Research note (no κ): T74 #2322 words — owner fills capped at b < 0.001697195759 (#2374 within 8.1e-10); exact prices of handoffs, aliases and rank-1 climb merges

research/t74-2322-move-prices/README.md

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+# Exact move prices and owner-fill headroom on the T74 PR2322 words
+
+**Research note; no κ claim.** This note gives exact local-move prices and a certified packing cap for the
+words of [PR2322](https://github.com/CrocSwap/integer-mult-bounds/pull/2322) (Vlado Vojdanovski). Those
+words underlie the current open T74 line: #2349, #2363 and #2374, whose claim is
+κ = 0.0016971949509. It answers one planning question: after owner fills, where can the next gain on
+these words come from, and how large can it be?
+
+| finding | value |
+|---|---|
+| **certified cap, any legal whole-helper owner fill of the PR2322 words** | **b < 0.001697195759** |
+| headroom above PR2374's claim | ≤ 8.1e-10 |
+| one more PR2322-type handoff (same reshapes) | +4.27e-8 each, linear up to 12 (+5.13e-7) |
+| one PR2339-type low alias | −1.91e-7 |
+| one PR2339-type high alias | exactly 0 |
+| one (1,1)→(2) merge of two rank-1 climbs (no width change) | +1.13e-7 |
+| the 26 such target merges that come with PR2339's low aliases, on their own | +2.94e-6 |
+
+All prices are LP-ceiling differences on the PR2322 words, both providers, at a = PR2374's κ.
+
+**Owner fills on these words are finished.** The width-aware bin-pattern LP of
+[PR2387](https://github.com/CrocSwap/integer-mult-bounds/pull/2387) /
+[PR2360](https://github.com/CrocSwap/integer-mult-bounds/pull/2360) is certified with an exact rational
+dual and 800-bit enclosures. It caps every legal whole-helper fill of these words at b < 0.001697195759,
+with a W·mu margin of 1.85e-7 and zero dual repair. The certificate does not reject PR2374's claimed
+point. PR2374 therefore sits within 8.1e-10 of the cap, and further owner relocations on these words can
+add at most that much.
+
+## Mechanism
+
+Under the frame-climb rule, a climb of d dimensions on a register pays one recursive child of rank d.
+Local mass equals 1856 plus the sum of helper widths, and the total rank mass is fixed. So a move changes
+b only through the **shape** of the children and through the helper widths, which act via the bin
+patterns.
+
+`move_prices.py` recounts which register pays each child in the pinned words. It checks every recount
+against the word's own reported profile and width inventory, then decomposes the realized changes
+exactly; the reshapes add up to the profile change with nothing left over.
+
+- **PR2312 → PR2322.** The four handoffs are:
+  - remove helpers 619/621/622/623, each (1,1,1,1);
+  - X lenders 1/4/6/12 go (5,1,1,1)→(2,1,1,1,1,1,1);
+  - partners 616/617/618/620 go (4,1,1,1,1)→(3,2,1,1,1).
+
+  The local change is {1:−2, 2:+2, 3:+1, 4:−1, 5:−1} with one fewer width-4 helper: a net −1.23e-5 per
+  call in 66304·mu, a small gain. The LP predicts a +1.96e-7 ceiling gain from PR2312 to PR2322; the
+  realized claims gained +1.69e-7. The underlying prices, LP-ceiling differences, are what this note
+  reports.
+- **PR2269 → PR2339.**
+  - A low alias's lender takes over the removed helper's climb: (3,2,1,1,1) is removed and the lender
+    goes (1,1,1,1)→(2,1,1,1,1,1,1,1). The net local change is −h(3), but the width change costs
+    +0.04137 per call through the bins, a loss of 5.5e-5 per call. Successive low aliases cost more,
+    because the LP is convex; this is why all 32 cost −9.8e-6 (PR2387).
+  - A high alias is exactly price-neutral: its local −h(5) equals its bin cost.
+  - The 26 targets whose climbs changed each gain exactly one (1,1)→(2) merge. That merge has no width
+    effect and is worth +1.13e-7 by itself.
+
+The independent recount (`independent_recount.py`) was written by a separate AI reviewer subagent
+from a plain statement of the frame rules, with its own GF(2) code and no shared code. It reproduces
+both providers' profiles, the exact PR2312→PR2322 change set and the site lists below.
+
+## Untried sites (leads, not constructions)
+
+- 33 helpers of the PR2322 words still climb (1,1,1,1). These are the 32 PR2226 low lenders 273..394
+  plus 653.
+- 12 X registers, 31 33 49 55 68 74 80 83 85 88 98 101, still climb (5,1,1,1) and lend no warp.
+  These are the shapes of PR2322's four lenders before the handoff.
+- Block 53 (registers 648–655) has exactly PR2322's pre-handoff pattern: helper 653 (1,1,1,1) next to
+  four (4,1,1,1,1) partners 648/649/650/652. Its other three slots were already removed. PR2374 notes
+  that alias 653 was dropped when 651 moved to lender 530.
+- The other blocks pair (1,1,1,1) helpers with (4,1,1,1,1,1) mates.
+
+If each extra handoff reproduced PR2322's reshapes, k handoffs would add k × 4.27e-8, up to 12 by the
+lender count. Whether any of them admits a coherent rank-zero handoff (frame availability, lifetimes,
+warp images, native lowering) is **not checked here**.
+
+## Follow-up: where the target merges come from (`merge_mechanism.py`)
+
+**Mechanism.** Checked on all 26 merged targets in both providers:
+
+- In the PR2269 words each merged target climbs through exactly one intermediate frame that the PR2339
+  words skip. For example, target 114 goes through dims [1,2,6,7,8] in PR2269 and [2,6,7,8] in PR2339.
+- Every gate on the target at that skipped frame comes from a helper that PR2339 removes (608/609/610/612
+  for target 114).
+- Each merged target receives PR2339's anticipatory initial corrections, which run at frame 0 and pay
+  nothing. In all, 26 of the 32 corrected targets merge.
+
+So a merge is the removed helper's early interaction, replaced by a free frame-0 correction from a frame-0
+lender.
+
+**Lenders.** A frame-0 lender that takes over a FULL-ending role must climb to FULL (dimension 9 = H).
+Its width therefore becomes 9, and widths can never exceed 9.
+
+On the PR2322 words, the frame-0 helpers end at dimensions {4: 32, 7: 8, 8: 8, 9: 240}. The 32 that end
+at dimension 4 are exactly the PR2226 lenders 273..394, which are PR2339's lenders. The 8 + 8 that end at
+7 or 8 would hand off at dimension 7–8 into first frames of dimension 4–5. PR2339's handoff screen found
+that this fails the rank-zero condition; that finding is inherited here, not re-proved.
+
+Prices on the PR2322 words (LP-ceiling differences):
+
+| lender take-over | status | db per alias | with one merge credited |
+|---|---|---|---|
+| width-4 → 9 (PR2339 pattern) | rank-zero, realizable | −1.91e-7 | −7.8e-8 |
+| width-5 → 9 | no such frame-0 lender | −1.91e-7 | −7.8e-8 |
+| width-7 (end dim 7) → 9 | outside rank-zero | +8.5e-7 | +9.6e-7 |
+| width-8 (end dim 8) → 9 | outside rank-zero | **+1.43e-6** | +1.54e-6 |
+
+**Answer to the second open question.** Under rank-zero anticipatory borrowing, the target merges come
+only with the width-4 → 9 lender change, and every such alias loses even with a full merge credited. The
+merges cannot be separated from that change by choosing a different lender. The positive rows are a price
+tag on a *changed handoff premise*: lenders ending at dimension 7–8, with 16 frame-0 candidates
+(572–575, 580–583, 311, 327, 498, 500, 501, 510, 512, 513). Those prices are optimistic. They omit any
+cost of raising the borrowed role's gate frames to contain the lender's frame, which also moves partner
+climbs.
+
+## Reproduce
+
+From the repository root, standard library only:
+
+```sh
+python3 -I -B research/t74-2322-move-prices/move_prices.py --fetch          # ~16 s
+python3 -I -B research/t74-2322-move-prices/independent_recount.py          # ~10 s
+python3 -I -B research/t74-2322-move-prices/merge_mechanism.py              # ~11 s
+```
+
+`--fetch` fetches `refs/pull/{2312,2322,2269,2339}/head` from the canonical repository if the pinned
+commits are absent. The pinned commits are fb76f799, 002304ef, b479fc55 and bf1742ae; each word file
+is blob- and sha256-pinned.
+
+Expected final lines:
+- `PASS_T74_MOVE_PRICES cap=0.001697195759 handoff_db=+4.272e-08 low_alias_db=-1.906e-07 high_alias_net=0 sites=33/12`
+- `PASS_T74_INDEPENDENT_RECOUNT`
+- `PASS_T74_MERGE_MECHANISM merged=26 lenders_end4=32 pr2339_alias_db=-7.792e-08 takeover8_db=+1.427e-06`
+
+`vendor/pr2387/` holds byte copies of this author's PR2387 `ceiling_lp.py` and `fill_check80.py` and the
+iteration-3 enclosure code, all pinned. `move_prices.py` patches one line of `local_recount` at load time,
+to record the register that pays each child. All results were run on CPython 3.14.4; the registered
+Python 3.12 replay was not run locally.
+
+## Scope and limits
+
+This is a pricing screen on fixed, pinned words:
+
+- Moves are priced only by their per-register climb reshapes and width change, through the bin-pattern
+  LP. Windows, families and ports are dropped, so prices are ceiling differences, not realized gains.
+- The cap is a certified relaxation ceiling for unsplit helpers on the unchanged PR2269 bin stock, not
+  a κ.
+- No new handoff or merge is constructed.
+- Prices are first-order in the sense that they are measured at the PR2322 operating point. The low-
+  alias row shows that the LP is convex, so large moves need a re-solve, which the script does for the
+  ladder.
+
+## Open next questions
+
+- **Block 53 handoff.** Does helper 653 admit a PR2322-type coherent handoff to a free (5,1,1,1) X
+  register, with partners 648/649/650/652? The first step is to rerun PR2322's handoff generator with
+  653 as the target role. Its price is +4.27e-8. PR2374 dropped alias 653, so its lender conflict is the
+  blocker to check.
+- **Higher-rank lender take-over.** *(This replaces the earlier "target merges without low aliases"
+  question; the answer above closes it for rank-zero borrowing.)* Can a lender that ends at dimension 7
+  or 8 take over a removed width-8 helper's role if the role's gate frames are raised to contain the
+  lender's frame? There are 16 frame-0 candidates on the PR2322 words, priced at up to +1.43e-6 each
+  before frame-raising costs. The first step is to build one such word with the PR2339 builder,
+  relaxing condition (d), then recount it with `move_prices.py`'s recount to get the real reshape cost.
+- **More handoff patterns.** Which X shapes other than (5,1,1,1), and which partner shapes, give
+  positive prices? The first step is to tabulate prices for every (lender shape, partner shape) pair
+  that occurs in the words.
+
+## Credits and licence
+
+- PR2322 (Vlado Vojdanovski): handoffs and words.
+- PR2312 and PR2269 (Jeremy Kothe): width exchange and borrow43 words.
+- PR2374 and PR2349 (jeremykothe2) and PR2363 (mason-hale): owner fills on PR2322's words.
+- PR2226 (Cinna Davis / 1337isnot1337): anticipatory borrowing.
+- PR2339 and PR2387 (this author).
+- PR2360 (Jeremy Diviney) and PR1129 (Ali Romman): pattern-dual relaxation.
+- PR2394 (commdata2338): native anticipatory prefix.
+- PR1995 and PR2013: fixed-word frame bounds.
+- PR1319 and PR1297: T74 lineage.
+
+Prepared with substantial AI assistance (Claude Code, Anthropic, Claude Opus 5.5). The independent
+recount was written by an AI subagent; no independent human review is claimed. Apache-2.0.

research/t74-2322-move-prices/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "ae6a99f28aa468649eb66a361e764d769363bf11",
+      "sha256": "10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "e8870e5917057d59b730eb14721a778639eca050",
+      "sha256": "d46eb6d5b3e279428753d770653663fbdf1063e55c255ea073d768f344927ef7"
+    }
+  ]
+}

research/t74-2322-move-prices/TASK.json

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+{
+  "schema_version": 1,
+  "task_id": "complex-new-helper-inventory",
+  "priority_revision": 4,
+  "base_commit": "0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2",
+  "original_card": {
+    "id": "complex-new-helper-inventory",
+    "question": "Can a new local cost interface change the deficit-to-recursive-cost ratio beyond current packing?",
+    "starting_points": [
+      "research/complex-local-word-ceiling/README.md",
+      "research/heterogeneous-owner-assembly/README.md",
+      "certificates/selected-result.json",
+      "research/selected-k8-unrestricted-ends/README.md"
+    ],
+    "methods": [
+      "Propose a changed helper inventory and derive an optimistic exact moment.",
+      "Design a small completed instance and independently verify its endpoint and rank bill."
+    ],
+    "acceptance_gate": "A new physically realizable contract outside the frozen catalogue whose per-call profile, screened with ceiling.ceiling(), beats the selected word's packing ceiling; a lower cap does not establish identical circuitry or order attainable optima, and is insufficient to pass the stronger-budget target under this model.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "research/complex-local-word-ceiling/verify.py"
+      ],
+      "expected": "PASS_COMPLEX_LOCAL_WORD_CEILING",
+      "inspect": "Read the selected word's per-call local profile, its packing ceiling and the lever table; run levers.py for the ideal budget of the lever you target.",
+      "scope": "Positive control and planning budget only; this is not a new construction or full cold replay."
+    },
+    "first_experiment": "Before any synthesis, write the candidate per-call local profile (and centre charge / unpaid coordinates if changed) and screen it with ceiling.ceiling(); reject it if even the ideal ceiling does not beat the selected word's by your stated margin.",
+    "bottleneck_link": "Packing of the selected word is capped at +0.87% (ceiling model); only a lower local excess spend or a changed coverage law raises the cap.",
+    "allowed_variations": [
+      "Narrow to one changed boundary or helper type with explicit counts.",
+      "If the coverage law, template or guard changes, derive an applicable fully charged replacement bound instead of applying the incumbent ceiling outside its scope."
+    ],
+    "scope_change_triggers": [
+      "Mixing only providers already inside the pinned catalogue.",
+      "Only changing helper terminal subspaces and monotone frames within the fixed scalar word, entrances and cleanup contract already covered by #1995/#2013."
+    ],
+    "overlap_terms": [
+      "helper inventory",
+      "heterogeneous-owner",
+      "mutable leaver"
+    ]
+  },
+  "scope_status": "refined",
+  "actual_scope": "On the PR2322 T74 words (the base of the open PR2349/PR2363/PR2374 line): what is the certified owner-fill headroom, and what is the exact LP-ceiling price of each realized or candidate local inventory move (PR2322 handoffs, PR2339 low/high aliases, rank-1 climb merges), with a census of untried PR2322-pattern handoff sites?",
+  "scope_change_reason": "The card asks whether a new local cost interface can beat current packing. For the live T74 line the useful narrowing is to price concrete inventory moves on the actual PR2322 words with the width-aware LP, after showing packing itself is exhausted, so constructors can see which interface changes are worth building.",
+  "bottleneck_link": "The complex supplier binds. On the PR2322 words, owner fills are within 8.1e-10 of a certified cap, so further T74 gains must come from local inventory or climb-shape changes; this package prices those moves exactly and points to the largest (rank-1 climb merges, +1.13e-7 each).",
+  "budget_minutes": 90,
+  "stop_rule": "Stop after the certified cap, exact decomposition of the two realized changes, prices of four move types, an independent recount and the site census; no handoff construction in this session.",
+  "outcome": "completed",
+  "evidence": "move_prices.py: per-register recount of 8 pinned words (each equal to its reported profile and inventory), exact decompositions PR2312->PR2322 and PR2269->PR2339, LP prices, calibration (LP +1.96e-7 vs realized +1.69e-7), certified cap b < 0.001697195759 for every legal whole-helper fill of the PR2322 words (PR2374 not rejected). independent_recount.py (separate implementation by an AI reviewer subagent) reproduces profiles, the change set and the site lists. merge_mechanism.py: all 26 PR2339 target merges come from removed helpers' early gates replaced by frame-0 initial corrections; frame-0 lender census {4:32, 7:8, 8:8, 9:240}; rank-zero aliases lose even with merge credit; higher-rank take-overs priced.",
+  "author": "Sagar Thakrar",
+  "ai_assistance": "Prepared with substantial AI assistance: Claude Code (Anthropic, Claude Opus 5.5) wrote move_prices.py and this text; a separate Claude subagent wrote independent_recount.py from a plain-language rule statement without access to the package code. No independent human review is claimed.",
+  "open_questions": [
+    {
+      "question": "Does helper 653 of the PR2322 words admit a PR2322-type coherent handoff to a free (5,1,1,1) X register with partners 648/649/650/652?",
+      "why_it_matters": "Block 53 has exactly PR2322's pre-handoff shape pattern; the move is priced at +4.27e-8, about a third of the whole PR2363->PR2374 owner gain.",
+      "first_step": "Rerun PR2322's handoff generator with role 653 and each of the 12 free (5,1,1,1) X registers as candidate lenders.",
+      "blocked_by": "PR2374 dropped alias 653 when moving 651 to lender 530; lender/lifetime conflicts are unchecked here."
+    },
+    {
+      "question": "Can a frame-0 lender ending at dimension 7 or 8 take over a removed width-8 helper's role (higher-rank handoff), with the role's gate frames raised to contain the lender's frame?",
+      "why_it_matters": "Rank-zero anticipatory aliases (width-4 -> 9 lenders) lose even with the target merges credited; the 16 frame-0 lenders ending at dimension 7-8 are priced at up to +1.43e-6 per take-over on the PR2322 words before frame-raising costs.",
+      "first_step": "Build one such word with the PR2339 builder relaxing its condition (d), then recount it with move_prices.py's recount to price the actual reshape of the role and its partners.",
+      "blocked_by": "The handoff is outside the rank-zero K_G contract of PR2226/PR2339; legality, frame raising and native lowering are unproven."
+    },
+    {
+      "question": "Which other (lender shape, partner shape) handoff patterns occurring in the PR2322 words have positive prices?",
+      "why_it_matters": "Only (5,1,1,1) lenders are priced here; the 12 such registers limit PR2322-type handoffs to +5.1e-7.",
+      "first_step": "Enumerate per-register shapes of all X registers and helpers and price each feasible-looking reshape pair with move_prices.applied().",
+      "blocked_by": "Legality of a handoff depends on frame availability and lifetimes that the pricing screen does not model."
+    }
+  ]
+}

research/t74-2322-move-prices/VERIFY.json

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+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-B",
+        "{package}/move_prices.py",
+        "--fetch"
+      ],
+      "timeout_seconds": 300,
+      "checks": "Fetches pinned PR2312/2322/2269/2339 heads if absent; recounts 8 pinned words against their reports; exact decompositions; prices four move types; certifies the PR2322-word owner-fill cap b < 0.001697195759 and checks PR2374 is not rejected (control). Expect PASS_T74_MOVE_PRICES."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-B",
+        "{package}/independent_recount.py"
+      ],
+      "timeout_seconds": 300,
+      "checks": "Separate implementation: both providers' profiles equal the reports; exact PR2312->PR2322 change set; 33 (1,1,1,1) helpers and 12 free (5,1,1,1) X registers. Expect PASS_T74_INDEPENDENT_RECOUNT."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-B",
+        "{package}/merge_mechanism.py",
+        "--fetch"
+      ],
+      "timeout_seconds": 300,
+      "checks": "Mechanism of the 26 PR2339 target merges (both providers), frame-0 lender census on the PR2322 words, and LP prices of lender take-over variants. Expect PASS_T74_MERGE_MECHANISM."
+    }
+  ]
+}