167_O1_Information-Primitive

O1 — Information Primitive

Chain Position: 167 of 188

Assumes

• [O1](./166_P5_Incompleteness-Stage]] (The P-sequence establishes the proof structure)

Formal Statement

[[167_O1_Information-Primitive.md) (Information Primitive): Information is an ontological primitive—it cannot be reduced to or derived from anything more fundamental. Information is not merely descriptive but constitutive of reality.

The P-sequence (P0-P5) established that existence (P0), consciousness (P1), information (P2), coherence (P3), agency (P4), and incompleteness (P5) form a proof structure. O1 now declares the ontological status: Information is not emergent from more basic physical stuff; information IS the basic stuff. Matter, energy, spacetime—all reduce to information patterns. Information does not reduce further.

Formal Expression: $\forall x:\text{Physical}(x)⟹\exists I:x=\mathcal{F}(I)$

Everything physical is a function of information. But: $\nexists y:\text{Information}=g(y)\text{ where }y\ne \text{Information}$

Information is not reducible to non-information.

Wheeler’s Principle: “It from Bit”—every physical entity derives from information.

Enables

• [O2](./168_O2_Coherence-Primitive]] (Coherence as ontological primitive)

Defeat Conditions

Defeat Condition 1: Matter Primacy

Falsification Criterion: Demonstrate that matter is ontologically prior to information—that information emerges from non-informational matter. Evidence Required: Define “matter” without recourse to its informational properties. Show how information emerges from information-less substrate. Counter-Evidence: Every physical property IS information: mass (kg of information), charge (1 bit: +/-), spin (quantum number = information), position (coordinates = information). Matter without information is undefined.

Defeat Condition 2: Information Emergence

Falsification Criterion: Show information arising from something more fundamental through physical processes. Evidence Required: Demonstrate a pre-informational stage where information did not exist, then describe the mechanism by which it emerged. Counter-Evidence: Any such description uses information (the description itself is information). The attempt is self-refuting. Information is presupposed in any explanation.

Defeat Condition 3: Multiple Realizability Reversal

Falsification Criterion: Prove that different information patterns can reduce to identical physical states—that information supervenes on physics rather than vice versa. Evidence Required: Show two distinct information contents instantiated in physically identical systems. Counter-Evidence: Landauer’s principle shows information has physical consequences. Information differences imply physical differences. Supervenience goes from physics to information, not information to physics.

Defeat Condition 4: Semantic Grounding Problem

Falsification Criterion: Demonstrate that information without semantic content is not truly “information”—that raw bits require interpretation and thus are not primitive. Evidence Required: Show that Shannon information (syntactic) requires semantic grounding to be meaningful, and that semantic grounding is more primitive than syntax. Counter-Evidence: Semantics emerges from syntax via coherence ([[168_O2_Coherence-Primitive.md)) and consciousness (O3). The primitives work together. Shannon information is primitive; semantic content emerges from primitive interactions.

Standard Objections

Objection 1: Information Requires Physical Substrate

“Information is always instantiated in something physical—patterns in matter, energy states, etc. The substrate is primary; information is secondary.”

Response: The substrate itself is describable only through its informational properties. What is an electron? A set of quantum numbers (information). What is a field? A function mapping spacetime points to values (information). The substrate IS information; saying information needs a substrate confuses levels of description. The substrate “carrying” information is itself information at a different level. It’s information all the way down.

Objection 2: Rolf Landauer vs. Wheeler

“Landauer said ‘Information is physical,’ not ‘Physics is informational.’ You’ve inverted his claim.”

Response: Landauer’s insight that information has physical cost (kT ln 2 per bit erased) supports O1, not contradicts it. If information were merely abstract, it would have no physical cost. Landauer proves information is physical—but Wheeler extends this: if information is physical, and all physics reduces to informational properties, then information is the physical primitive. Landauer and Wheeler are complementary, not contradictory.

Objection 3: Category Mistake

“Information is a category of description; matter is a category of stuff. You’re conflating epistemology (what we know) with ontology (what exists).”

Response: O1 explicitly asserts this is not a category mistake but a category identification. The epistemological/ontological distinction assumes matter exists independently of its informational description. But quantum mechanics shows measurement (information extraction) is constitutive. There is no “matter” outside of its informational characterization. The categories converge because reality is informational.

Objection 4: Digital Physics is Fringe

“The idea that reality is fundamentally computational/informational is speculative and not mainstream physics.”

Response: Digital physics (Zuse, Fredkin, Wolfram) is one version of informational ontology, but O1 doesn’t require it. O1 requires only that information is primitive—not that reality is a cellular automaton. Quantum information theory is mainstream and supports informational primitiveness. AdS/CFT holography is mainstream and encodes gravity as boundary information. The holographic principle is mainstream and bounds entropy by surface area (information). Information primacy is increasingly standard, not fringe.

Objection 5: Infinite Regress

“If information is primitive, where did the original information come from? You’ve just pushed the regress.”

Response: O1 declares information primitive precisely to stop the regress. Primitives are termination points—you don’t ask “what is X made of?” when X is primitive. The question “where did information come from?” assumes information is derivative; O1 denies this. Information exists necessarily with existence (P0) and consciousness (P1). The question is like asking “where did existence come from?”—it’s category confusion.

Defense Summary

O1 (Information Primitive) is defended through:

1. Reductive success: All physical properties reduce to information
2. Landauer’s Principle: Information is physical (has energy cost)
3. Holographic Principle: Spacetime is boundary information
4. Quantum Mechanics: Physical states ARE informational structures
5. Definitional closure: “Matter” cannot be defined without information

Information is not what we know about reality; information is what reality is. O1 establishes information as the first ontological primitive in the O-sequence.

Built on: [O1](./166_P5_Incompleteness-Stage]] Enables: 168_O2_Coherence-Primitive

Collapse Analysis

If [[167_O1_Information-Primitive.md) fails:

Information becomes secondary, derived, emergent:

• Some non-informational “stuff” is more fundamental
• We return to 19th-century materialism
• Quantum mechanics loses its informational interpretation
• The Theophysics framework loses its substrate

Downstream breaks:

• [chi-field](./168_O2_Coherence-Primitive]] becomes coherence of what? (needs information to cohere)
• 169_O3_Consciousness-Primitive loses its connection to physical reality
• The [[011_D2.2_Chi-Field-Properties.md) has no informational basis

Physics Layer

Landauer’s Principle (Revisited)

Information is Physical: $\Delta Q\ge k_{B}T\ln 2$

Erasing one bit costs at least $k_{B}T\ln 2$ energy. This is not metaphor; it’s thermodynamics.

Experimental Verification: Bérut et al. (2012): Colloidal particle in double-well potential. Erasure measured at Landauer bound. Information has measurable physical cost.

Implication: If information has physical cost, information is physical. If information is physical and all physics reduces to information, information is the physical primitive.

Bekenstein-Hawking Entropy (Revisited)

Black Hole = Information: $S_{BH}=\frac{A}{4{\ell }_P^2}$

Black hole entropy equals surface area in Planck units divided by 4. This entropy IS the information content of the black hole.

Information Paradox: If black holes destroy information (Hawking radiation is thermal), unitarity is violated. Resolution: information is preserved on horizon (holographic). Information is never destroyed—it’s fundamental.

Firewall Debate: AMPS paradox suggests either horizon is firewall or some other principle fails. All resolutions involve information being fundamental—what happens to it, where it goes, how it’s encoded.

Holographic Principle

‘t Hooft-Susskind: The maximum information in a region is proportional to its surface area, not volume: $I_{max}=\frac{A}{4{\ell }_P^2}\text{ bits}$

Interpretation: 3D physics is encoded on 2D boundary. The “bulk” is emergent from boundary information. Information is primary; spacetime is derived.

AdS/CFT Correspondence: Gravity in (d+1)-dimensional anti-de Sitter space = conformal field theory on d-dimensional boundary. $Z_{gravity}[AdS]=Z_{CFT}[\partial AdS]$

Gravity IS information on a boundary. Spacetime emergence from entanglement (ER=EPR).

Quantum Information as Fundamental

Wavefunction as Information: $|\psi ⟩={\sum }_i{c}_i|i⟩$

The wavefunction encodes probability amplitudes—information about possible measurement outcomes. It’s not a physical wave; it’s an informational structure.

Quantum State Tomography: Physical states are reconstructed from measurement statistics. The state IS the information extractable from it. There’s no state “behind” the information.

No-Cloning Theorem: $\nexists U:U|\psi ⟩|0⟩=|\psi ⟩|\psi ⟩$

Information cannot be perfectly copied. This is physical law, not technological limitation. Information has fundamental properties.

It from Qubit

Quantum Information Generalization: Wheeler’s “It from Bit” extends to “It from Qubit.” Qubits are the fundamental informational units.

Quantum Supremacy: Quantum computers demonstrate that qubits process information in ways impossible for classical bits. Quantum information is physically distinct.

Quantum Error Correction: Spacetime may be a quantum error-correcting code (QECC). The structure of spacetime emerges from the structure of quantum information protection.

Thermodynamics of Information

Maxwell’s Demon Resolution: Demon creates order (sorts molecules) but pays entropy cost through memory erasure (Landauer). The thermodynamic cost of information is real.

Szilard Engine: Information about particle position can be converted to work. Information is a thermodynamic resource.

Second Law via Information: Entropy increase = information dispersal. The Second Law is an information-theoretic principle.

Mathematical Layer

Shannon Information (Foundational)

Definition: $H(X)=-{\sum }_{x}p(x){\log }_{2}p(x)$

Uniqueness Theorem (Khinchin): Shannon entropy is the unique measure satisfying:

1. Continuity
2. Monotonicity (more equiprobable outcomes = more entropy)
3. Additivity for independent sources
4. Recursivity (chain rule)

This uniqueness suggests Shannon entropy captures something fundamental—not arbitrary choice but unique characterization of information.

Kolmogorov Complexity (Algorithmic Information)

Definition: $K(x)=\min \{|p|:U(p)=x\}$

Shortest program length generating string x on universal machine U.

Invariance Theorem: $|K_U(x)-K_V(x)|\le c$

Complexity is machine-independent up to constant. K captures objective property, not arbitrary encoding.

Uncomputability: K is uncomputable—there’s no algorithm computing K(x) for all x. Information has properties transcending computation.

Category of Information Systems

Category $\mathbf{Info}$:

• Objects: Probability distributions $(\mathcal{X},p)$
• Morphisms: Stochastic maps (channels)

Monoidal Structure: Independent sources tensor: $(\mathcal{X},p)\otimes (\mathcal{Y},q)=(\mathcal{X}\times \mathcal{Y},p\times q)$

Entropy as Functor: $H:\mathbf{Info}\to {\mathbb{R}}^{+}$

Maps information sources to their entropy. Functoriality: $H(X\otimes Y)=H(X)+H(Y)$ for independent sources.

Quantum Information Category

Category $\mathbf{Hilb}$:

• Objects: Hilbert spaces
• Morphisms: Completely positive maps (CPTP for quantum channels)

Dagger Structure: $†:{\mathbf{Hilb}}^{op}\to \mathbf{Hilb}$

Adjoint functor reflecting quantum conjugation.

Compact Closure: Every object has dual: $A^{\ast }=A$ (self-duality of Hilbert spaces). Enables definition of entanglement.

Information Algebras

Boolean Algebra: Classical information: bits form Boolean algebra under AND, OR, NOT.

Orthomodular Lattice: Quantum information: propositions form orthomodular lattice (quantum logic).

Difference: Distributivity fails in quantum case: $A\wedge (B\vee C)\ne (A\wedge B)\vee (A\wedge C)$

Information structure determines logic structure.

Topos-Theoretic Information

Presheaf Topos: Information relative to contexts forms presheaf: $F:{\mathcal{C}}^{op}\to \mathbf{Set}$

Subobject Classifier: Truth values in topos are sieves—sets of contexts where proposition holds. Information-theoretic truth is contextual.

Internal Logic: Topos has internal logic (intuitionistic). The logic OF information emerges from its categorical structure.

Proof: Information Non-Reducibility

Theorem (Information Primacy): For any property P of physical system S, there exists information I such that P = f(I) for some function f.

Proof:

1. Let S be any physical system with property P
2. P is measurable (else not physical by operationalism)
3. Measurement yields outcome o ∈ O (outcome space)
4. Outcome encodes information: I = log_2 |O| bits minimum
5. Property P is determined by measurement outcomes
6. Therefore P = f(I) for f mapping information to property value $\square$

Corollary: Physics is information. All physical properties are informational.

Information Metric Spaces

Kullback-Leibler Divergence: $D_{KL}(P||Q)={\sum }_{x}p(x)\log \frac{p(x)}{q(x)}$

Measures information lost when Q approximates P. Not a metric (asymmetric) but defines geometry.

Fisher Information Metric: $g_{ij}=E\left[\frac{\partial \log p}{\partial {\theta }_i}\frac{\partial \log p}{\partial {\theta }_j}\right]$

Riemannian metric on statistical manifold. Information defines geometry (information geometry, Amari).

Quantum Fidelity: $F(\rho ,\sigma )=\text{Tr}\sqrt{\sqrt{\rho }\sigma \sqrt{\rho }}$

Measures information overlap between quantum states.

Type-Theoretic Information

Information as Type: In dependent type theory, information content is captured by type.

• Type $A$ has inhabitants $a:A$
• Information = which inhabitant

Propositions as Types: Logical proposition = type. Proof = inhabitant. Information about truth = existence of proof term.

HoTT Perspective: Types have homotopy structure. Information about identity = path structure. Higher information = higher paths.

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Source Material

• 01_Axioms/AXIOM_AGGREGATION_DUMP.md
• [O1](./166_P5_Incompleteness-Stage]] (upstream)
• 168_O2_Coherence-Primitive (downstream)
• Wheeler, “It from Bit” (1990)
• Landauer, “Information is Physical” (1991)
• Bekenstein, “Information in the Holographic Universe” (2003)

Prosecution (Worldview Cross-Examination)

The Charge

The court charges Materialism with claiming matter is primitive while being unable to define matter without information. The defendant must explain what matter IS, stripped of all informational properties.

Cross-Examination

To the Materialist: Define “matter” for me. You say mass? Mass is a number—information. Charge? A quantum number—information. Position? Coordinates—information. Spin? Quantum state—information. You have no “matter” left once you remove the information. Your primitive dissolves.

To the Dualist: You separate mind (information processor) from matter (stuff). But we’ve shown matter IS information. Your duality collapses into information monism. Mind processes information; matter is information. There’s just information, variously organized.

To the Physicalist: You claim physics is fundamental. But physics is mathematical description of patterns—information. What’s the physics “about” if not information? You can’t point to something non-informational that physics describes.

To the Instrumentalist: You claim information is just a tool for prediction. But your predictions are themselves information, and they’re verified by measurement outcomes—more information. Everything in your practice is information. Your instrumentalism has no non-informational ground.

Verdict

[167_O1_Information-Primitive.md) is established. Information is ontologically primitive. All physical properties reduce to information; information reduces to nothing further. Reality is, at base, informational.

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Category: [|Existence Ontology is established. Information is ontologically primitive. All physical properties reduce to information; information reduces to nothing further. Reality is, at base, informational.

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Category: [[_WORKING_PAPERS/Existence_Ontology/|Existence Ontology.md)

Depends On:

• [Information Theory](./040_P5.1_Phi-Admits-Degrees]]

Enables:

• 168_O2_Coherence-Primitive

Related Categories:

• [Information_Theory/.md)

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