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"""ELYSIAEN: a standalone headless local-scattering capsule API.
Only NumPy and SciPy are required. No earlier ELYSIAEN file is imported.
Compilation may inspect the full reference world; evaluate() has only six
small retained arrays and performs no global query. A separately assembled
full sparse solve is used solely as the verification oracle in main().
This script makes no photonic timing, hardware, or energy claim.
"""
from dataclasses import dataclass
from pathlib import Path
from typing import Mapping
import argparse
import json
import platform
import numpy as np
import scipy
from scipy import sparse
from scipy.sparse.linalg import splu, spsolve
@dataclass(frozen=True)
class Capsule:
"""The complete retained model: six owning arrays and scalar metadata.
No full-world state, source, observation, coordinate set, inverse, sparse
factorization, callback, or external reference is retained.
"""
Cext: np.ndarray # k by k: T_ZR (I-T_RR)^-1 T_RZ
g: np.ndarray # k by s: T_ZR (I-T_RR)^-1 W_R
Y: np.ndarray # m by s: O_R (I-T_RR)^-1 W_R
M: np.ndarray # m by k: O_R (I-T_RR)^-1 T_RZ
WZ: np.ndarray # k by s: W_Z
OZ: np.ndarray # m by k: O_Z
metadata: Mapping[str, object]
def _own(a):
result = np.array(a, dtype=float, order="C", copy=True)
result.setflags(write=False)
assert result.base is None and result.flags.owndata
return result
def compile_capsule(T0, Z, sourcebasis, obs):
"""Compile a fixed source family b=W xi and fixed output map y=O x.
T0 must be nonnegative with a stable exterior principal block. For the
declared nonnegative transport family, checking a strict full reference
contraction is a simple sufficient compilation admission condition. A
production uncertainty contract must conservatively enclose these data.
No positivity clipping or approximation is performed here.
"""
t = sparse.csc_matrix(T0, dtype=float)
n = t.shape[0]
zz = np.asarray(Z, dtype=int)
if t.shape != (n, n) or len(zz) == 0 or len(np.unique(zz)) != len(zz):
raise ValueError("Expected a square reference and nonempty unique Z")
if min(zz) < 0 or max(zz) >= n:
raise ValueError("Z is outside the reference coordinates")
if np.min(t.data, initial=0.) < 0:
raise ValueError("The reference must be nonnegative")
W = np.asarray(sourcebasis, dtype=float)
O = np.asarray(obs, dtype=float)
if W.ndim != 2 or W.shape[0] != n or O.ndim != 2 or O.shape[1] != n:
raise ValueError("Source and output families have incompatible shapes")
rr = np.setdiff1d(np.arange(n), zz)
tzr = t[zz, :][:, rr]
trz = t[rr, :][:, zz]
trr = t[rr, :][:, rr]
if len(rr):
factor = splu(sparse.eye(len(rr), format="csc")-trr)
# Solve all compilation right hand sides together, once. factor,
# exterior arrays and the full input matrices die when this returns.
response_with_witness = factor.solve(np.column_stack((trz.toarray(), W[rr, :], np.ones(len(rr)))))
vext = response_with_witness[:, -1]
if not np.all(np.isfinite(vext)) or np.any(vext <= 0):
raise ValueError("The nonnegative exterior failed its positive stability witness")
# For a nonnegative exterior, a positive v with (I-T_RR)v=1
# is an exact mathematical stability witness. Here its residual is
# checked numerically, with the ordinary-float scope stated above.
witness_residual = (sparse.eye(len(rr), format="csc")-trr) @ vext-np.ones(len(rr))
if np.max(abs(witness_residual)) > 1e-8:
raise ValueError("The exterior stability witness residual is too large")
response = response_with_witness[:, :-1]
lift = response[:, :len(zz)]
background = response[:, len(zz):]
cext = tzr @ lift
incoming = tzr @ background
ybg = O[:, rr] @ background
outlift = O[:, rr] @ lift
else:
cext = np.zeros((len(zz), len(zz)))
incoming = np.zeros((len(zz), W.shape[1]))
ybg = np.zeros((O.shape[0], W.shape[1]))
outlift = np.zeros((O.shape[0], len(zz)))
arrays = [_own(a) for a in (cext, incoming, ybg, outlift, W[zz, :], O[:, zz])]
metadata = {"reference_dimension": n, "ports": len(zz),
"source_basis_dimension": W.shape[1], "outputs": O.shape[0],
"retained_array_bytes": sum(a.nbytes for a in arrays),
"retained_global_arrays": 0, "fresh_global_queries_per_evaluation": 0,
"exterior_stability_diagnostic": "positive resolvent witness",
"contract": "b=sourcebasis@xi; fixed source and observation families"}
return Capsule(*arrays, metadata)
def evaluate(capsule, B, L, R, xi, *, witness=None, eta=None):
"""Return (protected output, local state) using only retained arrays.
The nonnegative closure C=B+L Cext R must satisfy rho(C)<1. Passing a
positive witness and eta validates C v<=eta v in ordinary floating point;
without one the spectral-radius check is a diagnostic admission test.
This is a finite numerical API, not an outward-rounded hard certificate.
"""
k = capsule.Cext.shape[0]
B, L, R, xi = (np.asarray(a, dtype=float) for a in (B, L, R, xi))
if any(a.shape != (k, k) for a in (B, L, R)):
raise ValueError("B, L, R must be k by k")
if xi.shape != (capsule.WZ.shape[1],):
raise ValueError("xi is outside the declared source coordinate dimension")
if any(np.min(a, initial=0.) < 0 for a in (B, L, R)):
raise ValueError("Local scattering maps must be nonnegative")
C = B+L @ capsule.Cext @ R
if witness is not None:
v = np.asarray(witness, dtype=float)
if v.shape != (k,) or min(v) <= 0 or eta is None or not (0 <= eta < 1):
raise ValueError("Expected a positive local witness and 0<=eta<1")
if np.any(C @ v > eta*v+1e-12*np.maximum(1., abs(v))):
raise ValueError("Local positive stability witness failed")
elif float(np.max(abs(np.linalg.eigvals(C)))) >= 1:
raise ValueError("Local closure is unstable")
f = capsule.WZ @ xi+L @ (capsule.g @ xi)
z = np.linalg.solve(np.eye(k)-C, f)
y = capsule.Y @ xi+capsule.OZ @ z+capsule.M @ (R @ z)
return y, z
def _world(n, k, s, m, rng):
"""A sparse positive reference with a fixed six-state editable patch."""
rows, cols, vals = [], [], []
for i in range(n):
for offset, value in ((-1,.30),(1,.30),(-3,.08),(3,.08)):
rows.append(i); cols.append((i+offset) % n); vals.append(value)
t0 = sparse.csc_matrix((vals, (rows, cols)), shape=(n, n))
zz = np.arange(2, 2+k)
# The fixed source family is deliberately small and positive. The API
# is exact only for this span, however many configurations are evaluated.
W = np.zeros((n, s))
for j in range(s):
for i in ((j*11+1) % n, (j*17+13) % n):
W[i, j] += rng.uniform(.2, 1.)
O = np.zeros((m, n))
for j in range(m):
O[j, (j*19+4) % n] = 1.
O[j, (j*23+12) % n] += .2
return t0, zz, W, O
def _local_maps(k, cext, rng, frame):
# Includes positive noncommuting mixers and arbitrary local B additions.
L = rng.uniform(.0, 1., (k, k))
R = rng.uniform(.0, 1., (k, k))
L *= rng.uniform(.3, 1., (k, 1))/L.sum(axis=1, keepdims=True)
R *= rng.uniform(.3, 1., (k, 1))/R.sum(axis=1, keepdims=True)
B = rng.uniform(.0, 1., (k, k))
B *= rng.uniform(.1, .4, (k, 1))/B.sum(axis=1, keepdims=True)
C = B+L @ cext @ R
scale = np.minimum(1., .82/C.sum(axis=1))
B *= scale[:, None]; L *= scale[:, None]
if frame % 25 == 0:
B[:] = L[:] = R[:] = 0. # Fully opaque endpoint with local emitters.
return B, L, R
def _oracle(t0, zz, W, O, B, L, R, xi):
"""Independent global assembly and solve; never called by evaluate()."""
n = t0.shape[0]
rr = np.setdiff1d(np.arange(n), zz)
# Permute to [Z, exterior] so sparse block assembly is simple and explicit.
edited = sparse.bmat([[sparse.csc_matrix(B), L @ t0[zz, :][:, rr]],
[t0[rr, :][:, zz] @ R, t0[rr, :][:, rr]]], format="csc")
order = np.concatenate((zz, rr))
xperm = spsolve(sparse.eye(n, format="csc")-edited, (W @ xi)[order])
return O[:, order] @ xperm, xperm[:len(zz)]
def main():
parser = argparse.ArgumentParser(description=__doc__)
parser.add_argument("--quick", action="store_true", help="Run 12 frames per size instead of 100")
parser.add_argument("--output", default="v4_compiled_results.json")
args = parser.parse_args()
rng = np.random.default_rng(2026102541)
frames = 12 if args.quick else 100
k, m, s = 6, 4, 5
records = []
max_y, max_z, max_noncommute = 0., 0., 0.
bytes_values = []
for n in (64, 128, 256, 512):
t0, zz, W, O = _world(n, k, s, m, rng)
capsule = compile_capsule(t0, zz, W, O)
assert set(capsule.__dict__) == {"Cext","g","Y","M","WZ","OZ","metadata"}
arrays = [capsule.Cext,capsule.g,capsule.Y,capsule.M,capsule.WZ,capsule.OZ]
assert [a.shape for a in arrays] == [(k,k),(k,s),(m,s),(m,k),(k,s),(m,k)]
assert all(a.base is None and a.flags.owndata and not a.flags.writeable for a in arrays)
assert all(not isinstance(v, np.ndarray) for v in capsule.metadata.values())
array_bytes = sum(a.nbytes for a in arrays)
bytes_values.append(array_bytes)
this_y, this_z = 0., 0.
for frame in range(frames):
B, L, R = _local_maps(k, capsule.Cext, rng, frame)
xi = rng.uniform(.0, 1., s)
y, z = evaluate(capsule, B, L, R, xi, witness=np.ones(k), eta=.82)
yo, zo = _oracle(t0, zz, W, O, B, L, R, xi)
ey, ez = float(max(abs(y-yo))), float(max(abs(z-zo)))
assert ey < 1e-11 and ez < 1e-11
this_y, this_z = max(this_y, ey), max(this_z, ez)
max_noncommute = max(max_noncommute, float(max(abs((L @ R-R @ L).ravel()))))
max_y, max_z = max(max_y, this_y), max(max_z, this_z)
records.append({"reference_dimension":n,"frames":frames,"ports":k,
"source_basis_dimension":s,"protected_outputs":m,
"retained_six_array_bytes":array_bytes,
"retained_full_world_arrays":0,"fresh_global_queries_in_evaluate":0,
"max_output_error":this_y,"max_local_state_error":this_z})
assert len(set(bytes_values)) == 1 and bytes_values[0] == 1312
# Source-span negative control: a missing independent emitter cannot be
# represented by changing xi. This is outside evaluate()'s contract.
n = 64
t0, zz, W, O = _world(n, k, s, m, rng)
capsule = compile_capsule(t0, zz, W, O)
B, L, R = _local_maps(k, capsule.Cext, rng, 1)
xi = rng.uniform(.2, 1., s)
y, _ = evaluate(capsule, B, L, R, xi)
# Choose a protected observed exterior coordinate absent from source span.
source_index = 35
extra = np.zeros(n); extra[source_index] = 1.
span_residual = float(np.linalg.norm(extra-W @ np.linalg.lstsq(W,extra,rcond=None)[0]))
assert span_residual > .9
W_aug = np.column_stack((W,extra))
outside_source_y, _ = _oracle(t0,zz,W_aug,O,B,L,R,np.r_[xi,1.])
stale_source_error = float(max(abs(outside_source_y-y)))
assert stale_source_error > .01
recomp_source = compile_capsule(t0,zz,W_aug,O)
repaired_source_y, _ = evaluate(recomp_source,B,L,R,np.r_[xi,1.])
source_recompile_error = float(max(abs(repaired_source_y-outside_source_y)))
assert source_recompile_error < 1e-11
# A new observation also requires its compiled projections. Reusing y
# from the old protected map produces a visible negative control.
O_new = O.copy()
O_new[0,:] = 0.; O_new[0,13] = 1.
new_view_y, _ = _oracle(t0,zz,W,O_new,B,L,R,xi)
stale_view_error = float(max(abs(new_view_y-y)))
assert stale_view_error > 1e-3
new_view_cap = compile_capsule(t0,zz,W,O_new)
repaired_view_y, _ = evaluate(new_view_cap,B,L,R,xi)
view_recompile_error = float(max(abs(repaired_view_y-new_view_y)))
assert view_recompile_error < 1e-11
# Nonnegative but unstable local closure must be rejected before solving.
rejection = False
try:
evaluate(capsule,1.1*np.eye(k),np.zeros((k,k)),np.zeros((k,k)),xi)
except ValueError:
rejection = True
assert rejection
exterior_rejected = False
try:
compile_capsule(1.1*sparse.eye(n, format="csc"),zz,W,O)
except ValueError:
exterior_rejected = True
assert exterior_rejected
summary = {"project":"ELYSIAEN standalone compiled local capsule",
"date":"2026-10-25","status":"All assertions passed","seed":2026102541,
"python":platform.python_version(),"numpy":np.__version__,"scipy":scipy.__version__,
"sizes":[64,128,256,512],"total_frames":4*frames,
"maximum_output_error":max_y,"maximum_local_state_error":max_z,
"retained_array_bytes_at_every_size":bytes_values[0],
"retained_array_shapes":[[k,k],[k,s],[m,s],[m,k],[k,s],[m,k]],
"retained_global_state_arrays":0,"fresh_global_queries_in_evaluate":0,
"maximum_noncommutation_LR":max_noncommute,
"outside_source_span_residual":span_residual,
"stale_source_family_error":stale_source_error,
"source_recompilation_error":source_recompile_error,
"stale_observation_error":stale_view_error,
"observation_recompilation_error":view_recompile_error,
"unstable_local_closure_rejected":rejection,
"unstable_exterior_reference_rejected":exterior_rejected,
"scope":"Exact finite source/view families; compilation includes full-world work. Zero evaluation queries is a code property, not a measured optical advantage.",
"records":records}
Path(args.output).write_text(json.dumps(summary,indent=2)+"\n")
print(json.dumps(summary,indent=2))
if __name__ == "__main__":
try:
from threadpoolctl import threadpool_limits
except ImportError:
main()
else:
with threadpool_limits(limits=1):
main()
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