Autonomy from the External Clock: Clock-Ladder Lemma, Relational Autonomy Theorem for P2 and the Predictive/Generative Memory Balance over the BMR-1 Bank — RCD Program, Technical Notes REP-1 and REP-1STAT (v0.1)

Technical Note REP-1STAT of the RCD program, with the adjudicated summary of REP-1 (continues REP-0, DOI 10.5281/zenodo.23774093; BMR-1, DOI 10.5281/zenodo.23717139; JANO-BR-0, DOI 10.5281/zenodo.23785719; and the Adjudication Record, DOI 10.5281/zenodo.23717352). REP-1 asked whether any erasure-closed family in the BMR-1 bank needs less quantum than classical memory to be reproduced sequentially over the local Rideout–Sorkin alphabet with a clock: it remains inconclusive, with three accredited findings — 77% of the 2,045 closures are not faithfully resolvable as families over the local alphabet; 93 closures show a quantum predictive advantage (Gram rank below the number of causal states), always at the terminal stage; and 92 of them admit an explicit classical generator of dimension at most the Gram rank, so predictive advantage is not generative advantage. REP-1STAT asked how much memory the rule needs when it is not told from outside which stage it is at (No External Clock). Results: (i) Clock-Ladder Lemma — if infinitely many prefix/letter pairs satisfy f(u_j x_j) ≠ 0 and f(u_k x_j) = 0 for k > j, no finite-dimensional homogeneous linear representation exists; over the literal RS alphabet P2 has such a ladder, hence the stationary dimension left open by REP-0 is infinite; (ii) Relational Autonomy Theorem for P2 — removing absolute cardinality while keeping the relational predicate FULL (“the newborn covers the whole current order”) yields a finite, classical, positive homogeneous realization with 6 states: autonomy does not require knowing the stage, it requires a relation between the birth and the present; (iii) Gate 0 — the only projectively consistent growth process for P2/D2 has constant class weights, so the erasure flow of BMR-1 is coarse-graining, not growth, and star+isolated and fan families are not closed under growth; (iv) Finite-No-Clock Lemma recorded for TIME-0: an unbounded clock cannot live only in a finite internal memory. Emergent time is not claimed; the standard Gram construction did not compress and no quantum minimality certificate is given. Scripts with SHA-256 manifest. English and Spanish versions; the Spanish text prevails.