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module module moderate

IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.Grow.SignedOrbitLeProductRightFactorIffOfBalancedChoiceFree

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Foundation module for right-factor cancellation of signed-orbit order under balanced, choice-free hypotheses. It packages a right multiplicative congruence and the iff that reduces a product inequality on the right to the corresponding factor inequality. Anyone building ordered ratio-orbit arithmetic in the primitive calculus would cite it. Arguments are algebraic reductions from the left-congruence import and the cross-multiplication order on ratio orbits.

claimIn the balanced choice-free fragment of signed-orbit order: multiplicative congruence on the right, and the equivalence that a product inequality with fixed left factor is equivalent to the inequality on the right factor alone (i.e. $x\cdot y \le x\cdot z$ iff $y\le z$ under the standing positivity/balance hypotheses).

background

Primitive Recognition Calculus grows integers and rationals from orbits of distinctions before classical field axioms are available. Orbit supplies the positive distinction carriers; IntegerRational and IntegerOrder give the signed integer layer. Signed orbits embed positive dens via SignedOrbit.ofOrbit.

Order on ratio orbits is the cross-multiplication relation from RatioOrbitLeReflTotal: leQ p q holds iff the signed orbit $p.num\cdot q.den$ is $\le$ the signed orbit $q.num\cdot p.den$. The companion module SignedOrbitLeCongrLeftOfBalancedChoiceFree already handles left congruence under the same balanced choice-free discipline; this module supplies the matching right-hand side.

"Balanced choice-free" means comparisons stay inside a fragment that avoids classical choice and works with balanced representatives, so cancellation lemmas must be proved directly from the orbit order rather than from a total ordered field.

proof idea

Definition-and-lemma module, not a single theorem. It imports the left balanced congruence and the ratio-orbit cross-multiplication order, then establishes two sibling results: right multiplicative congruence under balanced CF hypotheses, and the right-factor product iff that rewrites a product inequality into an inequality on the free right factor. Proofs are algebraic reductions and congruence rewrites against those imports, not a long tactic script.

why it matters in Recognition Science

Without right cancellation, the cross-multiplication order on ratio orbits cannot support the usual ordered-semiring manipulations needed later in the forcing chain's arithmetic layer. This module closes the right-hand half of the balanced CF congruence package begun by SignedOrbitLeCongrLeftOfBalancedChoiceFree, so product inequalities can be reduced factorwise on either side.

No downstream edges are recorded yet in the mirror graph; the natural consumers are further Grow lemmas that treat leQ as a partial ordered-monoid relation on ratio orbits, and any development that lifts signed-orbit order toward the rational comparison used in J-cost and ladder constructions. It is scaffolding for the ordered arithmetic underwriting T5–T8 rather than a landmark theorem itself.

scope and limits

depends on (5)

Lean names referenced from this declaration's body.

declarations in this module (2)