Pith. sign in
module module moderate

IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.Grow.SignedOrbitLeOfProductRightFactorIffOfBalancedChoiceFree

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Choice-free module for signed-orbit order: an inequality involving a right product is equivalent to the corresponding right-factor comparison once a balanced hypothesis is in force. Also records right-congruence of the order under multiplication by balanced quantities. Anyone building the cross-multiplication order on ratio orbits from integer data would cite it. Argument is algebraic reduction via the imported right-congruence and integer-order facts.

claimIn the signed-orbit order, under a balanced choice-free hypothesis, $a \le b\cdot c$ if and only if the corresponding right-factor comparison holds. Separately, right multiplication by a balanced quantity is an order congruence: $x \le y$ iff $x\cdot m \le y\cdot m$ when $m$ is balanced.

background

This sits in the Grow layer of Primitive Recognition Calculus, which assembles order structure on orbits and ratio orbits from bare integer and rational scaffolding, avoiding classical choice.

Upstream, RatioOrbitLeReflTotal defines the cross-multiplication order on ratio orbits: $\mathrm{leQ},p,q$ holds iff the signed orbit $p.\mathrm{num}\cdot q.\mathrm{den}$ is $\le$ the signed orbit $q.\mathrm{num}\cdot p.\mathrm{den}$, with positive denominators cast in via $\mathrm{SignedOrbit.ofOrbit}$. Integer order and integer-rational modules supply the underlying $\le$ and arithmetic. The sibling import on signed-orbit right congruence under balanced choice-free hypotheses is the immediate algebraic engine.

Balanced quantities are those for which left/right multiplication behaves symmetrically enough to transport inequalities without picking representatives.

proof idea

Two sibling declarations carry the content. The congruence lemma reduces right multiplication by a balanced factor to an order-preserving map on signed orbits, using the imported choice-free right-congruence module and integer-order facts. The main iff then rewrites a product-on-the-right comparison as a right-factor comparison by applying that congruence in both directions and cancelling the balanced factor in the signed-orbit order. No analytic estimates; pure equational and order transport.

why it matters in Recognition Science

Cross-multiplication order on ratio orbits is the bridge from discrete orbit data to a total preorder usable in the recognition calculus. This module supplies the cancellation/factorization step that lets product inequalities on signed orbits drop a balanced right factor, which is exactly what cross-multiplication needs when denominators are cast as positive orbits. It closes a choice-free gap between raw signed-orbit $\le$ and the reflexive-total $\mathrm{leQ}$ developed next door. No downstream edges are recorded yet in the mirror graph; the natural parents are further Grow lemmas that assemble $\mathrm{leQ}$ totality and the passage toward J-cost comparisons on positive ratios (forcing-chain T5 territory once the cost functional is attached).

scope and limits

depends on (5)

Lean names referenced from this declaration's body.

declarations in this module (2)