Research-Stack/6-Documentation/papers/OTOM/00_Master_References.md

Master References: OTOM Research Stack

Authors: Research Stack Team Date: May 2026 Domain: TTM Layer A (Compression/Routing) + Linguistic Invariants Purpose: Master reference document mapping all equations, concepts, and structures to their source files

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Core Formalism (Lean 4)

Information Manifold

Source: 0-Core-Formalism/lean/Semantics/Core/InformationManifold.lean

Equations/Concepts:

• Fisher-Rao metric: g_{ij}(\theta) = E_p[\partial_i \log p \cdot \partial_j \log p]
• Affine connection: \nabla = \nabla^{LC} + T (with torsion T)
• S1 Fisher-Geometric: Torsion-free, genus-3 topology
• S2 Alcubierre Warp: dI^2 = -d\tau^2 + (dH - \beta d\tau)^2
• S3 Sovereign Informatic Manifold (SIM): Torsion active, anisotropy M^{ij} \neq g^{ij}
• S4 Behavioral/MOIM: Discrete lattice, Genome18 addressing
• Bind primitive: \text{Bind}(A, B, g) : A \times B \times \text{Metric} \rightarrow \mathbb{R}

Used in:

• 17_Meta_Manifold_Language_Merging.md (manifold structure, bind primitive)
• 13_Language_as_Inverted_Manifold.md (manifold topology)

Manifold Flow

Source: 0-Core-Formalism/lean/external/OTOM/ManifoldFlow.lean

Equations/Concepts:

• Gradient flow for \phi: \partial_t \phi = \nabla_i(M^{ij} \nabla_j \delta F/\delta \phi) - \sigma \partial \phi/\partial I_{lock}
• Gradient flow for X^A: \partial_t X^A = -\Gamma^A_{BC} \partial_i X^B \partial_i X^C - \Lambda^{AB}(X^B - X_0^B) - \delta F/\delta X^A + \tau T^A
• Torsion tensor: T^k_{ij} = \Gamma^k_{ij} - \Gamma^k_{ji}
• Anisotropy tensor: M^{ij} (directional information flow)
• Fold-back-lock invariant: I_{lock}

Used in:

• 17_Meta_Manifold_Language_Merging.md (gradient flow, torsion, anisotropy)
• 06_Manifold_Flow.md (core equations)

Mass Number

Source: 0-Core-Formalism/lean/Semantics/Core/MassNumber.lean

Equations/Concepts:

• Mass Number structure: (AdmissiblePacket, ResidualReceipt, BoundaryMarker)
• Admissibility gate: \text{MassLe}(m, \tau) := A \leq \tau \cdot (R + \varepsilon)
• Depth policy: max depth = 3
• Promotion readiness: \text{MassLeDefault} \land \text{depthPolicyOk} \land \text{boundCheck}
• Underverse rule: quarantine for failed gates

Used in:

• 17_Meta_Manifold_Language_Merging.md (admissibility gates, numerical calculations)

FiveD Torus Topology

Source: 0-Core-Formalism/lean/Semantics/Semantics/FiveDTorusTopology.lean

Equations/Concepts:

• Torus distance: d_{torus} = \sum_{i=0}^{4} \min(|x_i - y_i|, k_i - |x_i - y_i|)
• Torus diameter: D_{torus} = \sum_{i=0}^{4} \lfloor k_i/2 \rfloor
• Bisection bandwidth: B = k_0 \cdot k_1 \cdot k_2 \cdot k_3 \cdot k_4 / 2
• Node degree: 10 (2 neighbors per dimension)
• Total connectivity: C_{total} = k_0 \cdot k_1 \cdot k_2 \cdot k_3 \cdot k_4

Used in:

• 17_Meta_Manifold_Language_Merging.md (5D torus topology, routing)

Menger Sponge Fractal Addressing

Source: 0-Core-Formalism/lean/Semantics/Semantics/MengerSpongeFractalAddressing.lean

Equations/Concepts:

• Hausdorff dimension: d_H \approx 2.7268
• Fractal occupancy: |P_{occ}| = \rho_{occ} \cdot N^{d_H}
• State space reduction: R_{reduction} = N^{d_H - 3}
• Menger hash: \text{menger\_hash}(x, y, z) = x \oplus (y \ll 1) \oplus (z \ll 2)
• Fractal offset: \text{fractal\_offset}(x, y, z) = (x + y + z) \cdot d_H / 65536
• Menger address: \text{address} = \text{menger\_hash} \oplus \text{fractal\_offset}

Used in:

• 17_Meta_Manifold_Language_Merging.md (fractal addressing, state space reduction)

Hybrid TSM-PIST-Torus

Source: 0-Core-Formalism/lean/Semantics/Semantics/HybridTSMPISTTorus.lean

Equations/Concepts:

• PIST phase classification: Grounded/Drift/Seismic
• Normalized tension ratio: \rho(n) = 4m(n)/(2k+1)^2
• Genetic optimization score: I = (H \times G) \times (1 - D/64)
• Lyapunov functional: \Lambda(S) = m(n) + \lambda f + \mu c(rej)
• Mirror involution: \sigma_k(k^2+t) = (k+1)^2-t
• Resonance check: m(\sigma_k(n)) = m(n)

Used in:

• 17_Meta_Manifold_Language_Merging.md (PIST phase classification, genetic optimization)

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GCL Documentation

Mass Number Surface Translation

Source: 0-Core-Formalism/otom/docs/gcl/MassNumberSurfaceTranslation.md

Equations/Concepts:

• Surface translation: Mass Number → Surface
• Surface fields: height, slope, curvature, basins, ridges, holes, seams, scar field
• MassSurfacePacket structure
• Surface translation rule (8 questions)

Used in:

• 17_Meta_Manifold_Language_Merging.md (surface fields, MassSurfacePacket)

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CAD Models

Gabriel's Horn

Source: 3-Mathematical-Models/cad_models/gabriels_horn.scad

Equations/Concepts:

• Horn profile: y = 1/x
• Horn volume: V = \int_1^\infty \pi (1/x)^2 dx = \pi
• Horn surface area: A = \int_1^\infty 2\pi (1/x) \sqrt{1 + 1/x^4} dx = \infty
• Byte container surface interpretation

Used in:

• 17_Meta_Manifold_Language_Merging.md (Gabriel's horn geometry, finite volume/infinite surface)

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OTOM Papers

04_Hutter_Prize_Equation.md

Source: 6-Documentation/papers/OTOM/04_Hutter_Prize_Equation.md

Equations/Concepts:

• Winning equation: C = (0.4 \cdot C_{comp} + 0.35 \cdot C_{phys} + 0.25 \cdot C_{geom}) \times (S/(G + F))
• Compression gain: -\rho (linear penalty)
• Decoder penalty: \tau^2 (quadratic penalty)
• Resource penalty: \sigma^2 + q^2 (quadratic penalty)
• Tradeoff theorem: sufficient compression gain can offset penalties

Used in:

• 17_Meta_Manifold_Language_Merging.md (Hutter Prize integration, penalty terms)

09_Cognitive_Load_Theory_Invariant_Enhanced.md

Source: 6-Documentation/papers/OTOM/09_Cognitive_Load_Theory_Invariant_Enhanced.md

Equations/Concepts:

• Invariant load: L_{inv} = \sum_{i=1}^{N_w} w_i \cdot L_S(w_i)
• NSM primes as invariants
• Cognitive load with invariants

Used in:

• 16_Minimal_Semantic_Loss_Cross_Linguistic_Conversion.md (cognitive load integration)
• 17_Meta_Manifold_Language_Merging.md (cognitive load calculations)

10_NSM_Primes_Expanded_for_Cognitive_Load.md

Source: 6-Documentation/papers/OTOM/10_NSM_Primes_Expanded_for_Cognitive_Load.md

Equations/Concepts:

• 64 NSM semantic primes (expanded list)
• Cognitive load mapping for each prime
• Invariant weights

Used in:

• 16_Minimal_Semantic_Loss_Cross_Linguistic_Conversion.md (NSM primes reference)
• 17_Meta_Manifold_Language_Merging.md (anchor points)

11_Language_Invariant_Catalog_Complete.md

Source: 6-Documentation/papers/OTOM/11_Language_Invariant_Catalog_Complete.md

Equations/Concepts:

• Language invariant catalog (27+ languages)
• Language family classification
• Prime realization patterns
• Compression implications
• Severity weights

Used in:

• 15_Public_Domain_Language_Exemplars.md (language list)
• 16_Minimal_Semantic_Loss_Cross_Linguistic_Conversion.md (language families)
• 17_Meta_Manifold_Language_Merging.md (language parameters)

12_Language_Prime_Equations_ReDerived.md

Source: 6-Documentation/papers/OTOM/12_Language_Prime_Equations_ReDerived.md

Equations/Concepts:

• Language prime equations
• Cognitive load matrix integration
• Prime-based compression

Used in:

• 17_Meta_Manifold_Language_Merging.md (language prime equations)

13_Language_as_Inverted_Manifold.md

Source: 6-Documentation/papers/OTOM/13_Language_as_Inverted_Manifold.md

Equations/Concepts:

• Inverted manifold hypothesis
• Language manifold: \mathcal{L}(t) = (M, g_{ij}(t), \tau(t))
• Stress-induced reconfiguration
• Manifold curvature changes

Used in:

• 17_Meta_Manifold_Language_Merging.md (manifold reconfiguration)

14_Phonon_Mediated_Languages_Mined.md

Source: 6-Documentation/papers/OTOM/14_Phonon_Mediated_Languages_Mined.md

Equations/Concepts:

• Cartesian phonon prime integration
• Manhattan distance metric
• Phonon force law
• Forgotten punctuation symbols
• Cognitive load tradeoff conjecture

Used in:

• 17_Meta_Manifold_Language_Merging.md (phonon-mediated language patterns)

15_Public_Domain_Language_Exemplars.md

Source: 6-Documentation/papers/OTOM/15_Public_Domain_Language_Exemplars.md

Equations/Concepts:

• Archive.org public domain exemplars
• 27+ languages with canonical works
• URLs and significance notes
• Resource constraint note

Used in:

• 17_Meta_Manifold_Language_Merging.md (language exemplars reference)

16_Minimal_Semantic_Loss_Cross_Linguistic_Conversion.md

Source: 6-Documentation/papers/OTOM/16_Minimal_Semantic_Loss_Cross_Linguistic_Conversion.md

Equations/Concepts:

• Semantic loss: L_S(w_{A \rightarrow B}) = 1 - |S(w_A) \cap S(w_B)| / |S(w_A) \cup S(w_B)|
• Cross-linguistic compression ratio: C_{cross} = 1 - \overline{L_S}
• Language distance: D_{ling} = 1 - E_{cross}
• Semantic loss tiers (T1-T4)
• Severity weights

Used in:

• 17_Meta_Manifold_Language_Merging.md (semantic loss matrix, compression ratios)

17_Meta_Manifold_Language_Merging.md

Source: 6-Documentation/papers/OTOM/17_Meta_Manifold_Language_Merging.md

Equations/Concepts:

• Language manifold: \mathcal{M}_L \subset \mathbb{R}^d
• Meta-manifold: \mathcal{M}_{meta} = \bigcup_{L \in \mathcal{L}} \mathcal{M}_L
• Geometric structure folding (Torus-Menger-Horn)
• Mass Number integration
• Numerical calculations

Used in:

• (This document is the synthesis of all above)

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CAD Models

MetaManifoldLanguageMerging.lean

Source: 0-Core-Formalism/lean/Semantics/Semantics/MetaManifoldLanguageMerging.lean

Equations/Concepts:

• Language manifold: \mathcal{M}_L \subset \mathbb{R}^d
• Meta-manifold: \mathcal{M}_{meta} = \bigcup_{L \in \mathcal{L}} \mathcal{M}_L
• Torus distance: d_{torus} = \sum_{i=0}^{4} \min(|x_i - y_i|, k_i - |x_i - y_i|)
• Menger hash: \text{menger\_hash}(x, y, z) = x \oplus (y \ll 1) \oplus (z \ll 2)
• Horn radius: r(x) = 1/x
• Fold energy: E_{fold} = \alpha E_{torus} + \beta E_{menger} + \gamma E_{horn}
• Mass Number gates for manifold merging
• Surface translation

Used in:

• 17_Meta_Manifold_Language_Merging.md (foundational formalization)
• 01_Cognitive_Load_Theory.md (geometric structure folding)
• 05_Bind_Primitive.md (manifold merging gates)
• 06_Manifold_Flow.md (fold dynamics)
• 04_Hutter_Prize_Equation.md (unified compression)
• 08_Invariant_Theory.md (anchor points)

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Citation Format

Inline citation format:

• For Lean files: [Source.lean]
• For documentation: [Source.md]
• For papers: [XX_Paper_Name.md]

Example:

• Fisher-Rao metric [InformationManifold.lean]
• Mass admissibility gate [MassNumber.lean]
• Hutter Prize equation [04_Hutter_Prize_Equation.md]

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Cross-Reference Matrix

Concept	Source	Used In
Fisher-Rao metric	InformationManifold.lean	17_Meta_Manifold_Language_Merging.md, MetaManifoldLanguageMerging.lean
Gradient flow	ManifoldFlow.lean	17_Meta_Manifold_Language_Merging.md, 06_Manifold_Flow.md, MetaManifoldLanguageMerging.lean
Mass Number gate	MassNumber.lean	17_Meta_Manifold_Language_Merging.md, MetaManifoldLanguageMerging.lean
5D torus topology	FiveDTorusTopology.lean	17_Meta_Manifold_Language_Merging.md, MetaManifoldLanguageMerging.lean
Menger sponge	MengerSpongeFractalAddressing.lean	17_Meta_Manifold_Language_Merging.md, MetaManifoldLanguageMerging.lean
Gabriel's horn	gabriels_horn.scad	17_Meta_Manifold_Language_Merging.md, MetaManifoldLanguageMerging.lean
Hutter Prize equation	04_Hutter_Prize_Equation.md	17_Meta_Manifold_Language_Merging.md, MetaManifoldLanguageMerging.lean
NSM primes	10_NSM_Primes_Expanded_for_Cognitive_Load.md	16_Minimal_Semantic_Loss_Cross_Linguistic_Conversion.md, 17_Meta_Manifold_Language_Merging.md, MetaManifoldLanguageMerging.lean
Language invariants	11_Language_Invariant_Catalog_Complete.md	15_Public_Domain_Language_Exemplars.md, 16_Minimal_Semantic_Loss_Cross_Linguistic_Conversion.md, MetaManifoldLanguageMerging.lean
Semantic loss	16_Minimal_Semantic_Loss_Cross_Linguistic_Conversion.md	17_Meta_Manifold_Language_Merging.md, MetaManifoldLanguageMerging.lean
Compression ratios	16_Minimal_Semantic_Loss_Cross_Linguistic_Conversion.md	17_Meta_Manifold_Language_Merging.md, MetaManifoldLanguageMerging.lean
Inverted manifold	13_Language_as_Inverted_Manifold.md	17_Meta_Manifold_Language_Merging.md, MetaManifoldLanguageMerging.lean
Phonon-mediated languages	14_Phonon_Mediated_Languages_Mined.md	17_Meta_Manifold_Language_Merging.md, MetaManifoldLanguageMerging.lean
Meta-manifold	MetaManifoldLanguageMerging.lean	01_Cognitive_Load_Theory.md, 02_KDA_Physics.md, 03_SSMS_Master_Recurrence.md, 04_Hutter_Prize_Equation.md, 05_Bind_Primitive.md, 06_Manifold_Flow.md, 07_Braid_Structure.md, 08_Invariant_Theory.md
Fold dynamics	MetaManifoldLanguageMerging.lean	02_KDA_Physics.md, 03_SSMS_Master_Recurrence.md, 06_Manifold_Flow.md, 07_Braid_Structure.md, 17_Meta_Manifold_Language_Merging.md
Surface translation	MassNumberSurfaceTranslation.md	MetaManifoldLanguageMerging.lean, 17_Meta_Manifold_Language_Merging.md