Datasets:

PureOne
/

elysian-noctilucent-algebra

#!/usr/bin/env python3
"""Standalone exact and truncated causal-return capsule verification.
Run: python v5_dynamic_verify.py [--quick] [--output-prefix v5_dynamic]
Requires NumPy and Matplotlib. No earlier ELYSIAEN modules are imported.
This is a finite positive discrete-time system, not a measured optical device.
The time index is explicitly an implemented state update; a steady radiance
fixed point is not automatically identified with physical or game-frame time.
"""
from __future__ import annotations
import os
os.environ["OPENBLAS_NUM_THREADS"] = "1"
os.environ["OMP_NUM_THREADS"] = "1"
import argparse
import json
import platform
from pathlib import Path
import numpy as np

SEED = 2026102553

def wnorm(x, weight):
    x = np.asarray(x)
    return float(np.max(np.abs(x)/weight.reshape((-1,)+(1,)*(x.ndim-1)), initial=0.))

def row_map(rng, rows, cols, out_weight, in_weight, budget):
    a = rng.uniform(.03, 1., (rows, cols))
    a *= (budget*out_weight/(a@in_weight))[:, None]
    return a

def precompute(U, D, V, W, OZ, OF, steps):
    K, F, J, Q = [], [], [], []
    power = np.eye(D.shape[0])
    for j in range(steps):
        K.append(U@power@V); F.append(U@power@W)
        J.append(OF@power@V); Q.append(OF@power@W)
        power = power@D
    return K, F, J, Q

def initial_traces(U, D, OF, exterior0, steps):
    phi=[]; psi=[]; initial=exterior0.copy()
    for t in range(steps+1):
        phi.append(U@initial); psi.append(OF@initial); initial=D@initial
    return phi, psi

def causal(K, F, J, Q, OZ, Bs, Ls, Rs, local_source,
           xi, z0, phi, psi, horizon, *, wrong_current_egress=False):
    """Exact when horizon>=steps; otherwise only last horizon impulses kept.
    Initial exterior traces phi/psi are compiled before this routine.
    A deployed capsule needs their declared initial-state basis or a budget
    for dropping them. No exterior-sized array is read in this evaluator.
    """
    steps, k = len(Bs), len(z0)
    z = np.zeros((steps+1, k)); z[0] = z0
    y = np.zeros((steps+1, OZ.shape[0]))
    for t in range(steps+1):
        incoming = phi[t].copy()
        y[t] = OZ@z[t]+psi[t]
        for j in range(min(t, horizon)):
            old = t-1-j
            egress = Rs[min(t, steps-1)] if wrong_current_egress else Rs[old]
            outgoing = egress@z[old]
            incoming += K[j]@outgoing+F[j]@xi[old]
            y[t] += J[j]@outgoing+Q[j]@xi[old]
        if t < steps:
            z[t+1] = Bs[t]@z[t]+local_source[t]+Ls[t]@incoming
    return z, y

def full_oracle(U, D, V, W, OZ, OF, Bs, Ls, Rs, source, xi, z0, f0):
    steps, k, nf = len(Bs), len(z0), len(f0)
    states = np.zeros((steps+1, k+nf)); states[0] = np.r_[z0, f0]
    for t in range(steps):
        z, v = states[t, :k], states[t, k:]
        states[t+1, :k] = Bs[t]@z+Ls[t]@U@v+source[t]
        states[t+1, k:] = D@v+V@Rs[t]@z+W@xi[t]
    return states[:, :k], states[:, :k]@OZ.T+states[:, k:]@OF.T

def trial(rng, number, steps):
    k = (2, 3, 4)[number % 3]
    nf = (7, 12, 20)[(number//3) % 3]
    delta = (.2, .6, .9)[(number//9) % 3]
    a = np.exp(rng.uniform(-.7, .7, k))
    w = np.exp(rng.uniform(-.7, .7, nf))
    D = row_map(rng, nf, nf, w, w, delta)
    U = row_map(rng, k, nf, a, w, .18*np.sqrt(1-delta))
    V = row_map(rng, nf, k, w, a, .25*np.sqrt(1-delta))
    W = rng.uniform(.001, .02, (nf, 2))*w[:, None]
    OZ = rng.uniform(-.5, .8, (3, k))
    OF = rng.uniform(-.2, .4, (3, nf))
    Bbar = row_map(rng, k, k, a, a, (.4, .7)[number % 2])
    Lbar = row_map(rng, k, k, a, a, .85)
    Rbar = row_map(rng, k, k, a, a, .9)
    H = np.linalg.solve(np.eye(nf)-D, np.eye(nf))
    C = U@H@V
    Abar = Bbar+Lbar@C@Rbar
    q = float(np.max(Abar@a/a))
    assert q < .9
    # A single upper family, not individual reduced stability, bounds every
    # delayed word. This criterion is sufficient and need not be necessary.
    full_upper = np.block([[Bbar, Lbar@U], [V@Rbar, D]])
    full_radius=float(max(abs(np.linalg.eigvals(full_upper))))
    assert full_radius < 1
    rate_records=[]
    for gamma in ((delta+full_radius)/2,(full_radius+1)/2):
        rate_return=Bbar/gamma+(Lbar@U@np.linalg.solve(gamma*np.eye(nf)-D,V)@Rbar)/gamma
        local_rate_radius=float(max(abs(np.linalg.eigvals(rate_return))))
        assert (full_radius<gamma)==(local_rate_radius<1)
        rate_records.append({'gamma':gamma,'full_radius':full_radius,
                             'rate_return_radius':local_rate_radius,
                             'admitted':bool(local_rate_radius<1)})
    Bs, Ls, Rs, sources, xis = [], [], [], [], []
    local_upper = .12*a
    for t in range(steps):
        Bs.append(Bbar*rng.uniform(.3, 1., (k, k)))
        Ls.append(Lbar*rng.uniform(.2, 1., (k, k)))
        Rs.append(Rbar*rng.uniform(.2, 1., (k, k)))
        # Abrupt changes and smooth variation both stay in fixed envelopes.
        xis.append(np.array([.4+.3*np.sin(t/7.)**2, .8 if t >= steps//3 else .1]))
        sources.append(local_upper*rng.uniform(.1, 1., k))
    z0 = rng.uniform(0, .1, k)*a
    exterior0 = np.zeros(nf) if number%2==0 else rng.uniform(0,.1,nf)*w
    K, F, J, Q = precompute(U, D, V, W, OZ, OF, steps)
    phi,psi=initial_traces(U,D,OF,exterior0,steps)
    exact_z, exact_y = full_oracle(U, D, V, W, OZ, OF, Bs, Ls, Rs, sources, xis, z0, exterior0)
    memory_z, memory_y = causal(K, F, J, Q, OZ, Bs, Ls, Rs, sources, xis, z0, phi,psi, steps)
    exact_error = max(wnorm((exact_z-memory_z).T, a), float(np.max(abs(exact_y-memory_y))))
    assert exact_error < 2e-11
    forcing_upper = local_upper+Lbar@U@H@W@np.ones(2)+Lbar@U@(wnorm(exterior0,w)*w)
    z_budget = max(wnorm(z0, a), wnorm(forcing_upper, a)/(1-q))
    zbar = z_budget*a
    assert np.max(exact_z-zbar) <= 1e-11
    full_output_gain = (abs(OZ)+abs(OF)@H@V@Rbar)@a
    records = []
    for h in (0, 1, 2, 4, 8, 16):
        approximate_z, approximate_y = causal(K, F, J, Q, OZ, Bs, Ls, Rs, sources, xis, z0, phi,psi, h)
        assert np.max(approximate_z-zbar) <= 2e-11
        assert np.max(approximate_z-exact_z) <= 2e-11
        tail_power = np.linalg.matrix_power(D, h)
        exterior_drive = V@Rbar@zbar+W@np.ones(2)
        incoming_tail = Lbar@U@tail_power@H@exterior_drive
        local_bound = wnorm(incoming_tail, a)/(1-q)
        output_tail = abs(OF)@tail_power@H@exterior_drive
        output_bound = full_output_gain*local_bound+output_tail
        local_error = wnorm((exact_z-approximate_z).T, a)
        output_error = np.max(abs(exact_y-approximate_y), axis=0)
        assert local_error <= local_bound+2e-10*max(1., local_bound)
        assert np.all(output_error <= output_bound+2e-10*np.maximum(1., output_bound))
        records.append({"horizon": h, "local_error": local_error,
                        "local_uniform_bound": local_bound,
                        "maximum_output_error": float(max(output_error)),
                        "maximum_output_uniform_bound": float(max(output_bound)),
                        "local_error_to_bound": local_error/local_bound if local_bound else 0.,
                        "output_error_to_componentwise_bound": float(np.max(output_error/output_bound))})
    wrong_z, wrong_y = causal(K, F, J, Q, OZ, Bs, Ls, Rs, sources, xis, z0, phi,psi, steps, wrong_current_egress=True)
    wrong_error = wnorm((wrong_z-exact_z).T, a)
    # Compute an a-priori scalar horizon from ||D||<=delta. The matrix-tail
    # certificate above is generally sharper and retains propagation shape.
    alpha = float(np.max(U@w/a)); ell = float(np.max(Lbar@a/a))
    drive_norm = wnorm(V@Rbar@zbar+W@np.ones(2), w)
    prefactor = ell*alpha*drive_norm/((1-delta)*(1-q))
    target = .001
    required_h = max(0, int(np.ceil(np.log(target/prefactor)/np.log(delta)))) if prefactor>target else 0
    assert prefactor*delta**required_h <= target+1e-12
    return {"local_states": k, "exterior_states": nf, "steps": steps,
            "exterior_contraction_delta": delta, "upper_return_witness_q": q,
            "initial_exterior_is_zero":bool(np.all(exterior0==0)),
            "rate_threshold_checks":rate_records,
            "exact_causal_maximum_error": exact_error,
            "wrong_current_egress_error": wrong_error,
            "scalar_horizon_for_local_error_0_001": required_h,
            "retained_kernel_scalars_per_history_step": k*k+k*2+3*k+3*2,
            "records": records}, (exact_z, memory_z, records)

def counterexamples():
    # Stable reference with coupling .5. Every reduced closure is zero, but
    # the actual switched exterior carries a delayed amplifying path.
    U=V=np.array([[.5]]); D=np.zeros((1,1)); W=np.zeros((1,1))
    OZ=np.array([[1.]]); OF=np.array([[1.]])
    steps=16
    Bs=[np.zeros((1,1)) for _ in range(steps)]
    Ls=[np.array([[4. if t%2==0 else 0.]]) for t in range(steps)]
    Rs=[np.array([[0. if t%2==0 else 4.]]) for t in range(steps)]
    zero=[np.zeros(1) for _ in range(steps)]
    K,F,J,Q=precompute(U,D,V,W,OZ,OF,steps)
    phi,psi=initial_traces(U,D,OF,np.ones(1),steps)
    z,y=full_oracle(U,D,V,W,OZ,OF,Bs,Ls,Rs,zero,zero,np.zeros(1),np.ones(1))
    causal_z,causal_y=causal(K,F,J,Q,OZ,Bs,Ls,Rs,zero,zero,np.zeros(1),phi,psi,steps)
    wrong_z,wrong_y=causal(K,F,J,Q,OZ,Bs,Ls,Rs,zero,zero,np.zeros(1),phi,psi,steps,wrong_current_egress=True)
    assert np.max(abs(y-causal_y))==0
    assert y[-1,0]==65536. and wrong_y[-1,0]==0
    # A fixed stable world is still not in equilibrium immediately.
    fixed=np.array([[.1,.2],[.2,.8]]); source=np.array([.3,.1])
    equilibrium=np.linalg.solve(np.eye(2)-fixed,source)
    return {"switching": {"reference_couplings": .5,
            "reference_spectral_radius": .5,
            "each_reduced_closure": 0., "two_step_full_gain": 4.,
            "steps": steps, "final_full_output": float(y[-1,0]),
            "final_exact_causal_output": float(causal_y[-1,0]),
            "final_wrong_current_egress_output": float(wrong_y[-1,0]),
            "common_upper_reduced_return": 4.,
            "correct_envelope_admission_rejects": True},
            "equilibrium_is_not_first_update": {"equilibrium_local_state":float(equilibrium[0]),
            "first_dynamic_local_state_from_zero":.3,
            "absolute_difference":float(equilibrium[0]-.3)}}

def main():
    parser=argparse.ArgumentParser(description=__doc__)
    parser.add_argument("--quick",action="store_true")
    parser.add_argument("--output-prefix",default="v5_dynamic")
    args=parser.parse_args()
    rng=np.random.default_rng(SEED)
    ncases=12 if args.quick else 81
    steps=32 if args.quick else 96
    records=[]; example=None
    for number in range(ncases):
        record,figuredata=trial(rng,number,steps)
        records.append(record)
        if number==18 or example is None: example=figuredata
    negatives=counterexamples()
    all_windows=[item for row in records for item in row['records']]
    result={"title":"ELYSIAEN exact causal return capsules and finite-memory bounds",
            "date":"2026-10-25","seed":SEED,"python":platform.python_version(),
            "numpy":np.__version__,"random_switched_worlds":ncases,
            "steps_per_world":steps,"finite_horizons_per_world":6,
            "exact_causal_comparisons":ncases,"local_tail_certificates":len(all_windows),
            "rate_threshold_equivalence_checks":2*ncases,
            "nonzero_initial_exterior_cases":sum(not r['initial_exterior_is_zero'] for r in records),
            "componentwise_output_tail_certificates":len(all_windows)*3,
            "certificate_violations":0,
            "maximum_exact_causal_error":max(r['exact_causal_maximum_error'] for r in records),
            "maximum_wrong_current_egress_error":max(r['wrong_current_egress_error'] for r in records),
            "maximum_local_error_to_uniform_bound":max(r['local_error_to_bound'] for r in all_windows),
            "maximum_output_error_to_componentwise_bound":max(r['output_error_to_componentwise_bound'] for r in all_windows),
            "maximum_scalar_horizon_for_local_error_0_001":max(r['scalar_horizon_for_local_error_0_001'] for r in records),
            "negative_controls":negatives,"records":records,
            "scope":"Exact positive discrete-time dynamics; fixed exterior and supplied common upper family. CPU double-precision tests within declared tolerances, not outward-rounded proof, physical photonic timing, hardware energy or universal game dynamics."}
    prefix=Path(args.output_prefix); prefix.parent.mkdir(parents=True,exist_ok=True)
    prefix.with_name(prefix.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':10,'axes.spines.top':False,'axes.spines.right':False})
    fig,axes=plt.subplots(1,3,figsize=(13,3.7),constrained_layout=True)
    zs,zm,rs=example
    axes[0].plot(zs[:,0],color='#6b51b8',label='Full carried state')
    axes[0].plot(zm[:,0],'--',color='#27a9b8',label='Exact causal capsule')
    axes[0].set(xlabel='Implemented update',ylabel='Local state',title='History preserves chronology');axes[0].legend(fontsize=8)
    axes[1].semilogy([r['horizon'] for r in rs],[max(r['local_error'],1e-16) for r in rs],'o-',color='#6b51b8',label='Actual local error')
    axes[1].semilogy([r['horizon'] for r in rs],[r['local_uniform_bound'] for r in rs],'s--',color='#27a9b8',label='Uniform tail bound')
    axes[1].set(xlabel='Retained history length',ylabel='Weighted maximum error',title='Certified memory truncation');axes[1].legend(fontsize=8)
    eta=[];h=[]
    for row in records:eta.append(row['exterior_contraction_delta']);h.append(row['scalar_horizon_for_local_error_0_001'])
    axes[2].scatter(eta,h,color='#6b51b8',alpha=.45)
    axes[2].set(xlabel='Exterior contraction bound',ylabel='Sufficient history length',title='Near-critical memory costs more')
    fig.suptitle('ELYSIAEN / causal return capsules — finite numerical model',fontsize=12)
    fig.savefig(prefix.with_suffix('.png'),dpi=170);plt.close(fig)
    print(json.dumps({k:result[k] for k in ['random_switched_worlds','exact_causal_comparisons','local_tail_certificates','componentwise_output_tail_certificates','certificate_violations','maximum_exact_causal_error','maximum_scalar_horizon_for_local_error_0_001']},indent=2))

if __name__=='__main__':main()

	#!/usr/bin/env python3
	"""Standalone exact and truncated causal-return capsule verification.

	Run: python v5_dynamic_verify.py [--quick] [--output-prefix v5_dynamic]
	Requires NumPy and Matplotlib. No earlier ELYSIAEN modules are imported.
	This is a finite positive discrete-time system, not a measured optical device.
	The time index is explicitly an implemented state update; a steady radiance
	fixed point is not automatically identified with physical or game-frame time.
	"""
	from __future__ import annotations
	import os
	os.environ["OPENBLAS_NUM_THREADS"] = "1"
	os.environ["OMP_NUM_THREADS"] = "1"
	import argparse
	import json
	import platform
	from pathlib import Path
	import numpy as np

	SEED = 2026102553

	def wnorm(x, weight):
	x = np.asarray(x)
	return float(np.max(np.abs(x)/weight.reshape((-1,)+(1,)*(x.ndim-1)), initial=0.))

	def row_map(rng, rows, cols, out_weight, in_weight, budget):
	a = rng.uniform(.03, 1., (rows, cols))
	a = (budgetout_weight/(a@in_weight))[:, None]
	return a

	def precompute(U, D, V, W, OZ, OF, steps):
	K, F, J, Q = [], [], [], []
	power = np.eye(D.shape[0])
	for j in range(steps):
	K.append(U@power@V); F.append(U@power@W)
	J.append(OF@power@V); Q.append(OF@power@W)
	power = power@D
	return K, F, J, Q

	def initial_traces(U, D, OF, exterior0, steps):
	phi=[]; psi=[]; initial=exterior0.copy()
	for t in range(steps+1):
	phi.append(U@initial); psi.append(OF@initial); initial=D@initial
	return phi, psi

	def causal(K, F, J, Q, OZ, Bs, Ls, Rs, local_source,
	xi, z0, phi, psi, horizon, *, wrong_current_egress=False):
	"""Exact when horizon>=steps; otherwise only last horizon impulses kept.

	Initial exterior traces phi/psi are compiled before this routine.
	A deployed capsule needs their declared initial-state basis or a budget
	for dropping them. No exterior-sized array is read in this evaluator.
	"""
	steps, k = len(Bs), len(z0)
	z = np.zeros((steps+1, k)); z[0] = z0
	y = np.zeros((steps+1, OZ.shape[0]))
	for t in range(steps+1):
	incoming = phi[t].copy()
	y[t] = OZ@z[t]+psi[t]
	for j in range(min(t, horizon)):
	old = t-1-j
	egress = Rs[min(t, steps-1)] if wrong_current_egress else Rs[old]
	outgoing = egress@z[old]
	incoming += K[j]@outgoing+F[j]@xi[old]
	y[t] += J[j]@outgoing+Q[j]@xi[old]
	if t < steps:
	z[t+1] = Bs[t]@z[t]+local_source[t]+Ls[t]@incoming
	return z, y

	def full_oracle(U, D, V, W, OZ, OF, Bs, Ls, Rs, source, xi, z0, f0):
	steps, k, nf = len(Bs), len(z0), len(f0)
	states = np.zeros((steps+1, k+nf)); states[0] = np.r_[z0, f0]
	for t in range(steps):
	z, v = states[t, :k], states[t, k:]
	states[t+1, :k] = Bs[t]@z+Ls[t]@U@v+source[t]
	states[t+1, k:] = D@v+V@Rs[t]@z+W@xi[t]
	return states[:, :k], states[:, :k]@OZ.T+states[:, k:]@OF.T

	def trial(rng, number, steps):
	k = (2, 3, 4)[number % 3]
	nf = (7, 12, 20)[(number//3) % 3]
	delta = (.2, .6, .9)[(number//9) % 3]
	a = np.exp(rng.uniform(-.7, .7, k))
	w = np.exp(rng.uniform(-.7, .7, nf))
	D = row_map(rng, nf, nf, w, w, delta)
	U = row_map(rng, k, nf, a, w, .18*np.sqrt(1-delta))
	V = row_map(rng, nf, k, w, a, .25*np.sqrt(1-delta))
	W = rng.uniform(.001, .02, (nf, 2))*w[:, None]
	OZ = rng.uniform(-.5, .8, (3, k))
	OF = rng.uniform(-.2, .4, (3, nf))
	Bbar = row_map(rng, k, k, a, a, (.4, .7)[number % 2])
	Lbar = row_map(rng, k, k, a, a, .85)
	Rbar = row_map(rng, k, k, a, a, .9)
	H = np.linalg.solve(np.eye(nf)-D, np.eye(nf))
	C = U@H@V
	Abar = Bbar+Lbar@C@Rbar
	q = float(np.max(Abar@a/a))
	assert q < .9
	# A single upper family, not individual reduced stability, bounds every
	# delayed word. This criterion is sufficient and need not be necessary.
	full_upper = np.block([[Bbar, Lbar@U], [V@Rbar, D]])
	full_radius=float(max(abs(np.linalg.eigvals(full_upper))))
	assert full_radius < 1
	rate_records=[]
	for gamma in ((delta+full_radius)/2,(full_radius+1)/2):
	rate_return=Bbar/gamma+(Lbar@U@np.linalg.solve(gamma*np.eye(nf)-D,V)@Rbar)/gamma
	local_rate_radius=float(max(abs(np.linalg.eigvals(rate_return))))
	assert (full_radius<gamma)==(local_rate_radius<1)
	rate_records.append({'gamma':gamma,'full_radius':full_radius,
	'rate_return_radius':local_rate_radius,
	'admitted':bool(local_rate_radius<1)})
	Bs, Ls, Rs, sources, xis = [], [], [], [], []
	local_upper = .12*a
	for t in range(steps):
	Bs.append(Bbar*rng.uniform(.3, 1., (k, k)))
	Ls.append(Lbar*rng.uniform(.2, 1., (k, k)))
	Rs.append(Rbar*rng.uniform(.2, 1., (k, k)))
	# Abrupt changes and smooth variation both stay in fixed envelopes.
	xis.append(np.array([.4+.3np.sin(t/7.)*2, .8 if t >= steps//3 else .1]))
	sources.append(local_upper*rng.uniform(.1, 1., k))
	z0 = rng.uniform(0, .1, k)*a
	exterior0 = np.zeros(nf) if number%2==0 else rng.uniform(0,.1,nf)*w
	K, F, J, Q = precompute(U, D, V, W, OZ, OF, steps)
	phi,psi=initial_traces(U,D,OF,exterior0,steps)
	exact_z, exact_y = full_oracle(U, D, V, W, OZ, OF, Bs, Ls, Rs, sources, xis, z0, exterior0)
	memory_z, memory_y = causal(K, F, J, Q, OZ, Bs, Ls, Rs, sources, xis, z0, phi,psi, steps)
	exact_error = max(wnorm((exact_z-memory_z).T, a), float(np.max(abs(exact_y-memory_y))))
	assert exact_error < 2e-11
	forcing_upper = local_upper+Lbar@U@H@W@np.ones(2)+Lbar@U@(wnorm(exterior0,w)*w)
	z_budget = max(wnorm(z0, a), wnorm(forcing_upper, a)/(1-q))
	zbar = z_budget*a
	assert np.max(exact_z-zbar) <= 1e-11
	full_output_gain = (abs(OZ)+abs(OF)@H@V@Rbar)@a
	records = []
	for h in (0, 1, 2, 4, 8, 16):
	approximate_z, approximate_y = causal(K, F, J, Q, OZ, Bs, Ls, Rs, sources, xis, z0, phi,psi, h)
	assert np.max(approximate_z-zbar) <= 2e-11
	assert np.max(approximate_z-exact_z) <= 2e-11
	tail_power = np.linalg.matrix_power(D, h)
	exterior_drive = V@Rbar@zbar+W@np.ones(2)
	incoming_tail = Lbar@U@tail_power@H@exterior_drive
	local_bound = wnorm(incoming_tail, a)/(1-q)
	output_tail = abs(OF)@tail_power@H@exterior_drive
	output_bound = full_output_gain*local_bound+output_tail
	local_error = wnorm((exact_z-approximate_z).T, a)
	output_error = np.max(abs(exact_y-approximate_y), axis=0)
	assert local_error <= local_bound+2e-10*max(1., local_bound)
	assert np.all(output_error <= output_bound+2e-10*np.maximum(1., output_bound))
	records.append({"horizon": h, "local_error": local_error,
	"local_uniform_bound": local_bound,
	"maximum_output_error": float(max(output_error)),
	"maximum_output_uniform_bound": float(max(output_bound)),
	"local_error_to_bound": local_error/local_bound if local_bound else 0.,
	"output_error_to_componentwise_bound": float(np.max(output_error/output_bound))})
	wrong_z, wrong_y = causal(K, F, J, Q, OZ, Bs, Ls, Rs, sources, xis, z0, phi,psi, steps, wrong_current_egress=True)
	wrong_error = wnorm((wrong_z-exact_z).T, a)
	# Compute an a-priori scalar horizon from \|\|D\|\|<=delta. The matrix-tail
	# certificate above is generally sharper and retains propagation shape.
	alpha = float(np.max(U@w/a)); ell = float(np.max(Lbar@a/a))
	drive_norm = wnorm(V@Rbar@zbar+W@np.ones(2), w)
	prefactor = ellalphadrive_norm/((1-delta)*(1-q))
	target = .001
	required_h = max(0, int(np.ceil(np.log(target/prefactor)/np.log(delta)))) if prefactor>target else 0
	assert prefactordelta*required_h <= target+1e-12
	return {"local_states": k, "exterior_states": nf, "steps": steps,
	"exterior_contraction_delta": delta, "upper_return_witness_q": q,
	"initial_exterior_is_zero":bool(np.all(exterior0==0)),
	"rate_threshold_checks":rate_records,
	"exact_causal_maximum_error": exact_error,
	"wrong_current_egress_error": wrong_error,
	"scalar_horizon_for_local_error_0_001": required_h,
	"retained_kernel_scalars_per_history_step": kk+k2+3k+32,
	"records": records}, (exact_z, memory_z, records)

	def counterexamples():
	# Stable reference with coupling .5. Every reduced closure is zero, but
	# the actual switched exterior carries a delayed amplifying path.
	U=V=np.array([[.5]]); D=np.zeros((1,1)); W=np.zeros((1,1))
	OZ=np.array([[1.]]); OF=np.array([[1.]])
	steps=16
	Bs=[np.zeros((1,1)) for _ in range(steps)]
	Ls=[np.array([[4. if t%2==0 else 0.]]) for t in range(steps)]
	Rs=[np.array([[0. if t%2==0 else 4.]]) for t in range(steps)]
	zero=[np.zeros(1) for _ in range(steps)]
	K,F,J,Q=precompute(U,D,V,W,OZ,OF,steps)
	phi,psi=initial_traces(U,D,OF,np.ones(1),steps)
	z,y=full_oracle(U,D,V,W,OZ,OF,Bs,Ls,Rs,zero,zero,np.zeros(1),np.ones(1))
	causal_z,causal_y=causal(K,F,J,Q,OZ,Bs,Ls,Rs,zero,zero,np.zeros(1),phi,psi,steps)
	wrong_z,wrong_y=causal(K,F,J,Q,OZ,Bs,Ls,Rs,zero,zero,np.zeros(1),phi,psi,steps,wrong_current_egress=True)
	assert np.max(abs(y-causal_y))==0
	assert y[-1,0]==65536. and wrong_y[-1,0]==0
	# A fixed stable world is still not in equilibrium immediately.
	fixed=np.array([[.1,.2],[.2,.8]]); source=np.array([.3,.1])
	equilibrium=np.linalg.solve(np.eye(2)-fixed,source)
	return {"switching": {"reference_couplings": .5,
	"reference_spectral_radius": .5,
	"each_reduced_closure": 0., "two_step_full_gain": 4.,
	"steps": steps, "final_full_output": float(y[-1,0]),
	"final_exact_causal_output": float(causal_y[-1,0]),
	"final_wrong_current_egress_output": float(wrong_y[-1,0]),
	"common_upper_reduced_return": 4.,
	"correct_envelope_admission_rejects": True},
	"equilibrium_is_not_first_update": {"equilibrium_local_state":float(equilibrium[0]),
	"first_dynamic_local_state_from_zero":.3,
	"absolute_difference":float(equilibrium[0]-.3)}}

	def main():
	parser=argparse.ArgumentParser(description=__doc__)
	parser.add_argument("--quick",action="store_true")
	parser.add_argument("--output-prefix",default="v5_dynamic")
	args=parser.parse_args()
	rng=np.random.default_rng(SEED)
	ncases=12 if args.quick else 81
	steps=32 if args.quick else 96
	records=[]; example=None
	for number in range(ncases):
	record,figuredata=trial(rng,number,steps)
	records.append(record)
	if number==18 or example is None: example=figuredata
	negatives=counterexamples()
	all_windows=[item for row in records for item in row['records']]
	result={"title":"ELYSIAEN exact causal return capsules and finite-memory bounds",
	"date":"2026-10-25","seed":SEED,"python":platform.python_version(),
	"numpy":np.__version__,"random_switched_worlds":ncases,
	"steps_per_world":steps,"finite_horizons_per_world":6,
	"exact_causal_comparisons":ncases,"local_tail_certificates":len(all_windows),
	"rate_threshold_equivalence_checks":2*ncases,
	"nonzero_initial_exterior_cases":sum(not r['initial_exterior_is_zero'] for r in records),
	"componentwise_output_tail_certificates":len(all_windows)*3,
	"certificate_violations":0,
	"maximum_exact_causal_error":max(r['exact_causal_maximum_error'] for r in records),
	"maximum_wrong_current_egress_error":max(r['wrong_current_egress_error'] for r in records),
	"maximum_local_error_to_uniform_bound":max(r['local_error_to_bound'] for r in all_windows),
	"maximum_output_error_to_componentwise_bound":max(r['output_error_to_componentwise_bound'] for r in all_windows),
	"maximum_scalar_horizon_for_local_error_0_001":max(r['scalar_horizon_for_local_error_0_001'] for r in records),
	"negative_controls":negatives,"records":records,
	"scope":"Exact positive discrete-time dynamics; fixed exterior and supplied common upper family. CPU double-precision tests within declared tolerances, not outward-rounded proof, physical photonic timing, hardware energy or universal game dynamics."}
	prefix=Path(args.output_prefix); prefix.parent.mkdir(parents=True,exist_ok=True)
	prefix.with_name(prefix.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':10,'axes.spines.top':False,'axes.spines.right':False})
	fig,axes=plt.subplots(1,3,figsize=(13,3.7),constrained_layout=True)
	zs,zm,rs=example
	axes[0].plot(zs[:,0],color='#6b51b8',label='Full carried state')
	axes[0].plot(zm[:,0],'--',color='#27a9b8',label='Exact causal capsule')
	axes[0].set(xlabel='Implemented update',ylabel='Local state',title='History preserves chronology');axes[0].legend(fontsize=8)
	axes[1].semilogy([r['horizon'] for r in rs],[max(r['local_error'],1e-16) for r in rs],'o-',color='#6b51b8',label='Actual local error')
	axes[1].semilogy([r['horizon'] for r in rs],[r['local_uniform_bound'] for r in rs],'s--',color='#27a9b8',label='Uniform tail bound')
	axes[1].set(xlabel='Retained history length',ylabel='Weighted maximum error',title='Certified memory truncation');axes[1].legend(fontsize=8)
	eta=[];h=[]
	for row in records:eta.append(row['exterior_contraction_delta']);h.append(row['scalar_horizon_for_local_error_0_001'])
	axes[2].scatter(eta,h,color='#6b51b8',alpha=.45)
	axes[2].set(xlabel='Exterior contraction bound',ylabel='Sufficient history length',title='Near-critical memory costs more')
	fig.suptitle('ELYSIAEN / causal return capsules — finite numerical model',fontsize=12)
	fig.savefig(prefix.with_suffix('.png'),dpi=170);plt.close(fig)
	print(json.dumps({k:result[k] for k in ['random_switched_worlds','exact_causal_comparisons','local_tail_certificates','componentwise_output_tail_certificates','certificate_violations','maximum_exact_causal_error','maximum_scalar_horizon_for_local_error_0_001']},indent=2))

	if __name__=='__main__':main()