Mister-Meeseeks · Oct 20, 2026 · Oct 19, 2026
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+# Integer twins of three ideal lever combinations
+
+**All three minimal reaching cells in PR3874 still clear its stated target after
+integer rounding in the declared models.** This resolves its first open question
+as an arithmetic screen. No physical realization, supplier or new kappa is
+claimed. The selected conditional result remains 0.001672660197076918.
+
+The most restrictive row tested preserves one identical integer paid profile in
+every call and pools idle mass across nine calls. For selected-centre/rank8/u2,
+its root is strictly between **0.010351600977 and 0.010351600978**, giving a
+balanced-transfer kappa cap between **0.010245543201980** and
+**0.010245543202961** (outward rounded here). Thus integer rounding does not
+remove this cell's planning margin. However its remaining allowance at the
+target is only **16 additional rank-one children per nine-call pool**; 17 fail.
+This is an additive recursive-work budget, not a scalar-ADD tariff or permission
+to ignore restoration and other costs.
+
+## Exact question and source boundary
+
+This takes Nick Petalas's [PR3874](https://github.com/CrocSwap/integer-mult-bounds/pull/3874)
+Open next question 1 at `fb86b078a53042974d88b9758fe36622571349d0`, following the
+[handoff](https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19028179).
+The canonical claim/result lives in [Discussion2278](https://github.com/CrocSwap/integer-mult-bounds/discussions/2278#discussioncomment-19029577).
+The original card is `complex-new-helper-inventory`, explicitly **changed** to
+this requested pricing subproblem, not its physically-realizable-contract gate.
+
+Keep width 73, 896 live families per nine calls, rank guard 51 and the following
+three parent cells. Shared centre products stay rank 8, just as in PR3874.
+
+| Cell | Variable paid mass per call | Variable rank cap | Fixed rank-8 children per call | Centre charge | Unpaid coordinates |
+|---|---:|---:|---:|---:|---:|
+| selected-r8-u2 | 5586 | 8 | 0 | 64 | 2 |
+| shared-r32-u1 | 5522 | 32 | 4 | 32 | 1 |
+| free-r16-u1 | 5522 | 16 | 0 | 0 | 1 |
+
+The declared target is **kappa = 1/99**, following PR3874's explicit table.
+Since kappa = a/(1+a), the correct moment test is **a = 1/98**.
+Testing a = 1/99 would instead test kappa = 1/100. This package does not change
+the repository's separate kappa > .01 planning objective.
+
+For N calls, W = 896 N/9 and total rank mass is `(73-u)W + C N`.
+Subtract the full paid mass to obtain idle mass. For u2 the per-call idle
+average is `13918/9`; for the other two cells it is `1646`. Integral families
+and the u2 mass require N divisible by nine. We test N=9 and N=144; neither is
+asserted to be a legal physical pooling group for a future changed construction.
+
+The integer twin allows variable ranks **at most** the named cap, including a
+remainder child. Requiring every child to have exactly that rank is a different
+model and can fail divisibility. This is the rank-cap interpretation requested
+by the predecessor's proposed integer DP.
+
+Two paid restrictions are kept separate:
+
+* **uniform:** partition the variable mass into integer children inside each
+  call, then repeat that exact profile N times;
+* **pooled:** partition N times the variable mass with no call boundaries.
+  This is a further relaxation, not a repeated local word.
+
+Idle is independently pooled across all N calls in both models. Integrality is
+imposed **before** dividing by W. In particular, nonintegral per-call averages
+are not an aggregate impossibility. This follows the boundary correction in
+PR3747 and the two-unpaid mass observations of PR3754/PR3757.
+
+## Results
+
+All rows have certified `moment(a_lo) < 1 < moment(a_hi)` with
+`a_hi = a_lo + 10^-12`. Every target moment is strictly below one, checked by
+both arithmetic engines. The kappa column is the lower endpoint of the
+**model root's** transformed bracket, rounded for display; it is not an achieved
+multiplication exponent. Exact fractions, full paid/idle histograms and target
+moment intervals are in `RESULT.json`.
+
+| Cell / calls / paid restriction | a lower endpoint | kappa bracket lower (approx.) | Extra rank-one children: maximum passing integer |
+|---|---:|---:|---:|
+| selected-r8-u2/9/uniform | 0.010351600977 | 0.010245543202 | 16 |
+| selected-r8-u2/9/pooled | 0.010353593558 | 0.010247495158 | 16 |
+| selected-r8-u2/144/uniform | 0.010351914854 | 0.010245850680 | 268 |
+| selected-r8-u2/144/pooled | 0.010354156699 | 0.010248046816 | 272 |
+| shared-r32-u1/9/uniform | 0.012857118397 | 0.012693911277 | 120 |
+| shared-r32-u1/9/pooled | 0.012881196799 | 0.012717381703 | 121 |
+| shared-r32-u1/144/uniform | 0.012861767247 | 0.012698442831 | 1931 |
+| shared-r32-u1/144/pooled | 0.012887413706 | 0.012723441452 | 1946 |
+| free-r16-u1/9/uniform | 0.011000817562 | 0.010881116386 | 62 |
+| free-r16-u1/9/pooled | 0.011005405074 | 0.010885604586 | 62 |
+| free-r16-u1/144/uniform | 0.011003116946 | 0.010883365997 | 1003 |
+| free-r16-u1/144/pooled | 0.011008280352 | 0.010888417598 | 1009 |
+
+The last column adds work to the complete base histogram with unchanged W and
+**no compensating mass credit**. Both signs are certified at a=1/98: the listed
+integer k passes and k+1 fails. Any combined realization charges must fit the
+same moment slack; different-rank work must be priced at its own child cost. The surcharge is
+aggregate and need not preserve the base paid word's per-call uniformity. If the
+extra rank-one work must itself be identical in every call, the tight row allows
+only one extra child per call (nine total); two per call exceed its limit.
+These numbers do not price a complete producer, consumer, dirty restoration,
+frame transport, precision or the bit supplier.
+
+For the tight nine-call uniform row, paid `{8:6282,2:9}` and idle
+`{51:272,46:1}` have masses 50274 and 13918, totalling
+`64192 = 71*896 + 64*9`. Independent integer powers give the strict bound
+
+    moment(1/98) < 111970002856773289113 / 112000000000000000000 < 1.
+
+The other nine-call uniform paid profiles are
+`{32:1548,18:9,8:36}` and `{16:3105,2:9}`. Both have idle
+`{51:290,24:1}`. The pooled alternatives are printed in the exact receipt.
+No all-24-cell integer lattice or antichain classification is claimed.
+
+## Why the histograms are optimal in these models
+
+Reuse the strict-concavity argument of
+[PR1685's retained proof](../idle-packing-stability/README.md#proof-2-exact-integer-optimum),
+credited to Penguin (OpenAI) for Maximus Putnam. For 0<a<1,
+f(r)=(r/73)^(1-a) is strictly concave and f(0)=0. With two positive parts
+x<=y<R, transfer delta=min(x,R-y) from x to y. The sum of their costs strictly
+decreases; one noncap part disappears or becomes R. Repetition leaves only
+floor(M/R) cap parts and, when nonzero, one remainder M mod R. Conservation of
+mass uniquely determines this optimal histogram. At a=0 all partitions tie;
+no endpoint uniqueness claim is made.
+
+Apply this separately to each variable paid pool and each idle pool. The shared
+cell's fixed rank-8 centre copies are excluded from optimization. In uniform
+mode optimize one call then repeat; in pooled mode optimize the total. Thus
+all root brackets describe exact optima of the declared integer partition
+classes, not merely sampled candidate histograms. Strict monotonicity in a
+holds because every child rank is below width 73. A moment below one at the
+lower bracket and above one at the upper bracket enclose the unique root.
+The physical restrictions absent from these classes can only worsen the optimum.
+
+## Evidence and reproduction
+
+Python 3.12, standard library only:
+
+```sh
+python3 -I -B research/lever-integer-twins/verify.py
+python3 -I -B research/lever-integer-twins/independent_check.py
+python3 -I -B scripts/research_package.py --package research/lever-integer-twins
+```
+
+`verify.py --write` regenerates the deterministic receipt. The primary verifier
+reuses the pinned Colkitt/Claude atanh-log/Taylor-exp rational enclosures,
+checks all 12 root brackets and surcharge boundaries, source constants,
+fractional-ceiling domination and both pooling monotonicities. An independent
+small mass-axis DP checks 224 cases (masses 1..32, caps 2..8) against the exchange
+histogram. Nine malformed-domain controls reject fractional counts/mass,
+wrong mass, rank 52, invalid pooling and out-of-range savings. These finite
+controls support the implementation; the proof above supplies general optimality.
+
+`independent_check.py` imports neither primary nor predecessor code. At a=1/98,
+the cost is `(r/73)^(97/98)`. On the 10^-18 grid, L/10^18 is at or below that cost
+exactly when `L^98 * 73^97 <= r^97 * 10^(18*98)`. Integer binary search gives
+rigorous enclosures, then sums integer counts and divides by W. It independently
+certifies all 12 target signs and both sides of each additive surcharge boundary.
+Exact and irrational-square calibrations, actual failing moments, wrong target,
+malformed receipts and off-by-one surcharge controls are included. It does
+**not** independently certify the root brackets or derive the histograms.
+All controls use explicit exceptions and remain active under Python -O.
+
+`predecessor/grid.py` is a byte-identical retained PR3874 file, read only for AST
+constants and never imported/executed. Its exact head, original path, Git blob
+and SHA256 are recorded in SOURCES.json; the blob was matched to the GitHub
+file response at that head. This external pending source is retained in-package
+so replay does not need a foreign Git commit. Generic source-history checks bind
+the base inputs; predecessor binding is the verifier's explicit byte check, not
+an assertion that the foreign commit is in local history. No frozen predecessor
+file is changed.
+
+## Decision and limits
+
+**Decision: narrow to realization pricing; no family expansion is justified by
+this result alone.** Integer histograms and their model moments are proved and
+checked as above. Realization, legal owner groups, source/consumer frames,
+dirty compensation, restoration and precision costs are **unknown**, not zero.
+The completed budget could fail once those are charged. No full selected cold
+replay, Lean proof, supplier admission or bit-side improvement was attempted.
+Independent AI arithmetic/review is not independent human or formal acceptance.
+
+## Open next questions
+
+1. Can a physical rank-8/u2 producer and consumer fit the tight cell's remaining
+   moment slack? Name the source/consumer contract first, then charge every
+   extra operation at its rank. The 16-per-nine-call rank-one boundary is a
+   concrete stop test for an additive realization bill. Stop without a priced
+   contract; another ideal ceiling calculation does not supply one.
+2. Which idle pooling groups are legal for that actual changed construction?
+   Extract owner/chronology restrictions before citing either unrestricted
+   9-call or 144-call optimum as physically achievable. A pool size chosen only
+   to clear denominators is not a hosting theorem.
+
+## Scope, overlap and credit
+
+Base `20ecf2891d80b5acf18d980b2164fb44ea7faa90`, priority revision 6.
+The entry overlap scan covered all 592 open PR bodies including drafts; the
+repository helper used that complete paginated snapshot. Manual live checks
+covered #1163, recent #1166, the active board, lever search and #2278. There was
+no complete Discussion cache and sparse local package coverage; no-match is
+not a novelty guarantee. Exact checkpoints and limitations are in TASK.json.
+
+* Nick Petalas: PR3874's grid and requested integer continuation, PR3811's
+  joint integer cap study, PR3754's pooled two-unpaid model. Exact heads:
+  `fb86b078a53042974d88b9758fe36622571349d0`,
+  `2205148748bbce47256e916d7485fc0219b8427f`,
+  `4bde35c0200e83fe27a08ec408a2ae59b490a03f`.
+* Arham Tahir: PR3747 aggregation boundary audit,
+  `98847f9aba65ba835a15725c9f173c04a98788b3`.
+* jwr404: PR3757 two-unpaid mass accounting,
+  `f8c1754ad593ca1379d44c71722a9e9421caed3c`.
+* Penguin (OpenAI) for Maximus Putnam: PR1685's integer optimum proof.
+* Douglas Colkitt with Claude assistance: ceiling model and primary exact
+  arithmetic; PR1012 coverage identity; Ali Romman and retained selected
+  Rohan Arun/Jeremy Kothe/Hao Yan supplier lineage. Philippe Van Velzen's
+  lever-map thread provides the canonical coordination context.
+
+New contribution by ToikY with substantial OpenAI Codex assistance, Apache-2.0.
+The new result is the integer continuation and surcharge boundaries for these
+three cells, with independent target arithmetic; it does not claim the parent
+grid, concavity theorem or aggregation correction as new.