"""CC0 diagram and exact finite checks for LF/GJ/FD normal connections.""" from pathlib import Path from fractions import Fraction as Q import argparse,json,math,os def finite_checks(): rows=[] def check(name,passed,detail): assert passed,name rows.append(dict(name=name,passed=True,detail=detail)) # Nontrivial transition, actual affine weighted origin and identity arrow. tests=[] for t in [Q(-5),Q(3),Q(7,2)]: for weight in [Q(0),Q(1,3),Q(2,3),Q(1)]: omega_j=-weight*t;omega_i=(1-weight)*t tests.append(t+omega_j-omega_i==0) check('identity_displacement_after_affine_origin',all(tests), '12 rational chart-transition tests; t=3, weight=2/3 gives omega_j=-2, omega_i=1, B_1=0.') tests=[] for rho in [Q(k,8) for k in range(1,9)]: tau=1+100/rho**4 for b in [Q(1,4),Q(1,2),Q(1)]: for lam in [Q(1),Q(2),Q(16),Q(256)]: tests.append(lam/(tau+b*lam)**2<=1/(4*tau*b)) check('scalar_resolvent_derivative_inequality',all(tests), '96 rational spectral samples; difference reduces to (tau-b*lambda)^2>=0.') vals=[] for rho in [Q(k,8) for k in range(9)]: bound=(4*rho**3+(2*rho+rho**4)/4+rho**4)/100 vals.append(0<=bound<=Q(23,400)) check('conditional_relative_bound',all(vals), '9 rational samples with L=J=H=1 and K0=100; endpoint 23/400, zero endpoint 0.') # Two frame points, three auxiliary points; exact pinned Haar marginal. mu=[Q(1,6)]*6 check('pinned_frame_marginal',sum(mu[:3])==sum(mu[3:])==Q(1,2), 'Finite toy frame fibre with two equally weighted points; three auxiliary choices per frame.') values=[Q(k) for k in range(6)] avg=sum(m*v*v for m,v in zip(mu,values)) check('probability_minimum_and_gram',min(v*v for v in values)<=avg and avg-sum(m*v for m,v in zip(mu,values))**2>=0, 'Exact minimum<=mean and positive two-vector Gram determinant; not an infinite-field compactness test.') # Direct Gaussian-integer sum of four characters of the cyclic group C4. # This finite illustration does not establish the infinite circle theorem. roots=[(1,0),(0,1),(-1,0),(0,-1)] fourier=[] for m in range(4): row=[] for n in range(4): terms=[roots[(n-m)*k%4] for k in range(4)] row.append((Q(sum(v[0] for v in terms),4),Q(sum(v[1] for v in terms),4))) fourier.append(row) check('fixed_auxiliary_finite_fourier_modes',fourier==[ [(Q(int(n==m)),Q(0)) for n in range(4)] for m in range(4)] and Q(1,2)**2==Q(1,4), 'Four C4 characters summed directly in Gaussian integers have Haar Gram identity; inverse eigenvalue 1/2 preserves squared norm 1/4. Infinite S1 noncompactness is proved in FD.7.') return dict(schema='normal-connection-frame-descent-finite-checks/v1',count=len(rows),all_passed=True,checks=rows) def draw(output,resources,description): import matplotlib matplotlib.use('Agg') import matplotlib.pyplot as plt from matplotlib.font_manager import FontProperties from matplotlib.patches import FancyBboxPatch regular=FontProperties(fname=str(resources/'fonts/DejaVuSans.ttf')) bold=FontProperties(fname=str(resources/'fonts/DejaVuSans-Bold.ttf')) blue,green,red,ink,gray='#17628f','#16745a','#b34638','#19384d','#627789' plt.rcParams.update({'svg.hashsalt':'normal-connection-frame-descent-lf-gj-fd-v1', 'svg.fonttype':'path','axes.edgecolor':ink,'axes.labelcolor':ink, 'xtick.color':ink,'ytick.color':ink}) def text(ax,x,y,s,size=12,strong=False,**kw): return ax.text(x,y,s,fontproperties=bold if strong else regular, fontsize=size,color=kw.pop('color',ink),**kw) def panel(rect,title): ax=fig.add_axes(rect);ax.set_xlim(0,1);ax.set_ylim(0,1);ax.axis('off') text(ax,0,1,title,15,True,va='top');return ax def box(ax,x,y,w,h,s,color=blue): ax.add_patch(FancyBboxPatch((x,y),w,h,boxstyle='round,pad=.012', facecolor='#edf4f8',edgecolor=color,linewidth=1.4)) text(ax,x+w/2,y+h/2,s,12,ha='center',va='center') def arrow(ax,start,end,color=blue): ax.annotate('',xy=end,xytext=start,arrowprops=dict(arrowstyle='->',color=color,lw=1.8)) fig=plt.figure(figsize=(17,12),facecolor='white') head=panel([.035,.94,.93,.05],'Normal connections survive gluing; compact inverse can fail under averaging') text(head,0,.23,'Actual affine labels, flat collapse, geometric graph weight and original-unit frame descent. LF.1–FD.7',12) ax=panel([.04,.62,.43,.27],'A Correct affine origins cancel the identity label') box(ax,.01,.59,.35,.24,'Chart j\nωj = −2');box(ax,.62,.59,.35,.24,'Chart i\nωi = 1') arrow(ax,(.39,.72),(.59,.72));text(ax,.49,.80,'tij = 3',11,ha='center') text(ax,.02,.43,'Local i-origin in j: −3; weights φi=2/3, φj=1/3',12) text(ax,.02,.29,'ωj=(2/3)(−3)+(1/3)0=−2; ωi=tij+ωj=1',12,color=green) text(ax,.02,.14,'Identity in different charts: B1 = 3+(−2)−1 = 0',12,True) text(ax,.02,.01,'Transitions are constant-normal. The flat connection agrees. LF.4–LF.6',10) ax=panel([.53,.62,.43,.27],'B A disappearing factor requires flat connection cores') box(ax,.02,.62,.36,.21,'ρa > 0: active Qa\n(wa, wa za)');box(ax,.63,.62,.33,.21,'ρa = 0\none vertex') arrow(ax,(.41,.73),(.59,.73));text(ax,.49,.84,'wa → 0',11,ha='center') text(ax,.02,.46,'βa = exp(−1/ρa²), ba = βa /(Σ βc²)½, wa = ba²',12) text(ax,.02,.30,'Core observable: d(x) f(za), with every jet of d zero at ρa=0',11) text(ax,.02,.15,'Differentiate on the potential product; every derivative descends.',11,color=green) text(ax,.02,.01,'No derivative of a nonexistent coordinate is assigned. GJ.1–GJ.3',10) ax=fig.add_axes([.075,.285,.37,.255]) rho=[k/200 for k in range(201)] inv=[r**4/100 for r in rho] deriv=[(4*r**3+(2*r+r**4)/4+r**4)/100 for r in rho] ax.plot(rho,inv,color=blue,lw=2,label='ρ⁴/100: inverse norm upper bound') ax.plot(rho,deriv,color=green,lw=2,label='relative derivative upper bound') ax.axhline(23/400,color=gray,ls='--',lw=1) ax.set_xlim(0,1);ax.set_ylim(0,.062);ax.grid(alpha=.18) ax.set_title('C Conditional sample: L=J=H=1, K0=100',fontproperties=bold,fontsize=14,color=ink,pad=14) ax.set_xlabel('ρa (numerical samples)',fontproperties=regular,fontsize=11) ax.set_ylabel('operator norm upper bound',fontproperties=regular,fontsize=11) ax.legend(prop=FontProperties(fname=str(resources/'fonts/DejaVuSans.ttf'),size=9),loc='upper left') for label in ax.get_xticklabels()+ax.get_yticklabels():label.set_fontproperties(regular) text(ax,0,-.25,'Ta=(1+K0 ρa⁻⁴)I+ba Θa; defect = ba Sg. GJ.7–GJ.11',11,transform=ax.transAxes) text(ax,0,-.37,'Both displayed bounds vanish at collapse; endpoint 23/400 is exact.',10,transform=ax.transAxes) ax=panel([.53,.285,.43,.27],'D Pinned Haar probabilities return to original units') box(ax,.01,.65,.95,.19,'XH → P → T; compact frame Px, Haar probability mx') text(ax,.02,.50,'XG,x = {μ ∈ Prob(XH|Px) : frame_anchor* μ = mx}',12) text(ax,.02,.35,'Eμ = L²(μ; EH); Ψ(g,μ) = ∫ ‖C(g,u)(z)‖² dμ(u,z)',12,color=blue) text(ax,.02,.20,'δv ∫ f dμ = ∫ δv̂ f dμ; v̂ is the horizontal normal lift.',12,color=green) text(ax,.02,.06,'Lift potential-product measures to prove actual relation and null independence.',10) text(ax,.02,-.02,'Dense closable metric connection; no generic KK representative is used. FD.1–FD.6',10) ax=panel([.04,.035,.92,.14],'E Pointwise averaging of a compact-inverse weight can lose compact inverse') text(ax,.01,.58,'L²(S¹) ⊗ H0; Θ0 e1=2e1; vn=exp(inu)⊗e1, n∈Z, orthonormal.',12) text(ax,.01,.25,'(1 ⊗ Θ0)⁻¹ vn = (1/2)vn. The images stay orthogonal at norm 1/2.',12,color=red,strong=True) text(ax,.01,-.03,'Thus the inverse is noncompact. A confining averaged weight and closed physical sum remain required. FD.7',11) output.mkdir(parents=True,exist_ok=True) fig.savefig(output/'normal-connection-frame-descent.png',dpi=150, metadata=dict(Title='Normal connection and compact-frame descent',Description=description,Software='normal-connection-frame-descent v1')) fig.savefig(output/'normal-connection-frame-descent.svg', metadata=dict(Title='Normal connection and compact-frame descent',Description=description,Date=None)) plt.close(fig) return dict(schema='normal-connection-frame-descent-data/v1', affine_origin=dict(transition='3',weights=['2/3','1/3'],omega_j='-2',omega_i='1',identity_displacement='0'), collapse=dict(beta='exp(-1/rho^2)',b='beta/sqrt(sum beta_c^2)',w='b^2',core='d(x)f(z_a), all derivatives of d zero where rho_a=0'), exact_weight='(1+K0*rho^-4)I+b*Theta', conditional_sample=dict(L=1,J=1,H=1,K0=100,endpoint_bound='23/400',samples=[dict(rho=r,inverse_upper=a,relative_derivative_upper=b) for r,a,b in zip(rho,inv,deriv)]), frame_base='probabilities on X_H above P_x with prescribed Haar marginal m_x', noncompactness=dict(space='L2(S1) tensor H0',Theta0_eigenvalue='2',inverse_norm_every_Fourier_mode='1/2'), scope='Panel C is a numerical rendering of conditional bounds, not a chosen metric or a proved physical operator; the other panels are exact algebra or typed schematics.') def main(): parser=argparse.ArgumentParser(description=__doc__) parser.add_argument('--output-dir',type=Path,default=Path(__file__).resolve().parent/'out') parser.add_argument('--resources',type=Path,default=Path(__file__).resolve().parent.parent/'labelled-geometric-kernel') parser.add_argument('--checks-only',action='store_true');args=parser.parse_args() args.output_dir.mkdir(parents=True,exist_ok=True) os.environ.setdefault('MPLCONFIGDIR',str(args.output_dir/'runtime-cache')) checks=finite_checks();(args.output_dir/'finite-checks.json').write_text(json.dumps(checks,indent=2)+'\n',encoding='utf-8') if not args.checks_only: terms=Path(__file__).with_name('COMPONENT-TERMS.md').read_text(encoding='utf-8') data=draw(args.output_dir,args.resources,'LF.4–LF.6, GJ.1–GJ.5, FD.1–FD.7. Exact algebra, typed schematics and explicitly conditional sampled bounds.\n\n'+terms) (args.output_dir/'figure-data.json').write_text(json.dumps(data,indent=2)+'\n',encoding='utf-8') print(json.dumps(dict(all_passed=checks['all_passed'],checks=checks['count'],figure_rendered=not args.checks_only))) if __name__=='__main__':main()