Axioms - Information V2, 2025 Edition

Welcome to the Axioms page for Information V2 (2025 Edition). This page provides a concise, structured overview of the fundamental axioms that guide information theory, data interpretation, and logical reasoning within modern information systems. Whether you are a student, researcher, or practitioner, the Axioms establish a shared foundation for rigorous analysis, reproducible results, and clear communication.

What are Axioms?

Axioms are fundamental statements assumed to be true within a given framework. They form the building blocks from which theorems, models, and practical methods are derived. In Information V2, the axioms define how information is quantified, processed, and transmitted, ensuring consistency across disciplines and applications.

Core Axioms (Information V2, 2025 Edition)

• Quantification Axiom: Information is measurable and quantifiable using a consistent unit (e.g., bits) to compare, aggregate, and optimize data.
• Independence Axiom: Distinct information sources or components contribute independently unless explicitly coupled, enabling modular analysis.
• Non-Negativity Axiom: Information measures are non-negative; zero information corresponds to complete certainty or no variation.
• Additivity Axiom: For independent information sources, total information is the sum of individual information measures.
• Monotonicity Axiom: Processing or compression cannot increase the total information content; any change is a reduction or preservation of uncertainty.
• Invariance Axiom: Information measures remain consistent under equivalent representations and transformations that preserve semantic content.
• Optimality Axiom: Systems should be designed to maximize useful information while minimizing redundancy and noise.

Note: These axioms are designed to be broadly applicable across domains, including communications, data science, machine learning, and cognitive science. They provide a common language for evaluating information flows and decision-making processes.

Why These Axioms Matter

• Establish a clear foundation for theoretical work and practical tools.
• Enable rigorous comparisons between models, algorithms, and systems.
• Help practitioners reason about trade-offs in compression, encoding, and transmission.
• Support reproducibility by anchoring methods to well-defined principles.

How to Use This Page

1. Read the core axioms to understand the framework.
2. See how each axiom informs specific techniques (e.g., encoding schemes, entropy calculations, or data aggregation).
3. Apply the axioms to evaluate new methods or optimize existing pipelines.

Further Reading

• Information Theory: Fundamentals and Applications
• Entropy, Redundancy, and Efficiency in Data Systems
• Principles of Data Compression and Transmission

If you have questions or want to contribute elaborations, contact the knowledge engineering team. This edition emphasizes clarity, interoperability, and practical applicability of information axioms across modern technologies.

Introduction

An update of the 2020 article “Axioms - Information” with the help of multiple of today’s best LLMs including GPT5 pro , Grok4, Google Gemini 2.5 pro, that emerged from me trying to resolve my discomfort with v = d / t, that led to a relativty aligned restatement of velocity as progress through spatial data. My discomfort came from the presence of d in both the denominator and numerator. While v = d / t appears to be understood as a high-school level approximation, and most experts in the field would apparently say my discomfort is resolved through the Lorentz transformations used by Einstein, I was still, stubbornly and pig headedly not fully satisfied ;-) which ultimately led to the equation :

v_max(ρ) = min( v_cap , İ / ρ )

With the essential insight that velocity, in many domains, is not only limited by physical constraints but also by information constraints. This is not an improvement on Einstein, as if that would be possible, by me at least. Instead it is a perspective of a different type appears to have many interesting applications which I will write about.

Information Axioms - Version 2

Legend of symbols (used throughout)

• v = speed / rate of safe progress (e.g., m/s, deploys/day)

• v_cap = hard ceiling on speed (hardware/process/policy)

• İ = usable information rate (task-relevant, validated signal per second)

• ρ = spatial (or per-unit-progress) information density required at the chosen fidelity (bits/m or bits/change)

• C = channel capacity (bits/s) with İ ≤ C

• H(·), I(·;·) = entropy, mutual information

• R(D) = rate–distortion at fidelity D (bits needed per unit)

• K(·) = minimal (algorithmic) description length (up to constants)

• c = speed of light; k_B = Boltzmann constant; T = temperature

Intuition / Explanation followed by Equation capturing the idea

1. Finite propagation of effects. No usable information or influence exceeds the speed of light c.

signal_speed ≤ c

1. Measurements are physical. What you “know” comes from a physical measurement process producing random variables conditioned on the world’s state.

Y ~ P(Y | state)

1. Observers have finite resources. You cannot process infinite data: there’s a ceiling on usable rate.

İ ≤ C (bits/s)

1. Don’t conflate substrate with quantity.Time being measured by spatial devices (clocks) does not make time = space.

(Conceptual non-equivalence; no formula identity)

1. Information is task-relative. What matters is information about the question/parameter θ, not raw entropy.

Use I(θ; X) rather than H(X)

1. Lossless summaries (for the task) exist.If a summary keeps all task-relevant information, nothing is lost.

I(θ; T(X)) = I(θ; X) (if T is sufficient)

1. Any other summarization loses task info.Processing can only reduce task-relevant information unless it’s sufficient.

I(θ; Y) ≤ I(θ; X) for any mapping X → Y

1. Clarity = quality of the signal about the source. Noise reduces clarity.

“Clarity” ∝ I(X; Y); adding noise ⇒ I(X; Y) decreases

1. Accuracy needs a stated loss. “More accurate” means lower expected loss under a named loss function.

Minimize E[ L( θ̂(X), θ ) ]

1. Some complexity is irreducible. Tasks have lower bounds on computation/communication to reach a target error.

cost(task, ε) ≥ formal_lower_bound(task, ε)

1. Descriptions have minimal length. You can’t compress truly random structure beyond its algorithmic complexity (up to constants).

description_length(X) ≥ K(X) + O(1)

1. Erasing information has a thermodynamic cost. Locally resetting 1 bit dissipates heat; globally information disperses, not magically vanishes.

E_dissipated ≥ k_B · T · ln 2 per bit

1. Spatial coding density. To act at fidelity D, you must resolve a minimum number of task-relevant bits per meter (or per change).

ρ(D) ≈ R(D) / (meters or changes)

1. Information-limited speed bound. You can’t move faster than your pipeline can supply required bits.

v ≤ İ / ρ

1. Safe speed is the tighter of info-limit and hard cap.

v_max(ρ) = min( v_cap , İ / ρ )

1. Control needs information. Some systems require a minimum information rate to remain stable.

Example (data-rate flavor): İ ≥ Σ log |λ_i| (units depend on formulation)

1. Keep units consistent. Bits/s over bits/m gives m/s (or “units of progress per unit time”).

units(İ / ρ) = (bits/s) / (bits/m) = m/s

1. Spacetime metric links space and time without equating them.

Flat spacetime: ds² = −c² dt² + dx² + dy² + dz²

1. Proper time is what ideal clocks measure along a worldline.

dτ = √( dt² − (dx² + dy² + dz²)/c² )

1. Macroscopic arrow of time from coarse-graining/boundary conditions.

Coarse-grained entropy tends to increase: dS/dt ≥ 0

1. Declare conventions up front. Use SI by default; if you adopt an alternative (e.g., defining time from distance via fixed speed), state it and stick to it.

(Procedural; no single equation)

Two immediate corollaries (pulled together)

• Information-limited velocity (then clipped): v_max = min( v_cap , İ / ρ )

Read: safe progress is limited by how many task-relevant bits you can turn into action per second, relative to how many you must resolve per step, and also by any hard ceiling.

• When summarization doesn’t hurt: If T(X) is sufficient for the task, using T(X) achieves the same decision quality as using X: I(θ; T(X)) = I(θ; X).

Physical constraints (causality, locality, bounded resources)

Finite propagation (light-cone constraint). No influence or usable information available to an observer travels faster than c.

signal_speed ≤ c

Local observation (relativity of simultaneity). All measurements occur on the observer’s worldline; observers at spacelike separation can disagree on “when” and order.

Bounded observers. Every real observer has finite time, memory, energy, and communication rate İ; no omniscience or omnipotence.

İ ≤ C (C = channel capacity, bits/s)

Information is local, uncertain, and distributed in space; access is gated by distance, capacity, and time.

Measurement vs. ontology (how we know vs. what is)

Operational realism. Measurements are physical processes producing random variables whose distributions depend on underlying states.

Y ~ P(Y | state)

Non-entailment. That clocks are spatially realized doesn’t mean “time = space.” Substrates enable measurement without being the measured quantity.

Information is task-relative (sufficiency, clarity, accuracy)

Task relevance. Information is always about a question/parameter θ. Use: I(θ; X) rather than H(X)

Sufficiency (lossless for the task). A summary T(X) that is sufficient preserves task-relevant information. Equation: I(θ; T(X)) = I(θ; X)

Data-processing (lossy otherwise). Any processing that is not sufficient can only reduce task-relevant information. Equation: I(θ; Y) ≤ I(θ; X) for any mapping X → Y

Clarity = signal quality. Clarity rises with I(X; Y); added noise or irrelevant redundancy lowers it.

Accuracy requires a loss. “Accurate” means lower expected loss under a named loss.

minimize E[ L( θ̂(X), θ ) ]

Complexity & resource lower bounds (what cannot be simplified)

Irreducible complexity. For a stated task and error tolerance, there are formal lower bounds on computation/communication. Some complexity is unavoidable.

Minimal description length. Any lossless description length is bounded below (up to constants) by algorithmic complexity K(·).

description_length(X) ≥ K(X) + O(1)

Local erasure cost; global conservation.

Resetting a bit dissipates ≥ k_B · T · ln 2 heat (Landauer). In closed evolution, information disperses rather than vanishes.

E_dissipated ≥ k_B · T · ln 2 per bit

Spatial information & speed bounds (velocity through spatial data)

Spatial coding density. Given fidelity D, acting safely/competently requires ρ(D) bits per meter (or per unit of progress).

ρ(D) ≈ R(D) / (meters or changes)

Information-rate bound on motion. With usable rate İ (bits/s), attainable progress speed obeys: Equation: v ≤ İ / ρ

Safe speed with caps.

Hard ceilings (physics, process, policy) impose v_cap.

v_max(ρ) = min( v_cap , İ / ρ )

Control-theoretic causality (example).

Some systems require a minimum information rate to remain stable.

İ ≥ Σ log |λ_i| (units depend on the formulation)

Instead of “time = distance / velocity,” compute safe velocity from information required per unit distanceand usable information per unit time, then clip by hard caps.

Equivalence, re-parameterization, and units

F1 — Functional equivalence. Two formulations are equivalent for a task if they preserve (i) sufficient statistics for θ and (ii) relevant resource bounds.

Dimensional consistency. Keep bridges explicit:

İ in bits/s, ρ in bits/m → İ / ρ in m/s. For non-spatial tasks, replace “meters” with “units of progress per step.” Time & spacetime structure (what “time” means operationally)

Spacetime metric (link, not identity). Space and time are interwoven via the metric; they’re not the same quantity.

Proper time and clocks. Ideal clocks measure proper time along worldlines; real clocks approximate this physically.

Arrow of time (macroscopic). Temporal asymmetry arises from boundary conditions and coarse-graining; micro-laws are (mostly) time-reversible.

Corollaries & Examples (recasting your A4–A14 motifs)

Certainty rises with task-relevant information If the tuple you provide is sufficient for the task, certainty can match that of full data.

I(θ; T(X)) = I(θ; X) (when T is sufficient)

Summarization usually reduces certainty — unless sufficient General processing cannot increase task-relevant information.

I(θ; Y) ≤ I(θ; X) Exception: equality when Y is sufficient (C2).

Clarity vs. noise/redundancy Noise or irrelevant redundancy lowers I(X; Y) and thus clarity; de-noising and focusing on task-relevant fields raises it.

Information-limited motion Given ρ bits/m and İ bits/s, safe speed is İ / ρ, then clipped by v_cap.

v_max(ρ) = min( v_cap , İ / ρ )

Irreducible complexity Some computation/communication is unavoidable to meet a specified error/fidelity; design for those lower bounds rather than trying to wish them away.

Appendix A — Definitions of Information (operational)

Bit: unit of information; in practice, use mutual information I(·;·) to quantify task-relevant signal. Storage/processing/propagation: obey physical law; İ ≤ C (capacity). Propagation speed limit: c; minimum time to acquire distant information is bounded by distance/c plus system latencies. Equations and Values (with bridges)

Speed of light in vacuum: 299,792,458 m/s. Complete graph edges: n(n−1)/2 (useful when modeling communication links). Kinematic identity: time = distance / velocity (units sanity). Information-limited velocity: v ≤ İ / ρ; with caps: v_max(ρ) = min( v_cap , İ / ρ ). Landauer bound: E_dissipated ≥ k_B · T · ln 2 per bit. Description length: description_length(X) ≥ K(X) + O(1). Appendix C — Time (clarified)

Operational viewpoint: time is what clocks measure (proper time along a worldline). Relativity: measured time depends on frame and path; simultaneity is relative; order can differ for spacelike separations. Distance-based timing: using time = distance / c is a declared measurement convention, not an identity of time with space. Appendix D — Existing Axioms (compatibility)

Equality axioms (reflexive/symmetric/transitive) and Euclid’s primitives remain valid as mathematical structure. New framework overlays task-relevant information, resource limits, and units on top of these, ensuring claims are operational and measurable. One-page checklist (use this when you apply the axioms)

Specify the task (θ), the loss L(·), and the required fidelity D. Declare units/bridges: İ (bits/s), ρ(D) (bits/m or bits/step), v (m/s or steps/s), v_cap. Identify sufficient summaries; avoid lossy processing unless it’s still sufficient. Respect lower bounds (computation, communication, Landauer). For progress through spatial data, use v_max(ρ) = min( v_cap , İ / ρ ) and ask: should we lower ρ, raise İ, or (if binding) lift v_cap? Keep dimensional consistency and make conventions explicit.