03 · The nesting technique

The order is the technique

Every factoring method is a nestable morphism keyed by its own operator word, and the membrane is those morphisms nested in one carrier. Arms are tried cheapest-decisive first. Each bows out at once on a shape it cannot fit, so the cost collapses to the minimum over arms, not the sum. Send an input in and see where it stops.

Drag to orbit

Input shape

Outermost to innermost

FIX ⊙⊡latches the factor

Order of the HARD branch

Outermost to innermost

Height scale

Balanced semiprimes, self-initialized

Correct factors throughout. The MPQS arm began the day capped at 124 bits; the width of the hard input it closes has nearly doubled.

140 bit · about 3 s
160 bit · about 19 s
180 bit · about 2 min
just under 200 bit · about 4.5 min

Where the time goes. At 160 bits GF(2) elimination is about 3 s of a 20 s run; relation collection is the rest. A sparse solver is a fifth-of-the-cost optimization, not the lever.

A count means nothing until it is a distinct-relation count above the base width.

Timings are WSL2 wall clock: trust the large signals, not sub-second deltas.

The membrane also runs as a native-word constraint propagator: every (p_bits, q_bits) split, low arm hensel_unbraid then high arm range_prune_unbraid, each daughter pair gated by word-multiply and syzygy. Terminals: COMPOSITE-found, PRIME-INDECOMPOSABLE, UNIT, INVALID. Live: 100160063 closes to 10007 × 10009 with syzygy true. Shell sizes and bar heights here are schematic except where a figure is quoted.