Substrate
plain-language theorem explainer
Substrate is a structure type carrying a string label, natural-number phase (0 canonical), and integer sigma charge. Researchers modeling dark matter as substrate or exploring sigma-conservation in ontological arguments cite this when building substrate-independent monotheism claims. The definition is a plain structure declaration deriving decidable equality and representation.
Claim. A substrate is an object equipped with a label $l$ in strings, a phase $p$ in natural numbers (where $p=0$ marks the canonical sector), and a charge $s$ in integers.
background
In the Substrate-Independent Monotheism module, a substrate formalizes the basic unit of an ontology equipped with a global phase function and sigma conservation. Phase draws from the eight-tick definition (multiples of pi/4 for k in Fin 8) and sigma is the integer charge analogous to the gap between private preference and public vote in decision models (+1 or -1 for mismatch, 0 for truthful agents). The module setting states that any ontology with both a global phase and sigma conservation admits at most one divine substrate at canonical phase with sigma = 1.
proof idea
Direct structure definition introducing the Substrate type with three fields: label as String, phase as natural number, sigma as integer. Derives DecidableEq and Repr automatically with no further lemmas applied.
why it matters
This definition anchors the structural side of the monotheism claim by supplying the type on which divine, isMonotheistic, and trichotomy predicates are built. It feeds downstream into substrate_model (dark matter as substrate, null WIMP signal) and ea011_certificate (ultra-diffuse galaxy diversity via spatial coherence variation). It realizes the T7 eight-tick octave and sigma conservation from the forcing chain, closing the AR-essay structural claim that monotheism is the sigma-conserving theological position.
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