Why it is a question
THEORY-787 was rejected on 2026-10-30, because its own condition was met for a coarse-graining of one transition (THEORY-1167). LIT-1038 also reports emergence in macros of time. In one, a second-order Markov element answers a two-spike burst. In the other, nine heterogeneous elements are coarse-grained in space and time together (NOTE-726). There the macro effect is k micro steps ahead, and LIT-1038 compares it with the micro mechanism over one step. THEORY-1167’s “What this does not say” leaves this case out, and so does CLAIM-497.
What would answer it
There are two comparisons, and they need to be kept apart.
- Against the k-step micro mechanism. Take the macro map f applied to X_t and to X_{t+k}, scored with the same distribution over micro states. Data processing gives I(f(X_t); f(X_{t+k})) ≤ I(X_t; X_{t+k}), as in THEORY-1167. That is expected to settle this comparison against emergence. It has not been checked on LIT-1038’s examples, and the second-order element needs its state space defined so that the map is a function of the micro state.
- Against the one-step micro mechanism, LIT-1038’s comparison. The k-step and the one-step micro mechanisms can differ in effective information either way. So a macro gain over the one-step micro may measure the change of horizon, not the coarse-graining. The answer is a recomputation of LIT-1038’s two examples under a common distribution, against both the one-step and the k-step micro mechanism. Each comparison would then be credited to the horizon or to the coarse- graining.