Research: certify PR #2979's V45 final-word controls fail the complete packing budget

research/rank7-final-fill-controls/README.md

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+# V45 final-word controls fail the complete packing budget
+
+Prepared for commdata2338 (@commdata2338), with substantial OpenAI Codex assistance.
+Apache-2.0. No independent human or formal-kernel review is claimed.
+
+Every control that restores at least one of PR2979's four V45 endpoints from
+rank seven to rank eight fails its claimed complex saving, even after allowing
+arbitrary fractional repacking in the fixed capacity model. The control retaining
+only helper249 at rank seven has complete moment **at least 1.000005090277**.
+The fully coordinated final word passes at the same saving.
+
+This is a finite independent invoice audit and a scoped obstruction. It does
+not report a defect in the shipped joint construction, a new supplier, or a
+new kappa. The selected frontier is unchanged.
+
+## Exact object and difference from earlier checks
+
+The source is [PR2979](https://github.com/CrocSwap/integer-mult-bounds/pull/2979),
+head `be191346391661f4d35fc0d1508d0a6029572576`. The base is
+`463113d675af7410ac91d53a4fed51ec6494eb9a`, priority revision 6.
+All twelve live inputs in [SOURCES.json](SOURCES.json) match both the base
+checkout and the original PR head. No predecessor Python code is executed,
+imported or copied. Coordination is in [discussion #2612](https://github.com/CrocSwap/integer-mult-bounds/discussions/2612).
+
+The four original helper names are `249,251,265,267`; their compact allocation
+indices, derived from the actual removed-role lists, are `25,27,41,43`.
+Each has zero entrance and the common final seven-dimensional space
+`[256,128,64,35,16,11,5]`. Its own parent endpoint has dimension eight and
+contains that space.
+
+A control replaces any subset of these four terminal entries with their parent
+rank-eight entries. **All final scalar gates, partner frames, later extraction
+and rewrite stages, other endpoints, dispatch and private names stay fixed.**
+Both providers independently pass the resulting monotone frame recount. Each
+widened endpoint adds exactly one rank-one terminal child per invocation and
+one unit of helper residual width. The fixed T74 dispatch has 666 invocations.
+
+These are controls of the **final word**, not reconstructions of the original
+stage-one experiment. [PR2993](https://github.com/CrocSwap/integer-mult-bounds/pull/2993)
+and the [V45 profile audit](../rank7-v45-profile-audit/README.md) already check
+that original group's local profile and losing singleton guide price.
+[PR3014](https://github.com/CrocSwap/integer-mult-bounds/pull/3014) proves a complete
+cost increase when lowering V45 on the older PR2942 assignment. This package
+answers its changed-final-fill question: it checks the actual final assignment,
+then bounds all repackings of these explicitly different controls.
+[PR2987](https://github.com/CrocSwap/integer-mult-bounds/pull/2987)'s chronological
+packing audit and later allocation refinements retain their separate scope.
+The earlier [single-lowering](../rank7-v45-single-lowering/README.md) and
+[shipped-invoice](../rank7-v45-shipped-invoice/README.md) checks by jwr establish
+the recorded guide margins and endpoint/buffer inventory. They do not check
+these counterfactual final allocations or the complete packing lower bound.
+
+## Complete recursive invoice and literal capacity witnesses
+
+The checker reads all 271,728 placements and all 149,184 bins. It reconstructs
+source availability intervals from the source event schedule, checks each
+replica's stage order, unique roles, distinct physical families within a call,
+nonoverlapping coordinate masks and compact prefixes. Empty bins are charged.
+The two independently recounted local profiles are
+
+```text
+{1:2445, 2:706, 3:232, 4:108, 5:18, 8:8}.
+```
+
+The actual dispatch is 250 base and 416 extra calls. The independently rebuilt
+idle profile and complete profile are
+
+```text
+idle     {1:17310, 3:4956, 5:1728, 48:28190}
+complete {1:1645680, 2:470196, 3:159468, 4:71928,
+          5:13716, 8:5328, 48:28190}.
+```
+
+The complete rank mass is 4,816,512. The per-call local rank mass minus helper
+width mass is 1,856; widening an endpoint increases both by one. Consequently,
+a legal control allocation must replace its idle invoice as well. Retaining
+the final idle profile while adding the control's terminal children violates
+the complete rank-mass identity.
+
+Every one of the 2,664 V45 occurrences lies in a bin with zero idle remainder.
+Grant free compaction inside each bin and even arbitrary rearrangement of its
+disjoint axes. Restoring any endpoint to rank eight still exceeds capacity.
+
+| Widened endpoints | Overfull bins | Excess width | Whole-occurrence width that must leave those bins |
+|---|---:|---:|---:|
+| 249 only | 646 | 666 | at least 5,110 |
+| 251 only | 650 | 666 | at least 5,142 |
+| 265 only | 650 | 666 | at least 5,142 |
+| 267 only | 650 | 666 | at least 5,142 |
+| 251,265,267 (only 249 remains lowered) | 1,938 | 1,998 | at least 15,354 |
+| all four | 2,578 | 2,664 | at least 20,428 |
+
+For example, bin `(owner=2, child=0, port=99)` has capacity 16. It contains
+helper index 27 at time 0 with width 7 and helper index 177 at time 4 with width 9.
+Widening original helper251 makes the required width 17. The result contains
+an explicit witness and a digest of every witness for each of the sixteen
+subsets; `--witnesses` can emit them all.
+
+For any capacity-preserving reassignment, each overfull source bin must lose
+at least one of its original whole helper occurrences. In these actual bins
+the excess is at most two and every occupant is wider than the excess, so the
+minimum total width deleted from a bin is its shortest new occupant width.
+Summing those minima gives the last column. The bound is sharp in the
+relaxation that permits deletion into a free sink. It does not construct a
+legal relocation, price movement, or allow required occurrences to disappear.
+
+## Why no repacking rescues the controls at the source saving
+
+Use the exact claimed complex saving
+
+```text
+a = 1729033982646513517717 / 1000000000000000000000000.
+```
+
+Write `t_r = r^(1-a)`, with `t_0=0`. There are, per replica, 672 bins of capacity 8
+and 448 each of capacities 16, 32 and 48. The local word has nine invocations per
+replica and width inventory `{4:26,5:2,6:16,7:52,8:99,9:213}` per invocation.
+Its complete normalized moment is the per-replica tariff sum divided by
+`896*t_73`.
+
+For a capacity-c bin containing n_w helpers of width w, the idle tariff is
+`t_(c-sum(w*n_w))`. Define helper potentials
+
+```text
+p9 = (t3 - t48)/5       p8 = (3*p9 - t5)/4
+p7 = 2*p8 - p9          p6 = 3*p8 - 2*p9
+p5 = -t5               p4 = t1 + 4*p8 - 3*p9
+```
+
+and bin potentials
+
+```text
+alpha8 = -p8, alpha16 = -2*p8, alpha32 = -4*p8, alpha48 = t48.
+```
+
+These solve the ten equalities of the retained fractional basis. The checker
+independently enumerates **all 2,551 patterns** over widths 4 through 9 in the
+four capacities and verifies
+
+```text
+alpha_c + sum(n_w*p_w) <= t_(c-sum(w*n_w)).
+```
+
+Twelve patterns are symbolically tight. Every other inequality has a strictly
+positive rational lower bound. Summing these inequalities proves a lower bound
+for the entire idle bill, including fractional mixtures and allocations that
+ignore chronology. Adding every local child gives a lower bound for the
+complete recursive invoice, not only its idle component.
+Explicitly, for bin counts `B_c`, width inventory `I_w` per call and local
+profile `L_r` per call, the lower bill per replica is
+
+```text
+sum_c B_c*alpha_c + 9*sum_w I_w*p_w + 9*sum_r L_r*t_r.
+```
+
+The checker derives `B_c` from all source bins and independently binds the
+source fractional basis to these bin counts and to `9*I_w`.
+
+Widening one endpoint changes that bound per invocation by
+
+```text
+t1 + p8 - p7 = (20*t1 + t3 + 5*t5 - t48)/20.
+```
+
+This direction is strictly positive for every `a>0`: the positive ranks 1, 3, 5
+have total rank mass 48, and each has `r^(-a) > 48^(-a)`. The dual feasibility
+and the following numerical separations are certified at the stated source
+saving; no claim of dual optimality over every saving is needed.
+
+| Widened endpoint count | Certified lower bound on complete moment, rounded downward |
+|---:|---:|
+| 1 | 1.000001693386 |
+| 2 | 1.000003391832 |
+| 3 | 1.000005090277 |
+| 4 | 1.000006788723 |
+
+The baseline fractional bound has headroom at this saving, and the **actual**
+complete baseline profile independently satisfies the strict moment inequality.
+Thus rejecting the controls is not caused by testing above the joint word's
+own accepted numeric endpoint. No numerical root search or additional endpoint
+digits are supplied.
+
+`dual.py` uses integer arithmetic only. Successive floor/ceiling integer square
+roots enclose `r^(1/2^j)` on a 256-bit grid. The first 128 binary exponent bits
+of a give a lower exponent; one last root bounds the omitted tail. Outward
+rounded products enclose `r^a`, and division gives bounds on `r^(1-a)`.
+Rational linear combinations then enclose every dual slack and moment margin.
+All full rational bounds are retained in [RESULT.json](RESULT.json).
+
+## Physical ledger and remaining obligations
+
+The nominal stock arithmetic is preserved exactly:
+
+```text
+66,304 live families + 408 helper names + 8 centres
++ 91 private buffers + 2 coefficient families = 66,813.
+```
+
+The checker binds the 91 private names to the actual cleanup operand inventory
+and preserves all 5,207 cleanup operations per provider. This is namespace and
+invoice accounting; private zero-return and physical reuse are separate
+obligations. No scalar, wrapper, movement, prefix or precision charge is
+removed. Unpriced costs are not assumed to vanish: the packing relaxation
+rejects the controls already, before additional physical implementation costs.
+
+The scientific scope stops at the fixed whole-helper capacity model. Changed
+words, partner frames, helper splitting/coding, capacity pools, consumer
+contracts or local rank profiles are outside it. Native phases and full warp
+semantics, dirty-input operator identities, complete ownership/backing and
+movement, physical scalar implementations, prefix/precision/tape, outer release
+and all-size/compiler implications are not checked. Both supplier caps and the
+selected exponent remain unmodified; no new bit-side cap or outer transfer is
+asserted. Failure of the complex budget is sufficient for this control test.
+
+## Reproduce
+
+Python 3.12, standard library only, no network or solver:
+
+```sh
+python3 -B scripts/research_package.py --package research/rank7-final-fill-controls
+```
+
+The verifier prints
+
+```text
+PASS_RANK7_FINAL_FILL_CONTROLS cases=16 rejected_controls=15 roles=271728 bins=149184 packing_patterns=2551
+```
+
+The registered hook also runs thirteen behavioral groups. They include an
+explicit-point-set frame oracle, algebraic square/cube bounds for the interval
+engine, a separate pattern enumerator, all-subset axis-spill counts, exhaustive
+small deletion subsets, and semantic rejection of missing/duplicate roles,
+family conflicts, unavailable windows, overlapping/noncompact slices, omitted
+source bins, missing private names, unpaid children, an infeasible dual and an
+omitted pattern. An optimized Git-free export passes, and a changed source
+byte is rejected before decoding. Checks use exceptions and survive `-O`.
+
+`verify.py --emit` prints the deterministic result. `verify.py --witnesses
+WITNESSES.json` exclusively creates the optional full witness file. Default
+verification does not write repository artifacts. The full result requires
+about five seconds locally; behavioral controls take about twelve seconds.
+
+## Open next questions
+
+The two questions in [TASK.json](TASK.json) distinguish original stage-one
+attribution from supplier admission. A matched complete comparison of the
+original parent/joint/single words needs separately legal allocations and all
+charges; these final-word controls do not supply it. A physical admission audit
+should join the exact final fill to the native/private evidence and choose one
+remaining consumer cost. Another lowering or packing search is not started.
+
+## Predecessor credit
+
+Marius-Paul Dumitrean (mdumitrean) supplied PR2979's construction,
+PR2942 parent words, final allocation and fractional basis. Nick Petalas supplied
+PR2993's independent frame audit and PR3014's old-assignment obstruction and
+handoff. Ross Richardson supplied PR2987's independent chronological packing
+analysis. Jeremy Kothe supplied the frame/cleanup conventions and rewrite
+lineage; Mason Hale supplied PR2917's T74 call order and dispatch and, with
+Jeremy Diviney, the paired-extraction chain.
+Credit the retained V45 price/inventory checks by jwr (James Reed; PR3112 and
+PR3119) and Gonzalo Mier's V45 profile audit (PR3079) as complementary evidence.
+Retain Yiyang Lu's allocation lineage, Tekalign's source-window/owner conventions,
+Dugongue's packing and owner-engine lineage (PR2863, PR2839), icekylinx, Sagar
+Thakrar, themilkman216 and the geometry/native/owner contributors credited by
+the immutable source package. The earlier native endpoint audits, Nick Petalas's
+PR3006 and PR3008 and ToikY's PR3038, are credited as complementary evidence,
+not imported implementations or newly checked here.