#foundation

Overview — The Sweet-Potato-Vine Worldview

Philosophy and roadmap of RelationWorld Theory. Five foundational principles, the central sweet-potato-vine metaphor, and a correspondence dictionary between the discrete formalism and its classical counterparts.

Relation

The fundamental unit of RelationWorld — a relation tuple (i, j, w, g) carrying scalar intensity and group-valued transit, with symmetrised weight, degree, and their basic properties.

Relational Field

Gauge group and action on relational fields, scalar invariants, holonomy and discrete curvature, and the completeness of invariants in the connected case.

Fruit

Low-conductance clusters as the basic unit of cohesive existence. Cheeger conductance, the fruit definition with threshold θ, and Theorems A (Energy Isolation) and D (Metastability).

Stem

The connective tissue between fruits. Stem region, bridge edges, and the no-boundary principle that prevents the exterior from being defined explicitly.

Door

Internal singularities arising from anomalous external contact, detectable only within the fruit interior. The intrinsic data axiom A5, door definitions, and Theorems B, C, G.

Existence

Existence as a gauge-invariant triple combining the optimal gauge class, door locus, and residual energy. Flattening energy, optimal gauge, and Theorems E and H.

World

The world as a three-layer hierarchy of relational fields, fruits with existence triples, and their inter-fruit connectivity. Theorem F (Spectral Stability) closes Part I.

SCC Canonical Spec — Part 1: Foundations & Formal Universe

Sections 0, 2–5 of the SCC canonical specification (CV-1.5.2): the summation convention, the foundational orientation, the formal universe, the primacy of the soft form, and the derived geometric and morphological notions.

Chapter 0 — Formal State Space

Formalises the state space of RelationWorld — the mathematical object describing all possible world configurations and the structure of transitions between them.

T1 — Existence of Minimizers

The energy E attains its minimum on the constraint manifold Σ_m. The well-posedness foundation of SCC: every other result is a refinement of the bare existence statement.

T20 — Axiom Consistency (A1' resolves A1↔A3 incompatibility)

The axioms A1' / A2 / A3 / A4 of Group A (closure) are mutually consistent. The original A1 (weak extensivity) is incompatible with A3 (contraction) for the sigmoid closure realization; A1' (conditional extensivity) resolves the tension. The theory's foundational legitimacy.

T6 — Closure Fixed Point (Banach Contraction)

When the closure steepness parameter a_cl < 4, the sigmoid closure operator has a unique fixed point on [0,1]^n with geometric convergence rate a_cl/4. The convergence guarantee for the central self-referential operator.