Every-wide-width initialized callable product #
The width threshold depends only on original row/occurrence counts and the length/word bound of the protected caller prefix. The actual program never reads physical width. Generic execution does not assume query sparsity; only the final quantitative saving does.
The concrete threshold remains positive at all boundary parameters.
Physical admission covers both the computed virtual window and all caller-prefix words.
Every native caller prefix is physically word-bounded, including a loaded dirty word
before the following remainder. Here P bounds every initial register and memory word,
not merely the source input or normalized values. It is an initial-data bound, not a bound
on intermediates supplied by the caller.
Actual callable flat runs at every admitted physical width, with arbitrary dirty initial
workspace bounded by P. This is the interface for a real producer's reachable caller state:
P bounds all its initial words, including dirty pre-modulo loads. Original local input words
derive the concrete source execution; no solver, run, time, routing or layout certificate is
assumed. The same final y contains exact answers, restored input and the caller frame.
At every admitted physical width, the initialized callable program returns all requested exact product words and preserves caller/local-input memory. No sparsity premise is used.
Every initialized prefix, including raw dirty loads before normalization, fits the admitted
physical width. Outside the explicit original input the initialized memory is zero. Thus P
in this initialized interface need only bound the protected caller prefix.
Complete actual initialized thin saving, including base loading, all translated memory accesses, span normalization and halt. The occurrence bound is explicit and stronger than a distinct-set bound for lists containing duplicate requests.