costCompose_nonneg
plain-language theorem explainer
Cost composition preserves non-negativity in the Recognition Science algebra. Modelers of J-cost levels under the Recognition Composition Law cite this when verifying the non-negative reals form a monoid under the induced operation. The proof unfolds the explicit bilinear formula and closes via positivity plus linear arithmetic on the resulting terms.
Claim. If $a ≥ 0$ and $b ≥ 0$, then $a ⋆ b ≥ 0$, where $a ⋆ b := 2ab + 2a + 2b$.
background
The cost composition operation is induced by the Recognition Composition Law on J-costs. Its explicit form, from the upstream definition, is a ★ b = 2ab + 2a + 2b. This equips non-negative reals with a binary operation that captures how costs combine under ratio multiplication in the algebra module.
proof idea
Tactic-mode proof. Unfold the costCompose definition to obtain the expression 2ab + 2a + 2b. Establish three non-negativity facts via positivity on 2ab and linarith on 2a and 2b. Apply linarith to sum the inequalities and conclude non-negativity.
why it matters
The result closes a basic algebraic property required for the cost monoid that appears in the forcing chain after T5 J-uniqueness. It rests directly on the RCL-induced definition and supports later steps toward the eight-tick octave and D = 3. No downstream citations are recorded in the current graph.
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