Datasets:

PureOne
/

Lyotherin_Tia

#!/usr/bin/env python3
"""Exact finite checks of the manuscript's complete restricted realization.
Twenty-four seeded streams interleave legal clamped-affine actions, positive
invertible affine evolution, transported-prefix rebases, presentation-only
evaluations, and separately sampled native-history commits. The independent
reference directly executes every semantic event and the full affine image
recipe. The candidate uses a prepared base, ordered transported residual,
periodic source decoder, and compact effect context.
Only fractions.Fraction and Python's standard library are used. These finite
mathematical checks are neither a proof for all programs nor measurements of
hardware, floating-point engines, latency, or NVIDIA software.
"""

from __future__ import annotations

import argparse
import hashlib
import json
import random
from dataclasses import dataclass
from fractions import Fraction as F
from pathlib import Path
from typing import Optional


SEED = 20261030
STREAMS = 24
EVENTS_PER_STREAM = 40
PIXELS = 7
RGB = tuple[F, F, F]
Image = tuple[RGB, ...]


def require(condition: bool, message: str) -> None:
    if not condition:
        raise AssertionError(message)


def clip(value: F, lo: F, hi: F) -> F:
    return min(hi, max(lo, value))


@dataclass(frozen=True)
class ClampAffine:
    a: F
    b: F
    lo: F
    hi: F

    def __post_init__(self) -> None:
        require(self.a >= 0 and self.lo <= self.hi, "invalid action")

    def __call__(self, value: F) -> F:
        return clip(self.a * value + self.b, self.lo, self.hi)


# None is the expressly adjoined empty-word identity, not a finite-bound clip.
Residual = Optional[ClampAffine]


def apply(residual: Residual, value: F) -> F:
    return value if residual is None else residual(value)


def compose(outer: Residual, inner: Residual) -> Residual:
    if outer is None:
        return inner
    if inner is None:
        return outer
    return ClampAffine(
        outer.a * inner.a,
        outer.a * inner.b + outer.b,
        outer(inner.lo),
        outer(inner.hi),
    )


def transport(residual: Residual, alpha: F, beta: F) -> Residual:
    require(alpha > 0, "slow affine evolution must be invertible and positive")
    if residual is None:
        return None
    return ClampAffine(
        residual.a,
        alpha * residual.b + (1 - residual.a) * beta,
        alpha * residual.lo + beta,
        alpha * residual.hi + beta,
    )


def aggregate(word: list[tuple[int, ClampAffine]]) -> Residual:
    residual: Residual = None
    for _, action in word:
        residual = compose(action, residual)
    return residual


def source_reference(source: Image, coordinate: F) -> RGB:
    """Floor the unwrapped coordinate, then wrap the two endpoint indices."""
    lower = coordinate.numerator // coordinate.denominator
    weight = coordinate - lower
    left, right = source[lower % len(source)], source[(lower + 1) % len(source)]
    return tuple((1 - weight) * left[c] + weight * right[c] for c in range(3))


def source_candidate(source: Image, coordinate: F) -> RGB:
    """Wrap first; this also checks negative-coordinate periodic agreement."""
    position = coordinate % len(source)
    lower = position.numerator // position.denominator
    weight = position - lower
    left, right = source[lower], source[(lower + 1) % len(source)]
    return tuple(left[c] + weight * (right[c] - left[c]) for c in range(3))


def matvec(matrix: tuple[RGB, RGB, RGB], value: RGB) -> RGB:
    return tuple(sum((row[c] * value[c] for c in range(3)), F(0)) for row in matrix)


@dataclass(frozen=True)
class Recipe:
    source: Image
    scale: F
    weights: tuple[F, ...]
    matrix: tuple[RGB, RGB, RGB]
    bias: RGB
    hud: Image
    hud_alpha: tuple[F, ...]


@dataclass(frozen=True)
class Context:
    coefficient: tuple[F, ...]
    bias: Image
    history_tail: Image


def prepare(recipe: Recipe, history: Image) -> Context:
    tails = tuple(
        tuple((1 - recipe.weights[p]) * history[p][c] for c in range(3))
        for p in range(PIXELS)
    )
    coefficients = tuple(
        (1 - recipe.hud_alpha[p]) * recipe.weights[p] for p in range(PIXELS)
    )
    biases = []
    for p in range(PIXELS):
        graded_tail = matvec(recipe.matrix, tails[p])
        biases.append(tuple(
            recipe.hud[p][c]
            + (1 - recipe.hud_alpha[p]) * (graded_tail[c] + recipe.bias[c])
            for c in range(3)
        ))
    return Context(coefficients, tuple(biases), tails)


def render_reference(recipe: Recipe, history: Image, state: F) -> tuple[Image, Image]:
    visible, next_history = [], []
    for p in range(PIXELS):
        z = source_reference(recipe.source, F(p) + recipe.scale * state)
        mixed = tuple(
            recipe.weights[p] * z[c] + (1 - recipe.weights[p]) * history[p][c]
            for c in range(3)
        )
        next_history.append(mixed)
        graded = tuple(
            sum((recipe.matrix[c][j] * mixed[j] for j in range(3)), F(0))
            + recipe.bias[c]
            for c in range(3)
        )
        visible.append(tuple(
            recipe.hud[p][c] + (1 - recipe.hud_alpha[p]) * graded[c]
            for c in range(3)
        ))
    return tuple(visible), tuple(next_history)


def render_candidate(recipe: Recipe, context: Context, state: F) -> tuple[Image, Image]:
    visible, next_history = [], []
    for p in range(PIXELS):
        z = source_candidate(recipe.source, F(p) + recipe.scale * state)
        graded = matvec(recipe.matrix, z)
        visible.append(tuple(
            context.coefficient[p] * graded[c] + context.bias[p][c]
            for c in range(3)
        ))
        next_history.append(tuple(
            recipe.weights[p] * z[c] + context.history_tail[p][c]
            for c in range(3)
        ))
    return tuple(visible), tuple(next_history)


class NativeHistoryOwner:
    """A mathematical single-owner/version gate, not an engine adapter."""

    def __init__(self, history: Image):
        self.history = history
        self.version = 0
        self.committed_ids: set[int] = set()

    def commit(self, commit_id: int, expected_version: int, proposed: Image) -> bool:
        if commit_id in self.committed_ids or expected_version != self.version:
            return False
        self.history = proposed
        self.version += 1
        self.committed_ids.add(commit_id)
        return True


def random_image(rng: random.Random) -> Image:
    return tuple(tuple(F(rng.randint(0, 12), 12) for _ in range(3)) for _ in range(PIXELS))


def random_recipe(rng: random.Random) -> Recipe:
    source = random_image(rng)
    alpha = tuple(F(rng.randint(0, 6), 6) for _ in range(PIXELS))
    hud_color = random_image(rng)
    hud = tuple(tuple(alpha[p] * hud_color[p][c] for c in range(3)) for p in range(PIXELS))
    matrix = tuple(
        tuple(F(int(i == j)) + F(rng.randint(-2, 2), 12) for j in range(3))
        for i in range(3)
    )
    return Recipe(
        source=source,
        scale=F(rng.choice((-3, -2, -1, 1, 2, 3)), rng.choice((2, 3))),
        weights=tuple(F(rng.randint(0, 6), 6) for _ in range(PIXELS)),
        matrix=matrix,
        bias=tuple(F(rng.randint(-2, 2), 24) for _ in range(3)),
        hud=hud,
        hud_alpha=alpha,
    )


def random_action(rng: random.Random) -> ClampAffine:
    lo = F(rng.randint(-12, 0), 6)
    hi = F(rng.randint(0, 12), 6)
    return ClampAffine(
        rng.choice((F(0), F(1, 2), F(1), F(3, 2), F(2))),
        F(rng.randint(-12, 12), 6), lo, hi,
    )


def verify() -> dict:
    rng = random.Random(SEED)
    counts = {name: 0 for name in (
        "actions", "slow_affine_transitions", "identity_transport_events",
        "state_equalities", "native_rebases", "consumed_input_ids",
        "native_captures", "native_history_commits", "duplicate_commit_rejections",
        "explicit_presentations", "presentation_history_unchanged_checks",
        "visible_channel_equalities", "candidate_history_channel_equalities",
        "capture_vs_later_state_distinctions",
    )}
    stream_summaries = []

    for stream in range(STREAMS):
        recipe = random_recipe(rng)
        reference_history = random_image(rng)
        owner = NativeHistoryOwner(reference_history)
        context = prepare(recipe, owner.history)
        reference_state = F(rng.randint(-6, 6), 3)
        base = reference_state
        residual: Residual = None
        pending: list[tuple[int, ClampAffine]] = []
        next_input_id = 1
        consumed_prefix = 0
        consumed_ids: set[int] = set()
        captured = None

        def rebase(number: int) -> None:
            nonlocal base, residual, pending, consumed_prefix
            require(0 <= number <= len(pending), "invalid prefix length")
            old_state = apply(residual, base)
            prefix, suffix = pending[:number], pending[number:]
            # Execute the transported prefix directly, rather than by its summary.
            for input_id, action in prefix:
                require(input_id == consumed_prefix + 1, "noncontiguous consumed prefix")
                require(input_id not in consumed_ids, "duplicate input consumption")
                base = action(base)
                consumed_ids.add(input_id)
                consumed_prefix = input_id
                counts["consumed_input_ids"] += 1
            pending = suffix
            residual = aggregate(suffix)
            require(apply(residual, base) == old_state == reference_state, "rebase changed state")
            counts["native_rebases"] += 1

        # One cycle samples a native candidate before newer input arrives, and
        # commits that sampled candidate only at the cycle's final event.
        for event in range(EVENTS_PER_STREAM):
            phase, cycle = event % 10, event // 10
            if phase in (0, 3, 8):
                action = random_action(rng)
                reference_state = action(reference_state)
                residual = compose(action, residual)
                pending.append((next_input_id, action))
                next_input_id += 1
                counts["actions"] += 1
            elif phase in (1, 6):
                alpha = rng.choice((F(1, 2), F(2, 3), F(1), F(3, 2), F(2)))
                beta = F(rng.randint(-6, 6), 6)
                reference_state = alpha * reference_state + beta
                base = alpha * base + beta
                if residual is None:
                    counts["identity_transport_events"] += 1
                residual = transport(residual, alpha, beta)
                pending = [
                    (input_id, transport(action, alpha, beta))
                    for input_id, action in pending
                ]
                counts["slow_affine_transitions"] += 1
            elif phase == 2:
                require(captured is None, "native capture overwritten before commit")
                _, reference_next = render_reference(recipe, reference_history, reference_state)
                _, candidate_next = render_candidate(recipe, context, apply(residual, base))
                require(reference_next == candidate_next, "sampled native histories differ")
                captured = (cycle, owner.version, reference_next, candidate_next, reference_state)
                counts["native_captures"] += 1
            elif phase in (4, 7):
                counts["explicit_presentations"] += 1
            elif phase == 5:
                number = len(pending) if (stream + cycle) % 2 == 0 else max(1, len(pending) // 2)
                rebase(number)
            elif phase == 9:
                require(captured is not None, "native commit has no sampled candidate")
                commit_id, expected, reference_next, candidate_next, capture_state = captured
                if capture_state != reference_state:
                    counts["capture_vs_later_state_distinctions"] += 1
                reference_history = reference_next
                require(owner.commit(commit_id, expected, candidate_next), "valid native commit rejected")
                require(owner.history == reference_history, "native history continuation differs")
                counts["native_history_commits"] += 1
                committed_state = (owner.history, owner.version)
                require(not owner.commit(commit_id, expected, candidate_next), "duplicate history commit accepted")
                require((owner.history, owner.version) == committed_state, "rejected commit changed history")
                counts["duplicate_commit_rejections"] += 1
                context = prepare(recipe, owner.history)
                captured = None

            candidate_state = apply(residual, base)
            require(candidate_state == reference_state, "transported state differs from direct execution")
            counts["state_equalities"] += 1
            before = (owner.history, owner.version)
            reference_image, reference_next = render_reference(recipe, reference_history, reference_state)
            candidate_image, candidate_next = render_candidate(recipe, context, candidate_state)
            require(reference_image == candidate_image, "compact image differs from complete recipe")
            require(reference_next == candidate_next, "candidate next history differs")
            require((owner.history, owner.version) == before, "presentation advanced native history")
            counts["presentation_history_unchanged_checks"] += 1
            counts["visible_channel_equalities"] += PIXELS * 3
            counts["candidate_history_channel_equalities"] += PIXELS * 3

        rebase(len(pending))
        require(residual is None and base == reference_state, "final catch-up did not reach direct state")
        require(consumed_prefix == next_input_id - 1, "final catch-up lost an input")
        require(owner.version == 4 and len(owner.committed_ids) == 4, "wrong native-history cadence")
        require(captured is None, "uncommitted native candidate")
        digest = hashlib.sha256(json.dumps(
            [[str(value) for value in pixel] for pixel in owner.history],
            separators=(",", ":"),
        ).encode()).hexdigest()
        stream_summaries.append({
            "stream": stream,
            "final_state": str(reference_state),
            "consumed_input_prefix": consumed_prefix,
            "native_history_version": owner.version,
            "native_history_sha256": digest,
        })

    require(counts["identity_transport_events"] > 0, "identity transport not exercised")
    require(counts["capture_vs_later_state_distinctions"] > 0, "native sampling boundary not distinguished")
    return {
        "status": "passed",
        "scope": "Finite integrated mathematical witness for the declared periodic-strip model",
        "arithmetic": "fractions.Fraction; exact rational arithmetic",
        "seed": SEED,
        "streams": STREAMS,
        "events_per_stream": EVENTS_PER_STREAM,
        "total_stream_events": STREAMS * EVENTS_PER_STREAM,
        "pixels_per_image": PIXELS,
        "counts": counts,
        "reference": "Direct ordered state updates and full temporal/grade/HUD recipe",
        "candidate": "Transported clamped-affine residual, prefix rebase, periodic decoder, compact context",
        "native_history_rule": "One commit per sampled native boundary; intervening presentations cannot commit",
        "limitations": [
            "Finite examples supplement the analytical proofs; they are not exhaustive for all states.",
            "Exact rational equality does not imply equivalence to IEEE engine arithmetic.",
            "Periodic static texture and frozen affine effects do not cover general 3D visibility.",
            "No game process, GPU, physical input, optical detector, or NVIDIA method was tested.",
            "No latency or hardware-performance reduction is measured or asserted.",
        ],
        "stream_summaries": stream_summaries,
    }


def main() -> None:
    parser = argparse.ArgumentParser(description=__doc__)
    parser.add_argument("--output", type=Path, default=Path(__file__).with_name("constructive_system_validation.json"))
    args = parser.parse_args()
    result = verify()
    args.output.write_text(json.dumps(result, indent=2, sort_keys=True) + "\n", encoding="utf-8")
    print(json.dumps({key: result[key] for key in ("status", "streams", "total_stream_events", "counts")}, indent=2))


if __name__ == "__main__":
    main()

	#!/usr/bin/env python3
	"""Exact finite checks of the manuscript's complete restricted realization.

	Twenty-four seeded streams interleave legal clamped-affine actions, positive
	invertible affine evolution, transported-prefix rebases, presentation-only
	evaluations, and separately sampled native-history commits. The independent
	reference directly executes every semantic event and the full affine image
	recipe. The candidate uses a prepared base, ordered transported residual,
	periodic source decoder, and compact effect context.

	Only fractions.Fraction and Python's standard library are used. These finite
	mathematical checks are neither a proof for all programs nor measurements of
	hardware, floating-point engines, latency, or NVIDIA software.
	"""

	from __future__ import annotations

	import argparse
	import hashlib
	import json
	import random
	from dataclasses import dataclass
	from fractions import Fraction as F
	from pathlib import Path
	from typing import Optional


	SEED = 20261030
	STREAMS = 24
	EVENTS_PER_STREAM = 40
	PIXELS = 7
	RGB = tuple[F, F, F]
	Image = tuple[RGB, ...]


	def require(condition: bool, message: str) -> None:
	if not condition:
	raise AssertionError(message)


	def clip(value: F, lo: F, hi: F) -> F:
	return min(hi, max(lo, value))


	@dataclass(frozen=True)
	class ClampAffine:
	a: F
	b: F
	lo: F
	hi: F

	def __post_init__(self) -> None:
	require(self.a >= 0 and self.lo <= self.hi, "invalid action")

	def __call__(self, value: F) -> F:
	return clip(self.a * value + self.b, self.lo, self.hi)


	# None is the expressly adjoined empty-word identity, not a finite-bound clip.
	Residual = Optional[ClampAffine]


	def apply(residual: Residual, value: F) -> F:
	return value if residual is None else residual(value)


	def compose(outer: Residual, inner: Residual) -> Residual:
	if outer is None:
	return inner
	if inner is None:
	return outer
	return ClampAffine(
	outer.a * inner.a,
	outer.a * inner.b + outer.b,
	outer(inner.lo),
	outer(inner.hi),
	)


	def transport(residual: Residual, alpha: F, beta: F) -> Residual:
	require(alpha > 0, "slow affine evolution must be invertible and positive")
	if residual is None:
	return None
	return ClampAffine(
	residual.a,
	alpha * residual.b + (1 - residual.a) * beta,
	alpha * residual.lo + beta,
	alpha * residual.hi + beta,
	)


	def aggregate(word: list[tuple[int, ClampAffine]]) -> Residual:
	residual: Residual = None
	for _, action in word:
	residual = compose(action, residual)
	return residual


	def source_reference(source: Image, coordinate: F) -> RGB:
	"""Floor the unwrapped coordinate, then wrap the two endpoint indices."""
	lower = coordinate.numerator // coordinate.denominator
	weight = coordinate - lower
	left, right = source[lower % len(source)], source[(lower + 1) % len(source)]
	return tuple((1 - weight) * left[c] + weight * right[c] for c in range(3))


	def source_candidate(source: Image, coordinate: F) -> RGB:
	"""Wrap first; this also checks negative-coordinate periodic agreement."""
	position = coordinate % len(source)
	lower = position.numerator // position.denominator
	weight = position - lower
	left, right = source[lower], source[(lower + 1) % len(source)]
	return tuple(left[c] + weight * (right[c] - left[c]) for c in range(3))


	def matvec(matrix: tuple[RGB, RGB, RGB], value: RGB) -> RGB:
	return tuple(sum((row[c] * value[c] for c in range(3)), F(0)) for row in matrix)


	@dataclass(frozen=True)
	class Recipe:
	source: Image
	scale: F
	weights: tuple[F, ...]
	matrix: tuple[RGB, RGB, RGB]
	bias: RGB
	hud: Image
	hud_alpha: tuple[F, ...]


	@dataclass(frozen=True)
	class Context:
	coefficient: tuple[F, ...]
	bias: Image
	history_tail: Image


	def prepare(recipe: Recipe, history: Image) -> Context:
	tails = tuple(
	tuple((1 - recipe.weights[p]) * history[p][c] for c in range(3))
	for p in range(PIXELS)
	)
	coefficients = tuple(
	(1 - recipe.hud_alpha[p]) * recipe.weights[p] for p in range(PIXELS)
	)
	biases = []
	for p in range(PIXELS):
	graded_tail = matvec(recipe.matrix, tails[p])
	biases.append(tuple(
	recipe.hud[p][c]
	+ (1 - recipe.hud_alpha[p]) * (graded_tail[c] + recipe.bias[c])
	for c in range(3)
	))
	return Context(coefficients, tuple(biases), tails)


	def render_reference(recipe: Recipe, history: Image, state: F) -> tuple[Image, Image]:
	visible, next_history = [], []
	for p in range(PIXELS):
	z = source_reference(recipe.source, F(p) + recipe.scale * state)
	mixed = tuple(
	recipe.weights[p] * z[c] + (1 - recipe.weights[p]) * history[p][c]
	for c in range(3)
	)
	next_history.append(mixed)
	graded = tuple(
	sum((recipe.matrix[c][j] * mixed[j] for j in range(3)), F(0))
	+ recipe.bias[c]
	for c in range(3)
	)
	visible.append(tuple(
	recipe.hud[p][c] + (1 - recipe.hud_alpha[p]) * graded[c]
	for c in range(3)
	))
	return tuple(visible), tuple(next_history)


	def render_candidate(recipe: Recipe, context: Context, state: F) -> tuple[Image, Image]:
	visible, next_history = [], []
	for p in range(PIXELS):
	z = source_candidate(recipe.source, F(p) + recipe.scale * state)
	graded = matvec(recipe.matrix, z)
	visible.append(tuple(
	context.coefficient[p] * graded[c] + context.bias[p][c]
	for c in range(3)
	))
	next_history.append(tuple(
	recipe.weights[p] * z[c] + context.history_tail[p][c]
	for c in range(3)
	))
	return tuple(visible), tuple(next_history)


	class NativeHistoryOwner:
	"""A mathematical single-owner/version gate, not an engine adapter."""

	def __init__(self, history: Image):
	self.history = history
	self.version = 0
	self.committed_ids: set[int] = set()

	def commit(self, commit_id: int, expected_version: int, proposed: Image) -> bool:
	if commit_id in self.committed_ids or expected_version != self.version:
	return False
	self.history = proposed
	self.version += 1
	self.committed_ids.add(commit_id)
	return True


	def random_image(rng: random.Random) -> Image:
	return tuple(tuple(F(rng.randint(0, 12), 12) for _ in range(3)) for _ in range(PIXELS))


	def random_recipe(rng: random.Random) -> Recipe:
	source = random_image(rng)
	alpha = tuple(F(rng.randint(0, 6), 6) for _ in range(PIXELS))
	hud_color = random_image(rng)
	hud = tuple(tuple(alpha[p] * hud_color[p][c] for c in range(3)) for p in range(PIXELS))
	matrix = tuple(
	tuple(F(int(i == j)) + F(rng.randint(-2, 2), 12) for j in range(3))
	for i in range(3)
	)
	return Recipe(
	source=source,
	scale=F(rng.choice((-3, -2, -1, 1, 2, 3)), rng.choice((2, 3))),
	weights=tuple(F(rng.randint(0, 6), 6) for _ in range(PIXELS)),
	matrix=matrix,
	bias=tuple(F(rng.randint(-2, 2), 24) for _ in range(3)),
	hud=hud,
	hud_alpha=alpha,
	)


	def random_action(rng: random.Random) -> ClampAffine:
	lo = F(rng.randint(-12, 0), 6)
	hi = F(rng.randint(0, 12), 6)
	return ClampAffine(
	rng.choice((F(0), F(1, 2), F(1), F(3, 2), F(2))),
	F(rng.randint(-12, 12), 6), lo, hi,
	)


	def verify() -> dict:
	rng = random.Random(SEED)
	counts = {name: 0 for name in (
	"actions", "slow_affine_transitions", "identity_transport_events",
	"state_equalities", "native_rebases", "consumed_input_ids",
	"native_captures", "native_history_commits", "duplicate_commit_rejections",
	"explicit_presentations", "presentation_history_unchanged_checks",
	"visible_channel_equalities", "candidate_history_channel_equalities",
	"capture_vs_later_state_distinctions",
	)}
	stream_summaries = []

	for stream in range(STREAMS):
	recipe = random_recipe(rng)
	reference_history = random_image(rng)
	owner = NativeHistoryOwner(reference_history)
	context = prepare(recipe, owner.history)
	reference_state = F(rng.randint(-6, 6), 3)
	base = reference_state
	residual: Residual = None
	pending: list[tuple[int, ClampAffine]] = []
	next_input_id = 1
	consumed_prefix = 0
	consumed_ids: set[int] = set()
	captured = None

	def rebase(number: int) -> None:
	nonlocal base, residual, pending, consumed_prefix
	require(0 <= number <= len(pending), "invalid prefix length")
	old_state = apply(residual, base)
	prefix, suffix = pending[:number], pending[number:]
	# Execute the transported prefix directly, rather than by its summary.
	for input_id, action in prefix:
	require(input_id == consumed_prefix + 1, "noncontiguous consumed prefix")
	require(input_id not in consumed_ids, "duplicate input consumption")
	base = action(base)
	consumed_ids.add(input_id)
	consumed_prefix = input_id
	counts["consumed_input_ids"] += 1
	pending = suffix
	residual = aggregate(suffix)
	require(apply(residual, base) == old_state == reference_state, "rebase changed state")
	counts["native_rebases"] += 1

	# One cycle samples a native candidate before newer input arrives, and
	# commits that sampled candidate only at the cycle's final event.
	for event in range(EVENTS_PER_STREAM):
	phase, cycle = event % 10, event // 10
	if phase in (0, 3, 8):
	action = random_action(rng)
	reference_state = action(reference_state)
	residual = compose(action, residual)
	pending.append((next_input_id, action))
	next_input_id += 1
	counts["actions"] += 1
	elif phase in (1, 6):
	alpha = rng.choice((F(1, 2), F(2, 3), F(1), F(3, 2), F(2)))
	beta = F(rng.randint(-6, 6), 6)
	reference_state = alpha * reference_state + beta
	base = alpha * base + beta
	if residual is None:
	counts["identity_transport_events"] += 1
	residual = transport(residual, alpha, beta)
	pending = [
	(input_id, transport(action, alpha, beta))
	for input_id, action in pending
	]
	counts["slow_affine_transitions"] += 1
	elif phase == 2:
	require(captured is None, "native capture overwritten before commit")
	_, reference_next = render_reference(recipe, reference_history, reference_state)
	_, candidate_next = render_candidate(recipe, context, apply(residual, base))
	require(reference_next == candidate_next, "sampled native histories differ")
	captured = (cycle, owner.version, reference_next, candidate_next, reference_state)
	counts["native_captures"] += 1
	elif phase in (4, 7):
	counts["explicit_presentations"] += 1
	elif phase == 5:
	number = len(pending) if (stream + cycle) % 2 == 0 else max(1, len(pending) // 2)
	rebase(number)
	elif phase == 9:
	require(captured is not None, "native commit has no sampled candidate")
	commit_id, expected, reference_next, candidate_next, capture_state = captured
	if capture_state != reference_state:
	counts["capture_vs_later_state_distinctions"] += 1
	reference_history = reference_next
	require(owner.commit(commit_id, expected, candidate_next), "valid native commit rejected")
	require(owner.history == reference_history, "native history continuation differs")
	counts["native_history_commits"] += 1
	committed_state = (owner.history, owner.version)
	require(not owner.commit(commit_id, expected, candidate_next), "duplicate history commit accepted")
	require((owner.history, owner.version) == committed_state, "rejected commit changed history")
	counts["duplicate_commit_rejections"] += 1
	context = prepare(recipe, owner.history)
	captured = None

	candidate_state = apply(residual, base)
	require(candidate_state == reference_state, "transported state differs from direct execution")
	counts["state_equalities"] += 1
	before = (owner.history, owner.version)
	reference_image, reference_next = render_reference(recipe, reference_history, reference_state)
	candidate_image, candidate_next = render_candidate(recipe, context, candidate_state)
	require(reference_image == candidate_image, "compact image differs from complete recipe")
	require(reference_next == candidate_next, "candidate next history differs")
	require((owner.history, owner.version) == before, "presentation advanced native history")
	counts["presentation_history_unchanged_checks"] += 1
	counts["visible_channel_equalities"] += PIXELS * 3
	counts["candidate_history_channel_equalities"] += PIXELS * 3

	rebase(len(pending))
	require(residual is None and base == reference_state, "final catch-up did not reach direct state")
	require(consumed_prefix == next_input_id - 1, "final catch-up lost an input")
	require(owner.version == 4 and len(owner.committed_ids) == 4, "wrong native-history cadence")
	require(captured is None, "uncommitted native candidate")
	digest = hashlib.sha256(json.dumps(
	[[str(value) for value in pixel] for pixel in owner.history],
	separators=(",", ":"),
	).encode()).hexdigest()
	stream_summaries.append({
	"stream": stream,
	"final_state": str(reference_state),
	"consumed_input_prefix": consumed_prefix,
	"native_history_version": owner.version,
	"native_history_sha256": digest,
	})

	require(counts["identity_transport_events"] > 0, "identity transport not exercised")
	require(counts["capture_vs_later_state_distinctions"] > 0, "native sampling boundary not distinguished")
	return {
	"status": "passed",
	"scope": "Finite integrated mathematical witness for the declared periodic-strip model",
	"arithmetic": "fractions.Fraction; exact rational arithmetic",
	"seed": SEED,
	"streams": STREAMS,
	"events_per_stream": EVENTS_PER_STREAM,
	"total_stream_events": STREAMS * EVENTS_PER_STREAM,
	"pixels_per_image": PIXELS,
	"counts": counts,
	"reference": "Direct ordered state updates and full temporal/grade/HUD recipe",
	"candidate": "Transported clamped-affine residual, prefix rebase, periodic decoder, compact context",
	"native_history_rule": "One commit per sampled native boundary; intervening presentations cannot commit",
	"limitations": [
	"Finite examples supplement the analytical proofs; they are not exhaustive for all states.",
	"Exact rational equality does not imply equivalence to IEEE engine arithmetic.",
	"Periodic static texture and frozen affine effects do not cover general 3D visibility.",
	"No game process, GPU, physical input, optical detector, or NVIDIA method was tested.",
	"No latency or hardware-performance reduction is measured or asserted.",
	],
	"stream_summaries": stream_summaries,
	}


	def main() -> None:
	parser = argparse.ArgumentParser(description=__doc__)
	parser.add_argument("--output", type=Path, default=Path(__file__).with_name("constructive_system_validation.json"))
	args = parser.parse_args()
	result = verify()
	args.output.write_text(json.dumps(result, indent=2, sort_keys=True) + "\n", encoding="utf-8")
	print(json.dumps({key: result[key] for key in ("status", "streams", "total_stream_events", "counts")}, indent=2))


	if __name__ == "__main__":
	main()