This document specifies the Field Vector Architecture (FVA), a deterministic evaluation
framework that classifies the physical existence of information systems within a
defined field. FVA operates through six irreducible vertices arranged in a hexagonal
topology, producing a binary existence decision F ∈ {0,1} via a resonance
order parameter σ subject to a hard phase transition at threshold 0.33. The framework
is scale-invariant, strictly monotonic, and invariant under repeated analysis when
byte-level artifacts are preserved. All governing equations, axioms, mathematical
properties, and a complete validation test suite are specified herein.
B (Carrier). Raw bytes without semantic meaning. The non-interpreted input to the field.
V (Vectorization). Projection of B via four boolean predicates P₀, P₁, P₂, P₃, yielding a binary vector V ∈ {0,1}⁴.
κ (Coherence). Internal closure of a single artifact, defined as the arithmetic mean of its predicate vector components.
κ̄ (Mean Coherence). The arithmetic mean of κ across all artifacts in the field.
ρ (Density). Field consensus, defined as the ratio of the most frequently occurring vector to the total number of vectors.
σ (Resonance). The order parameter of the field, computed from κ̄, ρ, and the structural constant π₀.
F (Existence). Binary existence decision. F = 1 if σ > 0.33; F = 0 otherwise.
π₀ (Structural Constant). Fixed constant. π₀ = 1.8.
Threshold. The existence boundary σ = 0.33. This constitutes a hard phase transition: no gradation, no interpretation.
Each artifact is projected onto a four-dimensional binary vector via the following predicates, evaluated against the lowercased string representation of the artifact:
| Predicate | Name | Condition (yields 1 when true) |
|---|---|---|
P₀ |
Timeless | Artifact does not match setTimeout, history, or chronological |
P₁ |
Closure | Artifact matches closure, c(s), sum=1, or coherent |
P₂ |
Projection | Artifact matches svg, project, dom, or iframe |
P₃ |
Axiom | Artifact matches axiom or the pattern A followed by one or more digits |
An information system exists physically within the FVA field if and only if:
Truth value depends on ratio, not on quantity.
The resonance parameter σ governs field dynamics according to three conditions:
This is a hard phase transition. No gradation. No interpretation.
| Vertex | Symbol | Role | Specification |
|---|---|---|---|
| 1 | B | Carrier | Raw bytes without meaning |
| 2 | V | Vectorization | Projection via predicates P₀, P₁, P₂, P₃ |
| 3 | κ | Coherence | Internal closure: κ = Σpᵢ / 4 |
| 4 | ρ | Density | Consensus: ρ = |dominant_vector| / |total| |
| 5 | σ | Resonance | Order parameter: σ = (κ̄ × ρ × π₀) / 2 |
| 6 | F | Existence | Binary decision: F ∈ {0,1} at σ > 0.33 |
The vertices are traversed in a fixed, unidirectional sequence:
This flow is timeless: the evaluation order is invariant and admits no reordering.
For a single artifact with predicate vector [p₀, p₁, p₂, p₃]:
For a field of n artifacts:
Where |dominant_vector| is the count of the most frequently occurring
vector among all artifact vectors, and |total| is the total number of artifacts.
Where π₀ = 1.8 (structural constant).
The existence decision F constitutes a binary phase transition at the threshold σ = 0.33. The transition is hard: there exists no intermediate state between F = 0 (non-existent) and F = 1 (existent). The criterion is deterministic — identical byte inputs always produce identical existence decisions. The criterion is also scale-invariant — duplicating all artifacts by a constant factor k does not alter σ.
Formally, an information system is classified as physically existent within the FVA field if and only if all three conditions of the Existence Principle (Section 2.1) are satisfied simultaneously.
The following example demonstrates a complete hexagonal field evaluation on a mixed artifact set of four inputs. All intermediate values are shown.
Four raw byte sequences accepted without interpretation.
| Artifact | P₀ | P₁ | P₂ | P₃ | V |
|---|---|---|---|---|---|
| [0] closure sum=1 svg axiom A1 | 1 | 1 | 1 | 1 | [1,1,1,1] |
| [1] closure svg project axiom A2 | 1 | 1 | 1 | 1 | [1,1,1,1] |
| [2] setTimeout history chronological | 0 | 0 | 0 | 0 | [0,0,0,0] |
| [3] basic code | 1 | 0 | 0 | 0 | [1,0,0,0] |
| Artifact | κ |
|---|---|
| [0] | 1.00 |
| [1] | 1.00 |
| [2] | 0.00 |
| [3] | 0.25 |
Dominant vector: [1,1,1,1] (frequency: 2 of 4).
This document constitutes a prior art publication and defensive publication for the Field Vector Architecture (FVA) as described herein, including its hexagonal topology, governing equations, existence criterion, scale law, resonance law, and all mathematical properties specified in Sections 1 through 7.
Date of publication: 8 February 2026.
This document is publicly available for verification. Any party may independently reproduce the evaluation procedure and validate the mathematical properties using the test vectors specified in Appendix A.
The following test suite provides a deterministic, reproducible, and fully specified validation of the FVA governing equations and mathematical properties. Each test is defined by its input, the evaluation procedure as specified in Sections 3–4, and the expected outcome.
Input: Five identical artifacts, each consisting of the string "closure svg axiom".
Procedure: Complete hexagonal evaluation (Vertices 1–6).
Expected outcome:
| Parameter | Expected Value |
|---|---|
| κ̄ | ≈ 0.75 |
| ρ | 1.0 |
| σ | ≈ 0.675 |
| F | 1 (EXISTENT) |
Input A: N = 1, artifact: "closure svg axiom".
Input B: N = 100, all identical: "closure svg axiom".
Procedure: Evaluate σ for both inputs.
Expected outcome: σ(A) = σ(B). The resonance value must be invariant under uniform scaling.
Input A (above threshold): Three identical artifacts: "closure svg axiom".
Input B (below threshold): Three artifacts: "nothing", "basic", "code".
Procedure: Evaluate F for both inputs.
Expected outcome: F(A) = 1, F(B) = 0. The existence decision must reflect the binary phase transition at σ = 0.33.
Input: Three artifacts: "closure svg axiom", "setTimeout", "basic".
Procedure: Evaluate σ three times on identical input.
Expected outcome: σ(run 1) = σ(run 2) = σ(run 3). All evaluation runs must produce identical resonance values.
Input A (pure): Five identical artifacts: "closure svg axiom".
Input B (+1 violation): Input A with one additional artifact: "setTimeout".
Input C (+5 violations): Input A with five additional artifacts, each "setTimeout".
Procedure: Evaluate σ for all three inputs.
Expected outcome: σ(A) > σ(B) > σ(C). The resonance must decrease strictly with increasing violation dilution.
Input A (high κ̄): Three identical artifacts: "closure svg axiom".
Input B (medium κ̄): Three identical artifacts: "closure svg".
Input C (low κ̄): Three identical artifacts: "closure".
Procedure: Evaluate σ for all three inputs.
Expected outcome: σ(A) > σ(B) > σ(C). The resonance must increase strictly with increasing mean coherence, consistent with ∂σ/∂κ̄ > 0.
All six tests are deterministic and reproducible. Any implementation of the governing equations specified in Section 4 must pass all tests to be considered a conforming FVA implementation.
Reference implementation derived directly from Sections 3–7. Constants: π₀ = 1.8, threshold = 0.33. All equations, predicates, and test vectors are identical to those specified in the governing sections of this document. This appendix does not alter the canonical status of the main document.
Enter artifacts (one per line). Evaluation follows the hexagonal flow B → V → κ → ρ → σ → F.
Executes all six tests specified in Appendix A.
σ-landscape contour plot over (κ̄, ρ) parameter space, phase diagram showing the existence boundary at σ = 0.33, and per-artifact coherence distribution.