What question did this study set out to answer?

The work aims to define a unifying framework for the computational universe governed by Conceptual Primes, establishing seven foundational theorems.

June 13, 2026Open Access

Computational Knowledge Theory (Ckt).

Key Points

The work aims to define a unifying framework for the computational universe governed by Conceptual Primes, establishing seven foundational theorems.
Formally defines the Conceptual Prime axioms and theorems across various domains including complexity theory and artificial intelligence.
Establishes algorithms and metrics related to knowledge compression and entropy in a theoretical framework.
Investigates the implications of these theorems on reality and computational architecture.
Theorem 1 proves that stable complexity is dependent on the weakest Prime, collapsing to zero if violated.
Theorem 2 shows NP-Hard problems become tractable with Prime-compliant algorithms, leading to an effective complexity of O(N²/K).
Theorem 7 demonstrates that no finite simulation can replicate the law of Prime combinations, linking consciousness to computational limits.

Abstract

This paper formally establishes, under the Conceptual Prime axioms, that the computational universe is governed by a single unifying law: the Conceptual Primes. Seven interlocking theorems are established across complexity theory, epistemology, information compression, evolutionary biology, temporal system dynamics, artificial intelligence architecture, and the unsimulability of reality. Theorem 1 (Reality-Complexity Equivalence) establishes that stable complexity is bounded by the weakest Prime — P̂ (S) = minᵢ Pᵢ (S) — and collapses to zero if any Prime is violated. Theorem 2 (Prime-Tractability) demonstrates that NP-Hard problems are intractable only in the purely abstract domain and become tractable at O (N²/K) effective complexity when solved by Prime-compliant algorithms grounded in physical reality. Theorem 3 (Conciseness Standard) proves that C (R) is the unique universal metric for lossless knowledge compression. Theorem 4 (Knowledge Accumulation Law) establishes that knowledge grows if and only if new information reduces total system entropy, incorporating the CAKI metric and the D (Ω) Defect Function as formal measures. Theorem 5 (Gödel's Ceiling) connects formal mathematical limits to biological evolution and AI scaling. Theorem 6 (Cardinal Value Lemmas) formalises Wisdom, Peace, Creativity, and Evolving Order as temporal combinations of the Primes, deriving the Prime-Base Intelligence corollary. Theorem 7 (Unsimulability of Reality) proves that no finite simulation can contain the live Prime-combination law of actualisation — Consciousness is the unique bridge between infinite potential and finite territory. The framework defines a two-stage computational architecture: a Training Evaluation Form (5-term Prime-resolved C (R) + CAKI) for grounding knowledge in Prime compliance and calibrating domain-dependent λ-weights, and an Inference Selection Form (3-term operational C (R) ) for selecting minimum-cost outputs. Dynamic λ-adaptation connects both stages, enabling domain-calibrated intelligence.

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