Datasets:

PureOne
/

Lyotherin_Tia

#!/usr/bin/env python3
"""Exact, finite mathematical checks for Lyotherin_Tia.
This file checks declared algebra and tiny finite coding problems. It is not a
game adapter, timing benchmark, floating-point equivalence proof, or test of
NVIDIA software. Only Python's standard library is used. All algebra uses
fractions.Fraction; the seed and counts are deliberately modest and explicit.
"""

from __future__ import annotations

import argparse
import itertools
import json
import random
from dataclasses import dataclass
from fractions import Fraction as F
from pathlib import Path


SEED = 20261030


def fq(rng: random.Random, lo: int = -6, hi: int = 6) -> F:
    return F(rng.randint(lo, hi), rng.randint(1, 5))


def clip(x: F, lo: F, hi: F) -> F:
    assert lo <= hi
    return min(hi, max(lo, x))


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

    def __post_init__(self) -> None:
        assert self.a >= 0 and self.lo <= self.hi

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


def compose(outer: ClampAffine, inner: ClampAffine) -> ClampAffine:
    """Return outer o inner in the same four-parameter family."""
    return ClampAffine(
        outer.a * inner.a,
        outer.a * inner.b + outer.b,
        outer(inner.lo),
        outer(inner.hi),
    )


def random_clamp(rng: random.Random) -> ClampAffine:
    bounds = sorted((fq(rng), fq(rng)))
    return ClampAffine(F(rng.randint(0, 8), rng.randint(1, 5)), fq(rng), *bounds)


def check_clamps(rng: random.Random) -> dict:
    examples = [
        (ClampAffine(F(0), F(2), F(-1), F(1)),
         ClampAffine(F(3), F(-1), F(-2), F(2))),
        (ClampAffine(F(2), F(-1), F(0), F(0)),
         ClampAffine(F(0), F(7), F(0), F(3))),
        (ClampAffine(F(1), F(0), F(-10), F(10)),
         ClampAffine(F(2), F(1), F(-1), F(1))),
    ]
    examples.extend((random_clamp(rng), random_clamp(rng)) for _ in range(60))
    query_points = [F(i, 5) for i in range(-10, 11)]
    for outer, inner in examples:
        merged = compose(outer, inner)
        for x in query_points:
            assert merged(x) == outer(inner(x))

    word_lengths = []
    word_queries = [F(i, 3) for i in range(-6, 7)]
    for _ in range(24):
        word = [random_clamp(rng) for _ in range(rng.randint(1, 12))]
        word_lengths.append(len(word))
        merged = word[0]
        for fn in word[1:]:
            merged = compose(fn, merged)
        for x in word_queries:
            actual = x
            for fn in word:
                actual = fn(actual)
            assert merged(x) == actual

    # Aggregating only input deltas loses irreversible saturation information.
    x, u, v = F(9, 10), F(1, 5), F(-1, 5)
    sequential = clip(clip(x + u, F(0), F(1)) + v, F(0), F(1))
    total_only = clip(x + u + v, F(0), F(1))
    assert sequential == F(4, 5) and total_only == F(9, 10)
    return {
        "status": "passed",
        "composition_pairs": len(examples),
        "query_points_per_pair": len(query_points),
        "random_words": len(word_lengths),
        "word_lengths": word_lengths,
        "query_points_per_word": len(word_queries),
        "saturation_counterexample": {
            "initial": str(x), "deltas": [str(u), str(v)],
            "sequential": str(sequential), "sum_only": str(total_only),
        },
        "assumptions": [
            "Each slope is nonnegative; bounds are finite and ordered.",
            "These are exact rational identities, not bit-exact float claims.",
            "Constant parameter count does not bound integer bit complexity.",
        ],
    }


def check_transport(rng: random.Random) -> dict:
    for _ in range(100):
        x, k, d, u = (fq(rng) for _ in range(4))
        assert k * (x + u) + d == (k * x + d) + k * u

    lengths = []
    for _ in range(48):
        x = fq(rng)
        full = x
        K, D, delta = F(1), F(0), F(0)
        length = rng.randint(5, 30)
        lengths.append(length)
        for _ in range(length):
            if rng.randrange(2):
                k, d = fq(rng), fq(rng)
                full = k * full + d
                K, D, delta = k * K, k * D + d, k * delta
            else:
                u = fq(rng)
                full += u
                delta += u
            assert full == K * x + D + delta
    return {
        "status": "passed", "one_step_identities": 100,
        "interleaved_words": len(lengths), "word_lengths": lengths,
        "prefix_equalities": sum(lengths),
        "identity": "B o A_u = A_(k*u) o B, B(x)=k*x+d",
        "recurrence": "On B: (K,D,delta)<-(k*K,k*D+d,k*delta); on A_u: delta<-delta+u",
        "assumptions": [
            "Only the declared affine/translation dynamics are represented.",
            "Every input is consumed exactly once in its declared order.",
            "No hidden collision, branch, rounding or unobserved world update is inferred.",
        ],
    }


def matvec(matrix: list[list[F]], vector: list[F]) -> list[F]:
    return [sum((x * y for x, y in zip(row, vector)), F(0)) for row in matrix]


def rank(matrix: list[list[F]]) -> int:
    if not matrix:
        return 0
    a = [row[:] for row in matrix]
    m, n = len(a), len(a[0])
    pivot_row = 0
    for col in range(n):
        pivot = next((r for r in range(pivot_row, m) if a[r][col]), None)
        if pivot is None:
            continue
        a[pivot_row], a[pivot] = a[pivot], a[pivot_row]
        scale = a[pivot_row][col]
        a[pivot_row] = [v / scale for v in a[pivot_row]]
        for row in range(m):
            if row != pivot_row and a[row][col]:
                scale = a[row][col]
                a[row] = [v - scale * w for v, w in zip(a[row], a[pivot_row])]
        pivot_row += 1
        if pivot_row == m:
            break
    return pivot_row


def temporal_rows(weights: list[F], shift: int) -> list[list[F]]:
    n = len(weights)
    rows = []
    for p, weight in enumerate(weights):
        row = [F(0)] * (2 * n)
        row[(p + shift) % n] = weight
        row[n + p] = 1 - weight
        rows.append(row)
    return rows


def check_history(rng: random.Random) -> dict:
    # Arbitrary global RGB grade, fixed target history and premultiplied HUD.
    # Bilinear two-source mixing is just an explicit source sample here.
    for _ in range(80):
        source = [[fq(rng) for _ in range(3)] for _ in range(2)]
        grade = [[fq(rng) for _ in range(3)] for _ in range(3)]
        bias = [fq(rng) for _ in range(3)]
        for p in range(2):
            blend, hud_alpha, interp = (F(rng.randint(0, 8), 8) for _ in range(3))
            history, hud = [fq(rng) for _ in range(3)], [fq(rng) for _ in range(3)]
            z = [(1 - interp) * source[p][j] + interp * source[1-p][j]
                 for j in range(3)]
            next_history = [blend * z[j] + (1-blend) * history[j] for j in range(3)]
            transformed = matvec(grade, next_history)
            direct = [hud[j] + (1-hud_alpha) * (transformed[j] + bias[j])
                      for j in range(3)]
            k = (1-hud_alpha) * blend
            retained_history = [(1-blend) * v for v in history]
            early = matvec(grade, retained_history)
            b = [hud[j] + (1-hud_alpha) * (early[j] + bias[j]) for j in range(3)]
            mz = matvec(grade, z)
            compact = [k * mz[j] + b[j] for j in range(3)]
            assert direct == compact
            # The visible-only formula alone does not reconstruct next history.
            assert next_history == [blend * z[j] + retained_history[j] for j in range(3)]

    weights_grid = [F(0), F(1, 4), F(1, 2), F(3, 4), F(1)]
    two_pixel_rank_results = []
    for a0, a1 in itertools.product(weights_grid, repeat=2):
        weights = [a0, a1]
        C, R = temporal_rows(weights, 0), temporal_rows(weights, 1)
        extra = rank(C + R) - rank(C)
        assert extra in (0, 1)
        if F(0) < a0 < F(1) and F(0) < a1 < F(1):
            assert extra == 1
        c0, c1, h0, h1 = (fq(rng) for _ in range(4))
        u = [c0, c1, h0, h1]
        cached, required = matvec(C, u), matvec(R, u)
        delta = c1 - c0
        decoded = [cached[0] + a0 * delta, cached[1] - a1 * delta]
        assert decoded == required
        two_pixel_rank_results.append({"a0": str(a0), "a1": str(a1), "extra_rank": extra})

    cyclic_results = []
    for n in range(2, 7):
        for a in (F(1, 4), F(1, 2), F(3, 4)):
            C = temporal_rows([a] * n, 0)
            all_rows = [row for shift in range(n) for row in temporal_rows([a] * n, shift)]
            extra = rank(C + all_rows) - rank(C)
            assert extra == n - 1
            cyclic_results.append({"n": n, "weight": str(a), "cache_rank": rank(C),
                                   "joint_rank": rank(C + all_rows), "extra_rank": extra})
    return {
        "status": "passed", "rgb_configurations": 80, "pixels_per_configuration": 2,
        "two_pixel_rank_cases": two_pixel_rank_results,
        "cyclic_rank_cases": cyclic_results,
        "two_pixel_decoder": "delta=c1-c0; shifted=(cached0+a0*delta,cached1-a1*delta)",
        "assumptions": [
            "Linear RGB and the stated blend/grade/HUD order; no nonlinear tone mapping.",
            "HUD, target history, blend weights and grade are independent of admitted late input.",
            "The source samples are supplied; geometry/occlusion validity is not inferred.",
            "A candidate history is a value only; no canonical history is committed here.",
            "The cyclic-rank example is scalar; independent RGB gives three copies.",
        ],
    }


def partitions(n: int):
    """Enumerate canonical set partitions as restricted-growth label tuples."""
    if n == 0:
        yield ()
        return
    def visit(labels, largest):
        if len(labels) == n:
            yield tuple(labels)
            return
        for color in range(largest + 2):
            labels.append(color)
            yield from visit(labels, max(largest, color))
            labels.pop()
    yield from visit([0], 0)


def conflict_graph(cache, image, transition_record, eps):
    """Record equality includes successor, outcome, status and side effects.
    A hashable integer may be a shorthand label for a full exact transition
    record. Equal successor classes alone do not imply equal records.
    """
    n, queries = len(cache), len(image[0])
    adjacency = [set() for _ in range(n)]
    for i in range(n):
        for j in range(i + 1, n):
            if cache[i] != cache[j]:
                continue
            conflict = any(
                transition_record[i][q] != transition_record[j][q]
                or any(abs(a-b) > 2*eps for a, b in zip(image[i][q], image[j][q]))
                for q in range(queries)
            )
            if conflict:
                adjacency[i].add(j)
                adjacency[j].add(i)
    return adjacency


def direct_decoder(labels, cache, image, transition_record, eps):
    """Test image boxes and full exact-record agreement without a graph."""
    result = {}
    for known_cache, label in set(zip(cache, labels)):
        members = [i for i in range(len(cache)) if (cache[i], labels[i]) == (known_cache, label)]
        for q in range(len(image[0])):
            targets = {transition_record[i][q] for i in members}
            if len(targets) != 1:
                return None
            decoded = []
            for coordinate in range(len(image[0][q])):
                values = [image[i][q][coordinate] for i in members]
                # Independently intersect every admissible interval.
                lower = max(v - eps for v in values)
                upper = min(v + eps for v in values)
                if lower > upper:
                    return None
                decoded.append((min(values) + max(values)) / 2)
            result[(known_cache, label, q)] = (decoded, next(iter(targets)))
    return result


def chromatic_number(adjacency):
    n = len(adjacency)
    order = sorted(range(n), key=lambda v: (-len(adjacency[v]), v))
    for limit in range(1, n + 1):
        colors = [-1] * n
        def solve(at):
            if at == n:
                return True
            vertex = order[at]
            forbidden = {colors[w] for w in adjacency[vertex] if colors[w] >= 0}
            for color in range(limit):
                if color not in forbidden:
                    colors[vertex] = color
                    if solve(at + 1):
                        return True
                    colors[vertex] = -1
            return False
        if solve(0):
            return limit, tuple(colors)
    raise AssertionError("Finite graph must admit singleton coloring")


def check_code_instance(name, cache, image, transition_record, eps):
    adjacency = conflict_graph(cache, image, transition_record, eps)
    graph_min, graph_labels = chromatic_number(adjacency)
    direct_min, tested, valid = len(cache) + 1, 0, 0
    for labels in partitions(len(cache)):
        tested += 1
        decoder = direct_decoder(labels, cache, image, transition_record, eps)
        proper = all(labels[i] != labels[j] for i in range(len(cache)) for j in adjacency[i])
        assert proper == (decoder is not None)
        if decoder is not None:
            valid += 1
            direct_min = min(direct_min, max(labels) + 1)
    assert direct_min == graph_min
    decoder = direct_decoder(graph_labels, cache, image, transition_record, eps)
    assert decoder is not None
    for i in range(len(cache)):
        for q in range(len(image[0])):
            output, exact_record = decoder[(cache[i], graph_labels[i], q)]
            assert exact_record == transition_record[i][q]
            assert all(abs(a-b) <= eps for a, b in zip(output, image[i][q]))
    return {
        "name": name, "states": len(cache), "queries": len(image[0]),
        "image_coordinates": len(image[0][0]), "epsilon": str(eps),
        "partitions_examined": tested, "valid_partitions": valid,
        "minimum_code_alphabet": direct_min, "chromatic_number": graph_min,
        "edges": [[i,j] for i in range(len(cache)) for j in sorted(adjacency[i]) if i < j],
    }


def check_mixed_coding(rng: random.Random) -> dict:
    cases = []
    for n in range(1, 7):
        for trial in range(5):
            query_count = 1 + trial % 3
            cache = [rng.randrange(1 + trial % 2) for _ in range(n)]
            eps = (F(0), F(1, 4), F(1, 2), F(1), F(3, 2))[trial]
            image = [[[F(rng.randint(-3, 3), 2) for _ in range(2)]
                      for _ in range(query_count)] for _ in range(n)]
            # Integers abbreviate full exact transition records. Include
            # both identical records and per-query differing records.
            transition_record = [[0 if trial % 2 == 0 else rng.randrange(2)
                                  for _ in range(query_count)] for _ in range(n)]
            cases.append(check_code_instance(f"random_n{n}_case{trial}", cache, image, transition_record, eps))

    # An induced C5: each edge gets a coordinate with its endpoints +1 and -1.
    # Adjacent vertices differ by 2, nonadjacent ones by at most 1.
    edges = [(i, (i + 1) % 5) for i in range(5)]
    image = [[[F(1) if i == a else F(-1) if i == b else F(0)
               for a, b in edges]] for i in range(5)]
    c5 = check_code_instance("five_cycle", [0]*5, image, [[0]]*5, F(1,2))
    assert c5["chromatic_number"] == 3
    cases.append(c5)

    exact_continuation = check_code_instance(
        "same_image_distinct_successor", [0,0], [[[F(0)]],[[F(0)]]], [[0],[1]], F(1))
    assert exact_continuation["chromatic_number"] == 2
    cases.append(exact_continuation)

    # A shared future state does not erase required immediate nonvisual output.
    # Tuple order: (successor class, authoritative outcome, status, effects).
    record0 = (0, "pulse_committed", "accepted", ("emit_pulse",))
    record1 = (0, "pulse_blocked", "rejected", ())
    assert record0[0] == record1[0] and record0 != record1
    distinct_effects = check_code_instance(
        "same_image_same_successor_distinct_immediate_effect_status", [0,0],
        [[[F(0)]],[[F(0)]]], [[record0],[record1]], F(1))
    assert distinct_effects["chromatic_number"] == 2
    distinct_effects["same_successor_class"] = True
    distinct_effects["exact_transition_records"] = [record0, record1]
    cases.append(distinct_effects)

    cache_reuse = check_code_instance(
        "reuse_labels_across_cache_fibers", [0,0,1,1],
        [[[F(0)]],[[F(3)]],[[F(0)]],[[F(3)]]], [[0],[0],[0],[0]], F(1))
    assert cache_reuse["chromatic_number"] == 2
    cases.append(cache_reuse)

    # Pairwise distance <= 2 epsilon is not enough for arbitrary norms.
    # In L2, unit balls at (0,0),(2,0) meet only at (1,0), which is 3/2
    # from the third center (1,3/2). Every pair of centers is <= 2 apart.
    triangle = [(F(0),F(0)),(F(2),F(0)),(F(1),F(3,2))]
    distances_squared = [sum((x-y)**2 for x,y in zip(a,b))
                         for a,b in itertools.combinations(triangle,2)]
    assert max(distances_squared) <= 4
    assert (F(3,2))**2 > 1
    return {
        "status": "passed", "instances": cases,
        "total_instances": len(cases),
        "total_partitions_examined": sum(c["partitions_examined"] for c in cases),
        "l2_counterexample_squared_pair_distances": [str(x) for x in distances_squared],
        "assumptions": [
            "Finite state/cache/query sets with every listed query admissible for every listed state.",
            "The encoder sees state; decoder sees exact cache value, query and code label.",
            "Approximate image error uses an L-infinity ball, coordinate by coordinate.",
            "Exact transition records include successor continuation class, immediate authoritative outcomes, accept/reject status and required side-effect records.",
            "Generic integer labels abbreviate full exact transition records; successor classes require independent semantic certification and realization.",
            "Tests check the specified one-step decoder; they do not prove online approximate stability.",
            "Palette reuse across different cache fibers is permitted because the cache is decoder side information.",
            "No practical encoding or decoding time bound follows from chromatic minimality.",
        ],
    }


def main() -> None:
    parser = argparse.ArgumentParser()
    parser.add_argument("--output", type=Path, default=Path(__file__).with_name("verification_math_results.json"))
    args = parser.parse_args()
    rng = random.Random(SEED)
    result = {
        "project": "Lyotherin_Tia", "seed": SEED,
        "arithmetic": "Python standard-library fractions.Fraction",
        "evidence_type": "finite exact algebra and combinatorial sanity checks; not hardware measurement or formal proof assistant output",
        "clamped_affine": check_clamps(rng),
        "transported_translation": check_transport(rng),
        "history_compaction": check_history(rng),
        "mixed_image_exact_continuation": check_mixed_coding(rng),
        "overall_status": "passed",
    }
    args.output.write_text(json.dumps(result, indent=2) + "\n", encoding="utf-8")
    print(json.dumps({"overall_status": result["overall_status"], "output": str(args.output),
                      "clamp_pairs": result["clamped_affine"]["composition_pairs"],
                      "interleaved_words": result["transported_translation"]["interleaved_words"],
                      "rgb_configurations": result["history_compaction"]["rgb_configurations"],
                      "finite_coding_instances": result["mixed_image_exact_continuation"]["total_instances"],
                      "finite_partitions": result["mixed_image_exact_continuation"]["total_partitions_examined"]}))


if __name__ == "__main__":
    main()

	#!/usr/bin/env python3
	"""Exact, finite mathematical checks for Lyotherin_Tia.

	This file checks declared algebra and tiny finite coding problems. It is not a
	game adapter, timing benchmark, floating-point equivalence proof, or test of
	NVIDIA software. Only Python's standard library is used. All algebra uses
	fractions.Fraction; the seed and counts are deliberately modest and explicit.
	"""

	from __future__ import annotations

	import argparse
	import itertools
	import json
	import random
	from dataclasses import dataclass
	from fractions import Fraction as F
	from pathlib import Path


	SEED = 20261030


	def fq(rng: random.Random, lo: int = -6, hi: int = 6) -> F:
	return F(rng.randint(lo, hi), rng.randint(1, 5))


	def clip(x: F, lo: F, hi: F) -> F:
	assert lo <= hi
	return min(hi, max(lo, x))


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

	def __post_init__(self) -> None:
	assert self.a >= 0 and self.lo <= self.hi

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


	def compose(outer: ClampAffine, inner: ClampAffine) -> ClampAffine:
	"""Return outer o inner in the same four-parameter family."""
	return ClampAffine(
	outer.a * inner.a,
	outer.a * inner.b + outer.b,
	outer(inner.lo),
	outer(inner.hi),
	)


	def random_clamp(rng: random.Random) -> ClampAffine:
	bounds = sorted((fq(rng), fq(rng)))
	return ClampAffine(F(rng.randint(0, 8), rng.randint(1, 5)), fq(rng), *bounds)


	def check_clamps(rng: random.Random) -> dict:
	examples = [
	(ClampAffine(F(0), F(2), F(-1), F(1)),
	ClampAffine(F(3), F(-1), F(-2), F(2))),
	(ClampAffine(F(2), F(-1), F(0), F(0)),
	ClampAffine(F(0), F(7), F(0), F(3))),
	(ClampAffine(F(1), F(0), F(-10), F(10)),
	ClampAffine(F(2), F(1), F(-1), F(1))),
	]
	examples.extend((random_clamp(rng), random_clamp(rng)) for _ in range(60))
	query_points = [F(i, 5) for i in range(-10, 11)]
	for outer, inner in examples:
	merged = compose(outer, inner)
	for x in query_points:
	assert merged(x) == outer(inner(x))

	word_lengths = []
	word_queries = [F(i, 3) for i in range(-6, 7)]
	for _ in range(24):
	word = [random_clamp(rng) for _ in range(rng.randint(1, 12))]
	word_lengths.append(len(word))
	merged = word[0]
	for fn in word[1:]:
	merged = compose(fn, merged)
	for x in word_queries:
	actual = x
	for fn in word:
	actual = fn(actual)
	assert merged(x) == actual

	# Aggregating only input deltas loses irreversible saturation information.
	x, u, v = F(9, 10), F(1, 5), F(-1, 5)
	sequential = clip(clip(x + u, F(0), F(1)) + v, F(0), F(1))
	total_only = clip(x + u + v, F(0), F(1))
	assert sequential == F(4, 5) and total_only == F(9, 10)
	return {
	"status": "passed",
	"composition_pairs": len(examples),
	"query_points_per_pair": len(query_points),
	"random_words": len(word_lengths),
	"word_lengths": word_lengths,
	"query_points_per_word": len(word_queries),
	"saturation_counterexample": {
	"initial": str(x), "deltas": [str(u), str(v)],
	"sequential": str(sequential), "sum_only": str(total_only),
	},
	"assumptions": [
	"Each slope is nonnegative; bounds are finite and ordered.",
	"These are exact rational identities, not bit-exact float claims.",
	"Constant parameter count does not bound integer bit complexity.",
	],
	}


	def check_transport(rng: random.Random) -> dict:
	for _ in range(100):
	x, k, d, u = (fq(rng) for _ in range(4))
	assert k * (x + u) + d == (k * x + d) + k * u

	lengths = []
	for _ in range(48):
	x = fq(rng)
	full = x
	K, D, delta = F(1), F(0), F(0)
	length = rng.randint(5, 30)
	lengths.append(length)
	for _ in range(length):
	if rng.randrange(2):
	k, d = fq(rng), fq(rng)
	full = k * full + d
	K, D, delta = k * K, k * D + d, k * delta
	else:
	u = fq(rng)
	full += u
	delta += u
	assert full == K * x + D + delta
	return {
	"status": "passed", "one_step_identities": 100,
	"interleaved_words": len(lengths), "word_lengths": lengths,
	"prefix_equalities": sum(lengths),
	"identity": "B o A_u = A_(ku) o B, B(x)=kx+d",
	"recurrence": "On B: (K,D,delta)<-(kK,kD+d,k*delta); on A_u: delta<-delta+u",
	"assumptions": [
	"Only the declared affine/translation dynamics are represented.",
	"Every input is consumed exactly once in its declared order.",
	"No hidden collision, branch, rounding or unobserved world update is inferred.",
	],
	}


	def matvec(matrix: list[list[F]], vector: list[F]) -> list[F]:
	return [sum((x * y for x, y in zip(row, vector)), F(0)) for row in matrix]


	def rank(matrix: list[list[F]]) -> int:
	if not matrix:
	return 0
	a = [row[:] for row in matrix]
	m, n = len(a), len(a[0])
	pivot_row = 0
	for col in range(n):
	pivot = next((r for r in range(pivot_row, m) if a[r][col]), None)
	if pivot is None:
	continue
	a[pivot_row], a[pivot] = a[pivot], a[pivot_row]
	scale = a[pivot_row][col]
	a[pivot_row] = [v / scale for v in a[pivot_row]]
	for row in range(m):
	if row != pivot_row and a[row][col]:
	scale = a[row][col]
	a[row] = [v - scale * w for v, w in zip(a[row], a[pivot_row])]
	pivot_row += 1
	if pivot_row == m:
	break
	return pivot_row


	def temporal_rows(weights: list[F], shift: int) -> list[list[F]]:
	n = len(weights)
	rows = []
	for p, weight in enumerate(weights):
	row = [F(0)] * (2 * n)
	row[(p + shift) % n] = weight
	row[n + p] = 1 - weight
	rows.append(row)
	return rows


	def check_history(rng: random.Random) -> dict:
	# Arbitrary global RGB grade, fixed target history and premultiplied HUD.
	# Bilinear two-source mixing is just an explicit source sample here.
	for _ in range(80):
	source = [[fq(rng) for _ in range(3)] for _ in range(2)]
	grade = [[fq(rng) for _ in range(3)] for _ in range(3)]
	bias = [fq(rng) for _ in range(3)]
	for p in range(2):
	blend, hud_alpha, interp = (F(rng.randint(0, 8), 8) for _ in range(3))
	history, hud = [fq(rng) for _ in range(3)], [fq(rng) for _ in range(3)]
	z = [(1 - interp) * source[p][j] + interp * source[1-p][j]
	for j in range(3)]
	next_history = [blend * z[j] + (1-blend) * history[j] for j in range(3)]
	transformed = matvec(grade, next_history)
	direct = [hud[j] + (1-hud_alpha) * (transformed[j] + bias[j])
	for j in range(3)]
	k = (1-hud_alpha) * blend
	retained_history = [(1-blend) * v for v in history]
	early = matvec(grade, retained_history)
	b = [hud[j] + (1-hud_alpha) * (early[j] + bias[j]) for j in range(3)]
	mz = matvec(grade, z)
	compact = [k * mz[j] + b[j] for j in range(3)]
	assert direct == compact
	# The visible-only formula alone does not reconstruct next history.
	assert next_history == [blend * z[j] + retained_history[j] for j in range(3)]

	weights_grid = [F(0), F(1, 4), F(1, 2), F(3, 4), F(1)]
	two_pixel_rank_results = []
	for a0, a1 in itertools.product(weights_grid, repeat=2):
	weights = [a0, a1]
	C, R = temporal_rows(weights, 0), temporal_rows(weights, 1)
	extra = rank(C + R) - rank(C)
	assert extra in (0, 1)
	if F(0) < a0 < F(1) and F(0) < a1 < F(1):
	assert extra == 1
	c0, c1, h0, h1 = (fq(rng) for _ in range(4))
	u = [c0, c1, h0, h1]
	cached, required = matvec(C, u), matvec(R, u)
	delta = c1 - c0
	decoded = [cached[0] + a0 * delta, cached[1] - a1 * delta]
	assert decoded == required
	two_pixel_rank_results.append({"a0": str(a0), "a1": str(a1), "extra_rank": extra})

	cyclic_results = []
	for n in range(2, 7):
	for a in (F(1, 4), F(1, 2), F(3, 4)):
	C = temporal_rows([a] * n, 0)
	all_rows = [row for shift in range(n) for row in temporal_rows([a] * n, shift)]
	extra = rank(C + all_rows) - rank(C)
	assert extra == n - 1
	cyclic_results.append({"n": n, "weight": str(a), "cache_rank": rank(C),
	"joint_rank": rank(C + all_rows), "extra_rank": extra})
	return {
	"status": "passed", "rgb_configurations": 80, "pixels_per_configuration": 2,
	"two_pixel_rank_cases": two_pixel_rank_results,
	"cyclic_rank_cases": cyclic_results,
	"two_pixel_decoder": "delta=c1-c0; shifted=(cached0+a0delta,cached1-a1delta)",
	"assumptions": [
	"Linear RGB and the stated blend/grade/HUD order; no nonlinear tone mapping.",
	"HUD, target history, blend weights and grade are independent of admitted late input.",
	"The source samples are supplied; geometry/occlusion validity is not inferred.",
	"A candidate history is a value only; no canonical history is committed here.",
	"The cyclic-rank example is scalar; independent RGB gives three copies.",
	],
	}


	def partitions(n: int):
	"""Enumerate canonical set partitions as restricted-growth label tuples."""
	if n == 0:
	yield ()
	return
	def visit(labels, largest):
	if len(labels) == n:
	yield tuple(labels)
	return
	for color in range(largest + 2):
	labels.append(color)
	yield from visit(labels, max(largest, color))
	labels.pop()
	yield from visit([0], 0)


	def conflict_graph(cache, image, transition_record, eps):
	"""Record equality includes successor, outcome, status and side effects.

	A hashable integer may be a shorthand label for a full exact transition
	record. Equal successor classes alone do not imply equal records.
	"""
	n, queries = len(cache), len(image[0])
	adjacency = [set() for _ in range(n)]
	for i in range(n):
	for j in range(i + 1, n):
	if cache[i] != cache[j]:
	continue
	conflict = any(
	transition_record[i][q] != transition_record[j][q]
	or any(abs(a-b) > 2*eps for a, b in zip(image[i][q], image[j][q]))
	for q in range(queries)
	)
	if conflict:
	adjacency[i].add(j)
	adjacency[j].add(i)
	return adjacency


	def direct_decoder(labels, cache, image, transition_record, eps):
	"""Test image boxes and full exact-record agreement without a graph."""
	result = {}
	for known_cache, label in set(zip(cache, labels)):
	members = [i for i in range(len(cache)) if (cache[i], labels[i]) == (known_cache, label)]
	for q in range(len(image[0])):
	targets = {transition_record[i][q] for i in members}
	if len(targets) != 1:
	return None
	decoded = []
	for coordinate in range(len(image[0][q])):
	values = [image[i][q][coordinate] for i in members]
	# Independently intersect every admissible interval.
	lower = max(v - eps for v in values)
	upper = min(v + eps for v in values)
	if lower > upper:
	return None
	decoded.append((min(values) + max(values)) / 2)
	result[(known_cache, label, q)] = (decoded, next(iter(targets)))
	return result


	def chromatic_number(adjacency):
	n = len(adjacency)
	order = sorted(range(n), key=lambda v: (-len(adjacency[v]), v))
	for limit in range(1, n + 1):
	colors = [-1] * n
	def solve(at):
	if at == n:
	return True
	vertex = order[at]
	forbidden = {colors[w] for w in adjacency[vertex] if colors[w] >= 0}
	for color in range(limit):
	if color not in forbidden:
	colors[vertex] = color
	if solve(at + 1):
	return True
	colors[vertex] = -1
	return False
	if solve(0):
	return limit, tuple(colors)
	raise AssertionError("Finite graph must admit singleton coloring")


	def check_code_instance(name, cache, image, transition_record, eps):
	adjacency = conflict_graph(cache, image, transition_record, eps)
	graph_min, graph_labels = chromatic_number(adjacency)
	direct_min, tested, valid = len(cache) + 1, 0, 0
	for labels in partitions(len(cache)):
	tested += 1
	decoder = direct_decoder(labels, cache, image, transition_record, eps)
	proper = all(labels[i] != labels[j] for i in range(len(cache)) for j in adjacency[i])
	assert proper == (decoder is not None)
	if decoder is not None:
	valid += 1
	direct_min = min(direct_min, max(labels) + 1)
	assert direct_min == graph_min
	decoder = direct_decoder(graph_labels, cache, image, transition_record, eps)
	assert decoder is not None
	for i in range(len(cache)):
	for q in range(len(image[0])):
	output, exact_record = decoder[(cache[i], graph_labels[i], q)]
	assert exact_record == transition_record[i][q]
	assert all(abs(a-b) <= eps for a, b in zip(output, image[i][q]))
	return {
	"name": name, "states": len(cache), "queries": len(image[0]),
	"image_coordinates": len(image[0][0]), "epsilon": str(eps),
	"partitions_examined": tested, "valid_partitions": valid,
	"minimum_code_alphabet": direct_min, "chromatic_number": graph_min,
	"edges": [[i,j] for i in range(len(cache)) for j in sorted(adjacency[i]) if i < j],
	}


	def check_mixed_coding(rng: random.Random) -> dict:
	cases = []
	for n in range(1, 7):
	for trial in range(5):
	query_count = 1 + trial % 3
	cache = [rng.randrange(1 + trial % 2) for _ in range(n)]
	eps = (F(0), F(1, 4), F(1, 2), F(1), F(3, 2))[trial]
	image = [[[F(rng.randint(-3, 3), 2) for _ in range(2)]
	for _ in range(query_count)] for _ in range(n)]
	# Integers abbreviate full exact transition records. Include
	# both identical records and per-query differing records.
	transition_record = [[0 if trial % 2 == 0 else rng.randrange(2)
	for _ in range(query_count)] for _ in range(n)]
	cases.append(check_code_instance(f"random_n{n}_case{trial}", cache, image, transition_record, eps))

	# An induced C5: each edge gets a coordinate with its endpoints +1 and -1.
	# Adjacent vertices differ by 2, nonadjacent ones by at most 1.
	edges = [(i, (i + 1) % 5) for i in range(5)]
	image = [[[F(1) if i == a else F(-1) if i == b else F(0)
	for a, b in edges]] for i in range(5)]
	c5 = check_code_instance("five_cycle", [0]5, image, [[0]]5, F(1,2))
	assert c5["chromatic_number"] == 3
	cases.append(c5)

	exact_continuation = check_code_instance(
	"same_image_distinct_successor", [0,0], [[[F(0)]],[[F(0)]]], [[0],[1]], F(1))
	assert exact_continuation["chromatic_number"] == 2
	cases.append(exact_continuation)

	# A shared future state does not erase required immediate nonvisual output.
	# Tuple order: (successor class, authoritative outcome, status, effects).
	record0 = (0, "pulse_committed", "accepted", ("emit_pulse",))
	record1 = (0, "pulse_blocked", "rejected", ())
	assert record0[0] == record1[0] and record0 != record1
	distinct_effects = check_code_instance(
	"same_image_same_successor_distinct_immediate_effect_status", [0,0],
	[[[F(0)]],[[F(0)]]], [[record0],[record1]], F(1))
	assert distinct_effects["chromatic_number"] == 2
	distinct_effects["same_successor_class"] = True
	distinct_effects["exact_transition_records"] = [record0, record1]
	cases.append(distinct_effects)

	cache_reuse = check_code_instance(
	"reuse_labels_across_cache_fibers", [0,0,1,1],
	[[[F(0)]],[[F(3)]],[[F(0)]],[[F(3)]]], [[0],[0],[0],[0]], F(1))
	assert cache_reuse["chromatic_number"] == 2
	cases.append(cache_reuse)

	# Pairwise distance <= 2 epsilon is not enough for arbitrary norms.
	# In L2, unit balls at (0,0),(2,0) meet only at (1,0), which is 3/2
	# from the third center (1,3/2). Every pair of centers is <= 2 apart.
	triangle = [(F(0),F(0)),(F(2),F(0)),(F(1),F(3,2))]
	distances_squared = [sum((x-y)**2 for x,y in zip(a,b))
	for a,b in itertools.combinations(triangle,2)]
	assert max(distances_squared) <= 4
	assert (F(3,2))**2 > 1
	return {
	"status": "passed", "instances": cases,
	"total_instances": len(cases),
	"total_partitions_examined": sum(c["partitions_examined"] for c in cases),
	"l2_counterexample_squared_pair_distances": [str(x) for x in distances_squared],
	"assumptions": [
	"Finite state/cache/query sets with every listed query admissible for every listed state.",
	"The encoder sees state; decoder sees exact cache value, query and code label.",
	"Approximate image error uses an L-infinity ball, coordinate by coordinate.",
	"Exact transition records include successor continuation class, immediate authoritative outcomes, accept/reject status and required side-effect records.",
	"Generic integer labels abbreviate full exact transition records; successor classes require independent semantic certification and realization.",
	"Tests check the specified one-step decoder; they do not prove online approximate stability.",
	"Palette reuse across different cache fibers is permitted because the cache is decoder side information.",
	"No practical encoding or decoding time bound follows from chromatic minimality.",
	],
	}


	def main() -> None:
	parser = argparse.ArgumentParser()
	parser.add_argument("--output", type=Path, default=Path(__file__).with_name("verification_math_results.json"))
	args = parser.parse_args()
	rng = random.Random(SEED)
	result = {
	"project": "Lyotherin_Tia", "seed": SEED,
	"arithmetic": "Python standard-library fractions.Fraction",
	"evidence_type": "finite exact algebra and combinatorial sanity checks; not hardware measurement or formal proof assistant output",
	"clamped_affine": check_clamps(rng),
	"transported_translation": check_transport(rng),
	"history_compaction": check_history(rng),
	"mixed_image_exact_continuation": check_mixed_coding(rng),
	"overall_status": "passed",
	}
	args.output.write_text(json.dumps(result, indent=2) + "\n", encoding="utf-8")
	print(json.dumps({"overall_status": result["overall_status"], "output": str(args.output),
	"clamp_pairs": result["clamped_affine"]["composition_pairs"],
	"interleaved_words": result["transported_translation"]["interleaved_words"],
	"rgb_configurations": result["history_compaction"]["rgb_configurations"],
	"finite_coding_instances": result["mixed_image_exact_continuation"]["total_instances"],
	"finite_partitions": result["mixed_image_exact_continuation"]["total_partitions_examined"]}))


	if __name__ == "__main__":
	main()