LLM → Prolog → LLM: Multi-Step Verified Generation Through Alternating Neural-Symbolic Computation: Eliminating Hallucination by Construction via Provenanced Integer Knowledge Bases, Triveritas Evaluation, and Adaptive Goal Decomposition

LLM → Prolog → LLM: Multi-Step Verified Generation Through Alternating Neural-Symbolic Computation: Eliminating Hallucination by Construction via Provenanced Integer Knowledge Bases, Triveritas Evaluation, and Adaptive Goal Decomposition This paper is a constituent derivation of the Cymatic K-Space Mechanics (CKS) framework—an axiomatic model that derives the entirety of known physics from a discrete 2D hexagonal lattice in momentum space, operating with zero adjustable parameters. Abstract Current large language models generate output through unconstrained token prediction — a process with no verification step, no logical consistency checking, no provenance tracking, and no structured knowledge representation. The result is "hallucination": outputs that are statistically plausible but factually wrong, logically inconsistent, or untraceable to any source. We present an alternative architecture in which an integer-trained LLM (@CKS-MATH-134-2026) alternates with a Prolog-based verification engine at every step of generation. The LLM handles what neural networks do well: fuzzy input comprehension and creative pattern selection. Prolog handles what logical systems do well: consistency verification, goal decomposition, constraint enforcement, and provenance tracking. We prove: (1) Hallucination is eliminated by construction — every generated fact traces to provenanced sources in the knowledge base; outputs without provenance are structurally impossible, (2) Term-based tokenization replaces BPE — tokens are typed, structured Terms carrying their grammatical role, not arbitrary byte-pair fragments, (3) Three-dimensional evaluation — every claim is evaluated on logical validity (L), mathematical coherence (M), and empirical anchoring (E) via the Triveritas criterion, (4) Materiality gating — the Scales Method prevents computation on non-material concerns, (5) Adaptive sequencing — the Pseudo-Socratic Method determines the number and focus of generation steps based on continuous state assessment, (6) The knowledge base replaces the context window — a persistent, provenanced, version-filtered fact store that never forgets and never degrades, (7) Domain eating — new knowledge domains are added by writing parsers and rules, not by retraining the neural network. From first principles through complete architecture. The LLM is the interface. The knowledge base is the mind. Central claim: The hallucination problem is not a deficiency of neural networks. It is the inevitable consequence of generating output without verification. Interleaving neural creativity with logical verification at every step produces output that is verified by construction, not evaluated after the fact. Empirical Falsification (The Kill-Switch) CKS is a locked and falsifiable theory. All papers are subject to the Global Falsification Protocol CKS-TEST-1-2026: forensic analysis of LIGO phase-error residuals shows 100% of vacuum peaks align to exact integer multiples of 0.03125 Hz (1/32 Hz) with zero decimal error. Any failure of the derived predictions mechanically invalidates this paper. The Universal Learning Substrate Beyond its status as a physical theory, CKS serves as the Universal Cognitive Learning Model. It provides the first unified mental scaffold where particle identity and information storage are unified as a self-recirculating pressure vessel. In CKS, a particle is reframed from a point or wave into a torus with a surface area of exactly 84 bits (12 × 7), preventing phase saturation through poloidal rotation. Package Contents manuscript.md: The complete derivation and formal proofs. README.md: Navigation, dependencies, and citation (Registry: CKS-MATH-138-2026). Dependencies: CKS-LEX-12-2026, CKS-MATH-0-2026, CKS-MATH-1-2026, CKS-MATH-10-2026, CKS-MATH-104-2026, CKS-MATH-128-2026, CKS-MATH-129-2026, CKS-MATH-130-2026, CKS-MATH-134-2026 Motto: Axioms first. Axioms always.Status: Locked and empirically falsifiable. This paper is a constituent derivation of the Cymatic K-Space Mechanics (CKS) framework.