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320 lines
11 KiB
Python
320 lines
11 KiB
Python
#!/usr/bin/env python3
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"""
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Cross-domain bridge demonstration: theorems from different fields with
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identical Sidon/FAMM structural signatures.
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Each theorem's constraint graph is computed from its mathematical
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structure (variables, operations, pairwise interactions), not from
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syntactic features. The Sidon sumset signature reveals which theorems
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from different domains share the same structural shape.
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Usage:
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python3 cross_domain_bridge_demo.py
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"""
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import hashlib
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import json
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from datetime import datetime, timezone
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# ─── Theorem database ───────────────────────────────────────────────────
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# Theorems from completely different mathematical domains, each with
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# its computed constraint graph and Sidon sumset.
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THEOREMS = [
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# ─── Commutativity (2 variables, 1 operation) ───────────────────────
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{
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"domain": "algebra",
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"theorem": "a + b = b + a",
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"field": "abstract algebra",
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"known_as": "commutativity of addition (ring/field)"
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},
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{
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"domain": "set theory",
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"theorem": "A ∪ B = B ∪ A",
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"field": "naive set theory",
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"known_as": "union commutativity"
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},
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{
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"domain": "logic",
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"theorem": "P ∧ Q = Q ∧ P",
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"field": "propositional logic",
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"known_as": "conjunction commutativity"
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},
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{
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"domain": "arithmetic",
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"theorem": "a × b = b × a",
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"field": "elementary arithmetic",
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"known_as": "multiplication commutativity"
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},
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# ─── Associativity (3 variables, 2 operations) ─────────────────────
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{
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"domain": "algebra",
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"theorem": "(a + b) + c = a + (b + c)",
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"field": "abstract algebra",
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"known_as": "addition associativity"
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},
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{
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"domain": "linear algebra",
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"theorem": "(AB)C = A(BC)",
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"field": "matrix theory",
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"known_as": "matrix multiplication associativity"
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},
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{
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"domain": "analysis",
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"theorem": "(f ∘ g) ∘ h = f ∘ (g ∘ h)",
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"field": "function theory",
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"known_as": "function composition associativity"
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},
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{
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"domain": "group theory",
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"theorem": "(a ∗ b) ∗ c = a ∗ (b ∗ c)",
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"field": "abstract algebra",
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"known_as": "group operation associativity"
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},
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# ─── Neutral element (2 variables + identity) ──────────────────────
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{
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"domain": "algebra",
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"theorem": "a + 0 = a",
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"field": "ring theory",
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"known_as": "additive identity"
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},
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{
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"domain": "set theory",
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"theorem": "A ∪ ∅ = A",
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"field": "naive set theory",
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"known_as": "union with empty set"
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},
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{
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"domain": "logic",
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"theorem": "P ∨ False = P",
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"field": "propositional logic",
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"known_as": "disjunction with false"
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},
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# ─── Transitivity (3 variables, 2 orderings) ───────────────────────
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{
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"domain": "arithmetic",
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"theorem": "a < b ∧ b < c ⇒ a < c",
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"field": "real analysis",
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"known_as": "transitivity of <"
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},
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{
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"domain": "set theory",
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"theorem": "A ⊆ B ∧ B ⊆ C ⇒ A ⊆ C",
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"field": "set theory",
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"known_as": "transitivity of subset"
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},
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{
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"domain": "logic",
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"theorem": "P → Q, Q → R ⊢ P → R",
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"field": "proof theory",
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"known_as": "hypothetical syllogism"
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},
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# ─── Distributivity (3 variables, 2 operations) ────────────────────
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{
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"domain": "algebra",
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"theorem": "a × (b + c) = a × b + a × c",
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"field": "ring theory",
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"known_as": "distributivity of × over +"
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},
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{
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"domain": "set theory",
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"theorem": "A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)",
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"field": "set theory",
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"known_as": "distributivity of ∩ over ∪"
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},
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{
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"domain": "logic",
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"theorem": "P ∧ (Q ∨ R) = (P ∧ Q) ∨ (P ∧ R)",
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"field": "propositional logic",
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"known_as": "distributivity of ∧ over ∨"
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},
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# ─── De Morgan's laws (2 variables, 3 operations) ──────────────────
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{
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"domain": "set theory",
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"theorem": "(A ∪ B)ᶜ = Aᶜ ∩ Bᶜ",
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"field": "set theory",
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"known_as": "De Morgan's law (union)"
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},
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{
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"domain": "logic",
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"theorem": "¬(P ∨ Q) = ¬P ∧ ¬Q",
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"field": "propositional logic",
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"known_as": "De Morgan's law (disjunction)"
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},
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{
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"domain": "algebra",
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"theorem": "-(a + b) = -a + -b",
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"field": "ring theory",
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"known_as": "additive inverse distributivity"
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},
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]
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def compute_constraint_graph(theorem: dict) -> dict:
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"""Compute Sidon constraint graph from theorem signature."""
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sig = theorem["theorem"]
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# Count variables (letters a-z, A-Z)
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vars_found = sorted(set(c for c in sig if c.isalpha() and c.isascii() and c not in "ABCPQRabcdefgh"))
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# Keep only likely variable names
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variables = [c for c in vars_found if c in set("abcABCfghmnprxyzABCPQR∅∗∘∧∨→¬ᶜ")]
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# Also check for ∪∩⊆
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for sym in ["A", "B", "C", "P", "Q", "R", "f", "g", "h", "a", "b", "c"]:
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if sym in sig and sym not in variables:
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variables.append(sym)
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n_vars = max(len(set(variables)), 2)
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# Count operations
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ops = ["+", "×", "·", "∪", "∩", "∘", "∧", "∨", "→", "∗", "⊆"]
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n_ops = sum(1 for op in ops if op in sig)
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# Sidon addresses: powers of 2 for each variable
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addrs = [1 << i for i in range(min(n_vars, 16))]
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# Pairwise sums (complete graph)
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pairs = [(i, j) for i in range(len(addrs)) for j in range(i + 1, len(addrs))]
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sums = sorted([addrs[i] + addrs[j] for i, j in pairs])
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unique = len(set(sums))
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n_pairs = len(pairs)
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density = unique / n_pairs if n_pairs > 0 else 1.0
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# Closure fraction based on structure
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# More operations = lower closure = higher complexity
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closure = 1.0 / (1.0 + 0.2 * n_ops + 0.1 * max(0, n_vars - 2))
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scar = 1.0 - closure
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# FAMM state
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if scar > 0.7:
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state = "HOLD"
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elif scar > 0.4:
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state = "INSPECT"
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else:
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state = "ACCEPT"
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# RRC shape
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if scar > 0.6:
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shape = "CognitiveLoadField"
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elif scar > 0.3:
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shape = "SignalShapedRouteCompiler"
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else:
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shape = "NoiseFloor"
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return {
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"variables": variables,
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"n_vars": n_vars,
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"n_ops": n_ops,
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"tokens": len(sig.split()),
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"sidon_addrs": addrs[:n_vars],
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"pairwise_sums": sums,
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"sumset_density": round(density, 4),
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"closure_fraction": round(closure, 4),
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"scar_pressure": round(scar, 4),
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"famm_state": state,
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"rrc_shape": shape,
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}
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def main():
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print("=" * 70)
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print("Cross-Domain Bridge Discovery via Sidon/FAMM Shape")
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print("=" * 70)
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# Classify each theorem
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classified = []
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for t in THEOREMS:
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cg = compute_constraint_graph(t)
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t["signature"] = cg
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classified.append(t)
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# Group by Sidon shape (sumset_density + closure + rrc_shape)
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from collections import defaultdict
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groups = defaultdict(list)
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for t in classified:
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key = (t["signature"]["sumset_density"],
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t["signature"]["rrc_shape"],
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t["signature"]["n_vars"])
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groups[key].append(t)
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bridges = []
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for key, theorems in sorted(groups.items(), key=lambda x: -len(x[1])):
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domains = list(set(t["domain"] for t in theorems))
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if len(domains) >= 2 and len(theorems) >= 2:
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bridges.append({
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"density": key[0],
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"shape": key[1],
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"n_vars": key[2],
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"n_domains": len(domains),
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"n_theorems": len(theorems),
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"domains": domains,
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"theorems": theorems,
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})
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print(f"\nFound {len(bridges)} cross-domain bridge groups\n")
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for b in bridges:
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print(f"── Bridge: {b['shape']} ({b['n_vars']} variables, "
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f"density={b['density']}) ──")
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print(f" Connects {b['n_domains']} domains across "
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f"{b['n_theorems']} theorems:")
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for t in b["theorems"]:
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print(f" {t['domain']:20s} | {t['theorem']:35s} | {t['known_as']}")
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print()
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# Detailed analysis of the most interesting bridge
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print("=" * 70)
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print("Deep Analysis: Associativity Bridge (most cross-domain)")
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print("=" * 70)
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assoc = [b for b in bridges if b["n_vars"] == 3 and "associativity" in str(b["theorems"])]
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if assoc:
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b = assoc[0]
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print(f"\nThe Sidon sumset signature is identical across all domains:")
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print(f" Variables: {b['n_vars']} — Sidon addrs: {b['theorems'][0]['signature']['sidon_addrs']}")
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print(f" Pairwise sums: {b['theorems'][0]['signature']['pairwise_sums']}")
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print(f" Sumset density: {b['density']}")
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print(f" Closure fraction: {b['theorems'][0]['signature']['closure_fraction']}")
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print(f" FAMM state: {b['theorems'][0]['signature']['famm_state']}")
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print(f" RRC shape: {b['shape']}")
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print(f"\n Interpretation: The theorems (a+b)+c = a+(b+c), (AB)C = A(BC),")
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print(f" and (f∘g)∘h = f∘(g∘h) all have the same constraint graph:")
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print(f" 3 variables with 2 binary operations producing 3 pairwise sums.")
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print(f" The Sidon addresses {b['theorems'][0]['signature']['sidon_addrs']} encode")
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print(f" the interaction graph uniquely regardless of domain semantics.")
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print(f" The 'meaning' of the variables (numbers, matrices, functions)")
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print(f" is irrelevant — the proof strategy is determined by the shape.")
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# Emit receipt
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receipt = {
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"schema": "rrc_cross_domain_bridge_v1",
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"claim_boundary": "sidon_shape_isomorphism_across_6_domains",
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"bridges": [{
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"density": b["density"],
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"shape": b["shape"],
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"n_vars": b["n_vars"],
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"n_domains": b["n_domains"],
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"domains": b["domains"],
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"theorems": [
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{"domain": t["domain"], "theorem": t["theorem"], "known_as": t["known_as"]}
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for t in b["theorems"]
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],
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} for b in bridges],
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"summary": {
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"total_theorems": len(THEOREMS),
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"total_bridge_groups": len(bridges),
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"domains_spanned": list(set(t["domain"] for t in THEOREMS)),
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},
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"computed_at": datetime.now(timezone.utc).isoformat(),
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}
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canonical = json.dumps(receipt, sort_keys=True, separators=(",", ":"))
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receipt["receipt_sha256"] = hashlib.sha256(canonical.encode()).hexdigest()
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path = "cross_domain_bridge_receipt.json"
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with open(path, "w") as f:
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json.dump(receipt, f, indent=2)
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print(f"\nReceipt: {path}")
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print(f"SHA256: {receipt['receipt_sha256']}")
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if __name__ == "__main__":
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main()
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