mul_balanced_zero_of_balanced_zero_left
plain-language theorem explainer
If a signed orbit is balanced against zero, its product with any signed orbit is likewise balanced against zero. Developers of the Primitive Recognition Calculus integer order cite this when propagating the zero class under multiplication. The argument is a one-line term proof: reverse the multiplication-zero biconditional and inject the left disjunct.
Claim. For signed orbits $z$ and $w$, if $z$ is balanced with zero (equivalently $z.\mathrm{pos}=z.\mathrm{neg}$), then the product $z\cdot w$ is balanced with zero.
background
In the Primitive Recognition Calculus, integers are carried as signed orbits: pairs $(pos,neg)$ of distinction naturals, read as $pos-neg$ (K4.6). Two signed orbits $a,b$ are balanced when $a.pos+b.neg=b.pos+a.neg$; this is the internal integer equivalence, defined entirely on $\delta$-orbit positions (K4.9).
Zero is the balanced class of the zero signed orbit. Multiplication of signed orbits descends to these classes. An upstream biconditional records that the product is balanced with zero exactly when at least one factor is balanced with zero (via the integer interpretation of the product).
This module assembles the order surface on signed orbits (reflexivity, totality, trichotomy, sign flags, nonnegativity) that the integer-order certificate later packages.
proof idea
One-line term proof. Apply the reverse direction of the multiplication-zero biconditional, which states that $z\cdot w$ is balanced with zero if and only if $z$ is balanced with zero or $w$ is. Feed the hypothesis on $z$ through the left injection of that disjunction. No further unfolding of balanced or of the product is required at this site.
why it matters
Feeds the integer-order certificate, whose doc-comment states that the internal signed-orbit order surface is closed. That certificate packages display lemmas for truncated subtraction, absolute difference, nonnegativity, and sign flags on signed orbits.
Zero absorption under multiplication is part of the arithmetic backbone those order facts rest on: without it, the signed-orbit integers would fail to behave as an ordered ring fragment. The result lives in the Foundation layer of Recognition Science, inside the Primitive Recognition Calculus that builds discrete arithmetic before the forcing chain (T0–T8) and the continuous J-cost calculus. It does not itself touch J-uniqueness, $\varphi$, or the eight-tick octave; it only seals a zero-propagation step needed for the order certificate.
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