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D1.1 — Information Definition

Chain Position: 4 of 188

Assumes

A1.1 (Existence) - Something must exist to have uncertainty about
A1.2 (Distinction) - Reduction of uncertainty requires distinguishable states
A1.3 (Information Primacy) - Information is ontologically primitive, not derivative

Formal Statement

** Information ≡ that which reduces uncertainty about the state of a system.

Enables

D1.2 (Bit Definition) - The minimal quantum of uncertainty reduction
LN1.1 (Matter-Energy Derivative) - If information is defined, matter/energy derive from it
D5.2 (Integrated Information Φ) - Conscious information processing builds on this
All χ-field equations that quantify information content

Defeat Conditions

To falsify this definition, one would need to:

Provide an alternative definition of information that does not reference uncertainty reduction
Show a case where “information” exists but uncertainty is not reduced
Demonstrate that Shannon’s formalization is fundamentally flawed

Physical grounding: Shannon entropy H(X) = -Σ p(x) log p(x) quantifies exactly the uncertainty reduction upon learning X. Every experimental test of information theory confirms this definition operationally.

Standard Objections

Objection 1: Semantic vs Syntactic Information

“Shannon information ignores meaning. Your definition is purely syntactic.”

Response: Correct that Shannon entropy is syntax-only. But A1.3 establishes that χ carries semantic content. This definition captures the minimal operational criterion. Semantic information is uncertainty reduction about meaning, which still fits the definition. The χ-field adds the semantic layer; this definition provides the quantitative foundation.

Objection 2: Quantum Information Is Different

“Quantum information (qubits) doesn’t fit classical Shannon theory”

Response: Von Neumann entropy S(ρ) = -Tr(ρ log ρ) is the quantum generalization and still measures uncertainty reduction. The definition holds; only the mathematical formalism changes. A qubit’s information content is still “that which reduces uncertainty about the quantum state.”

Objection 3: Kolmogorov Complexity Alternative

“Algorithmic information (K-complexity) is more fundamental”

Response: Kolmogorov complexity K(x) measures the minimal description length of x—which is itself a measure of how much uncertainty is reduced by receiving x. They are complementary, not competing. Both confirm information as uncertainty reduction.

Defense Summary

This definition is the operational standard in physics, computer science, and communication theory. Shannon’s 1948 formalization remains unrefuted after 78 years of rigorous application. Every objection either:

Proposes a specialization (semantic, quantum, algorithmic) that still reduces to uncertainty reduction
Confuses the definition with specific formalisms (bits vs nats vs qubits)

The definition is mathematically minimal and experimentally confirmed across all scales from quantum cryptography to neural coding.

Collapse Analysis

If D1.1 fails:

No quantitative measure of information → χ-field becomes unmeasurable
Shannon entropy loses meaning → thermodynamics loses informational grounding
Quantum information theory collapses → no decoherence theory
D1.2 (Bit) has no parent definition → information hierarchy fails
Cannot define Φ (integrated information) → consciousness theory fails

Collapse radius: HIGH - Foundational definition for all quantitative claims about information

Physics Layer

Thermodynamic Information

Maxwell’s demon resolution (Szilard 1929, Bennett 1982): The demon must acquire information about particle positions. Landauer’s principle: erasing this information costs k_B T ln 2 per bit, exactly compensating the work extracted. Information acquisition/erasure is a physical process.

Entropy-information correspondence: $S = k_{B} H$ Boltzmann entropy S relates to Shannon entropy H by Boltzmann’s constant. Thermodynamic entropy IS information (measured in different units).

Black hole thermodynamics:

Bekenstein entropy: S = k_B A/(4ℓ_P²)
Hawking radiation carries information out
Information paradox resolution (via AdS/CFT): information is conserved, not destroyed

Quantum Information

Von Neumann entropy: $S (ρ) = - Tr (ρ lo g ρ)$ Quantum generalization preserves the operational meaning: it’s the minimum number of qubits needed to faithfully represent the state.

Holevo bound: $χ \leq S (ρ) - \sum_{i} p_{i} S (ρ_{i})$ Maximum classical information extractable from quantum ensemble ≤ quantum uncertainty reduction.

No-cloning theorem: Quantum information cannot be perfectly copied. This distinguishes quantum from classical information and proves information has physical constraints.

Communication Theory Confirmation

Shannon’s channel capacity theorem (1948): $C = max_{p (x)} I (X; Y) = max_{p (x)} [H (Y) - H (Y ∣ X)]$ Capacity = maximum uncertainty reduction per channel use. Experimentally confirmed across all communication systems.

Error correction: Information can be reliably transmitted through noisy channels iff rate < capacity. The physical implementation (fiber optic, radio, neural) is irrelevant; the information-theoretic bound is universal.

Connection to χ-Field

The χ-field’s information content is defined via this formalism:

Local χ-entropy: S_χ(x,t) = -∫ χ log χ dV
Coherence = mutual information between χ configurations
The Master Equation tracks information flow through χ

Mathematical Layer

Shannon’s Axiomatic Foundation

Shannon entropy is uniquely characterized by these axioms (Khinchin 1957):

Continuity: H(p₁,…,p_n) is continuous in all p_i
Maximum: H(1/n,…,1/n) = f(n) is monotonically increasing in n
Recursivity: H(p₁,…,p_n) = H(p₁+p₂,p₃,…,p_n) + (p₁+p₂)H(p₁/(p₁+p₂), p₂/(p₁+p₂))

Uniqueness theorem: Any function satisfying these axioms has the form: $H (p_{1}, ..., p_{n}) = - k \sum_{i} p_{i} lo g p_{i}$ for some constant k > 0. Shannon entropy is the ONLY consistent measure of uncertainty.

Rényi Entropy Family

Generalization: $H_{α} (X) = \frac{1}{1 - α} lo g \sum_{x} p (x)^{α}$

α → 1: Shannon entropy H(X)
α = 0: Hartley entropy (log of support size)
α = 2: Collision entropy (used in cryptography)
α → ∞: Min-entropy (worst-case uncertainty)

All satisfy the core property: uncertainty about distinguishable outcomes.

Mutual Information Decomposition

Chain rule: $I (X; Y, Z) = I (X; Y) + I (X; Z ∣ Y)$

Data processing inequality: $X \to Y \to Z ⟹ I (X; Z) \leq I (X; Y)$ Processing cannot create information—can only destroy it. This is the information-theoretic Second Law.

Category-Theoretic Information

Probability monad: The functor P: Set → Set sending X to probability distributions on X. Shannon entropy is a natural transformation measuring “spread” of distributions.

Divergence as morphism: KL-divergence D(p||q) measures how much q fails to describe p. It’s not symmetric, reflecting the asymmetry of information transfer.

Operational Definitions

Source coding theorem:

The average codeword length L ≥ H(X)
Equality achievable in the limit (Huffman, arithmetic coding)
H(X) IS the optimal compression rate

Channel coding theorem:

Error probability → 0 iff rate < capacity
Capacity C = max I(X;Y) over input distributions
Achievable via random codes (practical codes: turbo, LDPC, polar)

These operational theorems ground the definition in DOING, not just BEING.

Source Material

01_Axioms/_sources/Theophysics_Axiom_Spine_Master.xlsx (sheets explained in dump)
01_Axioms/AXIOM_AGGREGATION_DUMP.md

Category: Information_Theory/|Information Theory

Depends On:

[Information Theory](./001_A1.1_Existence]]
002_A1.2_Distinction
003_A1.3_Information-Primacy

Enables:

005_D1.2_Bit-Definition

Related Categories:

[Information_Theory/.md)

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