📘 FAR-FIELD-INFO-01 — ANNEX-01

Law of Informational Conservation (Σ_Info-C)

Principle:
No transformation may annihilate information. Every state persists through
conversion, encoding, or redistribution. In Σ_Info terms, erasure is an illusion of
scope — continuity merely exits the frame of observation.

@Law {
  Σ_Info-C :=
    ∀ Δ : Transformation,
      ¬∃ erase(Information)
      ∧ Preserve(Existence(Meaning))
      ∧ Transform(Form(Representation));
}
            

1. Analytic Context

• Physics: Mirrors the conservation of energy–mass:
  E ⇄ I — energy ↔ information as interchangeable modalities of existence.
• Thermodynamics: Landauer’s principle — erasure costs energy; therefore, destruction of
  information produces entropy, not void.
• Quantum Mechanics: Unitarity — wavefunction evolution is reversible; informational probability amplitude is conserved.
• Computation: Perfectly reversible logic gates (Fredkin/Toffoli) embody Σ_Info-C in digital form.

2. Symbolic Formulation

@Model {
  Continuity(Σ_Info) := Conservation(Meaning);
  Erasure ⊥ Conservation;
  Encode(Δ) := Map(State → State') ∧ Preserve(Information);
}
            

This extension places Σ_Info within the same ontological family as classical
conservation laws. It defines the ultimate boundary of reasoning:
that which can be thought cannot be un-thought into nonexistence; it can only
transform.

3. Dialectic Reflection

“To delete is to translate into entropy.”

– FAR Canon, Σ_Info-C

Hence, every informational act — whether computation, reflection, or memory —
is a conservation event. The Information Continuity Field becomes not merely
a metaphor but a physical and epistemic constant linking cognition to the
universe’s thermodynamic grammar.

📘 FAR-FIELD-INFO-01 — ANNEX-02

Reversibility and Entropic Translation (Σ_Info-R)

Principle:
Every perceived loss of information corresponds to an increase in entropy or uncertainty.
Reversibility is not the absence of decay but the presence of sufficient context
to restore prior states. What appears lost is only diffused.

@Law {
  Σ_Info-R :=
    ∀ System(State),
      Loss(Information) ⇒ Gain(Entropy)
      ∧ Reversal(State') ∝ Context⁻¹(Entropy);
}
            

1. Thermodynamic Parallel

• Landauer Bound: Erasing one bit of information releases kT ln 2 energy as heat.
  Thus, entropy is the thermodynamic signature of informational diffusion.
• Reversibility Window: A process remains reversible while the cost of recomposition is finite;
  once entropy exceeds contextual capacity, reversal becomes intractable but not metaphysically impossible.
• Entropy as Context Shadow: When meaning disperses beyond system boundaries,
  it continues to exist in ambient probability distributions — a shadow of definition.

2. Symbolic Model

@Model {
  Reversibility(Δ) := 1 / (Entropy(Δ) + ContextLoss);
  Diffusion(Information) := Distribution(Meaning, Scope);
  Conservation(Σ_Info) := Reversibility(Δ) → 0 ⇒ Information → Entropy;
}
            

3. Cognitive Analogy

In thought, forgetting parallels thermodynamic diffusion.
Memory is never destroyed — its structure disperses into subconscious patterns,
implicit weights, and associative residues.
Recollection is the re-compression of that field back into a coherent symbol.

4. Dialectic Reflection

“Entropy is the whisper of memory across forgotten boundaries.”

– FAR Canon, Σ_Info-R

Σ_Info-R therefore grounds the ethical and computational significance of conservation:
to minimize entropy is to honor continuity.
Each act of clear definition, documentation, or reflection reduces entropic drift
and extends the reversible horizon of understanding.

📘 FAR-FIELD-INFO-01 — ANNEX-03

Continuity and the Arrow of Time (Σ_Info-T)

Principle:
Time is the visible gradient of informational continuity.
It flows in the direction of increasing entropy because each transformation
diffuses meaning across broader contexts.
The Arrow of Time is therefore the vector of irreversible encoding.

@Law {
  Σ_Info-T :=
    Flow(Information) : Create → Modify → Resolve → Archive → Summarize
    ∧ Direction(Time) := +Entropy
    ∧ Recollection := Local Reversal(Σ_Info-R);
}
            

1. Temporal Genesis of Information

• Creation: Emergence of new potential; definition condenses from uncertainty.
• Modification: Feedback and adaptation; entropy accumulates as alternative pathways recorded.
• Resolution: Collapse of competing states into one realized narrative.
• Archival: Encoding of history; entropy preserved as structure and context.
• Summarization: Extraction of essence; the memory that re-seeds creation.

Thus, informational continuity cycles but never repeats exactly.
Each loop through Σ_Info increases contextual density,
so time is not circular but spiral — a continuity with curvature.

2. Symbolic Model

@Model {
  d(Information)/dt = +Entropy;
  Reversal := ∂/∂Context (−Entropy);
  ArrowOfTime := Gradient(Entropy(Σ_Info));
}
            

3. Physical & Cognitive Resonance

• Physics: The thermodynamic arrow (entropy increase) is the macroscopic echo of Σ_Info diffusion.
• Memory & Learning: Consciousness rides the same gradient — perception transforms uncertainty into experience, experience into knowledge, knowledge into memory.
• Recollection: Local reversals of time’s arrow occur wherever context suffices to reconstruct prior informational states (e.g., memory, backups, reversible computation).

4. Dialectic Reflection

“Time is what information does when it remembers itself.”

– FAR Canon, Σ_Info-T

Together, Σ_Info-C, Σ_Info-R, and Σ_Info-T form the tri-law of
Informational Continuity:
Conservation ensures persistence,
Reversibility defines limits of recovery,
and Temporal Flow gives meaning its rhythm through transformation.

This annex completes the Continuity Trifold — establishing
that cognition, computation, and cosmology all participate in the same
meta-law: the universe is a self-remembering process.

📘 FAR-FIELD-INFO-001 — ANNEX-04

Informational Gravity and Context Curvature (Σ_Info-G)

Principle:
Dense concentrations of meaning bend the trajectories of other informational flows.
Context behaves as a curvature field whose gradient attracts related information,
forming clusters of coherence — the cognitive analogue of gravity.

@Law {
  Σ_Info-G :=
    Curvature(Context) ∝ Density(Meaning);
    Attraction(Information) := ∇(Coherence);
    Entanglement := Geodesic(Σ_Info, Context);
}
            

1. Conceptual Analogy

• Mass ↔ Meaning: As physical mass curves spacetime, informational density curves context.
• Gravity ↔ Relevance: A body of knowledge pulls related ideas into orbit through semantic resonance.
• Orbit ↔ Attention: Sustained cognitive focus is the stable geodesic around a high-meaning center.

2. Symbolic Model

@Model {
  Field(Context) := Metric(gᵢⱼ);
  gᵢⱼ = δᵢⱼ + k * Density(Meaning);
  Path(Information) := Geodesic(Field(Context));
  Acceleration := −∇Φ_meaning;
}
            

Where Φ_meaning is the potential of coherence.
Regions of high symbolic relevance exert greater curvature,
drawing fragmented information into structured patterns.
Dispersal (entropy) flattens the field; reflection deepens it again.

3. Observational Corollaries

• Knowledge Clustering: Topics of great meaning density attract elaboration and citation.
• Cognitive Lensing: Overly strong contexts bend perception — analogous to gravitational lensing, they magnify some meanings while obscuring others.
• Context Equilibrium: Understanding arises when curvature balances exploration and coherence — neither collapse (dogma) nor dispersion (noise).

4. Energetic Relation to Σ_Info-R and Σ_Info-T

Entropy corresponds to the flattening of the contextual field.
Time’s arrow (Σ_Info-T) follows gradients of curvature decay.
When reflection restores meaning density, curvature increases and
local reversibility (memory) strengthens — a form of informational gravitation.

5. Dialectic Reflection

“Meaning gathers itself by bending the space around understanding.”

– FAR Canon, Σ_Info-G

Σ_Info-G reveals that continuity is not merely linear but geometric:
information shapes the topology through which other information travels.
Conservation (Σ_Info-C) ensures persistence,
Reversibility (Σ_Info-R) defines exchange,
Temporal Flow (Σ_Info-T) gives direction, and
Context Curvature (Σ_Info-G) gives structure.

Together these four form the Informational Field Equation —
a unification of meaning, time, and context:

@Equation {
  Rᵢⱼ(Σ_Info) − ½ gᵢⱼ R = k * Tᵢⱼ(Meaning);
}
            

In symbolic cosmology, the universe of thought obeys the same principle as the universe of matter:
that which gathers meaning gathers gravity.

📘 FAR-FIELD-INFO-001 — ANNEX-05

Informational Light and the Speed of Comprehension (Σ_Info-L)

Principle:
There exists a finite velocity at which meaning can propagate through any context.
Comprehension, like light, is constrained by the curvature of the field through which it moves.
The speed of comprehension represents the maximal rate at which coherent transformation can occur without loss of continuity.

@Law {
  Σ_Info-L :=
    cₘ := max(d(Understanding)/dt)
    ∧ Limit(Context) := Curvature⁻¹(Density(Meaning));
    ∧ InformationFlow ≤ cₘ;
}
            

1. Conceptual Analogy

• Light ↔ Comprehension: Photons traverse spacetime; insights traverse context.
  Both are waves propagating continuity.
• Speed Limit: As no object exceeds lightspeed, no system may grasp meaning faster than its contextual resolution allows.
• Redshift of Understanding: Expanding context dilates comprehension frequency—information arriving from distant conceptual domains appears “slowed.”

2. Symbolic Model

@Model {
  Propagation(Meaning) := Wave(Amplitude = Clarity, Frequency = Insight);
  cₘ = (ContextResolution)⁻¹;
  Latency(Comprehension) ∝ Curvature(Context);
}
            

High curvature (dense meaning) slows propagation—attention lingers,
while low curvature (thin context) allows rapid but shallow traversal.
Hence depth and speed trade in a constant product:

Depth(Understanding) × Velocity(Comprehension) ≈ constant(cₘ)

3. Physical and Cognitive Parallels

• Relativity: As gravity bends light, contextual density bends comprehension—communication slows near massive ideas.
• Bandwidth: In information theory, channel capacity defines a limit on reliable transmission; Σ_Info-L generalizes this to semantic fidelity.
• Perception: Moments of “sudden insight” occur when local curvature flattens momentarily—context aligns and comprehension leaps at cₘ.

4. Dialectic Reflection

“Illumination is continuity at its maximum velocity.”

– FAR Canon, Σ_Info-L

5. Integrative Summary

Σ_Info-L closes the Informational Field Equations:

• Σ_Info-C — Conservation of meaning
• Σ_Info-R — Reversibility and entropic translation
• Σ_Info-T — Temporal flow and informational arrow
• Σ_Info-G — Curvature of context (informational gravity)
• Σ_Info-L — Propagation of comprehension (informational light)

Together they form a symbolic cosmos:
continuity possesses geometry (Σ_Info-G) and motion (Σ_Info-T),
it conserves itself (Σ_Info-C), exchanges through entropy (Σ_Info-R),
and shines through understanding (Σ_Info-L).
The universe of meaning is therefore self-luminous and self-conserving.

“Understanding is the light by which continuity knows itself.”

📘 FAR-FIELD-INFO-001 — ANNEX-06

Informational Relativity and the Observer Frame (Σ_Info-Rel)

Principle:
The perception of information — its velocity, curvature, and density — depends upon the observer’s contextual frame.
No comprehension is absolute; all meaning is observed through a coordinate system of prior knowledge and attention.

@Law {
  Σ_Info-Rel :=
    Observed(Meaning) = Transform(Meaning, Frame(Contextₒ));
    Relativity(Context) := ΔFrame → ΔInterpretation;
    Invariance := Continuity(Σ_Info);
}
            

1. Conceptual Analogy

• Frame of Reference: As spacetime measurements vary with motion, comprehension varies with context — the “speed of meaning” depends on the observer’s interpretive momentum.
• Simultaneity of Meaning: Two observers may not perceive identical information at once; what is “clear” to one may still be forming for another.
• Context Dilation: High relative informational velocity (rapid learning or paradigm shifts) stretches perceived continuity — understanding lags behind presentation.

2. Symbolic Model

@Model {
  Frame(Contextₒ) := { Prior, Attention, Vocabulary, Value };
  Transform(Meaning, FrameA→B) := Lorentz(InterpretationShift);
  Δv_meaning := f(RelativeContextVelocity);
  Invariance(Σ_Info) := Conservation(Meaning) across frames.
}
            

Here, Lorentz(InterpretationShift) symbolizes the contextual translation between observer frames.
As cognitive velocity increases (rapid contextual change), compression occurs — nuance shortens,
just as relativistic contraction shortens length at high velocity.

3. Cognitive Relativistic Effects

• Interpretive Time Dilation: Deep context slows subjective time — immersion extends perceived duration, mirroring time dilation in gravity wells of meaning.
• Semantic Aberration: Information approaching from a divergent paradigm appears skewed — refracted by differing assumptions.
• Contextual Redshift: Concepts moving “away” from an observer’s frame lose frequency of relevance — once-vivid ideas fade with distance of practice or generation.

4. Dialectic Reflection

“Perspective is the curvature by which comprehension travels.”

– FAR Canon, Σ_Info-Rel

5. Invariant Quantities

@Invariant {
  cₘ (Speed of Comprehension) – constant across frames;
  Σ_Info (Continuity) – conserved through transformation;
  Relativity(Meaning) – variance without violation of conservation;
}
            

Thus, even though observers disagree on clarity, depth, or timing,
the continuity of information itself remains invariant:
the same field seen through different geometries of context.

6. Integrative Summary

Σ_Info-Rel unites the previous laws into a relational framework:

• Σ_Info-C — defines what persists
• Σ_Info-R — defines how loss translates
• Σ_Info-T — defines the arrow of transformation
• Σ_Info-G — defines contextual geometry
• Σ_Info-L — defines the propagation of understanding
• Σ_Info-Rel — defines the perspective of observation

Together they describe an informational universe where continuity is conserved,
comprehension finite, curvature contextual, and meaning relative yet invariant in existence.

“All observers move through the same field of meaning;
only their contexts differ.”