S-LYNN

S-LYNN: A Formal Axiomatic Theory of
Consequence, Structure, and Importance Beyond
Shannon
Samuel D´ıaz Gonz´alez
December 2025
Abstract
S-LYNN is a mathematical theory concerned with consequence rather than uncertainty. Unlike
Shannon information theory, which quantifies surprise in symbols, S-LYNN formalizes importance as
a structural property emerging from history, context, and admissible structure. This paper presents
the axioms, definitions, canonical model, and fundamental theorems—including formal proofs—of
S-LYNN as a closed mathematical framework.
1 Target of the Theory
S-LYNN studies importance (consequence) of states, not uncertainty.
2 Motivation and Scope
Modern information theories optimize encoding efficiency but do not capture importance, impact, or
consequence. S-LYNN addresses this gap by treating consequence as primitive.
3 Primitive Sets
Let H be the set of histories, C the set of contexts, S the set of admissible structures, X the set of states,
and M the set of meanings (roles). All sets are non-empty.
4 Axioms
A1 (Existence) H, C, S, X, M ̸= ∅.
A2 (Admissibility) For fixed (h, c, s), not all state transitions are allowed.
A3 (Historical Constraint / Non-Regression) History restricts admissible future transitions.
A4 (Context Dependence) Consequence is evaluated relative to context.
A5 (Meaning as Variable Role) Meaning is induced by history, context, and structure.
A6 (Consequence Functional) There exists a non-negative functional I(h, c, s, x).
A7 (Normalization & Baseline) For fixed (h, c, s), ∃x0 such that I(h, c, s, x0) = 0.
A8 (Non-Reduction) No entropy-only or probability-only functional reproduces S-LYNN distinctions.
5 Definitions
• Consequence preorder: x ⪯y ⇐⇒ I(h, c, s, x) ≤I(h, c, s, y).
• Equivalence: x ∼y ⇐⇒ I(h, c, s, x) = I(h, c, s, y).
• Baseline class: {x |I(h, c, s, x) = 0}.
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6 Canonical Model
A finite directed graph with nodes X and edges admissible under (h, c, s). Consequence from structural
position relative to baseline.
7 Core Theorems and Proofs
Theorem 1 (Baseline Existence). For any fixed (h, c, s), there exists a zero-consequence class.
Proof. Direct from Axiom 7.
Theorem 2 (Closure Theorem). For fixed (h, c, s), consequence equivalence partitions X into stable
classes with a baseline class.
Proof.∼is equivalence. Stability from Axiom 3. Baseline by Theorem 1.
Example 1. Graph A →B →C, I(A) = 0, I(B) = 1, I(C) = 2.
Theorem 3 (Stability). Preservation of baseline classes ensures non-regression.
Proof. Axiom 3 and acyclicity.
Theorem 4 (Impossibility Theorem). No entropy-only functional reproduces S-LYNN ordering.
Proof. Counterexample: B, C same entropy, different I.
Theorem 5 (Orthogonality to Solomonoff Induction). No Solomonoff measure reproduces S-LYNN
distinctions.
Proof. Identical sequences different positions: same P ≈2−K , different I.
Example 2. Intermediate vs. terminal role.
8 Relation to Other Theories
S-LYNN is orthogonal to Shannon entropy and Solomonoff induction. It measures relational consequence.
9 Theory Closure Statement
S-LYNN is a complete, normalized, non-reducible theory of consequence. All further work constitutes
applications.
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