code/v4_engine.py

#!/usr/bin/env python3
"""ELYSIAEN final: endpoint-safe local attenuation from one frozen Green capsule.

Standalone: no earlier ELYSIAEN files are imported. Requires Python 3.10+,
numpy, scipy and matplotlib. Run `python v4_engine.py`; `--quick` reduces the
randomized suite. This is a CPU finite-model correctness experiment, not an
optical-device, commercial-game-engine, energy or frame-rate benchmark.

Models: T_D = D_full T0 D_full, 0 <= D_full <= I diagonal; and the more
general fixed-interface local block [[B, Lin*T_ZR], [T_RZ*Rout, T_RR]].
These update a declared finite state basis. Scene-to-transport compilation,
new reflective surfaces and new scattering directions are not demonstrated.
"""
from __future__ import annotations

import os
# Keep this small reproducible experiment from oversubscribing BLAS threads.
os.environ["OPENBLAS_NUM_THREADS"] = "1"
os.environ["OMP_NUM_THREADS"] = "1"
os.environ["MKL_NUM_THREADS"] = "1"
import argparse
import json
import platform
from pathlib import Path
import numpy as np
import scipy
from scipy.linalg import solve

SEED = 20261025


def frozen_capsule(green, indices):
    """Compile a return matrix and harmonic lift using k coordinate columns.

    All acquisition/storage below is explicit numerical array access here.
    An optical device would have to acquire and validate these responses.
    """
    z = np.asarray(indices, dtype=int)
    columns = green[:, z]
    principal = columns[z]
    inverse = solve(principal, np.eye(len(z)))
    lift = columns @ inverse
    return {"indices": z, "inverse": inverse, "return": np.eye(len(z))-inverse,
            "lift": lift, "principal": principal}


def capsule_response(capsule, source, reference_response, attenuation):
    """Exact T_D response; finite at all-transparent and all-opaque endpoints.

    S=I-K^-1, L=G[:,Z]K^-1; u is the actual state on the edited coordinates.
    (I-D S D)u=(I-D)b_Z + D K^-1 q_Z.
    x=q + L(Du-q_Z) + E(I-D)u.
    No inverse attenuation or infinite penalty is formed.
    """
    z = capsule["indices"]
    d = np.asarray(attenuation, dtype=float)
    if d.shape != (len(z),) or np.any(d < 0) or np.any(d > 1):
        raise ValueError("attenuation must have k entries in [0,1]")
    small = np.eye(len(z)) - d[:, None]*capsule["return"]*d[None, :]
    rhs = (1-d[:, None])*source[z] + d[:, None]*(capsule["inverse"]@reference_response[z])
    boundary = solve(small, rhs)
    answer = reference_response + capsule["lift"]@(d[:, None]*boundary-reference_response[z])
    answer[z] += (1-d[:, None])*boundary
    return answer, boundary, small


def material_response(capsule, t0, source, reference_response, local, incoming, outgoing):
    """Compile arbitrary admitted local nonnegative mixing behind the same Z.

    Reordered as Z,R, the edited transport is
      [[local, incoming*T_ZR], [T_RZ*outgoing, T_RR]].
    The entire nonlocal return is cached: J=T_ZR*(I-T_RR)^-1*T_RZ.
    Stability must be checked for C=local+incoming*J*outgoing.
    """
    z = capsule["indices"]
    external_return = capsule["return"]-t0[np.ix_(z, z)]
    lift = capsule["lift"].copy()
    lift[z] -= np.eye(len(z))
    forcing = capsule["inverse"]@reference_response[z]-source[z]
    reduced_return = local+incoming@external_return@outgoing
    small = np.eye(len(z))-reduced_return
    boundary = solve(small, source[z]+incoming@forcing)
    answer = reference_response+lift@(outgoing@boundary-reference_response[z])
    answer[z] = boundary
    return answer, reduced_return, small


def weighted_infinity(matrix, row_weight, column_weight=None):
    if column_weight is None:
        column_weight = row_weight
    return float(np.max(np.sum(np.abs(matrix)*column_weight[None, :], axis=1)/row_weight))


def world_index(x, y):
    return (x//10)*100 + y*10 + x%10


def tile_world():
    adjacency = np.zeros((200, 200))
    for y in range(10):
        for x in range(20):
            for dx, dy in ((1, 0), (-1, 0), (0, 1), (0, -1)):
                xx, yy = x+dx, y+dy
                if not (0 <= xx < 20 and 0 <= yy < 10):
                    continue
                if (x < 10) != (xx < 10) and y not in (3, 4, 5, 6):
                    continue
                adjacency[world_index(x, y), world_index(xx, yy)] = 1
    return .72*adjacency/4


def obstacle(pose):
    x, y = pose
    return np.asarray([world_index(xx, yy) for yy in range(y, y+3)
                       for xx in range(x, x+2)], dtype=int)


def sources_for_pose(indices):
    base = np.full((200, 3), .004)
    for x, y, rgb in [(2, 2, [1.6, .25, .04]), (3, 2, [1.6, .25, .04]),
                      (16, 7, [.04, .85, 1.7]), (16, 8, [.04, .85, 1.7])]:
        base[world_index(x, y)] += rgb
    warm = base*np.asarray([.37, 1.2, .65])
    local = base.copy()
    local[world_index(8, 1)] += [.18, .4, .23]
    energized = base.copy()
    energized[indices] += (1+np.arange(len(indices))[:, None]/10)*np.asarray([.29, .31, .37])
    return np.concatenate([base, warm, local, energized], axis=1)


def scene_suite():
    t0 = tile_world()
    green = solve(np.eye(200)-t0, np.eye(200))
    rng = np.random.default_rng(SEED)
    patterns = [("opaque", np.zeros(6)), ("transparent", np.ones(6)),
                ("uniform_020", np.full(6, .2)), ("uniform_060", np.full(6, .6)),
                ("mixed_endpoints", np.asarray([0, .25, .5, .75, 1., 0.])),
                ("random_interior", rng.uniform(.05, .95, 6))]
    cache = set()
    records = []
    selected = None
    max_error = 0.
    max_condition = 0.
    for pose in [(x, 4) for x in range(5, 17)]:
        z = obstacle(pose)
        misses = [int(j) for j in z if int(j) not in cache]
        cache.update(map(int, z))
        capsule = frozen_capsule(green, z)
        source = sources_for_pose(z)
        reference = green@source
        min_lift = float(np.min(capsule["lift"]))
        lift_norm = weighted_infinity(capsule["lift"], np.ones(200), np.ones(6))
        return_norm = weighted_infinity(capsule["return"], np.ones(6))
        assert min_lift >= -1e-12 and lift_norm <= 1+1e-12 and return_norm <= .72+1e-12
        for name, d in patterns:
            full_d = np.ones(200)
            full_d[z] = d
            actual_t = full_d[:, None]*t0*full_d[None, :]
            truth = solve(np.eye(200)-actual_t, source)
            answer, boundary, small = capsule_response(capsule, source, reference, d)
            error = float(np.max(np.abs(answer-truth)))
            condition = float(np.linalg.cond(small, np.inf))
            assert error < 2e-11
            assert condition <= (1+.72)/(1-.72)+1e-11
            assert np.max(np.abs(answer[z]-boundary)) < 1e-12
            max_error = max(max_error, error)
            max_condition = max(max_condition, condition)
            record = {"pose": list(pose), "attenuation_pattern": name,
                      "attenuation": d.tolist(), "coordinate_ports": 6,
                      "RGB_source_families": 4, "scalar_source_RHS": 12,
                      "max_abs_RGB_coefficient_error": error,
                      "small_system_condition_infinity": condition,
                      "lift_weighted_infinity_norm": lift_norm,
                      "return_weighted_infinity_norm": return_norm,
                      "new_coordinate_columns_at_pose": len(misses) if name == "opaque" else 0,
                      "new_value_dependent_columns": 0,
                      "local_output_coordinates": 6}
            records.append(record)
            if pose == (9, 4) and name == "mixed_endpoints":
                selected = {"pose": pose, "attenuation": d, "reference": reference[:, 9:12],
                            "truth": truth[:, 9:12], "answer": answer[:, 9:12]}
    return {"transport_states": 200, "reference_rho": .72, "poses": 12,
            "attenuation_patterns_per_pose": 6, "RGB_source_families": 4,
            "RGB_world_states_checked": len(records)*4,
            "scalar_RHS_checked": len(records)*12,
            "maximum_absolute_RGB_error": max_error,
            "maximum_small_system_condition_infinity": max_condition,
            "uniform_small_system_condition_bound": (1+.72)/(1-.72),
            "response_work": {"unique_coordinate_columns": len(cache),
                "coordinate_columns_in_current_capsule": 6,
                "warm_attenuation_changes_requiring_new_coordinate_columns": 0,
                "value_dependent_columns": 0,
                "source_queries_per_pose_scalar_RHS": 12,
                "source_queries_total_scalar_RHS": 12*12,
                "source_queries_reused_across_attenuation_patterns": True,
                "full_RGB_coefficients_reconstructed_per_world_state": 200*3,
                "prototype_reference_storage_coefficients": 200*200,
                "prototype_computes_full_reference_inverse": True,
                "prototype_reassembles_capsule_from_dense_reference_each_pose": True,
                "scope": "Logical cache/acquisition counters, not optical measurements. Columns, source projection, small solves, calibration, and full output have costs."},
            "records": records}, selected


def material_suite():
    """Changing local scatter/routing matrices with a fixed six-port capsule."""
    t0 = tile_world()
    green = solve(np.eye(200)-t0, np.eye(200))
    rng = np.random.default_rng(SEED+3)
    records = []
    selected = None
    for pose in [(x, 4) for x in range(5, 17)]:
        z = obstacle(pose)
        r = np.asarray([j for j in range(200) if j not in set(z)], dtype=int)
        capsule = frozen_capsule(green, z)
        source = sources_for_pose(z)
        q = green@source
        p = np.roll(np.eye(6), 1, axis=0)
        h = np.eye(6)[::-1]
        random_mix = []
        for _ in range(3):
            m = rng.uniform(.1, 1, (6, 6))
            random_mix.append(m/m.sum(axis=1)[:, None])
        configs = [("reference", t0[np.ix_(z, z)], np.eye(6), np.eye(6)),
                   ("noncommuting_routes", .04*np.eye(6)+.08*p, .7*p+.1*np.eye(6), .6*h+.2*np.eye(6)),
                   ("dense_local_mixing", .15*random_mix[0], .8*random_mix[1], .9*random_mix[2])]
        for name, local, incoming, outgoing in configs:
            edited = t0.copy()
            edited[np.ix_(z, z)] = local
            edited[np.ix_(z, r)] = incoming@t0[np.ix_(z, r)]
            edited[np.ix_(r, z)] = t0[np.ix_(r, z)]@outgoing
            truth = solve(np.eye(200)-edited, source)
            answer, reduced, small = material_response(capsule, t0, source, q, local, incoming, outgoing)
            actual_rho = float(np.max(np.abs(np.linalg.eigvals(edited))))
            reduced_rho = float(np.max(np.abs(np.linalg.eigvals(reduced))))
            error = float(np.max(np.abs(answer-truth)))
            assert error < 2e-11 and actual_rho < 1 and reduced_rho < 1
            assert np.all(edited >= 0) and np.min(reduced) >= -1e-12
            records.append({"pose": list(pose), "configuration": name,
                            "coordinate_ports": 6, "RGB_source_families": 4,
                            "local_matrix_commutator_norm": float(np.linalg.norm(local@incoming-incoming@local, ord=np.inf)),
                            "incoming_outgoing_commutator_norm": float(np.linalg.norm(incoming@outgoing-outgoing@incoming, ord=np.inf)),
                            "full_transport_spectral_radius": actual_rho,
                            "small_return_spectral_radius": reduced_rho,
                            "small_system_condition_infinity": float(np.linalg.cond(small, np.inf)),
                            "maximum_RGB_coefficient_error": error,
                            "new_coordinate_columns_after_capsule_acquisition": 0,
                            "new_value_dependent_columns": 0})
            if pose == (9, 4) and name == "dense_local_mixing":
                selected = {"pose": pose, "attenuation": None, "reference": q[:, 9:12],
                            "truth": truth[:, 9:12], "answer": answer[:, 9:12]}
    return {"poses": 12, "local_configurations_per_pose": 3,
            "RGB_source_families": 4, "RGB_world_states_checked": len(records)*4,
            "maximum_absolute_RGB_error": max(a["maximum_RGB_coefficient_error"] for a in records),
            "scope": "Local internal B and incoming/outgoing mixing L,R vary behind a fixed six-coordinate interface; existing exterior transport remains fixed. Physical scene-to-matrix compilation is not demonstrated.",
            "records": records}, selected


def randomized_suite(count):
    rng = np.random.default_rng(SEED+1)
    rows = []
    for trial in range(count):
        n = (16, 32, 64)[trial % 3]
        rho = (.2, .7, .95, .999)[trial % 4]
        weight = np.exp(rng.uniform(-1.3, 1.3, n))
        base = rng.uniform(0, 1, (n, n))
        base *= rng.uniform(size=(n, n)) < .3
        base += np.eye(n)*.2
        base *= rho/base.sum(axis=1)[:, None]
        t0 = weight[:, None]*base/weight[None, :]
        green = solve(np.eye(n)-t0, np.eye(n))
        k = int(rng.integers(1, min(12, n)+1))
        z = rng.choice(n, k, replace=False)
        capsule = frozen_capsule(green, z)
        d = rng.uniform(0, 1, k)
        d[::3] = 0.
        d[1::4] = 1.
        source = rng.uniform(0, 1, (n, 3))*weight[:, None]
        reference = green@source
        full_d = np.ones(n)
        full_d[z] = d
        truth = solve(np.eye(n)-full_d[:, None]*t0*full_d[None, :], source)
        answer, boundary, small = capsule_response(capsule, source, reference, d)
        error = float(np.max(np.abs(answer-truth)/weight[:, None]))
        relative_error = error/float(np.max(np.abs(truth)/weight[:, None]))
        lift_norm = weighted_infinity(capsule["lift"], weight, weight[z])
        return_norm = weighted_infinity(capsule["return"], weight[z])
        condition = weighted_infinity(small, weight[z])*weighted_infinity(solve(small, np.eye(k)), weight[z])
        min_lift = float(np.min(capsule["lift"]))
        min_return = float(np.min(capsule["return"]))
        assert min_lift >= -2e-10 and min_return >= -2e-10
        assert error < 2e-9 and lift_norm <= 1+2e-10 and return_norm <= rho+2e-10
        assert condition <= (1+rho)/(1-rho)+2e-8
        # Opaque occupancy: a residual at the coordinates has a harmonic lift
        # with gain <=1. Check this independently of source response accuracy.
        qz = reference[z]
        theta = solve(capsule["principal"], qz)
        perturbation = rng.normal(size=(k, 3))*weight[z, None]*1e-5
        candidate = theta+perturbation
        residual = capsule["principal"]@candidate-qz
        exact_opaque = reference-capsule["lift"]@qz
        exact_opaque[z] += source[z]
        approximate = reference-green[:, z]@candidate
        approximate[z] += source[z]
        actual = float(np.max(np.abs(approximate-exact_opaque)/weight[:, None]))
        bound = float(np.max(np.abs(residual)/weight[z, None]))
        assert actual <= bound+2e-10
        rows.append({"n": n, "k": k, "rho": rho,
                     "weighted_absolute_response_error": error,
                     "relative_weighted_response_error": relative_error,
                     "minimum_lift_entry": min_lift, "minimum_return_entry": min_return,
                     "lift_weighted_infinity_norm": lift_norm,
                     "return_weighted_infinity_norm": return_norm,
                     "small_system_weighted_condition": condition,
                     "small_system_condition_bound": (1+rho)/(1-rho),
                     "occupancy_residual_error": actual, "occupancy_residual_certificate": bound})
    return {"cases": count, "response_failures": 0, "passivity_failures": 0,
            "occupancy_residual_certificate_violations": 0,
            "occupancy_residual_check_absolute_tolerance": 2e-10,
            "maximum_positive_floating_point_certificate_excess": max(0., max(r["occupancy_residual_error"]-r["occupancy_residual_certificate"] for r in rows)),
            "certificate_check_scope": "Double-precision checks within the declared tolerance; the theorem is exact but these numerical evaluations are not outward-rounded certificates.",
            "maximum_weighted_response_error": max(r["weighted_absolute_response_error"] for r in rows),
            "maximum_relative_weighted_response_error": max(r["relative_weighted_response_error"] for r in rows),
            "maximum_lift_weighted_norm": max(r["lift_weighted_infinity_norm"] for r in rows),
            "maximum_residual_error_to_bound": max(r["occupancy_residual_error"]/r["occupancy_residual_certificate"] for r in rows),
            "records": rows}


def endpoint_controls():
    # A one-state near-conservative world disproves a rho-independent bound
    # for residuals of the partial-attenuation boundary equation itself.
    values = []
    for rho in (.9, .99, .999, .9999):
        gain = 1/(1-rho)
        values.append({"rho": rho, "transparent_boundary_residual_gain": gain,
                       "occupancy_harmonic_lift_residual_gain": 1.})
    # Incorrectly treating a partial mask as a binary opaque obstacle fails.
    t0 = np.asarray([[0, .6], [.6, 0]])
    b = np.asarray([[1.], [.2]])
    green = solve(np.eye(2)-t0, np.eye(2))
    capsule = frozen_capsule(green, [0])
    q = green@b
    good, _, _ = capsule_response(capsule, b, q, [.5])
    binary, _, _ = capsule_response(capsule, b, q, [0.])
    return {"near_conservative_boundary_residual_control": values,
            "partial_mask_misclassified_as_opaque_maximum_error": float(np.max(np.abs(good-binary))),
            "source_convention": "An opaque isolated coordinate retains its local numerical source x_Z=b_Z. Mask b_Z to zero when the physical renderer requires a dark occupied coordinate."}


def stability_suite(count):
    rng = np.random.default_rng(SEED+4)
    cases = []
    for trial in range(count):
        n, k = 24, 5
        t0 = rng.uniform(.1, 1, (n, n))
        t0 *= .6/t0.sum(axis=1)[:, None]
        green = solve(np.eye(n)-t0, np.eye(n))
        z = rng.choice(n, k, replace=False)
        r = np.asarray([i for i in range(n) if i not in set(z)])
        capsule = frozen_capsule(green, z)
        m = rng.uniform(0, 1, (k, k)); m /= m.sum(axis=1)[:, None]
        h = rng.uniform(0, 1, (k, k)); h /= h.sum(axis=1)[:, None]
        incoming = .8*m
        outgoing = .8*h
        if trial % 2:
            local = 1.05*np.eye(k)+.01*m
        else:
            local = .1*m
        edited = t0.copy()
        edited[np.ix_(z, z)] = local
        edited[np.ix_(z, r)] = incoming@t0[np.ix_(z, r)]
        edited[np.ix_(r, z)] = t0[np.ix_(r, z)]@outgoing
        reduced = local+incoming@(capsule["return"]-t0[np.ix_(z, z)])@outgoing
        full_rho = float(np.max(np.abs(np.linalg.eigvals(edited))))
        small_rho = float(np.max(np.abs(np.linalg.eigvals(reduced))))
        assert (full_rho < 1) == (small_rho < 1)
        cases.append({"full_transport_spectral_radius": full_rho,
                      "reduced_return_spectral_radius": small_rho,
                      "stable": bool(full_rho < 1)})
    return {"cases": count, "stable_cases": sum(r["stable"] for r in cases),
            "unstable_cases": sum(not r["stable"] for r in cases),
            "classification_discrepancies": 0,
            "scope": "Only the threshold relation rho(T')<1 iff rho(C)<1 is asserted. Spectral radii need not be numerically equal.",
            "records": cases}


def image(values):
    return np.concatenate([values[:100].reshape(10, 10, -1), values[100:].reshape(10, 10, -1)], axis=1)


def render(selected, scenes, materials, randoms, path):
    import matplotlib
    matplotlib.use("Agg")
    import matplotlib.pyplot as plt
    from matplotlib.colors import LogNorm
    from matplotlib.patches import Rectangle
    plt.rcParams.update({"font.family": "DejaVu Sans", "font.size": 9})
    fig, axes = plt.subplots(2, 2, figsize=(11.4, 7.2), layout="constrained")
    fig.patch.set_facecolor("#f4f7fc")
    for ax, key, title in zip(axes.flat[:3], ("reference", "truth", "answer"),
        ("Frozen reference | energized source in edited region", "Dense local scatter/routing | direct 200-state oracle", "Six-column capsule | local material and route update")):
        display = (1-np.exp(-3*np.maximum(image(selected[key]), 0)))**(1/2.2)
        ax.imshow(display, interpolation="nearest")
        ax.set_title(title, loc="left", color="#1b2846", fontsize=10, pad=10)
        x, y = selected["pose"]
        ax.add_patch(Rectangle((x-.5, y-.5), 2, 3, fc="none", ec="#d7bdff", lw=1.5))
        ax.axvline(9.5, color="#cac0ee", ls="--", lw=.7)
        ax.set_xticks([]); ax.set_yticks([])
    ax = axes.flat[3]
    error = np.max(np.abs(selected["truth"]-selected["answer"]), axis=1)
    im = ax.imshow(image(np.maximum(error, 1e-18)[:, None])[:, :, 0], cmap="magma",
                   norm=LogNorm(1e-18, 1e-12), interpolation="nearest")
    ax.set_title(f"Absolute RGB coefficient error | max {error.max():.2e}", loc="left", color="#1b2846", fontsize=10, pad=10)
    ax.set_xticks([]); ax.set_yticks([])
    fig.colorbar(im, ax=ax, fraction=.035, pad=.025, label="Maximum RGB coefficient error")
    for ax in axes.flat:
        for spine in ax.spines.values(): spine.set_visible(False)
    fig.suptitle("ELYSIAEN final | one frozen capsule, changing local worlds", x=.01, ha="left", fontsize=15, weight="bold", color="#182441")
    fig.supxlabel(f"{scenes['RGB_world_states_checked']} attenuation + {materials['RGB_world_states_checked']} material RGB states; {randoms['cases']} weighted tests. CPU correctness only; acquisition and output have costs.", fontsize=9, color="#596680")
    fig.savefig(path, dpi=190)
    plt.close(fig)


def main():
    parser = argparse.ArgumentParser(description=__doc__)
    parser.add_argument("--quick", action="store_true", help="Run 24 rather than 180 randomized cases")
    parser.add_argument("--output-prefix", default="v4_engine")
    args = parser.parse_args()
    prefix = Path(args.output_prefix)
    prefix.parent.mkdir(parents=True, exist_ok=True)
    scenes, selected = scene_suite()
    materials, selected_material = material_suite()
    randoms = randomized_suite(24 if args.quick else 180)
    stability = stability_suite(20 if args.quick else 120)
    result = {"title": "Lh//?∞ — ELYSIÆN — The Noctilucent Algebra of Self-Evolving Worlds",
              "edition": "Final standalone research experiment", "seed": SEED,
              "model": "T_D=D_full*T0*D_full with diagonal D in [0,1]; and fixed-interface local block [[B,Lin*T_ZR],[T_RZ*Rout,T_RR]]; unchanged finite state basis",
              "software": {"Python": platform.python_version(), "NumPy": np.__version__, "SciPy": scipy.__version__},
              "scene_suite": scenes, "material_suite": materials,
              "randomized_suite": randoms, "stability_suite": stability,
              "negative_and_endpoint_controls": endpoint_controls(),
              "scope": "Exact finite transport identities and CPU checks. No optical hardware, game-engine deployment, timing, energy, historical-priority or T9 validation."}
    render(selected_material, scenes, materials, randoms, prefix.with_suffix(".png"))
    result_path = Path(str(prefix)+"_results.json")
    result_path.write_text(json.dumps(result, indent=2)+"\n", encoding="utf-8")
    print(json.dumps({"results": str(result_path), "figure": str(prefix.with_suffix('.png')),
                      "RGB_world_states_checked": scenes["RGB_world_states_checked"],
                      "maximum_absolute_RGB_error": scenes["maximum_absolute_RGB_error"],
                      "material_RGB_states_checked": materials["RGB_world_states_checked"],
                      "maximum_material_RGB_error": materials["maximum_absolute_RGB_error"],
                      "stability_classifications": stability["cases"],
                      "randomized_cases": randoms["cases"],
                      "maximum_weighted_response_error": randoms["maximum_weighted_response_error"],
                      "unique_coordinate_columns": scenes["response_work"]["unique_coordinate_columns"]}, indent=2))


if __name__ == "__main__":
    main()