Before You Begin

Let's start with a short detour.

This isn't just a longread. It's an entry point into a different frame of reference. You won't find a recap of familiar ideas here — instead, it's an attempt to build a new computational paradigm from the ground up: from a blinking dot to an emergent model of time.

The difficulty doesn't come from complexity, but from shifting perspective. The text explains everything from scratch — but "scratch" here doesn't mean what you're used to.

Almost all familiar terms are used differently. Words like “space,” “field,” or “rhythm” might sound metaphorical at first — but here, they’re not. Each one was chosen not for pleasantness or familiarity, but for precision. The geometric language in this theory isn’t poetic — it’s structural. Even what might seem like an edge case — division by zero, for example — is treated not symbolically, but as a formal consequence within the system.

If you're reading this with an LLM, ask it to defend the author's position, rephrase key ideas, and break them down step by step. LLMs are quick to spot patterns — but if you're human, you have something more valuable: the ability to be surprised, to rethink, and to return. That matters here.

What first feels abstract will gain structure — just not immediately. With each reading, the meaning shifts. What you understand at the start won't be what you see by the end.

Yes, there will be code — you'll be able to play with it, test things, challenge ideas. Some parts you'll disagree with. Others you'll return to later, and see in a new light.

You might be surprised, but the title is true. It just won't be the most important truth you'll find here.

This won't be an easy read.
But we promise a shift.

---

Introduction

This theory presents a fundamental reconceptualization of information and computation. Instead of viewing computation as state manipulation, we introduce a paradigm where:

1. Information exists as rhythmic sequences in multidimensional space
2. Computation occurs through directed flow of information waves
3. Results emerge as interaction patterns at specific coordinates
4. Efficiency arises from selectively processing significant states

What distinguishes this framework is its rigorous axiomatic foundation. Each theorem is explicitly derived from specific axioms, creating an unbroken chain of reasoning from first principles to practical implementations. This logical scaffolding ensures that each concept is not merely useful but necessarily true within the system.

The framework is organized into a unified structure where theorems build upon each other in clear progression:

• Fundamental Properties (Theorems 1-4): Establish the core nature of information
• Optimization Principles (Theorems 5-8): Enable efficiency through selective processing
• Spatial Structure (Theorems 9-12): Define how information exists in computational space
• Information Dynamics (Theorems 13-15): Describe how information flows to create results
• Geometric Complexity Theory (Theorems 16-19): Reinterpret computational complexity as geometric paths through information space

Each theorem's relationship to others is explicitly mapped, showing how concepts like "rhythmic sequences" enable "diagonal points," which in turn support "information routing." This interconnectedness transforms isolated concepts into a coherent theory where each component serves a specific role in the larger architecture.

The theory culminates in the General Formula of Information Emergence (Theorem 15), which unifies all previous concepts into a single mathematical expression. This formula doesn't merely summarize the theory—it reveals how seven distinct axioms manifest as aspects of a single underlying principle.

Building on this foundation, the Geometric Complexity Theory (Theorems 16-19) reframes algorithmic complexity as paths through information space rather than step counts. This perspective explains why O(n) notation is a projection—a shadow—of the true geometric nature of computation, where efficiency depends on minimizing the integral of information density along computational paths.

Unlike traditional computational models focused on state changes in memory, this theory reinterprets computation as movement through information space. A bit is not a storage location but a routing node directing information waves. This perspective shift enables new optimization strategies that bypass insignificant states, potentially transforming our approach to algorithmic complexity.

Throughout, we distinguish between formal correctness (logical derivability) and truth (stability under transformation), allowing us to identify which properties are artifacts of representation and which reflect objective features of information itself.

The theory bridges mathematical abstraction and practical implementation, as demonstrated in the accompanying diagonal algorithm code. Each theoretical concept has a direct counterpart in working code, showing how this paradigm translates to executable systems.

Basic Axioms

Core Axioms (Primary, Self-Validating)

Axiom 1: Distinction Existence In any information system, there exists a fundamental operation of distinguishing between states.

Explanation: Distinction is the foundational act from which all information flows. To deny this axiom is to perform it—the very act of disagreement requires distinguishing between "true" and "false." This self-validating principle manifests in every bit, every binary choice, every measurement, and every observation. It is both the origin and boundary of all information processes.

Axiom 2: Boundary Formation The act of distinction necessitates the establishment of boundaries between the distinguished states.

Explanation: Distinction cannot exist without boundaries—the very concept of difference requires a demarcation point. Any attempt to refute this principle requires drawing a conceptual boundary between refutation and acceptance. This axiom manifests as thresholds in measurements, decision criteria in logic, and type systems in programming.

Derived Axioms (Building on Core)

Axiom 3: Information Coherence Any system of information requires internal consistency in its application of distinctions and boundaries.

Explanation: Coherence emerges necessarily from the systematic application of distinctions. To argue against coherence would require a coherent argument—an inevitable self-reference that proves the axiom. This principle underlies formal systems, error-detection mechanisms, and the fundamental nature of truth.

Axiom 4: Temporal Discreteness Information transitions occur in discrete steps rather than continuously.

Explanation: The application of distinction across time creates discrete intervals—a natural consequence of Axioms 1 and 2 extended temporally. This discreteness manifests in digital state transitions, clock cycles, and the quantized nature of information change.

Axiom 5: Process Finiteness Any computational process has clearly defined initial and final states within a finite sequence of steps.

Explanation: This follows from the application of boundaries (Axiom 2) to temporal processes (Axiom 4). Even apparent exceptions like division by zero resolve through "zero-length rhythms"—instantaneous transitions to limiting states—preserving the axiom's universality.

Axiom 6: Spatial Interaction Multiple information sequences in spatial interaction generate emergent properties irreducible to their individual components.

Explanation: This axiom extends distinction and boundary concepts to multidimensional space, where relationships between distinctions create new information structures. This principle enables the formation of complex computational spaces and the emergence of operations' results.

Axiom 7: Self-Reference Capability Information systems must be capable of operations that reference other operations, including themselves.

Explanation: This axiom completes the system by allowing for recursive application of distinctions. The ability to apply distinction to distinction itself is necessary for information systems to achieve completeness. Any argument against self-reference must examine its own logical structure—confirming the axiom through its denial.

Theory Applicability

This theory is strictly applicable to:

• Discrete information systems that can be represented as projections of higher-dimensional movement
• Computable states and processes that follow geometric paths in information space
• Formal logical constructions that can be mapped to geometric structures
• Information-oriented computations where results emerge from spatial interactions

Definition of computability: A state is considered computable if there exists a geometric path in information space that leads to its determination in a finite number of steps. This definition extends the concept of Turing computability to include the geometric nature of computation.

The theory does not extend to:

• Physical mechanisms of computation implementation (which may not follow geometric paths)
• Non-computable objects (e.g., complete solution to the halting problem) that cannot be represented as geometric paths
• Continuous systems without a discretization procedure that preserves geometric properties
• Quantum systems in a superposition state (before measurement) where geometric paths are undefined

Relationship with Classical Computation Theory

This theory relates to classical computation theory as follows:

1. Equivalence of computational models: All operations in our theory can be implemented on a Turing machine, and conversely, any Turing machine can be described in terms of our theory as a system with state distinction.

2. Generalization of complexity concept: Asymptotic complexity in our theory is consistent with classical definitions but extends them, taking into account the relativity of representation and the possibility of optimization through skipping insignificant states.

3. Preservation of undecidability: Problems that are undecidable in classical algorithm theory (e.g., the halting problem) remain undecidable in our theory, which follows from the equivalence of computational models.

4. Structural interpretation: While classical computation theory focuses on states, our theory reinterprets computation as a dynamic process of information propagation through structural interactions.

Definitions

1. Information — a collection of distinguishable states and rules for transitions between them.
2. Discrete state — a state that is unambiguously separated by a boundary of distinction from other states.
3. Order of states — a relation that allows determining which state precedes another.
4. Computable state — a state that can be reached and uniquely identified in a finite number of elementary operations.
5. Coordinate system — a way of organizing information that defines the correspondence between states and their formal representation.
6. Significant state — a state whose change affects the result of a given operation.
7. Information independence — a property of regions in information space where changes in one region do not affect values in another, allowing for the division of computations into independent streams that can be processed in parallel without the need for synchronization between them.
8. Rhythmic sequence — a finite ordered sequence of bits representing a number or other information structure.
9. Information space — a multidimensional structure formed by the interaction of two or more rhythmic sequences, where each dimension corresponds to one sequence.
10. Event point — a node in the coordinate grid where information propagates to the surrounding space based on input data.
11. Transition table — a structure that defines how an event point distributes information depending on input parameters.
12. Chrono-energy — a metric characteristic of a computational process that determines the amount of resources needed to process information. Unlike classical computational complexity, chrono-energy is expended only on processing significant states, which allows for the optimization of computations through skipping insignificant states according to Theorem 5.
13. Diagonal point — a point with coordinates (x,x) in a two-dimensional information space, containing the result of the interaction of the corresponding bits of the original sequences.

Formal definition: In a two-dimensional information grid, where the X-axis contains bits of sequence A = [a₀, a₁, ..., aₙ], and the Y-axis contains bits of sequence B = [b₀, b₁, ..., bₘ], the diagonal point D_i with coordinates (i,i) contains the result of the interaction of bits aᵢ and bᵢ according to a given operation.

Mathematical expression: D_i = f(aᵢ, bᵢ), where f is the interaction operation (e.g., AND, OR, XOR, ADD, etc.).

Significance: The set of all diagonal points {D₀, D₁, ..., Dₖ}, where k = max(n,m), completely determines the result of the operation on sequences A and B.

14. Information routing — the process of directed propagation of information from event points to surrounding cells according to specified logic.

15. Information field of computation and emergence — a fundamental structure defining how ordered results emerge from interacting information elements, consisting of:

a. Intention (F) — a vector field defining information propagation directions at each point, formalizing transition tables in mathematical form (F: ℝ² → ℝ²).

b. Potential (P) — a scalar function of information significance, with its gradient ∇P indicating the direction of increasing significance.

c. Emergent effect (E) — the computation result, expressed as E = ∫(F·∇P)dM · Φ(V,O), where F·∇P represents information interaction density.

Practical interpretation: In computation, F corresponds to how bits propagate through circuits (e.g., carry bits in addition), P represents the importance of each bit position (e.g., more significant vs. less significant bits), and their interaction (F·∇P) determines which operations actually contribute to the result.

Invariance property: The formula remains valid under coordinate transformations, showing the objective nature of computation independent of specific representation.

16. Potential gradient — a vector field ∇P that is the first derivative of potential P. Mathematically expressed as ∇P = (∂P/∂x, ∂P/∂y) in two-dimensional space. The gradient has two key properties: (1) its direction at point (x,y) indicates the maximum increase in information significance, (2) its magnitude |∇P| is proportional to the rate of this increase. At points of zero gradient (∇P = 0), information is absent or insignificant for computation. The scalar product F·∇P determines the degree of consistency between the direction of information propagation and the direction of significance increase, creating a selective mechanism for the activation of information routes.

17. Truth — a characteristic of a statement that satisfies two criteria: (1) internal consistency (logical derivability within a formal system) and (2) external stability (preservation of significant properties under scaling or transformation of the coordinate system). Mathematically, truth is defined as a function T(s) that takes the value 1 if statement s simultaneously satisfies both criteria, and 0 otherwise.

Formal notation: T(s) = V(s) · S(s), where V(s) is the function of internal validity (equals 1 if the statement is derivable from the system's axioms), and S(s) is the function of structural stability (equals 1 if the statement is invariant with respect to admissible system transformations).

Significance: This definition of truth allows distinguishing formally provable theorems (V(s)=1) from theorems that have an objective reflection in information reality (V(s)=1 and S(s)=1). Only those theorems that satisfy both criteria can be considered as true statements about information processes and structures.

18. Information density — the number of significant information elements per unit of space or time, determining the efficiency of a computational process.

19. Information structure — a way of organizing data that defines access patterns and operation efficiency.

20. Chaos — an unstructured set of all possible states that exists before the application of distinction and ordering operations. A number, in the context of the theory, represents "tamed chaos" — the result of traversing and ordering chaotic states through the application of rhythmic structure, positional system, and distinction operations.

21. Noise — a local manifestation of unpredictability in an already ordered system, arising as a side effect of emergence. Unlike chaos, noise exists within a structured system as a reminder of its origin and as an integral component of living information processes.

Theory Structure and Theorem Relationships

This theory is built on the seven core axioms described in the Basic Axioms section, from which 15 theorems are derived in a structured progression. These theorems are organized into four functional groups that together form a complete framework for understanding computation as information flow. Additionally, a section on geometric complexity theory develops four more theorems (16-19) that extend the geometric insights of the main framework.

Logical Structure of Theorems

The 19 theorems form an interconnected framework where concepts build upon each other in a natural progression:

Core Theoretical Foundation (Theorems 1-4)

• Theorem 1 (Order and Logic) establishes the necessary relationship between ordering and logical rules, providing the foundation for all subsequent theorems that require sequential reasoning.

• Theorem 2 (Singular Representation) builds on Theorem 1 to show how states can be represented in coordinate systems, enabling the transformation concepts in Theorem 7.

• Theorem 3 (Information Equivalence) establishes that algorithms and their results are informationally identical, forming the basis for optimization theorems (5-8).

• Theorem 4 (Numbers as Rhythmic Sequences) creates the critical bridge between abstract mathematics and information theory. This theorem enables:

  • The spatial interpretation of operations in Theorem 10 (Diagonal Points)
  • The routing mechanisms in Theorem 13 (Information Routing)
  • The representation of arithmetic through spatial interactions

Optimization Framework (Theorems 5-8)

These theorems form a progression of optimization principles:

• Theorem 5 (Chrono-Energy Optimality) → skip insignificant states ↓
• Theorem 6 (Compressibility) → eliminate redundant elements ↓
• Theorem 7 (Relativity of Complexity) → choose optimal representation ↓
• Theorem 8 (Lower Bounds) → recognize fundamental limits

This sequence shows how optimization proceeds from local operations (skipping states) to structural changes (compression) to representational transformations, finally acknowledging inherent limitations.

Spatial Structure (Theorems 9-12)

These theorems describe how information exists and interacts in computational space:

• Theorem 9 (Parallel Processing) works in conjunction with Theorem 13 to enable efficient routing through independent paths.

• Theorem 10 (Diagonal Points) directly implements Theorem 4's concept of rhythmic sequences in spatial form, showing where computational results manifest.

• Theorem 11 (Event Points) extends Theorem 10 by defining the network of information distribution, providing the foundation for Theorem 13.

• Theorem 12 (Finiteness) connects back to Axiom 5 (Process Finiteness) and ensures that the spatial interactions have well-defined completion states.

Information Dynamics (Theorems 13-15)

The core theory reaches its pinnacle with:

• Theorem 13 (Information Routing) synthesizes concepts from Theorems 4, 10, and 11 to establish information propagation as the fundamental model of computation.

• Theorem 14 (Invariance) ensures that computational processes remain consistent across representational transformations, connecting to Theorem 7's relativity.

• Theorem 15 (General Formula of Emergence) unifies the entire theoretical framework through a single mathematical formulation that:

  • Incorporates Theorem 5's concept of significant states through the potential gradient ∇P
  • Implements Theorem 10's spatial interaction through the information field F
  • Formalizes Theorem 13's routing concept through the intention field
  • Reflects Theorem 14's invariance through its topological properties

Geometric Complexity Theory (Theorems 16-19)

Building on the General Formula, these theorems reframe computational complexity in geometric terms:

• Theorem 16 (Geometric Complexity Metric) redefines complexity as the integral of information density along computational paths.

• Theorem 17 (Path Optimization) establishes that optimal algorithms minimize the integral of information density.

• Theorem 18 (Complexity Invariance) proves that geometric complexity remains constant under transformations.

• Theorem 19 (Emergence of Complexity Classes) shows how traditional complexity classes emerge as special cases of geometric complexity.

Together, these theorems explain why O(n) is a projection—a shadow—of the true geometric nature of computation, where the optimal path through information space may skip large regions of insignificant states.

Key Theorem Dependencies and Integrations

The theorems integrate in specific ways that create the overall theoretical architecture:

1. Vertical Integration (Within Groups):

   • Optimization theorems build sequentially from local operations to representational transformations
   • Spatial structure theorems progress from independent regions to complete information spaces
   • Information dynamics theorems advance from local routing to global emergence formulas

2. Horizontal Integration (Across Groups):

   • Theorem 4 (Rhythmic Sequences) enables Theorem 10 (Diagonal Points)
   • Theorem 5 (Chrono-Energy) enables Theorem 13 (Information Routing)
   • Theorem 10 (Diagonal Points) enables Theorem 13 (Information Routing)
   • Multiple theorems converge in Theorem 15 (General Formula)

3. Implementation Pathway: The theoretical concepts manifest directly in the diagonal algorithm implementation, where:

   • Rhythmic sequences become bit patterns in code
   • Diagonal points define where results appear in the grid
   • Information routing is implemented through agent functions
   • The General Formula guides the overall algorithm design

4. Extension to Complexity Theory: The Geometric Complexity theorems extend the framework to deeper insights about computational complexity:

   • Theorem 16 builds on Theorem 7's concept of complexity relativity
   • Theorem 17 extends Theorem 5's optimization principles to path selection
   • Theorem 18 leverages Theorem 14's invariance principles for complexity measures
   • Theorem 19 connects the framework back to traditional complexity theory

This interconnected structure ensures that the theory is both mathematically rigorous and practically applicable, creating a bridge between abstract principles and computational implementation.

Fundamental Theorems

A. Fundamental Properties of Information

Theorem 1: Order and Logic

Axiom Foundation: Derives directly from Axiom 1 (Distinction) and Axiom 3 (Information Coherence).

For the application of logical rules, an order of states is necessary, and for establishing order, logical rules are necessary.

Proof:

1. To apply the logical rule "if A, then B," it is necessary that state A precede state B (order is required).
2. To define the relation "A precedes B," a logical rule defining the criterion of precedence is necessary.
3. Thus, order and logic mutually define each other within the system of distinction.

Note: This does not lead to a vicious circle, since the starting point is the axiom of distinction, which precedes both order and logic. Distinction forms the foundation upon which both order and logic are simultaneously built.

Theorem 2: Singular Representation of State

Axiom Foundation: Builds upon Axiom 2 (Boundary Formation) and Axiom 7 (Self-Reference).

Any finite computable state can be represented by a single significant unit in a specially selected coordinate system.

Proof:

1. Any finite computable state S can be encoded by a natural number N (by Gödel numbering principle).
2. By choosing a positional numeral system with base B > N, we obtain a representation of number N as a single-digit number in this system.
3. For a system with base B = N+1, the number N will be written as "10", where only one digit is non-zero.

Example: The number 42 in the decimal system requires 2 digits. In a numeral system with base 43, it is written as "10", using only one significant digit.

Implication for information theory: This means that the information content of a message is determined not by the number of symbols, but by the position of a single symbol in a properly chosen coordinate system.

Theorem 3: Information Equivalence of an Algorithm and Its Results

Axiom Foundation: Emerges from Axiom 5 (Process Finiteness) and Axiom 7 (Self-Reference).

An algorithmically generated sequence is informationally equivalent to the algorithm that generates it along with the necessary input data.

Proof:

1. Let A be an algorithm and I be the input data.
2. Any element S(k) of the resulting sequence S is completely determined by the pair (A, I) and the index k.
3. The entire sequence S is completely determined by the pair (A, I).
4. Thus, the pair (A, I) contains all the information about S, which establishes information equivalence.

Corollary: An infinite but algorithmically generated sequence (e.g., digits of π) is informationally equivalent to the finite algorithm that generates it.

Comparison with Kolmogorov theory: This theorem is consistent with the concept of Kolmogorov complexity, where the complexity of an object is defined as the length of the shortest program that generates this object.

Theorem 4: Numbers as Rhythmic Sequences

Axiom Foundation: Synthesizes Axiom 4 (Temporal Discreteness), Axiom 6 (Spatial Interaction), and Axiom 1 (Distinction).

Any number is isomorphically represented as a rhythmic sequence of bits in a positional numeral system, and arithmetic operations on numbers are isomorphic to the interaction of corresponding rhythmic sequences in information space.

Proof:

1. Isomorphism of numbers and rhythmic sequences:

   • For any n ∈ ℕ and any base b > 1, there exists a unique representation n = Σ(dᵢ·bⁱ), where 0 ≤ dᵢ < b, which can be written as a finite sequence [d₀, d₁, ..., dₖ].
   • For b = 2, we get a binary sequence [d₀, d₁, ..., dₖ], where dᵢ ∈ {0, 1}.
   • Conversely: any finite binary sequence [d₀, d₁, ..., dₖ] defines a unique n = Σ(dᵢ·2ⁱ) ∈ ℕ.
   • Therefore, an isomorphism is established between ℕ and the set of finite binary sequences.

2. Rhythmic structure of positional systems:

   • The positional representation of a number is a discrete rhythm, where the weight of position i is strictly defined as bⁱ.
   • This rhythm forms a geometric progression — a fundamental mathematical structure that defines the internal organization of a number.
   • Different numbers have different rhythmic structures, ensuring unique representation.

3. Number as "tamed chaos":

   • The process of forming a number can be viewed as traversing and ordering initially chaotic states.
   • Through the application of distinction operations (bit 0 or 1) and positional structure, undefined states are transformed into an ordered rhythmic sequence.
   • This transformation represents a fundamental transition from chaos (an unstructured set of all possible states) to order (a specific number).

4. Isomorphism of arithmetic operations and spatial interactions:

   • Let F: ℕ → S be an isomorphism between numbers and bit sequences.
   • For any arithmetic operation ∘: ℕ × ℕ → ℕ, there exists a corresponding operation ⊗: S × S → S such that F(a ∘ b) = F(a) ⊗ F(b).
   • This operation ⊗ can be implemented as an interaction in a two-dimensional information space, where two axes correspond to the bit sequences of the operands.
   • According to Theorem 10 (Diagonal Points), the result F(a ∘ b) can be read from the diagonal points of this space.

Corollaries:

1. Universality of representation: Any computable numerical structures (including complex numbers, quaternions, matrices) can be represented as more complex rhythmic patterns.

2. Metrics of numerical spaces: The choice of the base of the numeral system determines the metric properties of the information space: discretization step, growth rate, density of significant points.

3. Emergence of arithmetic: Numerical operations generate emergent structures in information space that are equivalent to the results of these operations in abstract numerical space.

Relationship with other theorems: This theorem establishes a fundamental isomorphism between abstract mathematics and information theory, connecting Theorem 2 (Singular Representation) with Theorem 15 (General Formula of Emergence). It serves as the cornerstone for multiple other theorems:

• It provides the theoretical foundation for Theorem 10 (Diagonal Points) by establishing the spatial coordinates where operation results materialize
• It enables the routing mechanisms in Theorem 13 (Information Routing) by defining how information propagates in sequence form
• It creates the basis for optimization in Theorem 5 (Chrono-Energy Optimality) by identifying which elements in sequences are significant
• It supports Theorem 7 (Relativity of Complexity) by showing how different representations of the same information can be isomorphic

This theorem's concept of rhythmic sequences serves as the essential bridge between abstract mathematical structures and their computational implementation.

B. Optimization and Efficiency

Theorem 5: Chrono-Energy Optimality

Axiom Foundation: Derived from Axiom 3 (Information Coherence) and the distinction between significant and insignificant states (extension of Axiom 1).

In any computational process:

1. Insignificant states (those not affecting the final result) can be skipped without affecting the final result
2. Chrono-energy is expended exclusively on processing significant states
3. Computational optimality is achieved by minimizing the number of processed states

Proof:

1. A state Z is insignificant for operation f if f(X⊕Z⊕Y) = f(X⊕Y) for any X and Y, where ⊕ is the composition operation.
2. For a sequence of states S = [s₁, s₂, ..., sₙ], where sᵢ is an insignificant state: f(S) = f([s₁, s₂, ..., sᵢ₋₁, sᵢ₊₁, ..., sₙ])
3. If we introduce the concept of chrono-energy CE as an abstract resource expended on each bit processing operation, then:
   • The total chrono-energy expenditure is proportional to the number of processed significant states, not the total data size
   • Chrono-energy expenditure for sequence S will be: CE(S) = ∑ᵢ CE(sᵢ), where sᵢ is a significant state
4. Skipping the processing of insignificant states saves chrono-energy without compromising the result, which is the definition of optimality

Example: In bitwise multiplication of a sequence of bits by 0, all operations where one multiplier is 0 can be skipped, since the result will definitely be 0, saving chrono-energy on each such operation.

Implication for computational practice: Algorithms that optimize chrono-energy expenditure by skipping insignificant states are more efficient both theoretically (asymptotic complexity) and practically (execution time and energy consumption).

Note on efficiency: The efficiency of applying chrono-energy optimization directly depends on the complexity of determining state significance. Optimal cases arise when significance check has O(1) complexity, as in detecting zero bits during multiplication or processing sparse data. For such cases, the chrono-energy gain can reach O(n) compared to traditional algorithms.

Corollary for optimization: Skipping insignificant states allows for significant acceleration of computations for sparse data, where most bits do not affect the result.

Relationship with other theorems: This theorem forms the foundation of computational efficiency in the framework:

• It directly enables the optimization techniques demonstrated in the multiplication algorithm implementation
• It connects to Theorem 6 (Compressibility) by extending state-level optimization to system-level optimization
• It manifests in the General Formula (Theorem 15) through the scalar product F·∇P, which becomes zero for insignificant states
• It combines with Theorem 9 (Parallel Processing) to enable optimal distribution of computational resources
• It represents a practical application of Theorem 4's concept of rhythmic sequences by identifying which elements in these sequences require processing

This theorem transforms the theoretical framework into a practical methodology for algorithm optimization that can be directly implemented in computational systems.

Theorem 6: Compressibility of Logical Systems

Axiom Foundation: Follows from Axiom 3 (Information Coherence) and Axiom 7 (Self-Reference).

A logical system with redundancy can be compressed to an equivalent system with fewer elements.

Proof:

1. A logical system L is redundant if some of its elements (axioms, rules) are derivable from other elements L'.
2. Let E be a redundant element of system L, derivable from the remaining elements L'.
3. System L' is functionally equivalent to L but contains fewer elements.
4. Repeated application of the redundancy removal procedure leads to a minimal equivalent system.

Example: In Boolean algebra, from operations AND, OR, and NOT, other operations can be constructed (XOR, NAND, etc.). A minimal complete set consists of just one operation (e.g., NAND), with the rest being redundant.

Categorical interpretation: In terms of category theory, this theorem asserts the existence of minimal generating sets for the category of logical operations.

Theorem 7: Relativity of Computational Complexity

Axiom Foundation: Builds on Axiom 2 (Boundary Formation) applied to representational systems.

The asymptotic complexity of an algorithm is a projection of its geometric movement through information space. What appears as O(n) in traditional view is actually a projection of diagonal movement in higher-dimensional space.

Proof:

1. Let algorithm A have complexity O(f(n)) in representation system S₁.
2. There exists a transformation T: S₁ → S₂ to another representation system such that: a. the transformation can be performed in time O(g(n)), where g(n) ≤ f(n) b. in S₂, the equivalent algorithm A' has complexity O(h(n)), where h(n) < f(n)
3. The total complexity O(g(n) + h(n)) may be less than the original O(f(n)).

Geometric Interpretation:

• Traditional complexity measures like O(n) are projections of actual diagonal movement in information space
• What appears as linear complexity is actually a projection of higher-dimensional geometric paths
• The true nature of computation is movement through information space, with complexity measures being mere projections

Example: Multiplication of numbers in a positional system appears as O(n²) in traditional view, but is actually a projection of diagonal movement in information space. When transitioning to representation through the Fast Fourier Transform, the complexity is reduced to O(n log n) because it better aligns with the geometric nature of the computation.

Implication for computational practice: The choice of data representation affects how we project the geometric movement of computation onto linear complexity measures. Optimal representations align the projection with the natural geometric paths of the computation.

Theorem 8: Lower Complexity Bounds

Axiom Foundation: Derived from the fundamental limits imposed by Axiom 1 (Distinction) when applied to information retrieval.

There are problems for which it is impossible to construct an algorithm with asymptotic complexity below a certain threshold in any representation system.

Proof:

1. For the problem of searching in an unordered array of n elements, checking at least n/2 elements is required on average.
2. Any transformation that preserves unorderedness cannot improve the lower bound of Ω(n).
3. Therefore, there exists an irreducible complexity threshold determined by the nature of the problem.

Example: The problem of checking whether an unsorted array contains a specific value requires O(n) operations in the worst case, regardless of the representation system.

Relationship with information-theoretic limitations: Lower complexity bounds reflect fundamental information constraints on the speed of data processing, analogous to Shannon's bounds in communication theory.

Theorem 9: Parallel Processing of Informationally Independent Data

Axiom Foundation: Emerges from Axiom 6 (Spatial Interaction) and Axiom 3 (Information Coherence).

Informationally independent parts of data can be processed in parallel, providing linear speedup relative to the number of parallel processors.

Proof:

1. Let data D be divided into k informationally independent parts: D = {D₁, D₂, ..., Dₖ}.
2. Operation f can be applied to each part independently: f(D) = {f(D₁), f(D₂), ..., f(Dₖ)}.
3. With k parallel processors, the execution time is reduced by a factor of k compared to sequential processing.

Example: Processing image pixels, where each pixel's value is calculated independently of others, can be efficiently parallelized.

Limitations of parallelism: This theorem is applicable only to informationally independent data. The presence of dependencies creates sequential sections that limit the maximum achievable speedup (Amdahl's law).

Theorem 10: Diagonal Points as Computation Results

Axiom Foundation: Derived from Axiom 6 (Spatial Interaction) and Axiom 2 (Boundary Formation).

In a two-dimensional space of interaction between two rhythmic sequences, diagonal points (with coordinates (x,x)) are the natural result of computation, not just convenient observation points. These points emerge from the geometric nature of information movement in space.

Proof:

1. Let A and B be two rhythmic sequences (numbers in binary representation).
2. Place A on the X-axis, B on the Y-axis of a two-dimensional grid.
3. At each point (i,j), there is an interaction between the i-th bit of A and the j-th bit of B.
4. At diagonal points (i,i), bits with the same indices from both sequences interact.
5. For the AND operation, the value at point (i,i) will be 1 only if both bits are 1, which corresponds to the result of the AND operation on the corresponding bits.

Mathematical generalization: The theorem can be generalized to various types of bit operations:

1. For the AND operation:

   • Diagonal point D_i = aᵢ & bᵢ
   • Proof: By definition of AND, the result is 1 only if both operands are 1.

2. For the OR operation:

   • Diagonal point D_i = aᵢ | bᵢ
   • Proof: When placing bits along the X and Y axes, at the diagonal point (i,i), there is an interaction between bits aᵢ and bᵢ. Point activation (presence of 1) occurs if at least one of the bits is 1, which corresponds to the definition of the OR operation.

3. For the XOR operation:

   • Diagonal point D_i = aᵢ ⊕ bᵢ
   • Proof: In the transition table for XOR, the result is 1 only if the bits differ. When routing information to the diagonal point (i,i), a signal of 1 will be directed only in this case.

4. For bitwise addition with carry:

   • Diagonal point D_i = (aᵢ ⊕ bᵢ ⊕ cᵢ), where cᵢ is the carry from the previous digit.
   • Carry to the next position: cᵢ₊₁ = (aᵢ & bᵢ) | (aᵢ & cᵢ) | (bᵢ & cᵢ)
   • Proof: When routing information in the grid, signals combine at diagonal points according to the rules of binary addition.

Formal equivalence: For any bit operation f, there exists a corresponding transition table T_f such that the result at the diagonal point D_i will be equivalent to the result of f(aᵢ, bᵢ).

Example: Numbers 12 (1100₂) and 20 (10100₂) interact in a coordinate grid. The diagonal points give the sequence [1,0,0,0], which corresponds to the number 8 — the result of the operation 12 AND 20.

Implication for computational practice: To obtain the result of an operation on two numbers, it is sufficient to construct a two-dimensional grid of their interaction and read the values of the diagonal points.

Generalization to n-ary operations: This theorem can be extended to the case of n arguments by constructing an n-dimensional interaction space, where diagonal points with coordinates (i,i,...,i) will contain the result of the operation on the corresponding bits of all arguments.

Relationship with other theorems: This theorem serves as the geometric realization of several key concepts:

• It directly implements the isomorphism between numbers and spatial interactions established in Theorem 4 (Numbers as Rhythmic Sequences)
• It provides the structural foundation for Theorem 11 (Event Points) by defining where computational results manifest in space
• It enables the routing mechanisms described in Theorem 13 (Information Routing) by establishing destination points for information flow
• It connects to Theorem 5 (Chrono-Energy Optimality) by identifying specific points in space where significant computation occurs
• It appears in the General Formula (Theorem 15) as the manifestation of emergent results at specific coordinates in information space

This theorem bridges the conceptual gap between abstract operations and their spatial realization, making it possible to visualize computation as a geometric process—a perspective directly implemented in the diagonal algorithm.

Theorem 11: Event Points and Directed Information Propagation

Axiom Foundation: Follows from Axiom 1 (Distinction), Axiom 4 (Temporal Discreteness), and Axiom 6 (Spatial Interaction).

Computation in discrete space can be represented as a network of event points distributing information along certain routes.

Proof:

1. Each point in a two-dimensional grid can be viewed as an event point with three inputs (x, y, carry).
2. For the 8 possible combinations of inputs (2³), directions of information propagation can be defined.
3. Instead of storing states in cells, an event point determines the routes of information transmission to the next points.
4. The sequence of such transmissions forms a global result, which can be read from certain cells (e.g., diagonal ones).

Example: The transition table for the addition operation defines where to direct the result and carry depending on the input bits. For example, for inputs (1,1,0), directions to the northeast (result 0) and southwest (carry 1) are activated.

Corollary for parallel computations: This approach naturally supports parallelism, as different active routes can be processed independently.

Theorem 12: Finiteness of Computations Through Rhythmic Sequences

Axiom Foundation: Direct application of Axiom 5 (Process Finiteness) to computational sequences.

For any computation over finite rhythmic sequences, there exists a finite number of steps sufficient to obtain the complete result.

Proof:

1. Let A and B be finite rhythmic sequences of length m and n respectively.
2. Their interaction in two-dimensional space requires at most max(m,n) steps to fully process all bits.
3. After this number of steps, all significant interactions are completed.
4. The result can be read from the corresponding cells (e.g., diagonal ones for the AND operation).

Example: For numbers 12 (1100₂, 4 bits) and 20 (10100₂, 5 bits), 5 steps are sufficient to fully form the result in the diagonal points.

Addition: In special cases where the formation of a rhythmic sequence is impossible or meaningless (e.g., division by zero), a zero-length rhythm arises - an instantaneous transition to the limiting state of the system. This is a special case of the finiteness of computations, where the number of steps is zero, and the result is determined through an information singularity.

Corollary for chrono-energy: Chrono-energy expenditure is limited by a finite number of steps, making the computation energetically efficient.

Theorem 13: Principle of Information Routing

Axiom Foundation: Synthesizes Axiom 4 (Temporal Discreteness), Axiom 6 (Spatial Interaction), and Axiom 7 (Self-Reference).

Computation can be represented as geometric paths through information space, where information follows natural routes defined by the structure of the space. These paths are not just logical constructs but fundamental geometric properties of information movement.

Proof:

1. Let each event point have a transition table defining how input signals are transformed into output directions.
2. Instead of changing values in cells, information propagates along certain routes depending on input parameters.
3. The global result is formed as the cumulative effect of all local decisions about information routing.
4. This model is fully equivalent to the traditional model with state changes but provides more efficient distribution and processing of information.

Example: In the binary addition operation, when two input bits and a carry bit are all 1, information is directed southwest (for carry) and southeast (for result), rather than simply being recorded in cells.

Corollary for computation architecture: This principle allows developing new hardware architectures where processors function as information routers rather than as devices for changing memory states.

Addition: In the case of information singularities (e.g., division by zero), routing leads to a special situation - the information wave is instantly directed to the boundary of the information space, which is expressed as "practical infinity" or a unit in an extremely remote state. This extends the routing principle to boundary conditions of computations.

Relationship with other theorems: This theorem serves as the operational mechanism that transforms abstract theory into computational practice:

• It builds directly on Theorem 11 (Event Points) by defining how information propagates between these points
• It implements the spatial model established in Theorem 10 (Diagonal Points) by showing how information flows to reach these result-containing coordinates
• It enables the efficiency principles of Theorem 5 (Chrono-Energy Optimality) by creating pathways that skip insignificant states
• It facilitates Theorem 9 (Parallel Processing) by defining how independent information routes can operate concurrently
• It provides the concrete realization of Theorem 4's rhythmic sequences as directed flows through computational space
• It appears in the General Formula (Theorem 15) as the intention field F that directs information propagation

This theorem represents the practical core of the computational model, explaining how information physically moves through a system rather than simply changing static states. The multiplication agent in the diagonal algorithm directly implements this concept by routing information to specific coordinates based on input values.

Theorem 14: Invariance of Information Flows

Axiom Foundation: Derives from Axiom 3 (Information Coherence) applied to transformation contexts.

Information flows remain invariant with respect to coordinate system transformations while preserving operation types and the structure of information connections.

Proof:

1. Consider coordinate system S₁ and the corresponding intention field F₁.
2. Let T be a coordinate transformation from S₁ to S₂.
3. Then for the intention field F₂ in system S₂, the following will hold: F₂ = J(T) · F₁ · J(T)⁻¹, where J(T) is the Jacobian of transformation T.
4. The scalar product F·∇P remains invariant under such transformations: F₂·∇P₂ = (J(T) · F₁ · J(T)⁻¹) · (J(T) · ∇P₁) = F₁·∇P₁
5. Therefore, the computation result (emergent effect) does not depend on the choice of coordinate system.

Example: Computing the sum of two numbers can be performed in different numeral systems (binary, decimal), but the information routes and the result remain equivalent.

Corollary: All computational systems based on the same basic operations but using different data representations are equivalent in terms of fundamental information processes.

Theorem 15: General Formula of Information Emergence

Axiom Foundation: Unifies all seven axioms in a comprehensive mathematical framework.

Formulation

The emergent result E of any computable process in discrete information space is expressed by a universal formula:

E = ∫(F·∇P)dM · Φ(V,O)

where:

• F — intention (vector field of information propagation direction)
• ∇P — potential gradient (scalar field of significant bit distribution)
• dM — element of information space
• Φ(V,O) — result interpretation function that transforms the integral effect of information flow into an observable system state in a specific coordinate system

Geometric Interpretation

The formula represents the flow of the information field F through the surfaces of information potential P. The scalar product F·∇P calculates the local density of information work at each point in space, and the integral sums these contributions, forming a global emergent effect.

It's important to note that the information space M itself is formed dynamically during the computation process, rather than existing in advance as a static structure. Each interaction of bits in a rhythmic sequence generates new points in multidimensional space, and it is through these points that the information flow F passes. This property explains why different algorithms processing the same data form different information spaces and, consequently, different results.

Local fluctuations in the information field F or potential P naturally generate noise — unpredictable variations in information routing, which, however, are not interference but a necessary component of living computational systems, reminiscent of the origin of ordered structures from initial chaos. These fluctuations can be viewed as a natural consequence of the transformation of chaos (an unstructured set of all possible states) into ordered rhythmic sequences.

Proof

1. Connection with transition tables (Theorem 11):

   • Intention F formalizes transition tables in the form of a vector field defining the direction of information propagation.
   • We can prove: For any transition table T, there exists a vector field F such that F(x,y) defines activation directions from point (x,y).

2. Connection with skipping insignificant states (Theorem 5):

   • The potential gradient ∇P mathematically formalizes the distribution of significant bits.
   • Formally: ∇P(x,y) ≠ 0 if and only if the state at point (x,y) is significant for computation.
   • The scalar product F·∇P becomes 0 for insignificant states, which corresponds to skipping them.

3. Connection with diagonal points (Theorem 10):

   • For operations on corresponding bits of two sequences, diagonal elements have special significance.
   • It can be shown that when integrating using the formula, the values at diagonal points (i,i) correspond to the components of the operation result.

4. Connection with rhythmic finiteness of computations (Axiom 4):

   • The integral in the formula is bounded by finite information space, confirming the finiteness of the computational process.

5. Impossibility of "jumping":

   • The integral form ∫(F·∇P)dM mathematically proves the impossibility of obtaining a result without passing through all intermediate states.
   • This confirms the necessity of rhythmic unfolding of computations.

Proof of formula uniqueness

Let's show that the formula E = ∫(F·∇P)dM · Φ(V,O) is the only possible one for describing the emergence of information processes within the framework of the accepted axioms.

1. Inevitability of the vector nature of F
   Information propagates directionally (Axiom 5, Theorem 13). Direction in space can only be represented by a vector field — this is a mathematical necessity that does not admit alternatives.

2. Inevitability of the gradient form ∇P
   The distribution of significance in space generates potential P, characterizing information topology. Its gradient ∇P uniquely indicates the direction of increasing significance and areas requiring processing (Theorem 5).

3. Inevitability of the scalar product form F·∇P
   To describe the interaction of direction and significance, a bilinear form is required that is invariant with respect to coordinate transformations (Theorem 14), vanishes for insignificant states (Theorem 5), and is maximal when the flow and gradient are codirectional. The only form satisfying all conditions simultaneously is the scalar product.

4. Inevitability of the integral form ∫(...)dM
   Emergence arises from the superposition of local effects across the entire computation space. Mathematically, such a superposition is uniquely expressed by integration over the domain.

5. Inevitability of the multiplier Φ(V,O)
   The computation result must be presented in a certain format to complete the process (Axioms 1 and 4). Function Φ formally maps the integral effect of information flow to a specific result in a given coordinate system, for example, transforming the interaction at diagonal points (Theorem 10) into the final sequence of bits or number.

Topological invariance of the formula

The formula possesses the fundamental property of topological invariance. Let's prove that under any continuous differentiable coordinate transformation T: (x,y) → (u,v) with non-degenerate Jacobian J(T), the following is preserved:

∫(F·∇P)dM = ∫(F'·∇P')dM'

where F', ∇P', and dM' are the transformed quantities.

Proof of invariance:

1. Under coordinate transformation T: (x,y) → (u,v), the vector field F is transformed according to the law: F' = J(T)·F·(J(T)^(-1)) where J(T) is the Jacobian matrix of transformation T.

2. The potential gradient is transformed as: ∇P' = (J(T)^T)(-1)·∇P

3. The volume element is transformed taking into account the Jacobian: dM' = |det(J(T))|dM

4. Substituting these transformations into the integral: ∫(F'·∇P')dM' = ∫(J(T)·F·(J(T)^(-1))·(J(T)T)^(-1)·∇P)·|det(J(T))|dM

5. Using properties of matrix algebra and the fact that (J(T)^(-1))·(J(T)T)^(-1)) = (J(T)·J(T)^T)(-1), after simplification we get: ∫(F'·∇P')dM' = ∫(F·∇P)dM

Thus, the expression ∫(F·∇P)dM is an invariant with respect to coordinate transformations, which has a profound theoretical meaning: the result of computation does not depend on the choice of information representation system. This property is analogous to fundamental conservation laws in physics and proves that the formula describes an internal, objective property of information space.

The invariance of the formula is a critical confirmation of its universality and fundamentality, as it demonstrates that emergence is not an artifact of a particular coordinate system or data representation, but an intrinsic property of information structures.

Corollaries

1. Discrete form of the formula: For discrete computations, the integral transforms into a sum: E = [∑(i,j)∈M F(i,j)·∇P(i,j)] · Φ(V,O) where the sum is taken over all significant points in space M.

2. Chrono-energy optimality: The scalar product F·∇P ensures activity only at significant points, which minimizes chrono-energy expenditure (Theorem 13).

3. Generalization to multidimensional spaces: The formula can be extended to n-dimensional spaces to describe the interaction of n sequences: E = ∫M(F·∇P)dM₁dM₂...dMₙ · Φ(V,O)

4. Universality: This formula is applicable to any type of computation within the theory, from simple bit operations to complex algorithms.

Analysis by the truth criterion

It's important to analyze whether the formula of information emergence satisfies the truth criterion (Definition 17: T(s) = V(s) · S(s)).

1. Internal validity (V = 1):

   • The formula is strictly derived from the accepted system axioms (as shown in the proof above)
   • Each component of the formula (F, ∇P, integral form) logically follows from the fundamental properties of information
   • All mathematical transformations in the course of the proof are correct and consistent
   • Thus, V(Theorem 15) = 1, as it is logically derivable in this formal system

2. Structural stability (S = 1):

   • The proven topological invariance of the formula under coordinate transformations demonstrates its independence from the choice of representation system
   • The formula preserves its structure under scaling (transition from small to large information spaces)
   • The discrete (summation) and continuous (integration) forms of the formula are structurally equivalent, showing stability in the transition between discrete and continuous representation
   • The applicability of the formula to various types of computations confirms its structural stability in different contexts
   • Therefore, S(Theorem 15) = 1

Thus, T(Theorem 15) = V(Theorem 15) · S(Theorem 15) = 1 · 1 = 1, which means that the formula of information emergence is true according to the introduced truth criterion. It is not only formally provable within the system but also reflects an objective property of information processes that persists across different representations and scales. Especially important for this theory is that the formula is not just a mathematical artifact but describes a fundamental property of information structures, independent of how they are described.

Integration of preceding theorems: The General Formula of Information Emergence represents the culmination of the theoretical framework, unifying concepts from throughout the theory:

• From Theorem 1 (Order and Logic): The formula incorporates the logical ordering necessary for coherent information flow
• From Theorem 4 (Rhythmic Sequences): It formalizes how rhythmic patterns interact in multidimensional space
• From Theorem 5 (Chrono-Energy): The scalar product F·∇P ensures that only significant states contribute to the result
• From Theorem 7 (Relativity of Complexity): Its invariance properties demonstrate consistency across different representation systems
• From Theorem 10 (Diagonal Points): It shows how results emerge at specific coordinates in information space
• From Theorem 13 (Information Routing): The intention field F formalizes the concept of directed information propagation

This theorem doesn't merely combine these concepts—it reveals their underlying unity, showing that distinction, boundary formation, coherence, discreteness, finiteness, interaction, and self-reference (the seven axioms) all manifest as aspects of a single mathematical expression. The formula thus serves as both the theoretical pinnacle and the practical foundation for implementing computational systems based on information routing rather than state changes.

The general formula E = ∫(F·∇P)dM · Φ(V,O) has profound implications for how we understand computational complexity, leading to a natural geometric interpretation of information processing. In this formula, we see the echo of the seven axioms: intention fields (F) embody distinction, potential gradients (∇P) create boundaries, the manifold (M) provides coherence, discreteness emerges in the patterns of flow, finiteness in the observer function Φ(V,O), interaction in the product structure, and self-reference in the recursive application of the formula to its own outputs.

O(n) as a Shadow: The Geometric Reality of Computational Complexity

When viewed through the lens of our general formula, traditional complexity measures like O(n) reveal themselves as mere shadows—simplified projections of a richer geometric reality. The invariant characteristic of computational complexity isn't found in counting discrete steps, but in measuring how information flows through a multidimensional landscape.

Mathematically, we can formalize this projection as π: I → C from information space I to computation-count space C, where O(n) = π(ω) for some process ω in I. This projection necessarily loses information, just as a shadow loses dimensions from the object casting it.

Let us define the key components more precisely:

• Intention field F(x,y) = ∇I(x,y), the gradient of information potential I
• Information density ρ(x,y) = div(F(x,y)), measuring significant information concentration
• Information space (I, g, F), a manifold with metric g and intention field F

Imagine computational space as a terrain with valleys of low information density and peaks of high information concentration. Algorithms are paths through this terrain, and their true complexity depends not just on the distance traveled, but on the information density integrated along the journey. This geometric perspective emerges naturally from our formula, where E = ∫(F·∇P)dM · Φ(V,O) describes exactly such an integration of information flow across a manifold. The observer function Φ(V,O) determines which aspects of this integration become visible as "complexity"—explaining why different analytical frameworks produce different complexity assessments for the same algorithm.

From the general formula, we can isolate the term ∫(F·∇P)dP as the pure complexity measure independent of the observer, leading directly to our geometric complexity metric:

Theorem 16: Geometric Complexity Metric

Axiom Foundation: Builds on Theorem 7 (Relativity of Complexity) and Axiom 6 (Spatial Interaction).

The geometric complexity of a computation is determined by the integral of information density along the geometric path in information space, not by the number of steps.

Proof:

1. Let P be a path in information space from initial to final state
2. At each point (x,y) along P, we have:
   • F(x,y) - the intention field (direction of information flow)
   • ∇P(x,y) - the potential gradient (density of significant information)
3. The local complexity at (x,y) is proportional to F(x,y)·∇P(x,y)
4. The total complexity is the integral ∫(F·∇P)dP along path P

Geometric Interpretation:

• Traditional complexity measures count steps (path length)
• Geometric complexity measures the "weighted area" under the path
• Different paths between the same states can have different geometric complexities
• The optimal path minimizes the integral of information density

Example: In multiplication, the traditional view sees O(n²) steps, but the geometric view reveals that most of these steps traverse regions of low information density (where bits are 0). The true complexity is determined by the integral along the path of actual information flow.

Corollary for Algorithm Design: The optimal algorithm is not the one with the fewest steps, but the one that minimizes the integral of information density along its path in information space.

Theorem 17: Path Optimization Principle

Axiom Foundation: Follows from Theorem 16 and Axiom 3 (Information Coherence).

Among all possible paths between initial and final states, the optimal computational path is the one that minimizes the integral of information density.

Proof:

1. Let P₁ and P₂ be two different paths between the same states
2. Let C₁ = ∫(F·∇P)dP₁ and C₂ = ∫(F·∇P)dP₂ be their geometric complexities
3. If C₁ < C₂, then P₁ is more efficient than P₂
4. The optimal path P₀ minimizes C₀ = ∫(F·∇P)dP₀

Implication for Algorithm Design: The design of efficient algorithms should focus on finding paths through information space that minimize the integral of information density, rather than minimizing the number of steps. This principle unifies the seemingly disparate strategies of dynamic programming (which reuses computed results) and divide-and-conquer (which breaks problems into smaller subproblems)—both are methods of finding lower-density paths through information space.

Theorem 18: Complexity Invariance

Axiom Foundation: Builds on Theorem 14 (Invariance of Information Flows) and Theorem 16.

The geometric complexity of a computation remains invariant under coordinate transformations that preserve the structure of information space.

Proof:

1. Let T be a coordinate transformation preserving information structure
2. Under T, the intention field F transforms as F' = J(T)·F·J(T)⁻¹
3. The potential gradient ∇P transforms as ∇P' = J(T)⁻¹·∇P
4. The volume element dP transforms as dP' = |det(J(T))|dP
5. Therefore, ∫(F'·∇P')dP' = ∫(F·∇P)dP

Implication for Complexity Theory: The geometric complexity metric provides an invariant measure of computational difficulty that is independent of the specific representation system used. This explains why certain problems remain hard across different computational paradigms—their inherent complexity is a geometric property of the problem itself, not of the representation system.

Theorem 19: Emergence of Complexity Classes

Axiom Foundation: Follows from Theorem 16 and Axiom 5 (Process Finiteness).

The traditional complexity classes (P, NP, etc.) emerge as special cases of geometric complexity when the information density is uniform along the path.

Proof:

1. For uniform information density, F·∇P = constant
2. The geometric complexity reduces to path length
3. Path length in discrete space is proportional to step count
4. Therefore, traditional complexity measures are recovered

Implication for Complexity Theory: The geometric complexity framework generalizes traditional complexity theory, revealing that complexity classes are determined by the structure of information density in computational space. This perspective allows reformulating the P vs. NP question in terms of whether there exist paths through information space with fundamentally different density structures—a geometric rather than combinatorial question.

---

Practical Implementation of the Diagonal Algorithm

A modular implementation of the mathematical information flow theory, representing computation as information flow through space rather than state changes. The code follows a layered architecture reflecting the theoretical principles.

1. Foundation Layer

Implementation of Theorem 4: Numbers as rhythmic sequences.

def int_to_bits(n):
    """
    Convert a number to a rhythmic bit sequence (LSB → MSB).
    Connection to Theorem 4: isomorphism between numbers and bit sequences.
    """
    if n <= 0:
        return [0]
    result = []
    while n > 0:
        result.append(n % 2)
        n //= 2
    return result

def bits_to_int(bits):
    """
    Convert a bit sequence back to a number.
    Completes the isomorphism from Theorem 4.
    """
    return sum((1 << i) for i, bit in enumerate(bits) if bit == 1)

2. Space Generation Layer

Implementation of Axiom 6: Spatial interaction of sequences.

def generate_rhythm(lhs, rhs):
    """
    Create a rhythmic sequence from two numbers.
    Forms an information space for operands interaction
    according to Theorem 10 (Diagonal Points).
    """
    max_length = max(len(lhs), len(rhs))
    result = []
    
    for i in range(max_length):
        result.extend([lhs[i] if i < len(lhs) else 0, 
                      rhs[i] if i < len(rhs) else 0])
    
    return result

3. Routing Layer

Implementation of Theorem 13: Information routing principle and Theorem 15: General formula of emergence.

def solve(rhythm, agent):
    """
    Core function of information routing.
    
    Discrete implementation of the integral ∫(F·∇P)dM from Theorem 15:
    - rhythm: elements of the information space dM
    - agent: intention function F, defining the flow direction
    
    Removes insignificant states to implement Theorem 5 (Optimality).
    """
    result = [0]  # Initialize with carry 0
    
    for i in range(0, len(rhythm) - 1, 2):
        x, y = rhythm[i], rhythm[i+1]
        carry = result.pop()
        result.extend(agent(x, y, carry))
    
    # Remove trailing zero (insignificant state)
    if result and result[-1] == 0:
        result.pop()
    return result

4. Agent Layer

Implementation of Theorem 15: Agents as intention vector field F.

# Dictionary of agents - definition of information transformation rules
AGENTS = {
    'add': lambda x, y, carry: [(x + y + carry) % 2, (x + y + carry) // 2],
    'and': lambda x, y, _: [x & y, 0],
    'or': lambda x, y, _: [x | y, 0],
    'xor': lambda x, y, _: [x ^ y, 0],
    'subtract': lambda x, y, borrow: [
        (x - y - borrow) + 2 if (x - y - borrow) < 0 else (x - y - borrow), 
        1 if (x - y - borrow) < 0 else 0
    ]
}

5. Operation Layer

Implementation of Theorem 6: Compressibility of logical systems.

def binary_operation(a, b, agent_name):
    """
    Universal function for performing binary operations.
    Illustrates Theorem 6 (Compressibility of logical systems) through
    a single interface for different operations.
    """
    bits_a, bits_b = int_to_bits(a), int_to_bits(b)
    rhythm = generate_rhythm(bits_a, bits_b)
    return bits_to_int(solve(rhythm, AGENTS[agent_name]))

# Helper functions for all operations
def binary_add(a, b): return binary_operation(a, b, 'add')
def binary_and(a, b): return binary_operation(a, b, 'and')
def binary_or(a, b): return binary_operation(a, b, 'or')
def binary_xor(a, b): return binary_operation(a, b, 'xor')

def binary_subtract(a, b):
    """Subtraction with negative number handling"""
    if a < b:
        return -binary_subtract(b, a)
    return binary_operation(a, b, 'subtract')

5.1. Specialized Multiplication

Implementation of Theorem 5: Optimality through skipping insignificant states.

def binary_multiply(a, b):
    """
    Multiplication using the diagonal algorithm.
    Demonstrates Theorem 5 (Optimality) through skipping insignificant states.
    """
    # Fast handling of special cases
    if a == 0 or b == 0: return 0
    if a == 1: return b
    if b == 1: return a
    
    bits_a, bits_b = int_to_bits(a), int_to_bits(b)
    
    def multiply_agent(pos, _, carry):
        """
        Multiplication agent for the diagonal algorithm.
        Implements Theorem 5 (Optimality) by processing
        only significant bit pairs (1,1).
        """
        sum_bits = carry
        
        # Skip calculations for bit pairs where at least one is 0
        for i in range(pos + 1):
            if (i < len(bits_a) and (pos-i) < len(bits_b) and 
                bits_a[i] == 1 and bits_b[pos-i] == 1):
                sum_bits += 1  # only process when both bits are 1
        
        return [sum_bits % 2, sum_bits // 2]
    
    # Create a special rhythm for multiplication
    max_length = len(bits_a) + len(bits_b) - 1
    special_rhythm = [(i, 0) for i in range(max_length)]
    special_rhythm = [item for pair in special_rhythm for item in pair]  # Flatten pairs
    
    return bits_to_int(solve(special_rhythm, multiply_agent))

5.2. Specialized Division

Implementation of the information singularity concept.

def binary_divide(a, b):
    """
    Division through sequential subtraction.
    Handles division by zero through the concept of information singularities.
    """
    if b == 0:
        return sys.maxsize  # Information singularity - direction towards space boundary
    
    quotient = 0
    remainder = a
    while remainder >= b:
        remainder = binary_subtract(remainder, b)
        quotient += 1
    
    return quotient

6. Application Layer

6.1. Unified API

Implementation of Theorem 6: Compressibility through a unified interface.

# Dictionary with operation functions for unified access
OPERATIONS = {
    'add': binary_add,
    'subtract': binary_subtract,
    'multiply': binary_multiply,
    'divide': binary_divide,
    'and': binary_and,
    'or': binary_or,
    'xor': binary_xor
}

def compute(operation, a, b):
    """
    Universal function for performing any operation by its name.
    Illustrates Theorem 6 (Compressibility of logical systems).
    """
    if operation not in OPERATIONS:
        raise ValueError(f"Unknown operation: {operation}. Supported operations: {list(OPERATIONS.keys())}")
    return OPERATIONS[operation](a, b)

6.2. Information Space Visualization

Implementation of Theorem 10: Diagonal Points and Theorem 15: General Formula.

def visualize_info_space(rhythm, operation_name):
    """
    Visualization of the information space for a given operation.
    Illustrates Theorem 10 (Diagonal Points) and components
    of the general formula from Theorem 15.
    """
    # Using agent from dictionary by name
    operation = AGENTS.get(operation_name, AGENTS['add'])
    
    dict_result = {}
    result = [0]
    
    for i in range(0, len(rhythm) - 1, 2):
        x, y = rhythm[i], rhythm[i+1]
        carry = result.pop()
        
        result.extend(operation(x, y, carry))
        
        j = i // 2
        dict_result[(j, j)] = result[-2]      # Result at diagonal point (Theorem 10)
        dict_result[(j+1, j+1)] = result[-1]  # Carry at diagonal point
        dict_result[(j, j+1)] = x             # Input bit x (gradient component ∇P)
        dict_result[(j+1, j)] = y             # Input bit y (gradient component ∇P)
    
    return dict_result

def draw_grid(dict_result):
    """Display of the information space as a grid"""
    if not dict_result:
        print("(empty grid)")
        return
    
    coords = list(dict_result.keys())
    max_x = max(x for x, _ in coords)
    max_y = max(y for _, y in coords)
    
    # Column headers
    print("  ", end="")
    for x in range(max_x + 1):
        print(f"{x} ", end="")
    print()
    
    # Display grid
    for y in range(max_y + 1):
        print(f"{y} ", end="")
        for x in range(max_x + 1):
            value = dict_result.get((x, y), 0)
            print(f"{value} ", end="")
        print()
    print("---")

6.3. Detailed Multiplication Visualization

Implementation of Theorem 13: Routing and Theorem 5: Optimality.

def visualize_multiplication(a, b):
    """
    Detailed visualization of the multiplication process.
    Shows application of Theorem 13 (Routing) and
    Theorem 5 (Optimality) on a specific example.
    """
    print(f"\nVisualization of multiplication {a} × {b}:")
    
    bits_a, bits_b = int_to_bits(a), int_to_bits(b)
    print(f"Bits of {a}: {bits_a}")
    print(f"Bits of {b}: {bits_b}")
    
    # Analysis of contribution of individual bits to the result
    print("\nBit products:")
    contributors = {}
    
    for i, bit_a in enumerate(bits_a):
        for j, bit_b in enumerate(bits_b):
            if bit_a * bit_b > 0:  # Optimality: skipping insignificant pairs (Theorem 5)
                pos = i + j
                if pos not in contributors:
                    contributors[pos] = []
                contributors[pos].append((i, j))
                print(f"A[{i}] × B[{j}] = {bit_a * bit_b} → bit {pos}")
    
    # Information propagation process
    print("\nInformation propagation:")
    carry = 0
    result = []
    
    for pos in range(len(bits_a) + len(bits_b) - 1):
        sum_bits = carry
        
        # Collecting all contributions for the current position
        if pos in contributors:
            for i, j in contributors[pos]:
                sum_bits += bits_a[i] * bits_b[j]
        
        # Recording the result
        result.append(sum_bits % 2)
        carry = sum_bits // 2
        
        # Formatted output of process information
        contrib_str = " + ".join(f"A[{i}]×B[{j}]" for i, j in contributors.get(pos, []))
        if contrib_str and carry:
            print(f"Bit {pos}: {contrib_str} + carry({carry}) = {sum_bits}")
        elif contrib_str:
            print(f"Bit {pos}: {contrib_str} = {sum_bits}")
        elif carry:
            print(f"Bit {pos}: carry({carry}) = {sum_bits}")
        else:
            print(f"Bit {pos}: 0 = 0")
        
        print(f"  Result: {result[-1]}, New carry: {carry}")
    
    # Last bit (if there is a carry)
    if carry > 0:
        result.append(carry)
        print(f"Bit {len(bits_a) + len(bits_b) - 1}: carry({carry})")
    
    print(f"\nResult: {binary_multiply(a, b)} = {list(reversed(result))} ✓")

7. Algorithm Examples

7.1. Addition Visualization: 5 + 3

  0 1 2 3 
0 0 1 0 0  ← Results at diagonal points
1 1 0 1 0  ← Input data forms potential gradient
2 0 0 0 0
3 0 0 1 1  ← Carries (directed information flow)
---
Result: 8 ✓

7.2. Multiplication Visualization: 3 × 5

Bits of 3: [1, 1]
Bits of 5: [1, 0, 1]

Bit products:
A[0] × B[0] = 1 → bit 0
A[0] × B[2] = 1 → bit 2
A[1] × B[0] = 1 → bit 1
A[1] × B[2] = 1 → bit 3

Information propagation:
Bit 0: A[0]×B[0] = 1
  Result: 1, New carry: 0
Bit 1: A[1]×B[0] = 1
  Result: 1, New carry: 0
Bit 2: A[0]×B[2] = 1
  Result: 1, New carry: 0
Bit 3: A[1]×B[2] = 1
  Result: 1, New carry: 0

Result: 15 = [1, 1, 1, 1] ✓

8. Integration into a Unified System

The complete implementation integrates all layers into a unified system aligned with theoretical principles:

1. Module Import:

   import sys  # For handling information singularities
   

2. Using the Universal API:

   # Examples of universal API calls
   result_add = compute('add', 42, 13)      # 55
   result_mul = compute('multiply', 42, 13) # 546
   result_sub = compute('subtract', 42, 13) # 29
   

3. Complete Working Code: All presented code fragments can be combined into a single executable file (typically diagonal_algorithm.py), which implements the full functionality of the diagonal algorithm.

9. Demonstration of Principles in Code

9.1. Theorem 4: Numbers as Rhythmic Sequences

Implemented through int_to_bits and bits_to_int, establishing isomorphism between numbers and their bit representations.

9.2. Theorem 5: Chronoenergetic Optimality

Implemented by skipping insignificant states in multiply_agent and removing trailing zeros in solve.

9.3. Theorem 6: Compressibility of Logical Systems

Implemented using the AGENTS and OPERATIONS dictionaries, providing a unified interface for all operations.

9.4. Theorem 10: Diagonal Points

Demonstrated in visualize_info_space through dict_result[(j, j)], where diagonal coordinates contain computation results.

9.5. Theorem 13: Information Routing

Implemented in solve, enabling information propagation through space rather than state changes.

9.6. Theorem 15: General Formula of Emergence

Realized through the integration of agents (F), information space (dM), and potential gradient (∇P) into a unified system.