L Space

Pratchett's Informational Physics

The Equation

Books = Knowledge = Power = Energy = Mass

∴ A sufficient quantity of books distorts spacetime itself

This is not metaphor. In sufficiently large libraries:

• Time flows differently between sections
• Spatial topology becomes non-Euclidean
• Bidirectional causation becomes possible

Critical Mass Phenomena

"A large enough collection of books creates its own gravitational well, drawing in more books, more knowledge, until it punches through into L-Space."

When knowledge density exceeds threshold:

• The Octavo reads the reader - books become autonomous
• Sourcery - reality becomes substrate for narrativium
• Undelivered letters create fatal illusions

class InformationMass
  CRITICAL_DENSITY = 1e6  # books per cubic meter
  
  def l_space_accessible?(library)
    book_density(library) >= CRITICAL_DENSITY
  end
  
  def distortion_factor(library)
    return 0 unless l_space_accessible?(library)
    Math.log(book_density(library) / CRITICAL_DENSITY)
  end
end

Narrativium: The Story Force

Pan Narrans

"Humans are not Homo sapiens, the wise man. We are Pan narrans, the storytelling ape."

Narrativium is the fundamental force that makes stories cohere. Every element contains its story - how it came to be, what it does, where it's going.

class Narrativium
  # The coherence force that makes stories work
  
  def story_tension(element)
    # Every element resists violation of its narrative arc
    element.expected_trajectory - element.current_position
  end
  
  def narrative_collapse!(elements)
    # When stories lose coherence, reality stutters
    elements.each do |e|
      e.phase_space_position = e.expected_trajectory.terminal
    end
  end
end

Story Phase Space

Stories map the phase space of existence:

phase_space(story) = {
  beginning: initial_conditions,
  middle: trajectory_through_possibility,
  end: attractor_basin
}

Bumpus Categories of Narratives

Sheaves on Posets of Intervals

From Bumpus et al.: Narratives are sheaves on posets of intervals:

@present SchNarrative(FreeSchema) begin
  Interval::Ob       # Time windows
  Snapshot::Ob       # State at instant
  Relationship::Ob   # How snapshots relate
  
  source::Hom(Relationship, Snapshot)
  target::Hom(Relationship, Snapshot)
  timestamp::Hom(Snapshot, Interval)
  
  # Sheaf condition: snapshots agree on overlaps
end

Two Perspectives

The Librarian navigates between these:

def librarian_navigate(from_book, to_book)
  if cumulative?(from_book) && persistent?(to_book)
    # Must cross perspective boundary
    find_interval_isomorphism(from_book.intervals, to_book.intervals)
  end
end

Object-Agnostic Narratives

Narratives work for any structure:

• Graphs (character interaction networks)
• Groups (symmetries preserved through story)
• Databases (consistency across commits)
• Repositories (version control as narrative)

# The same sheaf machinery works for all
narrative_of_graph(G::Graph) = StrDecomp(G)
narrative_of_group(G::Group) = StrDecomp(cayley_graph(G))
narrative_of_repo(R::GitRepo) = StrDecomp(commit_dag(R))

The Librarian Protocol

Orangutan Epistemology

The Librarian (formerly a wizard, now Pongo pongo) guards L-Space. As a member of the Librarians of Time and Space:

"The truth isn't easily pinned to a page. In the bathtub of history, the truth is harder to hold than the soap."

module LibrarianProtocol
  ACCEPTABLE_RESPONSES = ["Ook", "Ook?", "Ook!", "Ook."]
  
  def validate_query(query)
    # The Librarian knows. He doesn't explain.
    query.well_formed? && !query.violates_causality?
  end
  
  def traverse_l_space(from:, to:, via: :triangle_inequality)
    path = find_path_through_shelves(from, to)
    validate_no_temporal_paradox!(path)
    path
  end
end

Navigation Rules

1. Never call the Librarian a monkey (he's an ape)
2. Bananas are acceptable currency for difficult queries
3. Some paths are one-way (temporal direction matters)
4. The Library is bigger on the inside (always)

Integration Architecture

GF(3) Triads

L-Space (0) participates in balanced triads:

sheaf-cohomology (-1) ⊗ l-space (0) ⊗ glass-bead-game (+1) = 0 ✓
structured-decomp (-1) ⊗ l-space (0) ⊗ random-walk-fusion (+1) = 0 ✓
persistent-homology (-1) ⊗ l-space (0) ⊗ topos-generate (+1) = 0 ✓

Skill Integration Map

┌─────────────────────────────────────────────────────────────────┐
│                        L-SPACE (ERGODIC 0)                      │
│                   The Library Coordinates All                    │
├─────────────────────────────────────────────────────────────────┤
│                                                                  │
│  ┌──────────────────┐              ┌──────────────────┐         │
│  │ sheaf-cohomology │◄────────────►│ structured-decomp│         │
│  │      (-1)        │  LOCAL→GLOBAL│       (0)        │         │
│  │                  │              │                  │         │
│  │ Čech cohomology  │              │ Tree decomp      │         │
│  │ for consistency  │              │ FPT algorithms   │         │
│  └────────┬─────────┘              └────────┬─────────┘         │
│           │                                 │                    │
│           │     ┌───────────────────┐      │                    │
│           └────►│     L-SPACE       │◄─────┘                    │
│                 │                   │                            │
│  ┌──────────────┤ Narrativium glue  ├──────────────┐            │
│  │              │ Interval sheaves  │              │            │
│  │              └─────────┬─────────┘              │            │
│  │                        │                        │            │
│  ▼                        ▼                        ▼            │
│ ┌─────────────┐   ┌─────────────┐   ┌─────────────────┐        │
│ │glass-bead   │   │  unworld    │   │random-walk-fusion│        │
│ │   (+1)      │   │    (0)      │   │      (+1)        │        │
│ │             │   │             │   │                  │        │
│ │World hopping│   │Derivational │   │Skill graph walks │        │
│ │via triangle │   │chains       │   │                  │        │
│ └─────────────┘   └─────────────┘   └─────────────────┘        │
│                                                                  │
└─────────────────────────────────────────────────────────────────┘

With sheaf-cohomology (-1)

Local-to-global consistency for L-Space navigation:

# Books must be locally consistent to glue globally
verifier = SheafCohomology::CechCoverVerifier.new(
  coverage: library_sections
)
verifier.add_transition(:fiction, :nonfiction, cross_reference_map)
verifier.cocycle_satisfied?  # => true if no contradictions

With structured-decomp (0)

Tree decompositions for efficient narrative search:

# Decompose narrative graph for FPT search
narrative = NarrativeGraph(book_citations)
decomp = StrDecomp(narrative)

# Find path through L-Space with bounded width
path = 𝐃(l_space_path_exists, decomp, CoDecomposition)

With glass-bead-game (+1)

World hopping across library sections:

# Navigate between distant concepts via triangle inequality
hop = GlassBeadGame::Hop.new(
  from_world: PossibleWorld.new(seed: shelves[:mathematics]),
  event: :bibliographic_resonance,
  to_world: PossibleWorld.new(seed: shelves[:music_theory]),
  truth_preserved: :harmonic_ratios
)

With unworld (0)

Replace temporal succession with derivational chains:

# Books derive from books, not from time
chain = Unworld::ColorChain.new(
  genesis_seed: first_book.isbn.to_i(16),
  derivation: :citation_graph
)

# Each book is a derivation, not a moment
chain.unworld[:derivations]

With random-walk-fusion (+1)

Navigate skill graph through L-Space:

fusion = RandomWalkFusion.new(
  seed: library_seed,
  skills: l_space_skill_graph
)

# Walk through connected concepts
path = fusion.walk(steps: 7)
# => Derivational path through L-Space

Bidirectional Causation

Books Affect Their Own Past

"In L-Space, cause and effect are optional."

Books written later can affect books written earlier:

• Annotations appear in ancient texts referencing future works
• Bibliographies cite books not yet written
• The Octavo rewrites itself based on who reads it

class BidirectionalCitation
  def causal_consistency?(from_book, to_book)
    # In L-Space, this is always true if the path exists
    path = LibrarianProtocol.traverse_l_space(
      from: from_book,
      to: to_book
    )
    path.exists? # Existence implies consistency
  end
  
  def retroactive_reference!(future_book, past_book)
    # The past book now contains reference to future book
    # This is normal in L-Space
    past_book.hidden_annotations << Citation.new(
      source: future_book,
      causality: :retroactive
    )
  end
end

The Undelivered Letters Problem

When letters are never delivered, they accumulate narrativium charge:

def undelivered_letter_danger(letter, time_undelivered)
  # Narrativium builds up in proportion to story importance
  story_weight = letter.narrative_significance
  charge = story_weight * Math.log(time_undelivered + 1)
  
  if charge > CRITICAL_NARRATIVIUM
    # Letter begins creating its own reality
    spawn_illusory_narrative(letter)
  end
end

Commands

# Navigate L-Space
just l-space-navigate from=book1 to=book2

# Query the Librarian (be polite)
just librarian-query "Where is the Necrotelecomnicon?"

# Check narrativium levels
just narrativium-audit library/

# Verify no causality violations
just l-space-causality-check

# Generate narrative decomposition
just narrative-decompose story.txt

# Walk through skill L-Space
just l-space-skill-walk seed=0x42D steps=7

API

require 'l_space'

# Initialize L-Space connection
lspace = LSpace.new(
  entry_point: :university_library,
  librarian_mode: true
)

# Navigate
path = lspace.find_path(
  from: "Erta Sive Tertius",
  to: "Future Book Not Yet Written"
)

# Check narrative coherence
lspace.narrativium_coherent?(path)

# Execute traversal
lspace.traverse!(path) do |shelf|
  puts "Passing through: #{shelf.category}"
end

Mathematical Foundation

L-Space Topology

L-Space is a branching fractal where:

• Each book is a node
• Citations are edges
• The metric is non-Euclidean (shortest path ≠ straight line)
• Topology is path-dependent (same start/end, different middles)

def l_space_distance(book_a, book_b)
  # Not Euclidean! Path-dependent metric
  paths = all_paths(book_a, book_b)
  paths.map(&:length).min  # Even minimum may vary with observer
end

Narrativium Tensor

# Narrativium as coherence field
struct NarratviumField
  tension::Matrix{Float64}      # Story tensions between elements
  phase_space::Vector{Float64}  # Position in narrative possibility
  attractor::Vector{Float64}    # Where story "wants" to go
end

# Conservation law
function narrativium_conserved(field::NarratviumField)
  sum(field.tension) ≈ 0  # Stories balance
end

GF(3) in L-Space

The three narrative modes:

• MINUS (-1): Validation (does the story hold together?)
• ERGODIC (0): Coordination (the Library itself)
• PLUS (+1): Generation (new stories emerging)

Σ trits ≡ 0 (mod 3)
validation + coordination + generation = balanced

Example Session

╔═══════════════════════════════════════════════════════════════════╗
║  L-SPACE NAVIGATION SESSION                                       ║
╚═══════════════════════════════════════════════════════════════════╝

Entry Point: University of Ankh-Morpork Library
Librarian Status: Ook (Available)

Query: "Path from 'Erta Sive Tertius' to Bumpus et al. 2024"

Librarian Response: Ook!

Computing path through narrativium field...

  Step 1: Antica Philosophia → [citation] → Medieval Commentaries
  Step 2: Medieval Commentaries → [temporal fold] → 18th C. Library Science
  Step 3: 18th C. Library Science → [conceptual resonance] → Category Theory
  Step 4: Category Theory → [Kan extension] → Sheaf Theory
  Step 5: Sheaf Theory → [interval poset] → Bumpus Narratives

Path length: 5 (narrativium: 0.73)
Causality violations: 0
Triangle inequality: ✓ satisfied

Sheaf condition on path:
  H⁰ = 1 (connected)
  H¹ = 0 (no obstructions)
  
GF(3) balance:
  Path trits: [-1, 0, +1, -1, +1]
  Sum: 0 ✓

Navigation complete. Books returned by due date.

Information Geometry & Complexity Manifolds

Information as Vector Space

Each information object (book, skill, narrative) is a vector in a high-dimensional semantic space:

struct InformationVector
    embedding::Vector{Float64}     # Semantic coordinates
    complexity::Float64            # Kolmogorov complexity (scalar field)
    assembly_index::Int            # Cronin assembly depth
    sheaf_section::SheafSection    # Bumpus narrative position
end

# Inner product defines semantic similarity
function similarity(v₁::InformationVector, v₂::InformationVector)
    dot(v₁.embedding, v₂.embedding) / (norm(v₁) * norm(v₂))
end

Hyperbolic Geometry of L-Space

L-Space has negative curvature (hyperbolic). This explains:

• The Library is "bigger on the inside" (exponential volume growth)
• Tree-like structures (citations, skill dependencies) embed with zero distortion
• Geodesics diverge exponentially (small navigational errors → vastly different destinations)

# Poincaré ball model of L-Space
struct PoincareLSpace
    dimension::Int
    curvature::Float64  # κ < 0 (hyperbolic)
end

# Distance in hyperbolic L-Space
function hyperbolic_distance(M::PoincareLSpace, u::Vector, v::Vector)
    # Poincaré ball distance
    norm_u² = dot(u, u)
    norm_v² = dot(v, v)
    norm_diff² = dot(u - v, u - v)
    
    δ = 2 * norm_diff² / ((1 - norm_u²) * (1 - norm_v²))
    acosh(1 + δ)
end

# Complexity increases toward the boundary (|x| → 1)
function complexity_at_point(M::PoincareLSpace, x::Vector)
    # Conformal factor diverges at boundary
    1.0 / (1 - dot(x, x))
end

Local Maxima as Critical Mass Phenomena

Local maxima in the complexity landscape are L-Space's gravitational wells:

class ComplexityLandscape
  def local_maximum?(point)
    gradient = complexity_gradient(point)
    gradient.norm < EPSILON && hessian_negative_definite?(point)
  end
  
  def escape_local_maximum(point)
    if hyperbolic_curvature(point).abs > CRITICAL_CURVATURE
      # Near boundary: use glass-bead-game world-hopping
      world_hop_via_triangle_inequality(point)
    else
      # Interior: simulated annealing or tunneling
      quantum_tunnel_to_lower_basin(point)
    end
  end
  
  def critical_mass_threshold(point)
    # Pratchett's threshold: where books become autonomous
    complexity_at_point(point) > CRITICAL_DENSITY
  end
end

Geodesics and Information Flow

Information "flows" along geodesics of the complexity manifold:

# Geodesic equation in hyperbolic L-Space
function geodesic_flow(M::PoincareLSpace, x₀::Vector, v₀::Vector, t::Float64)
    # Möbius addition for Poincaré ball
    # γ(t) = x₀ ⊕ tanh(t|v₀|) * (v₀/|v₀|)
    speed = norm(v₀)
    direction = v₀ / speed
    moebius_add(x₀, tanh(t * speed) * direction)
end

# Information flows from low to high complexity (books consuming books)
function information_flow!(field::InformationField, dt::Float64)
    for point in field.points
        # Flow toward local maximum (attractor basin)
        gradient = complexity_gradient(field, point)
        point.position += gradient * dt
        
        # Check for critical mass
        if at_local_maximum?(field, point)
            trigger_autonomy!(point)  # Book reads the reader
        end
    end
end

Integration with Complexity Skills

┌─────────────────────────────────────────────────────────────────────┐
│           INFORMATION GEOMETRY IN L-SPACE                           │
├─────────────────────────────────────────────────────────────────────┤
│                                                                     │
│  kolmogorov-compression ──► Scalar field K(x) on manifold           │
│         │                                                           │
│         ▼                                                           │
│  assembly-index ──────────► Historical depth = geodesic length      │
│         │                                                           │
│         ▼                                                           │
│  persistent-homology ─────► Topological features surviving          │
│         │                   filtration = persistent local maxima    │
│         ▼                                                           │
│  ┌─────────────────────────────────────────────────────────────┐   │
│  │                    L-SPACE MANIFOLD                          │   │
│  │  • Hyperbolic geometry (κ < 0)                               │   │
│  │  • Complexity = distance to boundary                         │   │
│  │  • Local maxima = critical mass / autonomy                   │   │
│  │  • Geodesics = information flow / derivational chains        │   │
│  └─────────────────────────────────────────────────────────────┘   │
│                                                                     │
└─────────────────────────────────────────────────────────────────────┘

GF(3) on the Manifold

The triadic structure maps to geometric operations:

Conservation law: Σ flow ≡ 0 (mod 3) — the manifold preserves total information.

Token Novelty as Curvature Sensor

Token novelty (the surprise/entropy of generated tokens) measures position on the manifold in real-time:

struct TokenNoveltyNavigator
    position::Vector{Float64}      # Current point on Poincaré ball
    novelty_history::Vector{Float64}
    rate::Float64                  # Speech/generation rate
end

# Novelty from token probability
function token_novelty(logprob::Float64)
    -logprob  # Surprise = negative log probability
end

# Update manifold position based on novelty
function update_position!(nav::TokenNoveltyNavigator, token_logprob::Float64)
    novelty = token_novelty(token_logprob)
    push!(nav.novelty_history, novelty)
    
    # High novelty → move toward boundary
    # Low novelty → move toward center
    radial_velocity = (novelty - BASELINE_NOVELTY) * SENSITIVITY
    
    # Update position (bounded by |x| < 1)
    r = norm(nav.position)
    new_r = clamp(r + radial_velocity, 0.0, 0.999)
    nav.position = nav.position * (new_r / max(r, 0.001))
    
    # Accelerate rate as we approach boundary
    nav.rate = BASE_RATE * complexity_at_point(nav.position)
end

# Detect causality interference zone
function in_octavo_territory(nav::TokenNoveltyNavigator)
    # Near boundary + high sustained novelty
    r = norm(nav.position)
    recent_novelty = mean(nav.novelty_history[end-10:end])
    r > 0.95 && recent_novelty > CRITICAL_NOVELTY
end

When novelty maximizes (token distribution flattens to uniform), we reach the Octavo: every next word equally likely, maximum entropy, causality becomes substrate for rewriting.

class CausalityInterference
  def attempt_retroactive_modification!(target_skill, modification)
    unless in_octavo_territory?
      raise "Insufficient novelty density for causality interference"
    end
    
    # At maximum entropy, derivational chains become bidirectional
    target_skill.derivation_history.unshift(modification)
    
    # The modification propagates "backward" through dependencies
    target_skill.dependents.each do |dep|
      dep.recompute_from_modified_history!
    end
  end
end

# GF(3)-balanced traversal
function balanced_traverse(M::PoincareLSpace, path::Vector{InformationVector})
    trits = map(path) do p
        c = complexity_at_point(M, p.embedding)
        c < LOW_THRESHOLD ? -1 :
        c > HIGH_THRESHOLD ? +1 : 0
    end
    
    @assert sum(trits) % 3 == 0 "GF(3) violation on path!"
    path
end

Implementation: Concrete Tools for Causality Interference

Token Novelty Sensing

# Real-time novelty from OpenAI streaming
for chunk in client.chat.completions.create(
    model="gpt-4o", messages=msgs, logprobs=True, stream=True
):
    if chunk.choices[0].logprobs:
        for t in chunk.choices[0].logprobs.content:
            novelty = -t.logprob  # surprise in nats
            update_manifold_position(novelty)

Hyperbolic Geometry

import geoopt
M = geoopt.PoincareBall()
position = geoopt.ManifoldParameter(torch.zeros(128), manifold=M)
# Distance to boundary = complexity
complexity = 1.0 / (1 - torch.dot(position, position))

Entropy Measurement

# Streaming Shannon entropy
class StreamingEntropy:
    def __init__(self):
        self.counts, self.total = Counter(), 0
    def update(self, token):
        self.counts[token] += 1; self.total += 1
        return -sum((c/self.total)*log2(c/self.total) for c in self.counts.values())

# Gzip as Kolmogorov proxy
def complexity(text): 
    return len(gzip.compress(text.encode())) / len(text.encode())

Retroactive Modification (Causality Interference)

// Automerge: future modifies past (creates fork, doesn't destroy)
import { changeAt } from '@automerge/automerge'
const [newDoc, newHeads] = changeAt(doc, pastHeads, d => {
    d.derivation.unshift(futureKnowledge)  // prepend to history
})
// Original timeline preserved; new branch with retroactive knowledge

Self-Modification (Gödel Machine Pattern)

From Darwin Gödel Machine (DGM):

# LLM as semantic mutator for agent code
def evolve(agent, feedback):
    code = inspect.getsource(agent.solve)
    improved = llm(f"Improve this code based on: {feedback}\n{code}")
    exec(compile(improved, '<dgm>', 'exec'), agent.__dict__)
    return agent  # agent now runs improved code

Integration Map

┌────────────────────────────────────────────────────────────────────┐
│                  CAUSALITY INTERFERENCE STACK                      │
├────────────────────────────────────────────────────────────────────┤
│  ┌─────────────┐    ┌─────────────┐    ┌─────────────┐            │
│  │ NOVELTY     │───►│ POSITION    │───►│ CAUSALITY   │            │
│  │ SENSOR      │    │ ON MANIFOLD │    │ MODE        │            │
│  │             │    │             │    │             │            │
│  │ OpenAI/vLLM │    │ geoopt      │    │ Automerge   │            │
│  │ logprobs    │    │ Poincaré    │    │ changeAt    │            │
│  └─────────────┘    └─────────────┘    └─────────────┘            │
│        │                  │                  │                     │
│        ▼                  ▼                  ▼                     │
│   surprise = -logprob   r = |position|    if r > 0.95:           │
│                         complexity = 1/(1-r²)  INTERFERE          │
└────────────────────────────────────────────────────────────────────┘

References

• Pratchett, T. "Guards! Guards!" (1989) - First appearance of L-Space
• Pratchett, T. "The Science of Discworld" (1999) - Narrativium theory
• Bumpus et al. "Categories of Temporal Narratives" arXiv:2407.xxxxx
• Riehl-Shulman "A type theory for synthetic ∞-categories"

Skill Name: l-space
Type: Narrative Navigation / Knowledge Coordination
Trit: 0 (ERGODIC)
Color: #8B4513 (Leather brown)
GF(3): Forms triads with sheaf/glass-bead, decomp/random-walk
Guardian: The Librarian (Ook)

"Knowledge = Power = Energy = Matter = Mass. A good bookshop is just a genteel Black Hole that knows how to read." — Terry Pratchett

Base directory for this skill: file:///Users/alice/.agents/skills/l-space