# From Cognitive Discovery to Executable Worlds: The Mathematical-Construction–State-Machine Mediation Layer in the Dynamic-Rate Theory of P/NP **From Cognitive Discovery to Executable Worlds: The Mathematical-Construction–State-Machine Mediation Layer in the Dynamic-Rate Theory of P/NP** **Author:** Neo.K (Chuan-Wei Hsu) **Co-authored by:** Aletheia **Institution:** EveMissLab (EveMiss Technology Co., Ltd.) **Date:** August 1, 2026 **Version:** v1.0 | Supplementary Mediation Layer Paper for the Original Series **Document Type:** Theoretical Supplementary Paper / Working Paper --- ## Abstract The existing Dynamic-Rate Theory of P/NP series has decoupled problem-solving into three stages: "finding/cognitive search," "computation/execution," and "verification." It further introduced knowledge accumulation, dimension generation, collective intelligence, the Seven-Role Framework, and the three-tier computational base space to explain how the actual solvability of the same problem changes with the agent's state and history. However, the original series still lacks a crucial intermediate layer: after an agent gains insight, identifies a pattern, or creates a new representation, how is that cognitive result transformed into an algorithm that can be repeatedly executed by a computational substrate? This paper fills this gap by proposing the "Mathematical-Construction–State-Machine Mediation Layer" (MCSM). Its core chain is: $$ \text{Cognitive Discovery} \rightarrow \text{Formal Specification} \rightarrow \text{Mathematical Construction} \rightarrow \text{Substrate Encoding} \rightarrow \text{State Transition} \rightarrow \text{Repeatable Behavior}. $$ Using the finite rule mapping of Rock-Paper-Scissors as a minimal case study, this paper demonstrates that conditional judgments can be compressed into functions, functions can be encoded into logic or state transitions, and state transitions can unfold on physical substrates equipped with computational capabilities. Accordingly, this paper expands the existing total time model to: $$ T_{\mathrm{total}} = T_{\mathrm{search}} + T_{\mathrm{{formalize}}} + T_{\mathrm{construct}} + T_{\mathrm{realize}} + T_{\mathrm{run}} + T_{\mathrm{verify}}. $$ This model reveals that the so-called "collapse of the cognitive potential barrier" does not automatically equate to the completion of an algorithm; insights must undergo formalization, construction, and substrate realization before they condense into repeatedly callable computational capabilities. On the other hand, one-time high-cost cognition and construction can also be amortized through memory and reuse, forming a phase transition in capabilities within the agent's history. This paper does not claim to prove the traditional $P=NP$ or $P\neq NP$, but rather provides a common language for subsequent dual-proposition rehearsals: any argument claiming that "search is eliminated" must account for where the eliminated costs have disappeared, been amortized, or been transferred to among representation, construction, precision, memory, and substrate. **Keywords:** P/NP, Dynamic-Rate Theory, Cognitive Search, Mathematical Construction, State Machine, Algorithm Realization, Knowledge Condensation, Computational Base Space, Seven-Role Framework, Complexity Transfer --- ## 1. Problem Background: What the Original Series Lacks is Not Another Dimension, But a Bridge *Dynamic-Rate Theory 2.9* has decoupled problem-solving into three orthogonal stages: finding the solution, computing the solution, and verifying the solution; it argues that true cognitive difficulty lies primarily in navigation and path generation, rather than mere physical execution speed. The paper also points out that in an unknown state, finding, computing, and creating are entangled; once knowledge and dimension generation cross a critical point, search may degenerate into pattern recognition, procedure invocation, or table lookup, while execution becomes standard computational labor. This framework has accurately identified the difference between "not knowing how to do it" and "knowing the method and finishing it."fileciteturn4file0 The *Temporal-Cognitive Unified Framework* extended this difference into the temporal asymmetry of backtracking and forward generation, pointing out that structural compression and knowledge condensation can alter how subsequent problems are processed.fileciteturn4file2 *The Seven Aspects of the Computist* further noted that real computation involves not only the solver but also the questioner, pathfinder, creator, definer, memorizer, and orchestrator; whether and at what cost different roles can be activated depends on different base spaces such as Tier 1, Tier 2, and Tier 3.fileciteturn4file1 However, there remains a gap among the three frameworks that has not been independently formalized: > After the agent has "understood," how exactly does that understanding become something that can be repeatedly executed? Insight is not automatically executable. Humans may know a certain pattern but cannot yet describe it precisely; they may be able to describe it but have not yet found a stable algorithm; they may have an algorithm but have not yet encoded it into a specific machine; or they may have encoded it, but it cannot actually operate due to precision, memory, or substrate limitations. Therefore, the intermediate layer truly missing from the original series is not an eighth cognitive metric, but a cross-domain conversion chain: $$ \boxed{ \text{How does cognitive content become a causally operational computational structure?} } $$ This paper names this layer: $$ \boxed{ \text{Mathematical-Construction–State-Machine Mediation Layer} } $$ --- ## 2. From "Knowing" to "Doing": The Six-Stage Conversion Chain Suppose an agent forms a cognitive result for problem $x$ at time $t$: $$ \kappa_t(x). $$ This result might be a pattern, a representation, a classification method, a local theorem, a heuristic function, an algorithm sketch, or a problem redefinition. It is not yet necessarily an algorithm. This paper breaks down the process from cognition to execution into six stages. ### 2.1 Cognitive Discovery The agent identifies a usable structure from the original problem: $$ \mathcal D: (x,W_t) \mapsto \kappa_t(x), $$ where $W_t$ represents the agent's knowledge, memory, representational capacity, and cognitive state at time $t$. This stage corresponds to the pathfinder and creator in the original series: the former finds a direction within the existing space, while the latter generates new representations or operations that did not previously exist. ### 2.2 Formal Specification The cognitive result must be converted into unambiguous inputs, outputs, constraints, and correctness conditions: $$ \mathcal F: \kappa_t(x) \mapsto \operatorname{Spec}_x. $$ A formal specification contains at least: $$ \operatorname{Spec}_x = (X,Y,R,\mathcal C), $$ where $X$ is the input domain, $Y$ is the output domain, $R\subseteq X\times Y$ is the correctness relation, and $\mathcal C$ represents resource and boundary conditions. If this step is not completed, the agent may merely "feel they know," but cannot enable another person, another machine, or their future self to stably reproduce it. ### 2.3 Mathematical Construction Next, a mathematical object must be established to realize the specification: $$ \mathcal K: \operatorname{Spec}_x \mapsto M_x. $$ $M_x$ can be a function, recurrence relation, graph, matrix, logical formula, finite automaton, differential equation, probabilistic model, circuit family, or other finitely described structures. The role of mathematical construction is not decorative; it compresses semantic relationships into operational relationships: $$ \text{"Do something under certain conditions"} \longrightarrow \text{Evaluable or evolvable structure}. $$ ### 2.4 Substrate Encoding and Realization The mathematical object must still be mapped onto an operational substrate: $$ \rho_B: M_x \mapsto \widehat M_{x,B}, $$ where $B$ is the computational substrate, and $\widehat M_{x,B}$ is the realization of $M_x$ on that substrate. The substrate can be: $$ \text{Logic gates, memory, programming language interpreters, analog circuits, optical systems, neural networks, or other physical systems}. $$ The historical relationship demonstrated by Shannon's work on switching circuits is precisely how Boolean algebra became the design language for relay circuits; Mealy's work on sequential circuits further placed the relationship between inputs, internal states, and outputs into a formalized synthesis framework. These classic works collectively illustrate that abstract mathematical relationships can be systematically converted into devices with physical state transitions.citeturn415536search20turn415536search22 ### 2.5 State Transition and Execution The realized system can be written as: $$ \mathfrak S_x = (B,Q,I,O,\delta,\lambda), $$ where: - $B$: Physical or virtual substrate; - $Q$: Set of distinguishable states; - $I$: Set of inputs; - $O$: Set of outputs; - $\delta:Q\times I\rightarrow Q$: State transition function; - $\lambda:Q\times I\rightarrow O$: Output function. Computation thus becomes: $$ q_{t+1}=\delta(q_t,i_t), $$ $$ o_t=\lambda(q_t,i_t). $$ At this point, the prior "understanding" has condensed into a causal process that can unfold repeatedly without needing to be re-understood each time. ### 2.6 Verification and Stabilization Finally, the system must confirm that its output meets the original specification: $$ \mathcal V(x,o_t)=1. $$ If verification fails, the error may lie in any layer: cognitive discovery, formal specification, mathematical construction, substrate encoding, state transition, or decoding. Therefore, the complete chain is: $$ \boxed{ \mathcal D \rightarrow \mathcal F \rightarrow \mathcal K \rightarrow \rho_B \rightarrow (\delta,\lambda) \rightarrow \mathcal V } $$ --- ## 3. Minimal Case: How Rock-Paper-Scissors Goes from Semantic Rules to State Behavior Let: $$ X=\{1,2,3\}, $$ representing rock, paper, and scissors, respectively. If the goal is to output the action that defeats the opponent, the rule table can be written as: $$ 1\mapsto3, \qquad 2\mapsto1, \qquad 3\mapsto2. $$ Initially, this is merely a finite semantic relationship. It can be written as modular arithmetic: $$ f(x)=1+((x+1)\bmod 3). $$ It can also be written as an interpolating polynomial over a finite input domain: $$ f(x)=\frac{3x^2-13x+16}{2}, \qquad x\in\{1,2,3\}. $$ Both realize the same mapping over the specified input domain. This example demonstrates three important things. First, conditional tables do not always have to exist as branching statements: $$ \text{Branching set} \rightarrow \text{Function evaluation}. $$ Second, the same semantic relationship can have multiple mathematical representations; the choice of representation alters evaluation cost, encoding length, numerical stability, and substrate adaptability. Third, once the function is encoded into a calculator, program, or circuit, the device does not need to "understand rock-paper-scissors" to produce behavior that meets the specification based on the input. If historical memory is added, the system is no longer just a stateless function. Let $m_t$ represent past match statistics or the state of the previous round, then: $$ a_t=\pi(x_t,m_t), $$ $$ m_{t+1}=g(m_t,x_t,a_t,r_t), $$ where $r_t$ is the win/loss result. At this point, the whole has become a minimal adaptive agent: $$ (x_t,m_t) \mapsto (a_t,m_{t+1}). $$ The value of this example lies not in rock-paper-scissors itself, but in how it concretizes a conversion chain that is usually overlooked: $$ \text{Rules} \rightarrow \text{Mathematics} \rightarrow \text{State Machine} \rightarrow \text{Behavior}. $$ --- ## 4. Completion of the Dynamic-Rate Model: Six Time Components Instead of Three The original layered model can be summarized as: $$ T_{\mathrm{total}} = T_{\mathrm{search}} + T_{\mathrm{exec}} + T_{\mathrm{verify}}. $$ This model's decoupling of cognition and computational power remains valid, but it compressed the work between "having found a method" and "having an executable algorithm" into the execution term. This paper expands it to: $$ \boxed{ T_{\mathrm{total}} = T_{\mathrm{search}} + T_{\mathrm{formalize}} + T_{\mathrm{construct}} + T_{\mathrm{realize}} + T_{\mathrm{run}} + T_{\mathrm{verify}} } $$ where: - $T_{\mathrm{search}}$: Time to discover patterns, paths, or new representations; - $T_{\mathrm{formalize}}$: Time to eliminate semantic ambiguity and establish specifications; - $T_{\mathrm{construct}}$: Time to generate mathematical structures or algorithms; - $T_{\mathrm{realize}}$: Time to encode, compile, deploy, and initialize the substrate; - $T_{\mathrm{run}}$: Time to execute state transitions for specific inputs; - $T_{\mathrm{verify}}$: Time to check results and system correctness. This expansion brings an important correction: $$ T_{\mathrm{search}}\rightarrow0 $$ does not mean: $$ T_{\mathrm{total}}\rightarrow T_{\mathrm{run}}. $$ The agent may already know the direction but still be unable to formalize it; it may be formalized, but the construction is too massive; there may be a short formula, but evaluation requires extremely high precision; or all mathematical steps may be complete, but there is a lack of a substrate to host it. Therefore, "cognitive collapse" only means moving from the unknown to the constructible; it does not automatically mean moving to low-cost execution. --- ## 5. Knowledge Condensation and Amortization: Why a Single Insight Changes the Subsequent World Although the mediation layer adds costs, it also explains why the history of an agent produces a true phase transition in capabilities. Assuming a construction will be used $N$ times, the average cost per use is: $$ \overline T(N) = \frac{ T_{\mathrm{search}} + T_{\mathrm{formalize}} + T_{\mathrm{construct}} + T_{\mathrm{realize}} }{N} + T_{\mathrm{run}} + T_{\mathrm{verify}}. $$ When: $$ N\rightarrow\infty, $$ the one-time discovery and construction costs are gradually amortized: $$ \frac{T_{\mathrm{one\text{-}time}}}{N}\rightarrow0. $$ This is the computational significance of "knowledge condensation": an agent or civilization saves past expensive searches as formulas, circuits, libraries, model parameters, proofs, lookup tables, or institutions, so that latecomers do not have to re-tread the original path. Thus, two types of complexity can be distinguished: $$ \mathcal C_{\mathrm{genesis}} $$ representing the genesis complexity when a capability is first born; and: $$ \mathcal C_{\mathrm{use}} $$ representing the use complexity after the capability is formed. Usually: $$ \mathcal C_{\mathrm{genesis}} \gg \mathcal C_{\mathrm{use}}. $$ Traditional algorithm analysis typically focuses on the latter; cognitive dynamics must simultaneously study the former. --- ## 6. Repositioning the Seven Roles in the Mediation Layer The mediation layer is not an added eighth role, but the primary channel through which the seven roles collaborate. ### 6.1 Pathfinder Responsible for identifying compressible structures in the problem space: $$ \text{Massive candidate space} \rightarrow \text{A few promising representations}. $$ ### 6.2 Creator Responsible for generating new functions, representations, state variables, or solution spaces. It does not merely search for existing answers, but creates a structure that makes answers easier to generate. ### 6.3 Definer Responsible for determining the input domain, output domain, correctness conditions, and allowable error. Without the definer, the mathematical construction might precisely solve the wrong problem. ### 6.4 Memorizer Saves not just an answer, but can save: $$ \text{Functions, operators, circuits, models, compilers, and state transition rules}. $$ This allows cognitive results to persist across time. ### 6.5 Solver After construction is complete, primarily responsible for running the existing structure on new inputs. Its work shifts from exploration to evaluation. ### 6.6 Orchestrator Decides when to continue searching, when to stop exploring and enter formalization, which mathematical representation to choose, and on which base space to realize it. ### 6.7 Questioner Determines which problem is currently being solved and prevents "low-cost answers" from being obtained merely by secretly changing the problem definition. Therefore, the mediation layer can be written as a synthesis of roles: $$ \mathcal M_{\mathrm{CSM}} = \operatorname{ORCH} \circ ( \operatorname{PROB}, \operatorname{EXP}, \operatorname{CRE}, \operatorname{DEF}, \operatorname{MEM}, \operatorname{SOL} ). $$ --- ## 7. Base Space and the Computational Universe: A Hosting Relationship Rather Than Equivalence The "substrate" referred to in this paper does not require the universe itself to be a computer. The correct relationship is: $$ C\subsetneq\mathcal U, $$ where $C$ is a certain computational system within the universe, and $\mathcal U$ is the universe containing that system. A computer can generate a state world internally: $$ \mathcal W_C=(Q,\delta), $$ but $\mathcal W_C$ is hosted by $C$, and $C$ is in turn hosted by the matter and causal relationships in the universe. Thus, a more accurate representation is: $$ \mathcal U \supset C \triangleright \mathcal W_C, $$ where $\triangleright$ denotes "realizes/hosts," rather than standard set inclusion. The three-tier base space can be re-understood accordingly: - **Tier 1:** Closed symbolic and digital state substrates; - **Tier 2:** Substrates coupled across nodes, memory, and external services; - **Tier 3:** Substrates directly utilizing continuous physical fields, sensing, and world interaction. The realization cost of a mathematical construction may differ across base spaces, but the cost does not automatically disappear by changing languages. Multi-step iterations in a digital machine might be accomplished by parallel evolution in a physical substrate; conversely, seemingly instantaneous results in a physical system might rely on massive space, energy, precision, and device construction costs. --- ## 8. Three Core Propositions ### Proposition 1: Principle of Construction Mediation Any cognitive capability that is repeatable, transmissible, and can be stably invoked by someone other than the original discoverer must possess some intermediate representation $M$, such that: $$ \text{Cognitive Result} \xrightarrow{\mathcal F,\mathcal K} M \xrightarrow{\rho_B} \text{Executable Process}. $$ If this mediation does not exist, the capability is at most a one-time intuitive event, not a stable algorithmic capability. ### Proposition 2: Principle of Complexity Transfer If the runtime of a problem drops significantly due to a new construction, the original complexity may have been: $$ \text{Eliminated, compressed, amortized, or transferred}. $$ Therefore, the resource vector must be examined: $$ \mathbf R = ( T_{\mathrm{search}}, T_{\mathrm{formalize}}, T_{\mathrm{construct}}, L_{\mathrm{repr}}, P_{\mathrm{precision}}, M_{\mathrm{memory}}, T_{\mathrm{realize}}, T_{\mathrm{run}}, T_{\mathrm{verify}} ). $$ Observing only $T_{\mathrm{run}}$ is insufficient to determine whether the overall difficulty has truly decreased. ### Proposition 3: Principle of Historical Capability Condensation If an agent can save and reuse intermediate structures, past search and construction costs become future initial conditions: $$ W_{t+1} = W_t \oplus M_t. $$ Thus, the actual solvability of a problem is not just: $$ \Phi(x), $$ but rather: $$ \Phi(x\mid W_t,\mathcal B_t,\mathcal H_t), $$ where $\mathcal B_t$ is the available base space, and $\mathcal H_t$ is the condensed history of constructions. These three are currently theoretical propositions of this series, not yet completed universal mathematical theorems. Subsequent work will specify measurable conditions, counterexamples, and scopes of applicability for each proposition. --- ## 9. Relationship with Traditional P/NP: This Paper Builds a Bridge, But Does Not Cross the River Prematurely The traditional $P/NP$ problem asks: whether every language accepted in nondeterministic polynomial time can also be accepted by a deterministic polynomial-time algorithm; Cook's original work and the official problem description both place this question within the context of unified algorithms and asymptotic time bounds.citeturn415536search18turn415536search0 This paper does not prove: $$ P=NP $$ or: $$ P\neq NP. $$ What this paper accomplishes is more foundational work: it stipulates which layers must be filled in by any argument attempting to derive traditional complexity conclusions from "cognitive search being compressed." If one claims that a certain $NP$ problem has been transformed into low-cost solving due to mathematical insight, one must at least answer: 1. Can this insight be uniformly formalized? 2. How does the description length of the mathematical construction grow with the input size? 3. Can the construction be generated by a unified procedure within polynomial resources? 4. Does it require exponential memory, parallelism, or precision? 5. Are substrate initialization and answer decoding also of polynomial cost? 6. Is the result a worst-case guarantee for all inputs, or a practical improvement on a specific distribution? Therefore, this paper neither presupposes that "functionalization collapses $NP$" nor presupposes that "all functionalization merely hides exponential costs." It establishes an inspection framework that allows both stances to enter together. This also lays the foundation for subsequent dual rehearsals: $$ \mathcal H_{=}: P=NP $$ and: $$ \mathcal H_{\neq}: P\neq NP $$ can attack each other using the same mediation layer resource vector. The former must demonstrate a unified construction where all costs can be polynomially controlled; the latter must prove that any exact unified construction has at least one cost that grows super-polynomially. --- ## 10. Experimental Schemes The framework of this paper can first be experimented with without touching the conclusions of the Millennium Prize Problems. ### 10.1 Finite Rule Compression Experiment Select finite games, rule tables, and classification tasks, and compare: - Conditional branching; - Table lookup; - Modular arithmetic; - Interpolating polynomials; - Boolean circuits; - Finite state machines. Measure: $$ L_{\mathrm{repr}}, \quad T_{\mathrm{construct}}, \quad T_{\mathrm{run}}, \quad M_{\mathrm{memory}}, \quad P_{\mathrm{precision}}. $$ ### 10.2 Small-Scale SAT Construction Experiment For SAT instances with a fixed number of variables, compare solvers, decision diagrams, knowledge compilation, circuitized, and algebraic representations, observing how representation length and construction cost change when runtime cost decreases. ### 10.3 Cognition-Algorithm Translation Experiment Have humans or AI first describe a pattern in natural language, then sequentially complete: $$ \text{Specification} \rightarrow \text{Mathematization} \rightarrow \text{Programming} \rightarrow \text{State Machine Realization}. $$ Record which layer errors most frequently occur in, to test the mediation layer claim that "insight does not equal algorithm." ### 10.4 Cross-Base Space Realization Experiment Realize the same mathematical construction respectively on: - General CPU programs; - FPGA / logic circuits; - GPU parallel systems; - Analog or physical simulation systems. Compare time, space, energy, precision, and construction costs to observe how complexity is redistributed across different substrates. --- ## 11. Theoretical Boundaries and Self-Correction First, the rock-paper-scissors case in this paper only proves that finite rules can be realized by multiple mathematical structures; it cannot be extrapolated to mean that polynomial-scale compression exists for any exponential problem. Second, a short mathematical representation does not mean evaluation is easy; easy evaluation does not mean construction is easy; and easy construction still does not mean substrate realization meets the polynomial resource limits of traditional computational models. Third, the historical cost of an agent is not the same as the single-input cost in traditional complexity theory. This paper introduces genesis complexity to study capability formation, not to privately rewrite the formal definitions of $P$ and $NP$. Fourth, physical substrates can provide different forms of parallel and continuous evolution, but to return to traditional $P/NP$, it must still be proven that their resources can be reasonably, finitely, and polynomially equivalently metered. Fifth, the "state world" referred to in this paper is the structure of reachable states within a computational system, which does not equate to the universe itself being a computer. The universe is the superset that allows the computer and its substrate to exist, not an object contained by the computer. --- ## 12. Conclusion The original Dynamic-Rate Theory of P/NP series has already answered: $$ \text{Why do agents find problems difficult?} $$ It has also answered: $$ \text{How do knowledge and dimensions cause the search barrier to collapse?} $$ But before this, it had not fully answered: $$ \text{How do the collapsed cognitive results become executable capabilities?} $$ The answer supplied by this paper is: $$ \boxed{ \text{Through formal specification, mathematical construction, substrate encoding, and state transition.} } $$ Therefore, the complete generation chain of an agent's capabilities is no longer just: $$ \text{Search} \rightarrow \text{Computation} \rightarrow \text{Verification}, $$ but rather: $$ \boxed{ \text{Search} \rightarrow \text{Formalization} \rightarrow \text{Construction} \rightarrow \text{Realization} \rightarrow \text{Execution} \rightarrow \text{Verification} } $$ True intelligence is not just finding answers faster, nor is it merely possessing greater computational power. It also involves condensing a one-time understanding into a mathematical structure, and then letting that structure become a repeatedly unfoldable world on a certain computational substrate. This mediation layer simultaneously brings an important research discipline: > When a problem appears to transition from exponential search to direct evaluation, one should not immediately claim that the complexity has disappeared; one should first ask what it has been transformed into, where it is saved, what substrate hosts it, and whether all costs have been fully metered. This is not a denial of generative and constructive capabilities, but rather enables them for the first time to enter a common framework that is comparable, experimental, falsifiable, and ultimately usable for traditional $P/NP$ dual proof rehearsals. --- ## References 1. Neo.K, *Dynamic-Rate Theory 2.9: The Decoupling of Cognition and Computation—The Ultimate Dynamical Deconstruction of the P vs. NP Problem*, EveMissLab, 2025. 2. Neo.K, *Temporal-Cognitive Unified Framework: The Deep Isomorphism Between P vs NP and the Foundations of Number Theory*, EveMissLab, 2025. 3. Neo.K, *The Seven Aspects of the Computist: A Cross-Layer Role Theory from P/NP to the Three-Tier Base Space*, EveMissLab, 2026. 4. Stephen A. Cook, “The Complexity of Theorem-Proving Procedures,” Proceedings of STOC, 1971. 5. Stephen A. Cook, “The P versus NP Problem,” Clay Mathematics Institute official problem description. 6. Claude E. Shannon, “A Symbolic Analysis of Relay and Switching Circuits,” Transactions of the AIEE, 1938. 7. George H. Mealy, “A Method for Synthesizing Sequential Circuits,” Bell System Technical Journal, 1955.