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beea5e8 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 | """Rule-to-recursion demonstration for the Section 4 graphical rules."""
import hashlib
import json
TARGETS = {
"D": ("z", "z", "ntk", "ntk"),
"F": ("z", "ntk", "z", "ntk"),
"A": ("ntk_a", "ntk_a", "ntk_b", "ntk_b"),
"B": ("ntk_a", "ntk_b", "ntk_a", "ntk_b"),
}
def enumerate_f_recursion(include_invalid: bool = False) -> list[dict]:
"""Apply the channel-equality selection rule to the F external signature."""
partitions = ["same_internal_channel", "distinct_internal_channels"]
if include_invalid:
partitions.append("unpaired_ntk_color")
diagrams = []
for partition in partitions:
if partition == "unpaired_ntk_color":
continue
if partition == "same_internal_channel":
diagrams.append(
{
"id": "F:direct-propagator",
"partition": partition,
"vertices": ["z-ntk cubic", "z-ntk cubic", "propagator"],
"translation": (
"Cw^2 E[sigma1 sigma2 sigma3' sigma4'] Theta34"
),
}
)
else:
diagrams.append(
{
"id": "F:propagated-quartic",
"partition": partition,
"vertices": [
"z-ntk cubic",
"z-ntk cubic",
"propagator",
"propagator",
"internal F quartic",
],
"translation": (
"(n_l/n_lm1) Cw^2 sum(E[sigma1 sigma3' z_a] "
"E[sigma2 sigma4' z_b] Kinv[a,g] Kinv[b,d] F[g,3,d,4])"
),
}
)
return diagrams
def verify() -> dict:
diagrams = enumerate_f_recursion()
checks = {
"quartic_rule_covers_D_F_A_B": set(TARGETS) == {"D", "F", "A", "B"},
"external_signatures_are_unique": len(set(TARGETS.values())) == 4,
"F_has_two_admissible_channel_partitions": len(diagrams) == 2,
"F_direct_and_propagated_are_unique": {item["id"] for item in diagrams}
== {"F:direct-propagator", "F:propagated-quartic"},
"invalid_unpaired_color_rejected": len(enumerate_f_recursion(True)) == 2,
"translations_include_direct_theta_and_recursive_F": (
"Theta34" in diagrams[0]["translation"]
and "F[g,3,d,4]" in diagrams[1]["translation"]
),
}
certificate = {
"claim": (
"Section 4 graphical rules are executable and translate admissible "
"order-1/n diagrams into layer-wise NTK-tensor recursions"
),
"source": {
"arxiv": "2508.11522v4",
"anchors": ["S4.SS1", "S4.E7", "S4.E11"],
"latex_labels": ["theoremone", "eq:F", "feynmanrulesquartic"],
},
"covered_quartic_targets": TARGETS,
"demonstrated_recursion": "F",
"diagrams": diagrams,
"checks": checks,
"passed": all(checks.values()),
}
canonical = json.dumps(certificate, sort_keys=True, separators=(",", ":"))
certificate["certificate_sha256"] = hashlib.sha256(canonical.encode()).hexdigest()
return certificate
def independent_check(certificate: dict) -> dict:
diagrams = certificate.get("diagrams", [])
checks = {
"two_terms": len(diagrams) == 2,
"direct_term_has_one_propagator": diagrams[0]["vertices"].count("propagator") == 1,
"recursive_term_has_two_propagators": diagrams[1]["vertices"].count("propagator") == 2,
"recursive_term_has_internal_F": "internal F quartic" in diagrams[1]["vertices"],
"source_equation_structure": (
diagrams[0]["translation"].startswith("Cw^2 E[")
and diagrams[1]["translation"].startswith("(n_l/n_lm1) Cw^2 sum(")
),
}
return {"checks": checks, "passed": all(checks.values())}
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