Returns — what the lattice does that the block map does not, and in which directions

An inkling, recorded with conviction and unjudged. Nothing here is a result. The owner had it first, from the picture page (the plates on blocks crossed at crossing and on the three ways through a block), before the codifier named it on 2026-10-03.

The inkling. Every shortcut tried so far keeps routes that go through a block, in by one side and out by another, and drops routes that come back the way they came. What the lattice does that the block map does not is the returns.

Where it has shown. - The twins (READING-TWINS.md). Members of a polynomial group differ only by pendant parts, which a route enters and leaves by the same cell. That alone moves the threshold by about 0.0025 for four parts. In group B, trading a straight passage for a corner one moves it by less than 0.0003. - Blocks at crossing (runs 083, 085). The carpet crosses with about 38 of T's 64 blocks crossed where the block graph needs 50. The others are entered and left. - The table of side pairs (P-053). Its third class, the same side twice, is a return by definition. - The ladder (runs 084, 094), read this way and not shown: a few contact points per side thin the routes that leave a block and come back through its seam. - PG-02. A free coset flips at no cost: a detour that closes on itself.

Why a return is invisible at one level and not at the next. Inside one block a pendant part hangs on one cell, so no route between two ring cells can use it. One level up the same part is a whole block joined to its neighbour along a full side, and a route can leave by one bond of that seam and come back by another. This is the owner's Stokes inkling of 2026-10-01 (P-050, P-024) in concrete form: the detours cancel inside a level and fail to cancel at the seam between levels. Returns can only help a crossing, which would be why the lattice threshold sits below the block's fixed point on all 49 formulas (L8).

In which directions (the owner's question, 2026-10-03). Not left and right only. Sort a connected cluster inside a block by which sides it touches. Under the square's symmetry there are six kinds: no side, one side, two adjacent sides, two opposite sides, three sides, all four. The block map keeps one member of "two opposite". A return is a cluster that touches one side in two places, and all four sides are alike. A route crossing left to right can use returns in the blocks above and below it.

There is a second kind, the one the owner pictured as Pac-Man. Pattern #47 carries three-cell loops hanging on one core cell: a route leaves that cell by its east side, goes round the loop, and comes back in by its south side. It closes on a different side of the block it left. Inside one block this is still invisible; one level up it is a way round the block's own interior. Pattern #46 has two one-cell parts on the same cell in place of each loop, so #46 against #47 (run 098, reported) is the first comparison of the two kinds.

A third reading of the same picture, as a lead with nothing computed: glue a block's opposite sides, as on the Pac-Man screen. A return across the seam then becomes a route that winds round. The chance that a wrapped block is wound round, fed into itself, would be a shortcut that counts some returns as crossings, and its fixed point should sit nearer the lattice threshold than the plain block's does. Where it breaks: the wrapped formula is also a property of the core, so it cannot tell twins apart; it could only narrow the gap of L8 across the census.

With the centre pinned (the owner's second question, 2026-10-03: what of paths that lead to the centre or gather there, as a divergence or a curl would measure). The square's symmetries all fix the centre, and the centre cell is removed in every one of the 63 patterns (P-050). So nothing can gather at the centre: a path that heads for it is a dead end, and whatever goes in comes back out. Divergence at the centre is the returns again, and the spokes of a pattern are its pendant parts pointing inward. Curl is the other thing a hole forces: a route must pass the hole on one side or the other, so it goes round. The record already holds a curl measure under another name, the ring share of the pivotal set, and it is the best single predictor of the gap between the block's fixed point and the lattice threshold (L8, +0.83). That gives a three-way split of what a route does at a block, the split the owner named with Stokes, Gauss and Helmholtz: through (neither), round (curl, forced by holes), in and out (divergence, the dead ends). Put a battery across the block and only the first two carry current. The literature's names for the two parts of READING-TWINS.md are the backbone (the core) and the dangling ends (the pendant parts). What is particular to a carpet is that a dangling end at one level is not dangling at the next: it is a block joined along a whole side, and current can pass through it.

The twins as the eigenbasis experiment. The eigenbasis chase (L14) asked for a second coordinate beside the formula. A polynomial group holds the formula fixed and moves only the returns. On record the threshold moves; the rate may (1/ν differs by 3.4 errors on #62, #63 and by 1.5 on #58, #60, and by 0.6 on #23, #24 where only depth differs). Whether returns move the rate or only the threshold is the question under the programme: if only the threshold, one coordinate fixes the exponent and the rest is correction.

What tests it. Run 098: the size of a pendant part's effect on three cores and on cut 2. Run 097: the recorder keeps the same-side events on the lattice. A zero-compute reading owed: local slopes of the three measured pairs at 81, 729 and 6561 from their stored samples.

Run 098, scored 2026-10-03 (confirmed). What it did to the inkling. The place it could have broken, clause 1, held: four one-cell parts cost 0.0019, 0.0026 and 0.0027 on three different cut-4 cores. It is not one number: on group C four parts cost 0.027, and there they hang on the bottleneck; the run does not separate the part's place from the core. The return that closes on a different side, #47's loops, showed no cost beyond dead ends on the same cells (to about 0.0004 over the four). A second cell on a part added nothing measurable on G. Whether returns move the rate is still open: parts changed the width at 729 on four groups, which is one size.

Where it breaks. If the cost of a pendant part differs widely between cores (run 098, clause 1), the cost belongs to the core and "returns" is not one thing. If a block graph that keeps same-side events tracks the lattice no better than one without (a reading on run 097's files once it scores), returns are not what the map lacks.

Two things about method, from the same day. - The answer was in the picture. The case printed the twenty-two patterns; five seats, the House and the codifier reasoned about polynomials for a day; looking at which cells differ settled it in half an hour. The case gate now asks why a coincidence holds before a panel is asked about it (docs/v5-plan.md, G6). - The owner's reading of a picture ran ahead of the codifier's reading of the numbers. A hunch from a figure is recorded when it is had, unjudged, with the figure named, so the tools can catch it or kill it.