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"""Standalone certified return-compression experiments for ELYSIAEN.
NumPy/SciPy/Matplotlib only. No earlier project module is imported. The full
return matrix is retained by the independent oracle, never by CompressedReturn.
Arithmetic work models are counts, not CPU/GPU or photonic benchmarks.
"""
from __future__ import annotations
import os
os.environ.setdefault('OPENBLAS_NUM_THREADS', '1')
os.environ.setdefault('OMP_NUM_THREADS', '1')
from dataclasses import dataclass
from pathlib import Path
import argparse
import json
import platform
import numpy as np
import scipy
from scipy.linalg import solve
@dataclass(frozen=True)
class CompressedReturn:
U: np.ndarray
V: np.ndarray
weight: np.ndarray
return_on_weight: np.ndarray
delta: float
def weighted_inf(x, weight):
return float(np.max(np.abs(x)/weight, initial=0.0))
def compile_return(C, U, V, weight):
"""Oracle/offline compilation; measured deployment needs valid envelopes."""
a=np.asarray(weight, dtype=float)
if np.any(a<=0) or np.min(C, initial=0)<0:
raise ValueError('Expected positive weights and nonnegative true return')
E=np.abs(C-U@V)
delta=weighted_inf(E@a,a)
arrays=[np.array(z,copy=True) for z in (U,V,a,C@a)]
for z in arrays: z.setflags(write=False)
return CompressedReturn(*arrays,delta)
def evaluate_diagonal(capsule, bdiag, ldiag, rdiag, f):
"""Only compressed arrays; no true/dense return or full-state callback.
This optimized path assumes diagonal B,L,R. The true A is admitted by a
positive witness derived from a and C a. Any error certificate is ordinary
floating-point evaluation of an exact inequality, not outward rounding.
"""
U,V,a,c,delta=(capsule.U,capsule.V,capsule.weight,
capsule.return_on_weight,capsule.delta)
b,l,r=map(np.asarray,(bdiag,ldiag,rdiag))
if any(np.min(z,initial=0)<0 for z in (b,l,r)) or np.any(b>=1):
raise ValueError('Nonnegative diagonal maps and I-B invertible required')
tR=weighted_inf(r*a,a)
slack=a-b*a-tR*l*c
if np.min(slack)<=0:
raise ValueError('The retained true-class stability witness fails')
P=(l/(1-b))[:,None]*U
T=V*r[None,:]
q=f/(1-b)
K=np.eye(V.shape[0])-T@P
z=solve(K,T@q,assume_a='gen',check_finite=True)
u=q+P@z
residual=(1-b)*u-l*(U@(V@(r*u)))-f
forcing=np.abs(residual)+delta*weighted_inf(r*u,a)*l*a
beta=float(np.max(forcing/slack, initial=0.0))
return u,beta,{'small_rank':V.shape[0], 'true_weighted_slack_min':float(np.min(slack/a)),
'small_condition_2':float(np.linalg.cond(K)),
'retained_array_bytes':sum(z.nbytes for z in (U,V,a,c))}
def positive_world(k,rng):
q=10
u=rng.uniform(.2,1.2,(k,q))
v=rng.uniform(.2,1.2,(q,k))
u/=u.sum(axis=0,keepdims=True)
v/=v.sum(axis=1,keepdims=True)
decay=.55**np.arange(q)
u*=decay[None,:]
# The positive diagonal tail makes the return full rank. Compressibility
# is a stipulated generated property, not inferred for arbitrary scenes.
U0=np.column_stack((u,.001*np.eye(k)))
V0=np.vstack((v,np.eye(k)))
C0=U0@V0
scale=.60/np.max(C0.sum(axis=1))
U0*=scale
a=np.exp(rng.uniform(-1.5,1.5,k))
U=a[:,None]*U0
V=V0/a[None,:]
return U@V,U,V,a
def signed_svd_control(rng):
# Search a fixed seeded family for the well-known fact that truncating an
# asymmetric nonnormal stable positive operator need not preserve stability.
for attempt in range(1500):
k=5
C0=rng.uniform(0,1,(k,k))
C0*=.85/max(abs(np.linalg.eigvals(C0)))
w=np.exp(rng.uniform(-7,7,k))
C=w[:,None]*C0/w[None,:]
uu,ss,vv=np.linalg.svd(C,full_matrices=False)
for rank in (2,3):
Cr=(uu[:,:rank]*ss[:rank])@vv[:rank]
rho=float(max(abs(np.linalg.eigvals(Cr))))
if rho>1.02 and np.min(Cr)<-1e-7:
return {'found':True,'attempt':attempt,'rank':rank,
'true_spectral_radius':float(max(abs(np.linalg.eigvals(C)))),
'truncated_spectral_radius':rho,'minimum_truncated_entry':float(Cr.min()),
'C':C.tolist(),'C_rank':Cr.tolist(),
'meaning':'SVD rank truncation can destroy positivity and stability; certificate uses true class.'}
return {'found':False,'meaning':'No such counterexample found in this finite search; no preservation claim follows.'}
def main():
parser=argparse.ArgumentParser(description=__doc__)
parser.add_argument('--quick',action='store_true')
parser.add_argument('--output-prefix',default='v5_compression')
args=parser.parse_args()
rng=np.random.default_rng(2026102551)
count=12 if args.quick else 60
records=[]; max_reconstruction=0.; max_ratio=0.; violations=0
monotone_checks=0; monotone_violations=0; signed_checks=0
frontier=[]
for case in range(count):
k=(16,32,64)[case%3]
C,U,V,a=positive_world(k,rng)
b=rng.uniform(0,.14,k); l=rng.uniform(.25,1,k); r=rng.uniform(.25,1,k)
f=a*rng.uniform(.1,1,k)
O=rng.uniform(0,1,(3,k))/a[None,:]/k
A=np.diag(b)+l[:,None]*C*r[None,:]
exact=solve(np.eye(k)-A,f)
ranks=(1,2,4,8,10,k+10)
previous=None
for rank in ranks:
cap=compile_return(C,U[:,:rank],V[:rank],a)
approx,beta,info=evaluate_diagonal(cap,b,l,r,f)
approx_direct=solve(np.eye(k)-np.diag(b)-l[:,None]*(cap.U@cap.V)*r[None,:],f)
max_reconstruction=max(max_reconstruction,weighted_inf(approx-approx_direct,a))
err=weighted_inf(exact-approx,a)
out_err=np.abs(O@(exact-approx)); out_bound=(np.abs(O)@a)*beta
if err>beta+2e-10 or np.any(out_err>out_bound+2e-10): violations+=1
if beta>1e-13: max_ratio=max(max_ratio,err/beta)
if previous is not None:
monotone_checks+=1
if np.min(approx-previous)<-2e-10: monotone_violations+=1
previous=approx
rec={'case':case,'k':k,'factor_count':rank,'error':err,'certificate':beta,
'relative_return_envelope':cap.delta,**info}
records.append(rec)
if case==1: frontier.append(rec)
# Signed SVD factors use the very same true-class compression barrier.
uu,ss,vv=np.linalg.svd(C,full_matrices=False)
for rank in (2,5,min(10,k)):
cap=compile_return(C,uu[:,:rank]*ss[:rank],vv[:rank],a)
approx,beta,_=evaluate_diagonal(cap,b,l,r,f)
if weighted_inf(exact-approx,a)>beta+2e-10: violations+=1
signed_checks+=1
assert violations==0 and monotone_violations==0
near=[]
for eps in np.logspace(-1,-6,20):
delta=1e-4
exact=1/eps; approx=1/(eps+delta)
actual=abs(exact-approx)
certificate=delta*approx/eps
assert actual<=certificate+1e-5*max(1,actual)
near.append({'epsilon':float(eps),'return_error':delta,'actual_state_error':actual,
'certificate':certificate})
work=[]
for k in (32,64,128,256):
dense=2*k**3/3+2*k*k
for rank in (1,2,4,8,12):
# One dense r-core factorization per local configuration and one RHS.
low=2*k*rank**2+2*rank**3/3+8*k*rank+2*rank**2+12*k
work.append({'k':k,'rank':rank,'dense_digital_model_flops':dense,
'compressed_digital_model_flops':low,'modeled_ratio':dense/low})
signed_control=signed_svd_control(rng)
assert signed_control['found'], 'Seeded signed-SVD negative control was not reproduced'
# Simple exact impossibility control: any deficient-rank Cr has a kernel.
gamma=.6; k=8
Cr=np.zeros((k,k));Cr[:3,:3]=gamma*np.eye(3)
z=np.zeros(k);z[-1]=1
lower_bound_error=float(np.linalg.norm(z/(1-gamma)-solve(np.eye(k)-Cr,z)))
assert abs(lower_bound_error-gamma/(1-gamma))<1e-12
result={'project':'ELYSIAEN certified return compression','date':'2026-10-25',
'seed':2026102551,'software':{'Python':platform.python_version(),'NumPy':np.__version__,'SciPy':scipy.__version__},
'random_capsules':count,'positive_prefix_candidate_checks':len(records),
'signed_svd_candidate_checks':signed_checks,'certificate_violations':violations,
'monotone_prefix_checks':monotone_checks,'monotone_prefix_violations':monotone_violations,
'maximum_small_solve_reconstruction_error':max_reconstruction,
'maximum_actual_error_to_certificate_ratio':max_ratio,
'retained_contract':'U,V,a,Ca,delta only; no dense true return in evaluate_diagonal',
'signed_svd_negative_control':signed_control,
'full_rank_negative_control':{'gamma':gamma,'k':k,'rank':3,'observed_error':lower_bound_error,'lower_bound':gamma/(1-gamma)},
'rank_frontier':frontier,'near_critical':near,'digital_work_model':work,'records':records,
'scope':'Finite arithmetic identities, stated diagonal local maps, generated compressibility and oracle-derived offline envelopes. Flop models are not measured timing/energy or photonic speedups. Ordinary double precision, not outward-rounded certificates.'}
path=Path(args.output_prefix);path.parent.mkdir(parents=True,exist_ok=True)
path.with_name(path.name+'_results.json').write_text(json.dumps(result,indent=2)+'\n')
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt
plt.rcParams.update({'font.size':11,'axes.spines.top':False,'axes.spines.right':False})
fig,axes=plt.subplots(1,2,figsize=(9,4.2),constrained_layout=True)
# Plot practical nonexact prefixes. Exact reconstruction reaches ordinary
# floating-point noise, not an outward-rounded certificate floor.
shown=[x for x in frontier if x['factor_count']<=10]
ranks=[x['factor_count'] for x in shown]
axes[0].semilogy(ranks,[x['error'] for x in shown],'o-',label='Actual state error')
axes[0].semilogy(ranks,[x['certificate'] for x in shown],'s--',label='Certified barrier')
axes[0].set(xlabel='Retained positive factors',ylabel='Weighted infinity error',title='Compression versus accuracy');axes[0].legend()
axes[1].loglog([x['epsilon'] for x in near],[x['actual_state_error'] for x in near],'o-',label='Actual = barrier')
axes[1].invert_xaxis();axes[1].set(xlabel='True feedback slack',ylabel='State error for return error 10⁻⁴',title='Near-unit return remains sensitive')
for ax in axes: ax.grid(alpha=.2)
fig.suptitle('Finite generated transport; no hardware measurements',fontsize=12)
fig.savefig(path.with_suffix('.png'),dpi=170);plt.close(fig)
fig,ax=plt.subplots(figsize=(8,3.4),constrained_layout=True)
for rank in (2,4,8,12):
rr=[x for x in work if x['rank']==rank]
ax.plot([x['k'] for x in rr],[x['modeled_ratio'] for x in rr],'o-',label=f'r={rank}')
ax.axhline(1,color='gray',lw=.7);ax.set(xlabel='Capsule dimension k',ylabel='Dense / compressed work model',title='Same algorithm, digital baseline');ax.legend();ax.grid(alpha=.2)
fig.savefig(path.with_name(path.name+'_work').with_suffix('.png'),dpi=170);plt.close(fig)
print(json.dumps({k:result[k] for k in ['random_capsules','positive_prefix_candidate_checks','signed_svd_candidate_checks','certificate_violations','monotone_prefix_checks','maximum_small_solve_reconstruction_error','maximum_actual_error_to_certificate_ratio']},indent=2))
if __name__=='__main__': main()
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