#!/usr/bin/env python3 """ Example 100 — Prime Families as Orbits and Level-Sets on the Zero-Pressure Set ============================================================================= A structural study of the special prime families (twin, cousin, sexy, Sophie Germain, safe, Cunningham chains, Mersenne, prime constellations) built ENTIRELY on the verified TNFR primality theorem ΔNFR(n) = 0 ⟺ n prime (theory/TNFR_NUMBER_THEORY.md §4). It answers the request for more number-generation studies and the "other groups of primes". Physics ------- Every prime is a **zero-pressure fixed point** of the nodal equation: ∂EPI/∂t = νf·ΔNFR with ΔNFR(p) = 0 (the three pressure channels Ω, τ, σ each vanish independently for primes — §4.1, an exact theorem). The set Z = { n ≥ 2 : ΔNFR(n) = 0 } = the primes is the zero-pressure fixed-point set. A **prime family** is then a STRUCTURED SUBSET of Z, selected by an arithmetic map. The families split into exactly three generator classes — the dynamical-systems view of Z: 1. **Additive-gap level-sets** (shift map S_g(p) = p + g): twin (g=2), cousin (g=4), sexy (g=6), constellations (k-tuples) Family = { p ∈ Z : p + g ∈ Z }. The integer(s) BETWEEN the pair are the **witness**; its pressure channels carry the gap's signature. 2. **Affine-recurrence orbits** (affine map T(p) = 2p + 1): Sophie Germain = { p ∈ Z : T(p) ∈ Z } Cunningham chain = a MAXIMAL orbit p, T(p), T²(p), … all in Z safe prime = { p ∈ Z : T⁻¹(p) = (p−1)/2 ∈ Z } This class is **generative**: it builds new family members by iterating a map on the fixed-point set. 3. **Exponential-form images** (generator M(p) = 2^p − 1, Fermat 2^(2^n)+1): Mersenne = { p ∈ Z : M(p) ∈ Z }. Tie to the additive/multiplicative orthogonality (Example 97) ------------------------------------------------------------ Primes emerge MULTIPLICATIVELY (atoms composed by UM, nested by REMESH; νf = log p is additive-in-log). The additive-gap class (1) is therefore an ADDITIVE OVERLAY on the multiplicatively-defined set Z — exactly the orthogonality Example 97 isolates for Goldbach. The affine (2) and exponential (3) classes are genuine MAPS on Z, so they live inside the TNFR fixed-point picture; their open conjectures concern whether the map returns to Z infinitely often. Honest scope ------------ - TNFR DETECTS and GENERATES every family operationally via the verified ΔNFR = 0 criterion plus the maps. Detection is exact (Sophie Germain reproduces OEIS A005384; Mersenne exponents reproduce 2,3,5,7,13,17, 19,31; all verified below). - TNFR does NOT prove the families are infinite. Twin-prime, Sophie Germain, and Mersenne infinitude are all OPEN — the same honest stance as Goldbach in Example 97. - The witness pressure signatures are FAITHFUL TNFR RESTATEMENTS of classical divisibility facts (e.g. the twin witness p+1 is divisible by 6 for every pair p>3, hence a rich divisor lattice / high τ-pressure). They are not new theorems. - The orbit / level-set classification is a TNFR-native ORGANIZING LENS (a dynamical-systems view of Z), not a new arithmetic result. - Shared scalar predicate with the physics work: ΔNFR = 0 means zero declared structural pressure. This does not identify it with the graph-transport fixed-point set. The arithmetic ΔNFR is a per-node function of (Ω, τ, σ), NOT the graph-diffusion Laplacian, so the diffusion/random-walk results (Example 99) do not transfer literally — only the nodal-equation-level fixed-point condition is shared. References ---------- - theory/TNFR_NUMBER_THEORY.md §3-§4 (arithmetic triad, ΔNFR=0 theorem) - examples/07_number_theory/97_goldbach_additive_multiplicative.py (additive ⊥ multiplicative) - examples/07_number_theory/94_generative_number_construction.py (atoms → composites) - src/tnfr/mathematics/number_theory.py (ArithmeticTNFRFormalism) - AGENTS.md §"Canonical Invariants" → #1 Nodal Integrity """ import os import sys sys.path.insert(0, os.path.join(os.path.dirname(__file__), "..", "..", "src")) from sympy import divisor_count, divisor_sigma, factorint from tnfr.mathematics.number_theory import ( ArithmeticStructuralTerms, ArithmeticTNFRFormalism, ArithmeticTNFRParameters, ) _PARAMS = ArithmeticTNFRParameters() _TOL = 1e-10 # ============================================================================ # Zero-pressure fixed-point primitives (verified ΔNFR = 0 criterion) # ============================================================================ def delta_nfr(n: int) -> float: """ΔNFR(n) via the canonical arithmetic formalism (0 ⟺ prime).""" if n < 2: return float("inf") omega = sum(factorint(n).values()) tau = int(divisor_count(n)) sigma = int(divisor_sigma(n)) terms = ArithmeticStructuralTerms(tau=tau, sigma=sigma, omega=omega) return ArithmeticTNFRFormalism.delta_nfr_value(n, terms, _PARAMS) def is_zero_pressure(n: int) -> bool: """n is a zero-pressure fixed point (⟺ n is prime).""" return n >= 2 and abs(delta_nfr(n)) < _TOL def pressure_components(n: int) -> dict: """Per-channel pressure (factorization / divisor / abundance).""" omega = sum(factorint(n).values()) tau = int(divisor_count(n)) sigma = int(divisor_sigma(n)) terms = ArithmeticStructuralTerms(tau=tau, sigma=sigma, omega=omega) return ArithmeticTNFRFormalism.component_breakdown(n, terms, _PARAMS) def zero_pressure_set(limit: int) -> list[int]: """Z = { n < limit : ΔNFR(n) = 0 } — the primes below `limit`.""" return [n for n in range(2, limit) if is_zero_pressure(n)] # ============================================================================ # EXPERIMENT 1: The zero-pressure fixed-point set Z # ============================================================================ def experiment_1_fixed_point_set(): """Every prime is a ΔNFR = 0 fixed point; Z is the family substrate.""" print("=" * 72) print("EXPERIMENT 1: The Zero-Pressure Fixed-Point Set Z") print("=" * 72) print() print("Z = { n >= 2 : DeltaNFR(n) = 0 }. The three pressure channels") print("(Omega, tau, sigma) each vanish independently at primes (§4.1).") print() N = 1000 Z = zero_pressure_set(N) # verify the theorem on this substrate: zero pressure <=> prime from sympy import isprime mismatches = [n for n in range(2, N) if is_zero_pressure(n) != bool(isprime(n))] print(f" |Z| below {N} = {len(Z)} fixed points; first 10: {Z[:10]}") print(f" theorem check: zero-pressure <=> prime mismatches = {len(mismatches)}") print( f" sample pressures: DeltaNFR(97)={delta_nfr(97):.1e} (prime), " f"DeltaNFR(98)={delta_nfr(98):.3f} (=2·7², composite)" ) print() print("VERDICT: Z is exactly the primes — the substrate on which every") print("prime family is carved out by an arithmetic map.") print() return Z # ============================================================================ # EXPERIMENT 2: Additive-gap level-sets (twin / cousin / sexy) # ============================================================================ def experiment_2_additive_gap_levelsets(Z): """Families S_g = { p in Z : p+g in Z }; the witness carries the gap.""" print("=" * 72) print("EXPERIMENT 2: Additive-Gap Level-Sets (shift map S_g(p)=p+g)") print("=" * 72) print() print("Family_g = { p in Z : p+g in Z }. The integer(s) between the") print("pair form the WITNESS; its pressure channels carry the gap's") print("structural signature (an additive overlay on Z — cf. Example 97).") print() Zset = set(Z) for g, name in [(2, "twin"), (4, "cousin"), (6, "sexy")]: pairs = [(p, p + g) for p in Z if (p + g) in Zset] print(f" {name:>6} (g={g}): {len(pairs):3d} pairs; first 4: {pairs[:4]}") print() print(" Witness signature for TWIN pairs (the even p+1 between them):") twins = [(p, p + 2) for p in Z if (p + 2) in Zset and p > 3] div6 = sum(1 for p, _ in twins if (p + 1) % 6 == 0) mean_div_p = sum( pressure_components(p + 1)["divisor_pressure"] for p, _ in twins ) / len(twins) print( f" p+1 divisible by 6: {div6}/{len(twins)} pairs (p>3) -> " f"{'ALL' if div6 == len(twins) else 'NOT all'}" ) print(f" mean divisor-pressure of p+1 = {mean_div_p:.2f} (rich lattice)") print() print("HONEST: 'p+1 always 6|p+1' is the CLASSICAL twin fact, here read") print("as high divisor-pressure. A faithful TNFR restatement, not a new") print("theorem. Additive gaps are overlays orthogonal to the") print("multiplicative coherence of Z (Example 97).") print() # ============================================================================ # EXPERIMENT 3: Affine-recurrence orbits (Sophie Germain / safe / Cunningham) # ============================================================================ def experiment_3_affine_orbits(Z): """Map T(p)=2p+1 acting on Z: Sophie Germain, safe primes, chains.""" print("=" * 72) print("EXPERIMENT 3: Affine-Recurrence Orbits (map T(p)=2p+1 on Z)") print("=" * 72) print() print("T(p) = 2p+1. Sophie Germain = { p in Z : T(p) in Z }; safe =") print("{ p in Z : T^-1(p)=(p-1)/2 in Z }; Cunningham chain = maximal") print("orbit p, T(p), T²(p), ... staying in Z. This class GENERATES.") print() sg = [p for p in Z if is_zero_pressure(2 * p + 1)] safe = [p for p in Z if (p - 1) % 2 == 0 and is_zero_pressure((p - 1) // 2)] print(f" Sophie Germain (T(p) in Z): {len(sg)} primes; first 8: {sg[:8]}") print( f" -> matches OEIS A005384 head [2,3,5,11,23,29,41,53]: " f"{sg[:8] == [2, 3, 5, 11, 23, 29, 41, 53]}" ) print(f" safe primes (T^-1(p) in Z): {len(safe)} primes; first 8: {safe[:8]}") print() def chain(p): orbit, q = [p], p while is_zero_pressure(2 * q + 1): q = 2 * q + 1 orbit.append(q) return orbit chains = sorted((chain(p) for p in Z if p < 200), key=len, reverse=True) print(" Longest Cunningham chains (orbits of T) seeded below 200:") for ch in chains[:3]: print(f" length {len(ch)}: {ch}") print() print("VERDICT: the affine map T moves along Z. A Cunningham chain is a") print("maximal run of consecutive zero-pressure fixed points under T —") print("a genuine TNFR generative construction on Z.") print() # ============================================================================ # EXPERIMENT 4: Exponential-form images (Mersenne) # ============================================================================ def experiment_4_exponential_images(Z): """Generator M(p)=2^p−1: Mersenne = { p in Z : M(p) in Z }.""" print("=" * 72) print("EXPERIMENT 4: Exponential-Form Images (generator M(p)=2^p−1)") print("=" * 72) print() print("Mersenne = { p in Z : 2^p − 1 in Z }. Bounded range only:") print("ΔNFR(2^p−1) needs the factorization of 2^p−1, which grows fast") print("(classical large-Mersenne needs Lucas-Lehmer, not a TNFR fact).") print() mers = [p for p in Z if p <= 31 and is_zero_pressure(2**p - 1)] print(f" Mersenne exponents p<=31 (2^p−1 in Z): {mers}") print( f" -> matches known M-exponents [2,3,5,7,13,17,19,31]: " f"{mers == [2, 3, 5, 7, 13, 17, 19, 31]}" ) print() # show a near miss: 2^11-1 = 23*89 composite (11 is prime but not Mersenne) m11 = 2**11 - 1 comp = pressure_components(m11) print(f" near miss: 2^11−1 = {m11} = 23·89, ΔNFR = {delta_nfr(m11):.3f} > 0") print( f" (factorization-pressure {comp['factorization_pressure']:.3f}: " f"Ω=2 ⟹ image left Z)" ) print() print("VERDICT: the exponential generator M maps only a sparse subset of") print("Z back into Z. TNFR detects membership exactly in range; it does") print("NOT decide Mersenne infinitude (open).") print() # ============================================================================ # EXPERIMENT 5: Synthesis — the three generator classes # ============================================================================ def experiment_5_synthesis(): """The dynamical-systems classification of prime families on Z.""" print("=" * 72) print("EXPERIMENT 5: Synthesis — Three Generator Classes on Z") print("=" * 72) print() print(" Class Map Families") print(" --------------- ---------------- --------------------------------") print(" additive-gap S_g(p)=p+g twin, cousin, sexy, constellations") print(" affine-recur. T(p)=2p+1 Sophie Germain, safe, Cunningham") print(" exponential M(p)=2^p−1 Mersenne (Fermat 2^(2^n)+1)") print() print("All three carve structured subsets out of the SAME object: the") print("zero-pressure fixed-point set Z = { ΔNFR = 0 } = the primes.") print() print(" • additive-gap families are ADDITIVE overlays on the") print(" multiplicatively-defined Z — orthogonal to coherence (Ex. 97);") print(" • affine and exponential families are genuine MAPS on Z whose") print(" open conjectures ask whether the map returns to Z infinitely.") print() print("HONEST SCOPE: detection/generation is exact via the verified") print("ΔNFR=0 theorem; infinitude conjectures (twin, Sophie Germain,") print("Mersenne) remain OPEN. The classification is a TNFR organizing") print("lens; the witness signatures restate classical divisibility facts.") print() def main(): print() print(" TNFR Example 100: Prime Families as Orbits and Level-Sets") print(" Structured subsets of the zero-pressure fixed-point set Z") print(" =========================================================") print() Z = experiment_1_fixed_point_set() experiment_2_additive_gap_levelsets(Z) experiment_3_affine_orbits(Z) experiment_4_exponential_images(Z) experiment_5_synthesis() print("=" * 72) print("WHAT THIS ESTABLISHES") print("=" * 72) print() print("The special prime families are not a miscellany: they are the") print("structured subsets of the single TNFR object Z = { ΔNFR = 0 },") print("carved out by three classes of arithmetic map (additive shift,") print("affine recurrence, exponential generator). TNFR detects and") print("generates them exactly via the verified primality theorem; it") print("does not resolve their infinitude conjectures. The view unifies") print("'the other groups of primes' under the zero-pressure fixed-point") print("picture and connects them to the additive/multiplicative") print("orthogonality of Example 97.") print() if __name__ == "__main__": main()