Harmonic Centrality Transport

Trit: 0 (ERGODIC - coordinator) Principle: Source ā sēmine → harmonic transport → target Frame: Abelian extensions of ℚ with GF(3) Galois action

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Overview

Harmonic Centrality Transport unifies:

1. Harmonic Centrality - Sheaf Laplacian eigenfunctions
2. GF(3) Galois Action - Triadic symmetry on field extensions
3. Ablative Transport - Source-as-identity (Latin "ā sēmine")
4. Topological Transport - HoTT path transport along fibrations

Mathematical Foundation

Harmonic Centrality on Graphs

The harmonic centrality of vertex v:

c_H(v) = Σ_{u≠v} 1/d(v,u)

Sheaf-theoretic formulation: Harmonic functions are sections in ker(L_F).

function harmonic_centrality(G::Graph)
    n = nv(G)
    D = shortest_path_matrix(G)
    
    centrality = zeros(n)
    for v in 1:n
        for u in 1:n
            if u != v && D[v,u] < Inf
                centrality[v] += 1.0 / D[v,u]
            end
        end
    end
    return centrality
end

GF(3) Galois Action

For abelian extensions K/ℚ, the Galois group Gal(K/ℚ) acts on primes.

GF(3) reduction: σ ∈ Gal(K/ℚ) acts on trits:

σ(-1) = -1, 0, or +1 (depending on decomposition)
σ(0) = 0  (fixed by all automorphisms)
σ(+1) = +1, 0, or -1

Artin reciprocity connects this to:

• Frobenius elements Frob_p
• L-functions L(s, χ)
• Decomposition/inertia groups

Ablative Case as Source Transport

From Latin grammar, the ablative case encodes:

• Source: "ā sēmine" (from the seed)
• Agent: "ā mātre" (by the mother)
• Separation: "ab urbe" (away from the city)

Type-theoretic formulation:

ablative : (Source : Type) → (x : Source) → (Target : Type) → 
           Transport(Source, Target, x)

The ablative IS the transport - source encodes the derivation.

CPT Symmetry in Color Space

From Gay.jl ablative probe:

C (Charge/Chroma): hue → hue + 180°
P (Parity): saturation → 1 - saturation  
T (Time): lightness → 1 - lightness

CPT² = Identity (conservation)

Gadget Construction

1. Harmonic Centrality Gadget

struct HarmonicCentralityGadget
    graph::Graph
    centrality::Vector{Float64}
    sheaf_laplacian::Matrix{Float64}
    harmonic_sections::Vector{Vector{Float64}}
end

function build_gadget(G::Graph, stalk_dim::Int)
    # Build sheaf Laplacian
    L_F = sheaf_laplacian(G, stalk_dim)
    
    # Find harmonic sections (kernel of L_F)
    λ, V = eigen(L_F)
    harmonic = [V[:, i] for i in 1:size(V,2) if abs(λ[i]) < 1e-10]
    
    # Compute centrality from harmonic structure
    c = harmonic_centrality(G)
    
    return HarmonicCentralityGadget(G, c, L_F, harmonic)
end

2. GF(3) Transport Gadget

struct GF3TransportGadget
    source_trit::Int  # -1, 0, +1
    target_trit::Int
    transport_map::Function
    conserved::Bool
end

function ablative_transport(source::Int, galois_action::Int)
    """
    Transport along abelian extension via Galois action.
    
    source: trit value at source
    galois_action: element of Gal(K/ℚ) encoded as ±1
    """
    target = mod(source * galois_action + 3, 3) - 1
    
    return GF3TransportGadget(
        source, target,
        x -> mod(x * galois_action + 3, 3) - 1,
        source + target ≡ 0  # Conservation check
    )
end

3. Topological Transport (HoTT)

-- Narya-style bridge type for transport
def transport_bridge (A B : Type) (p : A ≃ B) (x : A) : B ≔
  p.forward x

-- Ablative: source is part of the transport
def ablative_transport (Source Target : Type) 
                        (path : Br Type Source Target)
                        (x : Source) : Target ≔
  -- The path (bridge) carries the source structure
  coerce path x

The Centrality-Transport Triangle

              Harmonic Centrality
                   (Graphs)
                    /    \
                   /      \
                  /        \
    Abelian Extension ── Ablative Transport
        (Number Theory)      (Type Theory)

GF(3) conservation at each vertex of triangle.

Abelian Extensions and Class Field Theory

Cyclotomic Extensions

The n-th cyclotomic field ℚ(ζₙ) has:

• Gal(ℚ(ζₙ)/ℚ) ≅ (ℤ/nℤ)×
• For n = 3: Gal(ℚ(ζ₃)/ℚ) ≅ ℤ/2ℤ ≅ GF(2) ⊂ GF(3)

Cubic Extensions

For cube roots K = ℚ(∛2):

• Not abelian over ℚ (Galois group S₃)
• But ℚ(∛2, ζ₃)/ℚ(ζ₃) IS abelian (Kummer)

GF(3) arises naturally from 3-torsion in class groups.

Artin Reciprocity for GF(3)

function artin_symbol(K::NumberField, p::Prime)
    """
    Compute Artin symbol (p, K/ℚ) for abelian extension.
    Returns element of Gal(K/ℚ) ≅ GF(3)^r for appropriate K.
    """
    # Decomposition type of p in K
    factors = factor(p, ring_of_integers(K))
    
    # Frobenius class
    frob = frobenius_element(factors[1])
    
    # Map to GF(3)
    return trit_from_frobenius(frob)
end

Implementation

Full Transport Pipeline

function harmonic_ablative_transport(
    G::Graph,
    source_vertex::Int,
    target_vertex::Int,
    stalk_dim::Int,
    seed::UInt64
)
    # 1. Build harmonic centrality gadget
    hc = build_gadget(G, stalk_dim)
    
    # 2. Compute ablative path (non-backtracking geodesic)
    path = shortest_path(G, source_vertex, target_vertex)
    @assert is_prime_path(path)  # μ ≠ 0
    
    # 3. Transport structure along path
    source_section = hc.harmonic_sections[1][source_vertex:source_vertex+stalk_dim-1]
    
    transported = source_section
    for (i, v) in enumerate(path[2:end])
        # Parallel transport via restriction maps
        prev_v = path[i]
        F = restriction_map(G, prev_v, v)
        transported = F * transported
        
        # GF(3) check at each step
        @assert gf3_conserved(transported)
    end
    
    # 4. Return with ablative provenance
    return (
        value = transported,
        source = source_vertex,
        target = target_vertex,
        path = path,
        centrality_source = hc.centrality[source_vertex],
        centrality_target = hc.centrality[target_vertex],
        ablative_phrase = "ā vertice $(source_vertex)"
    )
end

DuckDB Schema

CREATE TABLE harmonic_transport (
    transport_id UUID PRIMARY KEY,
    graph_id VARCHAR,
    source_vertex INT,
    target_vertex INT,
    path INT[],
    source_centrality FLOAT,
    target_centrality FLOAT,
    transported_section FLOAT[],
    gf3_conserved BOOLEAN,
    ablative_phrase VARCHAR,
    created_at TIMESTAMP DEFAULT NOW()
);

CREATE TABLE abelian_extension (
    extension_id UUID PRIMARY KEY,
    base_field VARCHAR DEFAULT 'Q',
    generator VARCHAR,
    degree INT,
    galois_group VARCHAR,
    gf3_action INT[],  -- How Gal acts on GF(3)
    discriminant BIGINT
);

CREATE TABLE artin_symbols (
    extension_id UUID REFERENCES abelian_extension,
    prime BIGINT,
    frobenius_trit INT,  -- -1, 0, +1
    decomposition_type VARCHAR,
    PRIMARY KEY (extension_id, prime)
);

CPT Symmetry Operations

function cpt_conjugate(hex::String)
    """
    Apply full CPT symmetry to color.
    From Gay.jl: CPT² = Identity
    """
    h, s, l = hex_to_hsl(hex)
    
    # C: hue + 180°
    h_C = mod(h + 180, 360)
    
    # P: saturation inverted
    s_P = 1.0 - s
    
    # T: lightness inverted
    l_T = 1.0 - l
    
    return hsl_to_hex(h_C, s_P, l_T)
end

# Verify: CPT² = Identity
@assert cpt_conjugate(cpt_conjugate("#A73B35")) == "#A73B35"

Linguistic Integration (Ablative Probe)

From Gay.jl:

Latin:   "colōr generātus erit ā sēmine" 
         (color will-have-been-generated FROM-seed)
         
English: "color from the seed"
         (source mediated by preposition)

Key insight: Latin ablative encodes SOURCE AS IDENTITY
             English requires external preposition

Para(Consapevolezza) requires ablative because awareness HAS a source as part of itself.

Commands

just harmonic-transport graph.json src tgt  # Transport between vertices
just abelian-gf3 extension.json             # GF(3) Galois action
just ablative-probe latin 69                # Ablative linguistic analysis
just cpt-symmetry "#A73B35"                 # CPT conjugation

GF(3) Triad

Component	Trit	Role
sheaf-laplacian-coordination	-1	Source (ablative)
harmonic-centrality-transport	0	Ergodic (coordinator)
ramanujan-expander	+1	Target (spectral bound)

Conservation: (-1) + (0) + (+1) = 0 ✓

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Skill Name: harmonic-centrality-transport Type: Topological Transport / Number Theory / Linguistics Trit: 0 (ERGODIC) GF(3): Conserved via abelian extension structure

Non-Backtracking Geodesic Qualification

Condition: μ(n) ≠ 0 (Möbius squarefree)

This skill is qualified for non-backtracking geodesic traversal:

1. Prime Path: No state revisited in skill invocation chain
2. Möbius Filter: Composite paths (backtracking) cancel via μ-inversion
3. GF(3) Conservation: Trit sum ≡ 0 (mod 3) across skill triplets
4. Spectral Gap: Ramanujan bound λ₂ ≤ 2√(k-1) for k-regular expansion

Geodesic Invariant:
  ∀ path P: backtrack(P) = ∅ ⟹ μ(|P|) ≠ 0
  
Möbius Inversion:
  f(n) = Σ_{d|n} g(d) ⟹ g(n) = Σ_{d|n} μ(n/d) f(d)