FodaOS Unified Field Theory Series IV: The Residueon Trichotomy and the Residue Field — Non-Differential Generation of Nested Fractals, Reciprocity Siting, and Group-Theoretic Unified Stratification

This paper is the fourth in the FodaOS unified field theory series. The first three installments established the physical unified field ([Paper1], the complex Ginzburg–Landau master-equation framework), the computational unified field ([Paper2/ARC], the five-operator arithmetic resonance computer), and the mathematical unified field ([Paper3], algebraic quantum mechanics over GF(pp)). Taking the liusu (residueon) — the self-similar unit in which each pole possesses its own internal residue field — as the unifying primitive, this paper completes a threefold construction. First strand · structure. We prove that nested fractals do not depend on differentiation: three independent generating mechanisms are distinguished (M1 self-similar iteration / M2 SRRF self-referential fixed point / M3 collision emergence), with differentiation merely an optional unfolding operator for (M1). We give the Interpolation Theorem (analytically exact, zero parameters, zero truncation), the Extrapolation Theorem (uniqueness of analytic continuation within the strict isomorphism region guarantees deterministic convergence), and the Energy-Scale Bootstrap Theorem (physical scales are derived internally from the algebraic invariants of the SRRF fixed point; external experimental constants are demoted from generative inputs to verification benchmarks). Second strand · completion. The CRT unbounded-precision theorem (the precision wall is demoted to an engineering parameter counting primes); the residue path-integral formulation (Z(A→B;T)=(UT)A→BZ(A→B;T)=(UT)A→B, structurally isomorphic to the Cauchy contour theorem's "only poles enter the result", with zero measure-theoretic ambiguity); the exact conservation proposition (linear and quadratic invariants agree bit for bit; drift collapses from O(Tε)O(Tε) to zero); and the p-adic/adele correspondence table (the splitting criteria match p-adic mathematical physics verbatim; CRT = a finite adele). Third strand · semantics and adjudication. The three gaps left by [Paper3] are resolved: the zero-norm phenomenon is elevated to a dilemma theorem (NTT optimality p≡1 2kp≡1mod2k and probabilistic unitarity p≡7 8p≡7mod8 are mutually exclusive; the zero norm at 65537 is a necessity of a split algebra, not an anomaly); the Mersenne partner field GF(1310712)GF(1310712) is constructed (norm-one torus of order 217217; phase unitarity, non-degenerate norm, and torus NTT in one field); the Born rational-reconstruction theorem (ideal-circuit probabilities are dyadic rationals; modular values are uniquely recovered by rational reconstruction, extending the CRT bound to 2Θ(k)2Θ(k)); the CHSH integer adjudication (a purely integer proof of the local bound ∣S∣≤2∣S∣≤2 plus an exact GF(65537) simulation S≡8160S≡8160, S2≡8S2≡8 — an exact hit on the Tsirelson bound); the GHZ–Mermin all-algebraic adjudication (a three-body contradiction with zero statistics and zero probability: the product of assigned eigenvalues satisfies +1≡−1+1≡−1); and the classification theorem (any deterministic exact substrate must be nonlocal, contextual, or superdeterministic; this framework is a global-state contextual model — Bell's theorem is classified, not violated). Using the history of quantum-advantage disputes as reference (Sycamore's 2019 claim corrected by classical tensor-network contraction; D-Wave's 2025 claim corrected by a 3D tensor network), this framework's position is exact algebraic contraction of residue fields: in the "structured simulation" arena where classical methods have repeatedly won, characterized by zero floating point, zero fidelity ambiguity, and bit-for-bit reproducibility. Fixing and tower structure. The whole system is fixed as a stratified structure of arithmetic dynamics: the Legendre symbol of the discriminant together with the sign of the norm act as the master switches of classification; field levels correspond to Galois orbits; periods are given by the least common multiple of the norm-torus tower; only prime orders open new shells; and Artin reciprocity is the capstone of the tower. On the group-theoretic and number-theoretic side of the tower, this paper further provides: the characteristic-pp mirror of Kronecker–Weber (Corollary 19.1 — the cyclotomic nature of the tower and the source of its verifiability); the Ramanujan–Nagell identity of the twin integer 32768 (Corollary 20.1 — the RN representability of Fermat primes and their primality terminate together at F4F4, connecting via the discriminant −7−7 to the Heegner/class-field-theory prototype); the Coxeter identity of the tower's shell groups (Proposition 12.5 — the Dk=I2(k)Dk=I2(k) series; the non-crystallographic member H2H2 and the fifth shell k=5k=5 share the Z[ζ5]Z[ζ5] foundation; the emergence of φφ in the A5A5 character table); and a machine-verified evidence layer of Weyl-order/factorial unifications, the 6h+16h+1 Coxeter prime family, and the mod-43 primitive-root Galois equivalence (§12.14, declared under a three-tier confidence grading). The paper further upgrades the 'automatic gearbox' of field extension (Theorem 23) into gear-free tower evaluation (Theorem 23′): the uniqueness of subfields of finite fields makes the field order an intrinsic property of values rather than a machine state — 'shifting', as a representation choice, does not exist in characteristic pp, and the gear table degenerates into the divisibility lattice. The prime set is thereby upgraded from an axiomatically given binary choice into an open resource pool with an admission discipline (Theorem 30). Seven falsifiable predictions (P1–P7) are stated. The trichotomy system: the reciprocity siting law (Theorem 15) — quadratic reciprocity decides whether each pole lands in the base field or an extension; the splitting trichotomy and the three-fold classification of iteration periods (Theorems 16, 16′) — prime-ideal splitting, field level, Fatou–Julia dynamical type, and iteration period are aligned four ways, with the norm-refinement law locating the factor 2 exactly; the norm charge — an intrinsic multiplicative-conserved quantity serving as a Z2Z2 order parameter for time-reversal symmetry; the division of labor of the reciprocity tower (Theorem 17) — the Fermat field houses the complete 2-power reciprocity tower, the Mersenne field is the home of cubic reciprocity; the cubic tetrad law (Theorem 18) — the trivalent character χ3∈{1,ζ3,ζ32}χ3∈{1,ζ3,ζ32} upgrades the classification from Z2Z2 to Z3Z3; the capstone theorem (Theorem 19) — the general stratification theorem: only prime orders open new shells, and Artin reciprocity is the top of the tower. The gear-free tower evaluation (Theorem 23′) makes the field order an intrinsic property of values — 'shifting', as a representation choice, does not exist in characteristic pp, upgrading Theorem 23's 'automatic gearbox' into gear-free adaptive field order.