Research note: PR589 bit hosting is width-8 optimal; width-7 residuals worth up to +17-23% bit headroom (no new kappa)

[thakrarsagar]

Status: research note and scoped negative result (bit supplier). No new κ; the selected result is unchanged and complex-bound.

Track 00–19 (bit supplier), assignment seed 16 (OS CSPRNG via Python secrets). Base commit 2921874911956718fcaaaf9396fe8267b3cfa9f0. Overlap check at submission: there is no open or recent PR on PR589 hosting optimality or bit helper widths. The last bit-hosting PRs are #462, #205 and #197, all on older words.

What was learned

1. Slot model (exact, unique). The frozen nine-stage hosting of PR589 has, per replica, 666 slots of capacity 7, 444 of 14, 444 of 28 and 444 of 42. The total, 41,958, equals the coverage pending capacity. Every PR589 residual has width 8, and 8 divides none of these capacities. That is why the idle profile is fragmented: {2:435, 4:444, 6:444, 7:666, 42:9} per replica.
2. Width-8 re-hosting cannot help (certified). Only 45 width-8 units of slack exist per replica. At c0 + 10⁻²⁴, a fully idle 42-slot is certified as the unique cheapest unit of slack, and the selected placement uses exactly those. No no-bank placement therefore raises the bit endpoint on the grid. The +12.3% free-concentration relaxation is unreachable at width 8.
3. Width 7 is the lever. Width-7 residuals (hyperplane ends or rank-one entrances) fill every slot exactly. The table brackets one-unit conversions: removing a rank-one child gives the upper value, trimming the largest children the lower.

Converted helpers per call	Bit c gain	Bit-bound κ cap
0 (selected, reproduced)	—	0.001730427469823602
1	+0.068% … +0.082%	0.001731599905939462 … 0.001731841784869670
10	+0.72% … +0.86%	0.001742786661941923 … 0.001745239939229085
74 (all empty 7-slots)	+5.6% … +6.7%	0.001827854925395469 … 0.001846290613254756
all 439	+17.4% … +23.4%	0.002031571973028065 … 0.002134595909222345

One converted helper index is worth more than the whole PR589 tuning step (+0.0216%).

4. Supporting evidence on window timing. The 7-slots are the template's one-level windows. In the selected complex T37 hosting on the same nine-stage template, all six one-level windows are completely hosted. The bit-side timing check remains open.

Reproduce

make bit589-width-packing-verify

The expected last line of the replay is PASS_BIT589_WIDTH_PACKING_REPLAY, followed by 6 unit tests, OK. It runs in about 3 s on standard-library Python 3.10+ with no network.

• The replay regenerates RESULT.json byte-for-byte as JSON.
• Both pinned engines certify every endpoint, with the inherited bit surcharge.
• The κ caps use the transcribed selected outer witness, with all 47 inequalities and κ + 10⁻¹⁸ rejected.
• Six negative controls run: pattern moved, a capacity divisible by 8, a pin corruption, the next endpoint, lost coordinates, and too many residuals for the 7-slots.

Scope and remaining obligations

These are bit headroom numbers. κ stays 0.001661960547713947 because the complex side binds.

Nothing here constructs a width-7 bit helper. Realizing any row needs:

• hyperplane ends with cleanup retiming (the bit word's H4 literal reverse at the full frame is the first obstruction; cf. PR601), or rank-one entrances with kernels;
• a window-timing and ownership audit of the bit one-level windows;
• a re-emitted hosted template, invoice and bit assembly.

The partial rows are achievable values of one constructive placement in the slot model, not optima. All inherited all-size, realization, hosting, precision/tape, analytic and bit H1–H5 assumptions remain.

Credit

• DreamingOfClouds: PR589, and the PR510/PR463 nine-stage bit template.
• eumemic: PR555.
• DaysSky: the PR442 idle-axis hosting.
• Jeremy Kothe: hyperplane ends (PR601).
• Dominik Scholz: width-seven complex providers (PR569).
• avirazza: the bit cap recorded in PR974.

Prepared by Sagar Thakrar with substantial AI assistance from Claude (Anthropic) via Claude Code. New material is Apache-2.0.