Mister-Meeseeks · Oct 13, 2026 · Oct 13, 2026
diff --git a/Makefile b/Makefile
@@ -604,3 +604,10 @@ frontier890-verify: selected-record-check
 frontier942-verify: selected-record-check
 	@work=$$(mktemp -d "$${TMPDIR:-/tmp}/integer-mult-frontier942.XXXXXX"); \
 	python3 -B scripts/verify_frontier942.py --work "$$work/replay"
+
+# Research note: width packing of the selected PR589 bit hosting (no new kappa).
+.PHONY: bit589-width-packing-verify
+bit589-width-packing-verify:
+	@work=$$(mktemp -d "$${TMPDIR:-/tmp}/integer-mult-bit589-packing.XXXXXX"); \
+	python3 -B research/bit589-width-packing/verify.py --work "$$work/replay" && \
+	python3 -B -m unittest discover -s research/bit589-width-packing -p test_package.py
diff --git a/research/bit589-width-packing/CLAIM.json b/research/bit589-width-packing/CLAIM.json
@@ -0,0 +1,41 @@
+{
+ "ai_assistance": "substantial: Claude (Anthropic) via Claude Code",
+ "assignment_seed": 16,
+ "base_commit": "2921874911956718fcaaaf9396fe8267b3cfa9f0",
+ "bit_bound_kappa_caps": {
+  "selected": "865213734911801/500000000000000000",
+  "width7_all_optimistic": "426919181844469/200000000000000000",
+  "width7_all_pessimistic": "406314394605613/200000000000000000",
+  "width7_k10_optimistic": "349047987845817/200000000000000000",
+  "width7_k10_pessimistic": "1742786661941923/1000000000000000000",
+  "width7_k1_optimistic": "173184178486967/100000000000000000",
+  "width7_k1_pessimistic": "865799952969731/500000000000000000",
+  "width7_k37_optimistic": "111651901215299/62500000000000000",
+  "width7_k37_pessimistic": "888674029610291/500000000000000000",
+  "width7_k74_optimistic": "461572653313689/250000000000000000",
+  "width7_k74_pessimistic": "1827854925395469/1000000000000000000",
+  "width8_free_concentration_42": "485875782209319/250000000000000000",
+  "width8_free_concentration_45": "1952100603477031/1000000000000000000"
+ },
+ "claims": [
+  "The frozen PR589 nine-stage hosting has the unique slot model {7:666, 14:444, 28:444, 42:444} per replica (41958 coordinates, equal to the coverage pending capacity).",
+  "No no-bank placement of the width-8 residuals raises the bit endpoint on the 1e-24 grid: at c0+1e-24 a fully idle 42-slot is the certified unique cheapest unit of the 45 slack units per replica.",
+  "Width-7 residuals fill all slots exactly; converting all 439 helpers gives bit c in [0.002035707659654698, 0.002139162156009728] with idle {42:340, 21:1} per replica; partial conversions k=1,10,37,74 give +0.068..0.082%, +0.72..0.86%, +2.7..3.2%, +5.6..6.7%.",
+  "Supporting evidence: all six one-level windows of the selected complex T37 hosting on the same nine-stage template are fully hosted."
+ ],
+ "contributor": "Sagar Thakrar (thakrarsagar)",
+ "expected_last_line": "PASS_BIT589_WIDTH_PACKING_REPLAY",
+ "license": "Apache-2.0",
+ "not_claimed": [
+  "a width-7 bit construction",
+  "window timing for bit one-level windows",
+  "a new selected kappa (kappa is complex-bound)",
+  "optimality of the partial-conversion placements",
+  "any inherited all-size, realization, hosting, precision/tape, analytic or bit H1-H5 assumption"
+ ],
+ "reproduce": "make bit589-width-packing-verify",
+ "seed_source": "Python secrets (OS CSPRNG)",
+ "selected_kappa_unchanged": "1661960547713947/1000000000000000000",
+ "status": "research note / scoped negative result (bit supplier); no new kappa",
+ "track": "00-19 bit supplier"
+}
diff --git a/research/bit589-width-packing/README.md b/research/bit589-width-packing/README.md
@@ -0,0 +1,153 @@
+# Width packing of the selected PR589 bit hosting
+
+**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`.
+
+## Question
+
+The selected bit supplier is DreamingOfClouds' PR589 on eumemic's PR555, with
+c = 0.001733427039589844234749. It hosts all 3,951 width-8 helper residuals per replica in the
+nine-stage idle windows, with no banks, yet its idle remainders are fragmented:
+`{2:435, 4:444, 6:444, 7:666, 42:9}` per replica. Can re-hosting improve the bit profile, and
+what would a different helper width be worth?
+
+## Findings
+
+1. **Slot model (exact, unique).** The nine slot indices of the frozen selection hold one pair's
+   222 families each. Their capacities are `{14, 7, 28, 7, 42, 7, 42, 28, 14}`, so per replica there
+   are 666 slots of capacity 7, 444 of 14, 444 of 28 and 444 of 42. The total, 41,958 coordinates,
+   equals the coverage-identity pending capacity `2v·42 + 6v·49`. These are the template's one-,
+   two-, four- and six-level windows at 7 coordinates per level, the same window structure the
+   complex template has at 8 per level. The deficit satisfies `live − 63·calls = 10,272`.
+2. **Width-8 re-hosting cannot help (certified).**
+   - 8 divides no capacity, so every slot keeps a remainder of at least 7, 6, 4 or 2. Every
+     no-bank placement therefore has exactly one idle child per slot.
+   - Only 45 width-8 units of slack exist per replica (1,440 in total).
+   - At c* = c0 + 10⁻²⁴, the per-unit cost of a fully idle capacity-42 slot is certified strictly
+     below every other (capacity, unused units) option. The runner-up is (42, 4), behind by a
+     certified 0.0026 in units of g(r) = r(64/r)^c.
+   - So the selected placement, with 9 fully idle 42-slots per replica, minimizes the moment.
+     No re-hosting raises the bit endpoint on the 10⁻²⁴ grid, and the +12.3%
+     "free-concentration" relaxation is unreachable at width 8.
+3. **Width-7 residuals fill every slot exactly.** A helper of width 7 (a hyperplane end or a
+   rank-one entrance) leaves no remainder in a 7k slot. Converting all 439 helpers per call leaves
+   2,043 units of slack per replica. The majorization optimum then leaves only `{42: 340, 21: 1}`
+   idle children per replica, against 1,998 now. The bit c rises by +17.4% to +23.4%.
+4. **Partial conversion already pays.** Converting k helper indices in every call, with their
+   width-7 residuals filling empty 7-slots and the freed width-8 slack going to fully idle
+   42-slots, gives the table below.
+5. **Window timing: supporting evidence only.** 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
+   (children 1, 3, 5, 7, 9 and 11) are completely hosted: 198,912 coordinates, 0 idle. That
+   suggests one-level windows admit hosted residuals, but the bit-side timing check is not done.
+
+## Exact prices
+
+Each bit endpoint is certified by both pinned engines, with the inherited bad-residue surcharge:
+the moment is below one at c and above one at c + 10⁻²⁴. The κ cap is the selected T37 outer
+witness with the bit side binding: 7 bootstrap steps, all 47 inequalities, and κ + 10⁻¹⁸
+rejected. The selected κ stays 0.001661960547713947 because the complex side binds.
+
+| Scenario (one unit removed from one local child per converted helper) | Bit c | Gain | Bit-bound κ cap |
+|---|---|---:|---|
+| selected PR589 profile (reproduced exactly) | 0.0017334270395898442 | — | 0.001730427469823602 |
+| 1 helper per call width 7, rank-one child removed / largest child trimmed | 0.0017348463 / 0.0017346035 | +0.082% / +0.068% | 0.001731841784869670 / 0.001731599905939462 |
+| 10 helpers per call | 0.0017482911 / 0.0017458293 | +0.857% / +0.715% | 0.001745239939229085 / 0.001742786661941923 |
+| 37 helpers per call | 0.0017896275 / 0.0017805126 | +3.24% / +2.72% | 0.001786430419444784 / 0.001777348059220582 |
+| 74 helpers per call (all 666 empty 7-slots used) | 0.0018497057 / 0.0018312021 | +6.71% / +5.64% | 0.001846290613254756 / 0.001827854925395469 |
+| all 439 helpers width 7, slot-exact optimum | 0.0021391622 / 0.0020357077 | +23.41% / +17.44% | 0.002134595909222345 / 0.002031571973028065 |
+| relaxation: width-8 idle mass in rank-42 children | 0.0019472877 | +12.34% | 0.001943503128837276 |
+
+The bracket gives two bounds for a conversion that removes exactly one unit from one local child.
+Removing a rank-one child gives the largest gain and trimming the largest children gives the
+smallest, because g is concave. The partial rows use one constructive slot-exact placement, so
+for those k they are achievable values in the slot model, not optima. The last row ignores the
+slot structure that finding 2 proves binding.
+
+## What this means
+
+- These numbers are **bit headroom**. They give the κ that the selected outer recipe would allow if
+  the complex side improved past the bit side. Today's bit cap, 0.001730427469823602, matches the
+  value recorded in PR974.
+- No hosting or owner work on the current bit word can raise c. The lever is helper width.
+  Converting one helper index in every call, with its residuals in empty 7-slots, is worth about
+  +0.07% to +0.08% bit c, more than the whole PR589 tuning step (+0.0216%).
+- Width-7 mechanisms already exist on the complex side: hyperplane-ended helpers (PR601, PR593)
+  and width-seven entrance promotions (PR569). Their bit analogues are the natural next
+  construction targets. The first obstruction is the bit word's literal reverse cleanup at the
+  full frame (interface hypothesis H4): a hyperplane end needs that helper's cleanup retimed.
+
+## Remaining obligations (none attempted)
+
+1. Construct width-7 bit helpers, either as hyperplane ends with cleanup retiming or as rank-one
+   entrances with kernels. Keep H1–H5 or state a replacement interface, with exact scalar and
+   dirty-state checks.
+2. Check window timing: the one-level bit windows must admit the converted residuals, with owner
+   inactivity and one helper per family per call, through the PR510/PR463 hosting audit.
+3. Re-emit the hosted template and rerun the hosted audit, the finite invoice (prefix bits, row
+   coefficient) and the bit assembly.
+4. A real conversion may change more than one unit of one child, which the bracket does not cover.
+
+## Scope and limits
+
+- The slot model comes from the frozen selection and the pinned record. It is the unique solution
+  consistent with the observed remainders and the coverage capacity. Findings 2–4 do not impose
+  window timing or per-call ownership rules; the baseline placement satisfies them as selected.
+- Finding 2 compares no-bank placements only. Banks add live stock without free coordinates.
+- All inherited assumptions remain: all-size compilation, common-label/mutable-source
+  realization, whole-call hosting, precision/tape, analytic transfer, bit H1–H5 and the bit
+  bad-residue surcharge, which every price above keeps. This is not a construction and not a
+  formal proof of the multiplication theorem.
+
+## Reproduce
+
+```sh
+make bit589-width-packing-verify
+```
+
+The same check without make:
+
+```sh
+python3 -B research/bit589-width-packing/verify.py --work /new/empty/dir
+python3 -B -m unittest discover -s research/bit589-width-packing -p test_package.py -v
+```
+
+The expected last line is `PASS_BIT589_WIDTH_PACKING_REPLAY`, followed by 6 tests OK. It runs in
+about 3 s on standard-library Python 3.10+ with no network.
+
+- `verify.py` regenerates `RESULT.json` and requires it to equal the committed file.
+- It checks the T37 one-level-window evidence from the pinned T37 archive, read in memory.
+- It runs six negative controls:
+
+| Control | Rejected because |
+|---|---|
+| C1 one residual moved between slot indices | stage and pattern totals disagree |
+| C2 a capacity divisible by 8 | the per-unit argument refuses exact width-8 fits |
+| C3 corrupted record pin | digest mismatch |
+| C4 bit endpoint c + 10⁻²⁴ | both engines exceed one, surcharge included |
+| C5 a width-7 profile that drops the freed coordinates | rank mass changes |
+| C6 more converted residuals than empty 7-slots | the constructive placement refuses |
+
+## Sources and pins
+
+See [`SOURCES.json`](SOURCES.json).
+
+- **Bit record.** `research/e8-bit-p111-tuned/tune/EXPECTED.json` (`b9f0e5ca…`). This digest is
+  the `package:inputs/PR589-EXPECTED.json` input of the selected `ASSEMBLY.json`.
+- **Hosting selection.** `research/e8-bit-p111-tuned/data/selection9-p111.json` (`4ab1ed45…`).
+- **Engines.** The pinned engines recorded in the selected assembly.
+- **Outer witness.** Transcribed from the selected T37 archive.
+- **Supporting evidence.** The selected T37 hosted template (payload `956e0c0e…`).
+
+## Credit
+
+- DreamingOfClouds: the PR589 word and the PR510/PR463 nine-stage bit template.
+- eumemic: the PR555 punctured unit and selection.
+- DaysSky: the idle-axis hosting of PR442.
+- Jeremy Kothe: hyperplane-ended helpers (PR601).
+- Dominik Scholz: width-seven complex providers (PR569, `research/larger-gain`).
+- avirazza: the bit cap value of PR974.
+- Prepared by Sagar Thakrar (thakrarsagar) with substantial AI assistance from Claude
+  (Anthropic) via Claude Code. New material is Apache-2.0.