substrate_monotheism_one_statement
plain-language theorem explainer
The declaration asserts that every theological model (a list of substrates) is precisely one of atheistic, monotheistic, or polytheistic, that a two-substrate model at canonical phase carries total sigma charge 2, that the single canonical substrate carries charge 1, and that the latter is monotheistic. Researchers working on sigma-conserving ontologies would cite this as the one-statement encapsulation of substrate-independent monotheism. The proof proceeds by term construction, directly assembling the trichotomy partition together with the 2-
Claim. Any list of substrates $T$ satisfies isAtheistic($T$) or isMonotheistic($T$) or isPolytheistic($T$), where isAtheistic holds when the number of divine substrates is zero, isMonotheistic when exactly one, and isPolytheistic when two or more. For the two-substrate corpus $[⟨god1,0,1⟩, ⟨god2,0,1⟩]$ the total sigma charge equals 2, while for the single canonical substrate $[⟨GOD,0,1⟩]$ the total sigma charge equals 1 and the model is monotheistic.
background
Theology is defined as a list of substrates, each carrying a phase and a sigma charge. The divine substrates are those at phase zero with sigma one (canonical sector). isAtheistic holds precisely when a theology contains zero divine substrates, isMonotheistic when it contains exactly one, and isPolytheistic when it contains two or more. The totalSigma function sums the sigma weights across a corpus, and sigma is additive under concatenation. This module formalizes the structural claim that sigma conservation across moves forces at most one divine substrate in any ontology with a global phase. The upstream totalSigma definition states that the total sigma charge of a corpus equals the sum of precedent sigma weights.
proof idea
The proof is a term-mode construction that returns a four-tuple. The first component is the trichotomy lemma establishing the exhaustive partition into the three categories. The second is the lemma showing that a two-divine-substrate model violates canonical sigma conservation by yielding total sigma equal to 2. The third is the lemma confirming that the single canonical occupant yields total sigma equal to 1. The fourth is the direct verification that the unique-occupant model satisfies the monotheistic predicate.
why it matters
This theorem supplies the one-statement summary of substrate-independent monotheism, directly supporting the module's claim that monotheism is the sigma-conserving theological position. It draws on the sigma-additivity from the Jurisprudence.PrecedentStabilityFromSigma module and the partition predicates defined in the same file. In the broader Recognition Science framework it illustrates how conservation laws constrain ontological structure, here forcing uniqueness of the divine substrate. The open question remains the falsifier of a consistent two-substrate canonical model that preserves sigma.
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