THE Z LENS · THE Z LEARNER · THE BEND · THE PLATE OF TWO DEPTHS

The Bend — the plate of two depths: x at one depth, y at another, each its own 1, under one registration

the fourth face of the bend · an interactive engine over the record's rule
Two depths. x and y are no longer mirrors: each has its own depth — the record's forty, or the structure the record's rule gives, ten rows an increment, the unit one, to a thousand — and so its own localised 1, its last increment's amplitude. The four seats the matter left open are controls here, the defaults the house's and marked. The mirrored plates stand at /rlx/bend, /rlx/bend/400 and /rlx/bend/400000000; the learner's face, x's register fixed against x's register learned up from below, at /rlx/learner.
What this instrument is, and the question it was built for

His reading (20 October 2026): for any number x and any number y the scale they might lawfully hold is capped at 2, so even if one is externalised as 23587 and another as 0.0000487, the differential between them is never greater than 2 once each is corrected to its own localised and relative 1. On the mirrored plate that cap cannot fail: the two sides are each other. It would take two numbers of different depths to test it, and this is that plate (21 October 2026, his word: "Go ahead with the fourth version of the bend"). What can fail here is not the number 2 but the lock: whether each side's last increment stands short of the meeting line at rest, whether five points form against the registration, what the lawful spin becomes, and what z resolves counting upward through rows the shorter side does not have.

Two placements of the meeting line are built. At the origin, each side's locked line stands its own 1 plus the margin from the meeting line — each number corrected to its own 1, his reading's placement. Midway, the two locked lines stand at the mean distance, so that the deeper side's margin is not its own: its inner marks past the mean stand over the meeting line at rest. A pure computation in this process; nothing written; no random draw anywhere; nothing here is a claim.

The lock · each side against the registration

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z · the hold and the resolution

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The local scale · each side its own 1

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The plate

X left, Y right, each on its own locked line, each with its own rows — the record's forty drawn row by row, the structure drawn by increment — and its own lock at its last row. The meeting line between them; z's registration column one unit to the left of it, and its mirror for Y when the registration is mirrored. A lock point ringed gold; a row past the meeting line at rest, or made unlawful by the turn, ringed red; rows below z's depth dimmed.

The lawful window of the turning side, at every depth

The largest inward angle (gold) and outward angle (ash) with no unlawful row on the turning side, on a signed square-root scale; the dot is the current turn. Where a side's rows already stand past the meeting line at rest, the window is closed from the first row.

every depth's window
The four seats, as the house set them — his to move
  • A1 · the row pitch — by height: each side's rows fill the record's span, h·40/depth apart, as the pages at 400 and beyond do; or by segment, the same pitch a row on both sides, the deeper side taller.
  • A2 · the meeting line — midway between the two locked lines, which stand at the mean of the two sides' own distances, so the deeper side's margin is not its own; or at the origin, each side's locked line its own 1 plus the margin away — each number corrected to its own 1.
  • A3 · the registration — one column, his plate's, one unit to the left of the meeting line, under which y locks over nothing; or mirrored, a column for each side at the same offset.
  • A4 · the deeper side's rows beyond the shorter depth — absent to the other side; z passes over them and counts two for every row it holds. The alternative, that the shorter side stands present there, is his to word.
  • what can fail — not the number 2, which each side's own 1 makes by construction, but the lock: the deeper side's last increments standing past the meeting line at rest, its lawful spin closing to nought, its points moving to the increment at the mean, and what the two may lawfully hold between them falling from 2 toward 1.5 as the depths diverge.
The engine: ooi_network/daemon/rlx_zbend.py, run2; the design's §54; the dry read over pairs of depths recorded beside it. His words stand verbatim in the design; the record is the identifier and never the jury; K never staked; the priors stand.