Appendix: Discreteness Arises Not from an Outer Bound but from Nodes — Discreteness as Nodes Generated by Binarity

This appendix develops section 3 of the related report: that discreteness arises intrinsically from the nodes generated by Axiom 2 (binarity), not from an outer bound (boundedness). The usual explanation discreteness from boundedness implicitly borrows the boundary condition of a string with fixed ends, but ends are privileged points forbidden by anonymity (the lemma of the related paper), so borrowing them is circular. Instead, a two-valued quantity (+/-) necessarily produces a node at each sign switch; nodes are points, hence discrete; anonymity makes the node count an integer; and closure (returning after one round) makes it even, corresponding to the integer mode numbering lambda_n = L/n. Discreteness is rooted in the nodes that binarity generates, not in the size bound. No physical identification is made; node refers to the sign switch of the interior binary, not a physical wave node.