Master Document

Canonical Authority for QUES, QUAN, and THETA Mathematics

Version 1.0
Status Canonical Edition
Release Date February 26, 2026
Authority QUES Research Initiative
Open Main Index Read Formal Research Open Mathematical Systems Jump to Implementation

1. Canonical Authority

This document defines the governing statements of the framework whenever wording, notation, or emphasis diverges across supporting sources.

Theta Constant: Θ = 1/2
King's Triad Equation: Θ + n = (1 + n) / 2
Equivalent Theta Forms: Θ = θ ☯ θ, Θ = |, and Θ ≡ ☯

The model is observer-anchored. In this canonical branch, Theta is not treated as zero or additive identity.

In this form interpretation, | and / are equivalent balancing operators that resolve through the Theta-form, and ☯ is interchangeable with Θ.

The dotted Theta glyph is defined as uppercase Theta with one dot just above the center line inside the circle and one dot just below the center line inside the circle.

Precedence Rule: If any source conflicts with this equation set, this master document governs interpretation and implementation decisions.

2. King's Triad

A unit is interpreted as three points: left half, center unit value, and right half. In compact notation:

K(n) = (Θ, n, Θ)
Form View: Θ = θ ☯ θ = | = ☯

Triangular Form

The triad forms the minimal triangle: two boundary half-points and one central value.

Θ n Θ

Recursive Law

Each point is itself a full triad. Recursion therefore applies to every node:

K(n) -> (K(Θ), K(n), K(Θ))

This yields self-similar triangular recursion and a non-linear view of infinity in which every point contains full triadic structure.

3. QUES Architecture

QUES remains modeled as a Dual-Tetrad architecture, with an intrinsic observer layer (Zone 5) preserving coherence across scale.

Layer Role Canonical Constraint
Left Structural Tetrad Defines what the unit is Must remain mirror-coherent with right side
Operational Tetrad (Zone 5) Observer functions: declarative, generative, evaluative, transformative Observer scaling anchored by Θ = 1/2
Right Structural Tetrad Completes symmetry and closure Generated or implied by triadic symmetry

4. Invariants

Canonical invariants remain the six coherence guarantees used for validation, composition, and operational integrity.

Invariant Canonical Meaning
Origin-Anchored Every unit is explicitly grounded through Theta observer anchoring.
Internal Coherence No contradiction inside the unit's own declarations and operations.
Structural Closure No essential external dependency is required for identity.
Referencability Units are named, versioned, and addressable for composition.
Composability Units combine into higher-order units without identity loss.
Continuity Identity is preserved through transformation and scaling.

5. Implementation Standard

Implementation must encode triadic anchoring and operational functions explicitly.

<que:unit xmlns:que="urn:essencience:que:1.0">
  <que:identity>
    <que:label>Example Unit</que:label>
    <que:theta>1/2</que:theta>
    <que:kingTriad>(Theta, n, Theta)</que:kingTriad>
  </que:identity>
  <que:functions>
    <que:function type="declarative">...</que:function>
    <que:function type="generative">...</que:function>
    <que:function type="evaluative">...</que:function>
    <que:function type="transformative">...</que:function>
  </que:functions>
</que:unit>

Validation should fail if a unit substitutes theta = 0 or uses theta + n = n as governing law.