χTopic tag
#foundation
13 documents indexed here. Mathematical objects, methods, systems, and recurring technical questions.
ℓNotes
13- Notes
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.
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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.
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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.
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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.
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Relational Field
Gauge group and action on relational fields, scalar invariants, holonomy and discrete curvature, and the completeness of invariants in the connected case.
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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).
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Stem
The connective tissue between fruits. Stem region, bridge edges, and the no-boundary principle that prevents the exterior from being defined explicitly.
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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.
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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.
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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.
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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.
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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.
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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.