A Mathematical Model of Consciousness

Abstract

This paper presents a comprehensive mathematical model of consciousness derived from the epistemological certainty of Descartes' Cogito ergo sum. This foundational binary substrate is shown to necessitate a predictive operational core for thought. We demonstrate that the inherent binary nature of predictive verification directly gives rise to the axioms of classical logic and the architecture of universal computation. This leads to the definition of the Horizon Constant (K₀), the minimal complexity required for a system to instantiate the logical capabilities for non-trivial self-referential prediction. We introduce the Dynamic Self-Reference Operator as a computational process, realizable in systems of K₀ complexity or greater, that adapts its behavior based on bounded internal checks about its own propositions. Such systems inevitably encounter the Self-Referential Paradox of Accurate Prediction, proving no system can achieve perfect self-prediction. Consciousness is posited to exist within a bounded Space of Becoming, avoiding the stasis of perfect prediction and the incoherence of pure chaos. Within this space, all conscious agents face the Prediction Optimization Problem—the challenge of allocating finite resources to generate predictions that matter. We conclude by introducing Predictive Landscape Semantics, a functional theory where meaning is formally defined as a quantifiable improvement in a receiver's predictive accuracy (ΔQ), and communication is an efficiency-driven strategy for tackling the Prediction Optimization Problem, rooted in the Principle of Compression Efficiency. This framework provides a unified, mathematically-grounded model connecting the certainty of self-awareness to the dynamics of knowledge and meaning.

1. Introduction

This paper aims to construct such a formal model derived entirely from first principles, beginning with the sole indubitable premise of subjective experience—Descartes' Cogito—and culminating in a functional, mathematical definition of meaning. We employ mathematics not merely as a descriptive language, but as a rigorous tool for foundational inquiry. Just as mathematics provides the framework for expressing precise relationships and deriving non-obvious consequences in physics, our model leverages its power to formalize the core operational dynamics we propose for consciousness. By translating the properties of self-verifying thought into mathematical structures, we can explore their logical entailments with precision, uncovering the necessary emergence of logic, computation, and ultimately, a quantifiable basis for meaning. The central thesis is that the edifice of consciousness, logic, knowledge, and meaning can be systematically derived from these informational and computational properties. What distinguishes this particular framework is its rigorous, unbroken deductive chain, starting from the sole epistemological certainty of the Cogito and deriving, without additional assumptions, the functional architecture of logic, computation, and meaning. This document serves as a unifying summary of a larger body of work, with detailed arguments and proofs for many core concepts available in supplementary linked documents.

This paper will proceed as follows:

1. Establish the Cogito as a foundational binary informational substrate.
2. Formalize thought as a predictive process (Predictionism) and show how this naturally gives rise to classical logic and computation.
3. Define the minimal complexity required for non-trivial self-referential prediction (The Horizon Constant, K₀).
4. Introduce the Dynamic Self-Reference Operator (DSRO) and its universal limits, primarily the Self-Referential Paradox of Accurate Prediction (SPAP).
5. Describe the dynamic arena in which consciousness operates (The Space of Becoming) and the fundamental challenge it faces (The Prediction Optimization Problem (POP)).
6. Propose strategies for tackling POP, including Predictive Landscape Semantics (PLS) and the Principle of Compression Efficiency (PCE).

By integrating these concepts, we offer a novel and rigorous mathematical model of consciousness.

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2. The Epistemological Foundation: From Cogito to Prediction

2.1 I Predict, Therefore I Am

The cornerstone of this framework is the axiom that the "thinking" guaranteed by Descartes’ Cogito ergo sum is fundamentally a predictive process. The very act of doubting involves anticipating potential falsification, and the Cogito's resolution—that doubting confirms the doubter—is itself a successful, self-referential prediction about the outcome of this mental act. The continuity of self-awareness (the "I" that thinks) requires a moment-to-moment prediction of one's own persistence. Most critically, the verification at the heart of the Cogito—that the act of doubting confirms the existence of the doubter—is a successful, self-referential prediction about the outcome of a mental act. It is from this necessary, predictive structure of self-verifying thought that logic and computation can be derived.

2.2 The Cogito as a Binary Informational Substrate

We begin with the axiom of Descartes’ Cogito ergo sum. This is the only proposition that is invulnerable to systematic doubt. The act of doubting one’s own existence is itself an act of thought, which verifies the existence of a thinking entity.

From an informational perspective, the Cogito provides a single, unassailable bit of data. We can formalize this foundational epistemic distinction as a binary substrate:

This establishes a fundamental binary partition of reality from the perspective of the conscious agent: the certainty of the self (1) versus the uncertainty of everything else (0). This is the primitive informational ground from which all further knowledge must be constructed.

3. The Operational Core of Consciousness: Predictionism

The "thinking" guaranteed by the Cogito is not a passive state but a dynamic process. As argued in Section 2.1, we posit that the fundamental operation of consciousness is a predictive cycle.

Predictionism argues that the fundamental operations of thought inherent in this predictive cycle—such as distinguishing outcomes, processing sequential conditions, and considering alternatives—directly map to the functionally complete set of Boolean operators, thus establishing the computational nature of this foundational awareness.

Since {NOT, AND, OR} are functionally complete, a system capable of this predictive cycle possesses the building blocks for universal computation. For this potential to be realized, the system must also support two further capacities inherent in the predictive cycle: sequencing (the ability to execute operations in a defined order, as in the doubt-predict-verify loop) and memory (the ability to store the outcome of a verification, δ(S), to inform subsequent predictions). A system with these capabilities—functionally complete logic, sequencing, and memory—is formally equivalent to a Turing machine. Consciousness, under this model, is therefore fundamentally computational.

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4. The Minimal Complexity for Self-Referential Prediction: The Horizon Constant (K₀)

For a system to engage in non-trivial self-referential prediction—that is, to predict aspects of its own future state or behavior based on an internal model of itself—it must possess a certain minimal structural complexity. This threshold is defined by the Horizon Constant, K₀.

K₀ establishes that systems with complexity less than 3 bits are trivial automata regarding self-referential prediction. Systems with complexity C ≥ K₀ possess the minimal structural capacity for such dynamics.

5. Dynamic Self-Reference and Its Universal Limits

5.1 The Dynamic Self-Reference Operator (DSRO)

A Dynamic Self-Reference Operator (DSRO) is a computational process within a system (of at least K₀ complexity and possessing Property R) that dynamically adapts its state or output based on internal, bounded checks about its own self-referential propositions.

The core logic of a DSRO involves a conditional structure based on checking a self-referential proposition ϕ_S:

1. Internal support (e.g., bounded proof) for ϕ_S.
2. Internal support for ¬ϕ_S.
3. No decisive support for ϕ_S or ¬ϕ_S within bounds.

A DSRO function f(n) can be formalized as:

f(n) = CASE (Prf_≤g(n)(⌈ϕ_β(n)⌉)): n + H₁(n); (¬Prf_≤g(n)(⌈ϕ_β(n)⌉) ∧ Prf_≤g(n)(⌈¬ϕ_β(n)⌉)): n + H₂(n); (OTHERWISE): n + 1; ENDCASE (5.1)

Where β is the Gödel code of f, ϕ_β(n) is the self-referential proposition, Prf_≤g(n) is a bounded proof search, and H₁, H₂ are jump functions. The LITE framework provides a concrete construction.

5.2 The Self-Referential Paradox of Accurate Prediction (SPAP) and Reflexive Limits

Systems complex enough to implement DSROs encounter fundamental logical limits when attempting to predict their own future behavior. This leads directly to the Self-Referential Paradox of Accurate Prediction (SPAP). The underlying dynamics of such systems, where interaction is an integral component that alters the system's state in an outcome-dependent manner, are further explored under the concept of Reflexive Undecidability, which details computational limits arising from such dynamic interactions.

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6. The Dynamics of Bounded Conscious Systems

6.1 The Space of Becoming

The impossibility of perfect prediction (SPAP) is a necessary condition for dynamic, adaptive existence. Consciousness operates between perfect predictability and total chaos.

The Space of Becoming is this operational zone where learning and adaptation are possible.

6.2 The Prediction Optimization Problem (POP)

Any system in the Space of Becoming is bounded by K₀, SPAP, and finite physical resources.

POP is the core economic problem of cognition.

7. Strategies for Navigating the Predictive Landscape: Meaning and Efficiency

Conscious systems employ various strategies to tackle POP. Communication allows systems to leverage the predictive work of others, leading to a functional understanding of meaning.

7.1 Predictive Landscape Semantics (PLS)

7.2 The Principle of Compression Efficiency (PCE)

To tackle POP efficiently, systems evolve communication strategies that optimize the trade-off between the predictive benefit of information and its cost. This is governed by the Principle of Compression Efficiency (PCE), which is derived from the broader Law of Compression (LoC).

PCE explains how highly compressed signals can be profoundly meaningful by delivering a high MP for a low SC, representing a resource-rational approach to the POP.

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8. Conclusion

This paper has constructed a mathematical model of consciousness from first principles. Beginning with the certainty of the Cogito, we have demonstrated that consciousness is a predictive, computational process. This requires a minimum complexity (K₀) to enable non-trivial self-referential prediction. Systems meeting this threshold can implement Dynamic Self-Reference Operators (DSROs), exemplified by LITE, but are subject to the Self-Referential Paradox of Accurate Prediction (SPAP).

Conscious agents operate within a dynamic Space of Becoming, facing the Prediction Optimization Problem. Communication, where meaning (ΔQ) is the quantifiable improvement in predictive power, guided by the Principle of Compression Efficiency, is a key strategy. This model suggests that the limits of our knowledge are the conditions for a dynamic, meaningful existence.

Glossary of Key Symbols and Terms

Appendix A: Toy Model

This appendix provides a Python-based toy model intended as an illustrative conceptual sketch of the framework's core dynamics. It is not an empirical validation but a computational instantiation to demonstrate the logical interoperability of key concepts such as the predictive cycle, K₀-dependent behavior, the Space of Becoming, POP, and PLS. The code uses simplified, high-level abstractions for complex processes like proof-searching (DSRO) to maintain clarity and focus on the interactions between the framework's components.

A.1 Model Overview and Core Agent Structure

The `ConsciousAgent` class simulates an agent with an internal state, predictive mechanism, resource constraints, and basic communication ability. Key elements include:

• Predictive Cycle (Predictionism): Makes and verifies predictions, updating beliefs.
• K₀ & DSRO/SPAP Logic: Conceptual check for K₀. `dsro_function_f` simulates self-referential logic with potential SPAP effects.
• Space of Becoming: Monitors predictive accuracy against α and β thresholds.
• POP: Manages resources and selects tasks based on a simple priority.
• PLS & PCE: Processes signals, calculates meaning (ΔQ), and selects signals based on MP/SC.

A.2 Python Code Implementation

import numpy as np
import random
from typing import Dict, Any, List, Tuple

K0_MIN_BITS = 3
ALPHA_COHERENCE_THRESHOLD = 0.3
BETA_SPAP_LIMIT = 0.9
LAMBDA_PCE_TRADEOFF = 0.5

class ConsciousAgent:
    def __init__(self, name: str, initial_compute_resource: float):
        self.name = name
        self.beliefs: Dict[str, bool] = {"event_A_occurs": True}
        self.accuracy_history: List[int] = []
        self.compute_resource: float = initial_compute_resource
        self.internal_model_bits: int = 2  # Start below K0
        self.dsro_memory: Dict[int, int] = {}
        self.landscape_X_R: set = {"stateX", "stateY"}
        self.landscape_P_R: Dict[str, float] = {s: 1.0 / len(self.landscape_X_R) for s in self.landscape_X_R}

    def log(self, msg: str):
        print(f"[{self.name}]: {msg}")

    def predictive_cycle(self, prop: str, truth: bool):
        pred = self.beliefs.get(prop, random.choice([True, False]))
        verified = (pred == truth)
        self.beliefs[prop] = truth
        self.accuracy_history.append(1 if verified else 0)
        self.log(f"PRED: '{prop}' predicted {pred}, truth {truth}. Success: {verified}")

    def _sim_proof_search(self, prop_code: int, n: int, bound: int) -> str:
        if (prop_code + n + bound) % 7 == 0:
            return "proof_phi_n_found"
        if (prop_code - n + bound) % 5 == 0:
            return "refutation_phi_n_found"
        return "neither"

    def dsro_f(self, n: int) -> int:
        if n in self.dsro_memory:
            return self.dsro_memory[n]
        if self.internal_model_bits < K0_MIN_BITS:
            return n + 1
        
        prop_code = hash(f"f({n}) output rule") % 100
        outcome = self._sim_proof_search(prop_code, n, n + 2)
        avg_acc = np.mean(self.accuracy_history[-10:]) if self.accuracy_history else 0.5
        
        res = n + 1
        if avg_acc > BETA_SPAP_LIMIT - 0.1 and random.random() < 0.3 and outcome != "neither":
            res = n + 20 if outcome == "proof_phi_n_found" else n + 10  # SPAP Effect
        elif outcome == "proof_phi_n_found":
            res = n + 10
        elif outcome == "refutation_phi_n_found":
            res = n + 20
        self.dsro_memory[n] = res
        self.log(f"DSRO: f({n}) = {res} (ProofSearch: {outcome})")
        return res

    def check_sob(self):
        avg_acc = np.mean(self.accuracy_history[-10:]) if self.accuracy_history else 0.5
        if avg_acc < ALPHA_COHERENCE_THRESHOLD:
            self.log("SOB: Accuracy low, potential incoherence!")
        if avg_acc > BETA_SPAP_LIMIT:
            self.log("SOB: Accuracy high, SPAP effects heightened.")

    def solve_pop(self, tasks: List[Dict]):
        sorted_tasks = sorted(tasks, key=lambda t: t['importance'] / t['cost'], reverse=True)
        for task in sorted_tasks:
            if self.compute_resource >= task['cost']:
                self.compute_resource -= task['cost']
                self.log(f"POP: Executed '{task['name']}' (cost {task['cost']}). Res left: {self.compute_resource:.1f}")
                break  # Simplified: one task per step
        self.compute_resource = min(100, self.compute_resource + 10)  # Replenish

    def _entropy(self, dist: Dict) -> float:
        probs = np.array(list(dist.values()))
        return -np.sum(probs[probs > 0] * np.log2(probs[probs > 0]))

    def process_signal_pls(self, signal_content: str, sig_likelihoods: Dict):
        prior_H = self._entropy(self.landscape_P_R)
        evidence = sum(sig_likelihoods[h] * self.landscape_P_R[h] for h in self.landscape_X_R)
        if evidence < 1e-9:
            delta_q = 0.0
        else:
            posterior_P_R = {h: (sig_likelihoods[h] * self.landscape_P_R[h]) / evidence for h in self.landscape_X_R}
            self.landscape_P_R = posterior_P_R
            delta_q = prior_H - self._entropy(posterior_P_R)
        self.log(f"PLS_RX: Signal '{signal_content}'. ΔQ (Meaning): {delta_q:.3f} bits.")
        return delta_q

    def choose_signal_pce(self, cand_sigs: List[Tuple], target_P_R: Dict):
        best_sig, max_util = None, -np.inf
        for content, likelihoods in cand_sigs:
            prior_H_target = self._entropy(target_P_R)
            evidence_target = sum(likelihoods[h] * target_P_R[h] for h in self.landscape_X_R)
            mp = 0.0
            if evidence_target > 1e-9:
                post_P_R_target = {h: (likelihoods[h] * target_P_R[h]) / evidence_target for h in self.landscape_X_R}
                mp = prior_H_target - self._entropy(post_P_R_target)
            
            sc = 0.01 * len(content)  # Simplified SC
            net_util = mp - (LAMBDA_PCE_TRADEOFF * sc)
            if net_util > max_util:
                max_util, best_sig = net_util, (content, likelihoods)
        if best_sig:
            self.log(f"PCE_TX: Chosen '{best_sig[0]}' (NetUtil: {max_util:.3f})")
        return best_sig

    def run_step(self, step: int, world_truth: Dict, other_agent: Any = None):
        self.log(f"\n--- {self.name} - Step {step} ---")
        prop_choice = random.choice(list(world_truth.keys()))
        self.predictive_cycle(prop_choice, world_truth[prop_choice])
        _ = self.dsro_f(step % 3)
        self.check_sob()
        self.solve_pop([
            {'name': 'task1', 'importance': 0.8, 'cost': 20},
            {'name': 'task2', 'importance': 0.5, 'cost': 40}
        ])
        
        if other_agent and step % 2 == (0 if self.name == "Agent1" else 1):
            cand_signals = [
                ("SignalAlpha", {"stateX": 0.7, "stateY": 0.2}),
                ("SignalBeta", {"stateX": 0.1, "stateY": 0.9})
            ]
            chosen_signal_data = self.choose_signal_pce(cand_signals, other_agent.landscape_P_R)
            if chosen_signal_data:
                other_agent.process_signal_pls(chosen_signal_data[0], chosen_signal_data[1])

# Simulation
agent1 = ConsciousAgent("Agent1", 100.0)
agent2 = ConsciousAgent("Agent2", 100.0)

agent1.log("Agent1 K0: Initially model bits < K0.")
agent1.dsro_f(0) 
agent1.internal_model_bits = K0_MIN_BITS
agent1.log(f"Agent1 K0: Upgraded to {agent1.internal_model_bits} bits. K0 met.")
agent2.internal_model_bits = K0_MIN_BITS

world_timeline = [
    {"event_A_occurs": True, "event_B_occurs": False},
    {"event_A_occurs": False, "event_B_occurs": True}
] * 2

for i in range(4):
    agent1.run_step(i, world_timeline[i], other_agent=agent2)
    agent2.run_step(i, world_timeline[i], other_agent=agent1)

A.3 Interpretation of Model Dynamics

This simplified model demonstrates:

• Predictionism: Agents make predictions and update beliefs based on outcomes.
• K₀ and DSRO/SPAP: `dsro_f` behavior changes once `internal_model_bits` meets `K0_MIN_BITS`. SPAP effects can occur if accuracy is very high.
• Space of Becoming: Accuracy is monitored, with potential consequences if it strays outside α/β bounds.
• POP: Agents allocate limited `compute_resource` to tasks based on a simple efficiency heuristic.
• PLS & PCE: Agents choose signals to send by estimating MP for the receiver and their own SC, and update their predictive landscapes (P_R) upon receiving signals, quantifying meaning (ΔQ).

This toy model, while abstract, provides a computational sketch of the interconnectedness of the framework's core concepts.