I(min(z)//Q,ceildiv(max(z),Q))
	__rmul__=__mul__
	def divint(self,n):return I(self.lo//n,ceildiv(self.hi,n))

	def expi(x):
	steps=0
	while max(abs(x.lo),abs(x.hi))*2>Q:x=x.divint(2);steps+=1
	n=48
	assert 2F(1,2)*(n+1)/math.factorial(n+1)<F(1,Q)
	term=I.f(1);ans=term
	for k in range(1,n+1):term=(term*x).divint(k);ans=ans+term
	ans=I(ans.lo-1,ans.hi+1)
	for _ in range(steps):ans=ans*ans
	return ans
	def sinhi(x):return (expi(x)-expi(-x)).divint(2)
	def coshi(x):return (expi(x)+expi(-x)).divint(2)
	def asfr(x):return [str(F(x.lo,Q)),str(F(x.hi,Q))]

	L=F(1);H=F(1,20);K=F(4,5);DELTA=F(1);LAM=F(-1);C=F(3)
	M=L-4*H
	NS=[1,2,4,5,8,16,41,128]

	def layer_stack(n):
	if F(n)<=L/(4*H):
	b=L/(4*n);out=[('A',b)]
	for j in range(n):
	out.append(('B',2*b))
	out.append(('A',b if j==n-1 else 2*b))
	return out,'low',b
	a=M/(4*(n-1));out=[('A',H),('B',H+a)]
	for _ in range(n-2):out.extend([('A',2a),('B',2a)])
	out.extend([('A',2*a),('B',H+a),('A',H)])
	return out,'high',a

	def F_interval(j,n=None):
	j=I.f(j)
	if n is None:return (4/K)sinhi(KHj)+Mj
	if F(n)<=L/(4H):return (4n/K)sinhi(K(L/(4n))j)
	a=M/(4(n-1));return (4/K)(sinhi(KHj)+(n-1)sinhi(Ka*j))

	def bracket(n=None):
	lo=F(0);hi=DELTA/L
	assert F_interval(lo,n).hi<DELTA*Q<F_interval(hi,n).lo
	for _ in range(96):
	mid=(lo+hi)/2;v=F_interval(mid,n)
	if v.hi<DELTA*Q:lo=mid
	elif v.lo>DELTA*Q:hi=mid
	else:
	# F' >= L for every branch, so ambiguity already gives a tiny
	# exact enclosure rather than requiring an equality decision.
	e=max(DELTA-F(v.lo,Q),F(v.hi,Q)-DELTA)/L
	lo=max(lo,mid-e);hi=min(hi,mid+e);break
	assert lo>0 and F_interval(lo,n).hi<DELTA*Q<F_interval(hi,n).lo
	return lo,hi

	def resistance(br):
	lo,hi=br
	assert LAM<0
	return C+LAMDELTA/lo,C+LAMDELTA/hi

	def occupation(stack):
	q=F(0);coef=Counter();qpoints=[q]
	for phase,h in stack:
	qq=q+(K if phase=='B' else -K)*h
	coef[max(q,qq)]+=1;coef[min(q,qq)]-=1
	q=qq;qpoints.append(q)
	assert q==0
	return sorted((x,c) for x,c in coef.items() if c),qpoints

	def main():
	infj=bracket();infr=resistance(infj)
	ji=I(I.f(infj[0]).lo,I.f(infj[1]).hi)
	fp=4Hcoshi(KHji)+M
	# Interval for the exact asymptotic leading constant.
	num=(-LAMDELTA)KKMMM
	astar_lo=numinfj[0]/(96F(fp.hi,Q));astar_hi=numinfj[1]/(96F(fp.lo,Q))
	cases=[];rows=[]
	for n in NS:
	st,branch,scale=layer_stack(n)
	assert len(st)==2*n+1
	assert sum((h for _,h in st),F(0))==L
	assert all(sum((h for p,h in st if p==phase),F(0))==L/2 for phase in ['A','B'])
	assert st==st[::-1] and st[0][1]>=H
	occ,qp=occupation(st)
	br=bracket(n);rr=resistance(br)
	gap=(infr[0]-rr[1],infr[1]-rr[0]);assert gap[0]>0
	scaled=((n-1)*2gap[0],(n-1)*2gap[1])
	case={'n_B_layers':n,'interfaces':2n,'layers':2n+1,'branch':branch,'scale':str(scale),
	'stack':[[p,str(h)] for p,h in st],'occupation_coefficients':[[str(q),v] for q,v in occ],
	'Q_extrema':[str(x) for x in qp],'flux_interval':list(map(str,br)),
	'resistance_interval':list(map(str,rr)),'unrestricted_regret_interval':list(map(str,gap)),
	'scaled_regret_interval':list(map(str,scaled))}
	cases.append(case)
	rows.append({'n_B_layers':n,'interfaces':2n,'layers':2n+1,'branch':branch,
	'flux_lower':float(br[0]),'flux_upper':float(br[1]),'resistance_lower':float(rr[0]),
	'resistance_upper':float(rr[1]),'regret_upper':float(gap[1]),'scaled_regret':float(sum(scaled)/2)})
	floor=F(1,100);nmax=math.floor(1+M/(2*floor));assert nmax==41
	st,_,_=layer_stack(nmax);assert min(h for _,h in st)>=floor
	cert={'format':'Nuyori VI exact finite-interface frontier v1','synthetic':True,'fixed_point_bits':BITS,
	'inputs':{'L':str(L),'H_A':str(L/2),'H_B':str(L/2),'skin':str(H),'kappa':str(K),
	'a_L':'0','a_H':str(DELTA),'alpha_A':'3','alpha_B':'3','psi':'-1','Lambda':str(LAM*DELTA),
	'r_A':'1-kappaa','r_B':'1+kappaa','q_A':'2+kappaa','q_B':'2-kappaa'},
	'unrestricted':{'flux_interval':list(map(str,infj)),'resistance_interval':list(map(str,infr)),
	'attained_by_pure_film':False,'asymptotic_constant_interval':[str(astar_lo),str(astar_hi)]},
	'cases':cases,'minimum_thickness':{'floor':str(floor),'n_max':nmax,'layers':2nmax+1,'interfaces':2nmax,
	'minimum_actual_layer':str(min(h for _,h in st))}}
	(ROOT/'certificate.json').write_text(json.dumps(cert,indent=2)+'\n')
	with (ROOT/'minimum_thickness_layers.csv').open('w',newline='') as f:
	w=csv.writer(f);w.writerow(['layer','phase','start_depth','end_depth','thickness','relative_thickness'])
	y=F(0)
	for idx,(phase,thick) in enumerate(st,1):
	w.writerow([idx,phase,str(y),str(y+thick),str(thick),str(thick/L)]);y+=thick
	result={'status':'PASS: exact analytic global frontier with validated synthetic root intervals',
	'synthetic':True,'finite_interface_cases':len(cases),'nonattainment_for_any_pure_measurable_film':True,
	'unrestricted_flux_interval':[float(x) for x in infj],'unrestricted_resistance_interval':[float(x) for x in infr],
	'asymptotic_constant_interval':[float(astar_lo),float(astar_hi)],'cases':rows,
	'minimum_thickness':{'floor':float(floor),'n_max':nmax,'layers':2nmax+1,'interfaces':2nmax,
	'resistance_interval':[float(x) for x in map(F,cases[NS.index(nmax)]['resistance_interval'])]},
	'limitations':['Synthetic ideal 1D model; no manufactured polymer claim.',
	'Exact frontier applies to equal inventories, opposing affine reciprocal water slopes, potential-compatible oxygen laws and the stated A skins.',
	'Unrestricted optimum is a supremum and requires a relaxed mixing limit, not a finite pure film.',
	'The numerical minimum-layer example has d<=h; the general theorem uses h_eff=max(h,d).']}
	(ROOT/'results.json').write_text(json.dumps(result,indent=2)+'\n')
	with (ROOT/'frontier.csv').open('w',newline='') as f:
	w=csv.DictWriter(f,fieldnames=list(rows[0]));w.writeheader();w.writerows(rows)
	fig,axs=plt.subplots(1,3,figsize=(12.3,3.7),layout='constrained')
	xs=[r['interfaces'] for r in rows];gaps=[r['regret_upper'] for r in rows]
	axs[0].loglog(xs,gaps,'o-',color='#117b85',label='Exact global finite-interface gap')
	axs[0].set_xlabel('Maximum physical interfaces');axs[0].set_ylabel('Gap to unrestricted supremum')
	axs[0].set_title('Manufacturing frontier');axs[0].grid(alpha=.2)
	sel=[r for r in rows if r['n_B_layers']>1]
	axs[1].semilogx([r['n_B_layers']-1 for r in sel],[r['scaled_regret'] for r in sel],'o-',color='#117b85')
	axs[1].axhline(float((astar_lo+astar_hi)/2),color='#b18b31',linestyle='--',label='Proved leading constant')
	axs[1].set_xlabel('n - 1');axs[1].set_ylabel('(n - 1)^2 × gap');axs[1].set_title('Sharp inverse-square law');axs[1].legend(fontsize=7);axs[1].grid(alpha=.2)
	y=0.
	for phase,h in st:
	axs[2].barh(0,float(h),left=y,height=.5,color='#173c49' if phase=='A' else '#d4b35b');y+=float(h)
	axs[2].set_xlim(0,float(L));axs[2].set_ylim(-.6,.6);axs[2].set_yticks([]);axs[2].set_xlabel('Physical depth / total thickness')
	axs[2].set_title('1% minimum layer: 83-layer optimum')
	fig.suptitle('Synthetic exact model: interface count and layer thickness are explicit constraints',fontsize=10)
	fig.savefig(ROOT/'frontier.png',dpi=190);plt.close(fig)
	(ROOT/'verification_summary.txt').write_text(json.dumps(result,indent=2)+'\n')
	print(json.dumps({k:v for k,v in result.items() if k not in ['cases','limitations']},indent=2))

	if __name__=='__main__':main()