Prove the sharp pooled optimum for fourth staging

research/staged-fourth-mixture-ceiling/NOTICE.md

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+# Sources and credit
+
+This contribution: Ross Richardson, with substantial OpenAI GPT-6.1 Sol
+assistance in derivation, implementation and mathematical specialist review.
+Apache-2.0. AI team review is not independent human or formal-kernel review.
+
+The exact source subset in source4088/ is from Cinna Davis's PR4088,
+head 76088d49e3ed4142a958659aa996b0ca4da5e88e. Its word and literal fill are
+predecessor contributions, not new constructions here. SOURCE-EXPORT.json
+records original paths and byte pins. source4088/NOTICE.md retains the complete
+source-specific credit and scope.
+
+The unchanged binary-basis ledger and rational enclosure engine were retained
+through Dugongue/PR3945 and Rohan Arun/PR4076. The primary certificate reuses
+that rational engine; the separate independent checker does not import it.
+The staged-entry mechanism credits Ross Richardson/PR3605 and dataisfire/PR3734;
+ChaloupeMusic and Erwan Secousses/PR3949/3815 supply the general staging and
+T360 pattern/Hall lineage. Receiver sharing, native/owner closure and the
+predecessor all-character proof credit Dugongue and David Ganz/PR4085,
+Nick Petalas, Marius-Paul Dumitrean, Sagar Thakrar, roger-sato, zxv and the
+earlier geometry/compiler/transfer contributors named in the source notices.
+
+The new result is the exact pooled mixture optimum and its complete-cost
+ceiling, with a piecewise primal/dual proof and a separately implemented
+arithmetic check. No chronological fill, supplier admission, kappa improvement
+or selected-frontier update is claimed.

research/staged-fourth-mixture-ceiling/PROOF.md

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+# Pooled whole-helper ceiling for the fixed PR4088 words
+
+Prepared with OpenAI Codex assistance. Apache-2.0.
+
+## Scope and source premises
+
+This result fixes PR4088, Cinna Davis, head
+`76088d49e3ed4142a958659aa996b0ca4da5e88e`, and its THREE/FOUR word interfaces.
+The source inherits the PR4085 receiver-sharing words and PR4076 staged-entry
+lineage. Its native, hosting, restoration, precision, controller and outer
+transfer premises remain source premises; this calculation does not discharge
+them. Source file digests used here are in `RESULT.json`.
+
+Let `a` be the exponent saving and `H_r(a)=r^(1-a)`, with `H_0=0`. There are
+3240 calls, and `x` of them use FOUR. The local profiles are fixed as
+
+```
+L3 = 2487 H1 + 713 H2 + 224 H3 + 102 H4 + 20 H5 + 8 H8,
+d  = L4-L3 = 2 H1 + 3 H3 - 3 H4.
+```
+
+For THREE the whole-helper widths per call are
+`{5:28,6:18,7:56,8:93,9:213}`. FOUR changes helper 601 from width 8 to width 7
+and leaves every other helper width unchanged. Thus the pooled demand is
+
+```
+(N5,N6,N7,N8,N9) = (90720,58320,181440+x,301320-x,690120).
+```
+
+The source TEMPLATE has 2016 entries, already including all 112 ports. Its
+capacity counts are `{8:672,16:448,32:448,48:448}`. Equivalently there are 18
+distinct child slots per port, with counts `{8:6,16:4,32:4,48:4}`. Multiplying
+the TEMPLATE counts by 360, once, gives
+
+```
+(B8,B16,B32,B48) = (241920,161280,161280,161280),
+total bins = 725760, total capacity = 17418240.
+```
+
+The pooled model ignores source windows, chronology, call identity, repeated
+physical-family exclusions and any other restrictions on where a helper may
+be hosted. It allows real nonnegative pattern multiplicities and real
+`0<=x<=3240`. Each pattern still contains whole helpers. This is an explicitly
+optimistic relaxation of a legal source owner fill. The result gives a lower
+bound on the paid bill of every such fill, including any nonuniform dispatch
+of these fixed words. It does not construct a changed supplier.
+
+## Complete finite pattern program
+
+A pattern is `(c,n5,n6,n7,n8,n9)` with
+`n_w>=0` integral and `sum_w w*n_w<=c`. Its residual rank is
+`r=c-sum_w w*n_w`. There are exactly 841 patterns: 5 at capacity8,21 at 16,
+158 at 32 and 657 at 48. Empty patterns are included.
+
+For pattern multiplicities `y>=0`, require exactly `B_c` bins of each capacity
+and exactly the demands above. The paid objective is the complete expression
+
+```
+3240 L3 + x d + sum_patterns y_pattern H_residual.
+```
+
+Every admissible source allocation maps to such a feasible point. No idle bin
+is discarded. The helper rank mass is `10695240-x`, the local rank mass is
+`16708680-x`, and the residual rank mass is `6723000+x`. Consequently the full
+paid rank mass is always 23431680, or `65088*360`. This is distinct from the
+comparison stock mass `(896*360)*73=23546880`.
+
+## Exact optimum and dual certificate
+
+For every `a` in `[0.00173,0.00174]`, the unique optimal mixture is `x=1080`.
+An integral optimum of the pooled program is:
+
+| Capacity | Whole helpers in each bin | Multiplicity | Residual |
+| --- | --- | ---: | ---: |
+| 8 | 8 | 241920 | 0 |
+| 16 | 7+9 | 161280 | 0 |
+| 32 | 9+9+9 | 1620 | 5 |
+| 32 | 7+7+9+9 | 10620 | 0 |
+| 32 | 6+8+9+9 | 58320 | 0 |
+| 32 | 5+9+9+9 | 90720 | 0 |
+| 48 | empty | 138492 | 48 |
+| 48 | 9+9+9+9+9 | 22788 | 3 |
+
+Thus the optimal residual profile is
+
+```
+I* = 22788 H3 + 1620 H5 + 138492 H48.
+```
+
+Here is a dual certificate, written as exact rational linear combinations of
+the functions `H_r`, rather than rounded decimal prices. Set
+
+```
+v9 = (H3-H48)/5,
+v7 = (v9-H5)/2,
+v8 = v7+d,
+v6 = 2v7-v8,
+v5 = -H5,
+u8 = -v8,
+u16 = -v7-v9,
+u32 = H5-3v9,
+u48 = H48.
+```
+
+For every pattern the checker proves
+
+```
+u_c + sum_w n_w v_w <= H_residual.
+```
+
+The eight displayed patterns give exact symbolic equality. All other 833
+constraints have strictly positive slack throughout the stated interval;
+the certified minimum of slack divided by `a` is greater than
+`0.696988114241`. The mixture coefficient is identically
+`d+v7-v8=0`. Weak duality therefore gives the same lower bound for every real
+`x` and every feasible pattern allocation. Equality is attained by the table.
+
+Because every other pattern has positive slack, an optimum uses only the
+eight displayed patterns. The nine bin/item equations, including variable
+`x`, have determinant `-10`. Their solution is the displayed table and
+`x=1080`, proving uniqueness in the pooled program. One can also see the
+mixture directly: width 5 fixes 90720 bins of pattern `5+9+9+9`; width 6 fixes
+58320 bins of `6+8+9+9`. Width 8 balance then forces
+`241920+58320=301320-x`, hence `x=1080`.
+
+## A genuine mixture breakpoint
+
+The fixed-mixture optimum can also be stated for every real `x` in the full
+range. Let `t=x-1080`, and begin with the table above. Keep the capacity8,
+capacity16 and `5+9+9+9` rows fixed.
+
+For `-1080<=t<=0`, replace the affected multiplicities by
+
+```
+cap32, 9+9+9:       1620-t/4,
+cap32, 7+8+8+9:          -t/2,
+cap32, 7+7+9+9:    10620+3t/4,
+cap32, 6+8+9+9:    58320,
+cap48, empty:       138492+t/20,
+cap48, five9:       22788-t/20.
+```
+
+For `0<=t<=2160`, use
+
+```
+cap32, 9+9+9:       1620+t/2,
+cap32, 7+7+9+9:    10620+t/2,
+cap32, 6+8+9+9:    58320-t,
+cap48, empty:       138492,
+cap48, five9:       22788-t/2,
+cap48, 6+6+9+9+9+9:     t/2.
+```
+
+All multiplicities are nonnegative on their segments, and all constraints
+hold as exact affine identities. These are witnesses in the relaxation.
+They do not assert chronological or physical-family feasibility.
+
+For the left segment replace `v8` by `(v7+v9)/2`; for the right segment set
+`v6=(-H48-4v9)/2` and `v8=2v7-v6`. Keep the other dual formulas. The checker
+certifies all 841 inequalities for each dual over the same exponent interval,
+and each corresponding affine witness is complementary to its dual.
+Consequently the optimum complete bill has slopes
+
+```
+s_left  = d-H3/20-H5/4+H48/20,
+s_right = d+(H5-H3)/2.
+```
+
+Throughout the exponent interval, exact bounds give
+`s_left/a` in `[-0.348857711147,-0.348494057120]` and `s_right/a` in
+`[4.365719222915,4.365911868442]`, using outward rounded displayed bounds.
+Thus the bill strictly decreases to 1080 fourth calls and strictly increases
+after 1080. Increasing the fourth-call frequency beyond PR4088's one-third
+mixture cannot improve even this optimistic pooled bill.
+
+## Exact root ceiling and remaining headroom
+
+The complete optimum profile is
+
+```
+8060040 H1 + 2310120 H2 + 751788 H3 + 327240 H4
+  + 66420 H5 + 25920 H8 + 138492 H48.
+```
+
+Compare this with the same source stock `(896*360) H73`. Rational logarithm
+and exponential remainder bounds, imported unchanged from the source engine,
+give a strict root bracket
+
+```
+0.001730980255989036959962 < a* < 0.001730980255989036959963.
+```
+
+The lower and upper signed moment intervals have signs negative and positive,
+respectively; the exact endpoints are in `RESULT.json`. Approximate values are
+`-1.73328*10^-17` and `+4.89396*10^-17`. Every positive rank in any feasible
+complete profile is less than 73. After division by `73^(1-a)`, its complete
+moment is strictly increasing in `a`. Therefore positivity of the pooled lower
+bound at the upper endpoint excludes any feasible fixed-word owner allocation
+having a larger moment root, including exponents outside the interval where
+the displayed dual was certified. The pooled table attains `a*`, so this is a
+sharp ceiling for the relaxation.
+
+PR4088's literal chronological fill already has 1080 FOUR calls. Its residual
+profile is `22785 H3+1625 H5+H16+2H32+138490 H48`. Literal minus pooled optimum
+is exactly
+
+```
+D(p) = -3*3^p + 5*5^p + 16^p + 2*32^p - 2*48^p,  p=1-a.
+```
+
+This difference is strictly positive for every `0<p<1`. Indeed, for its
+coefficients `c_r`, the piecewise linear function
+`S(t)=sum_r c_r min(r,t)` has values 9,21,32,32,0 at the knots 3,5,16,32,48;
+it is nonnegative everywhere and positive on an interval. The identity
+
+```
+D(p) = p(1-p) integral_0^infinity S(t) t^(p-2) dt
+```
+
+proves strict positivity. The source literal root bracket and the new rational
+bracket show that the entire remaining root headroom lies within
+
+```
+[1.950913151266531*10^-9, 1.950913151266533*10^-9].
+```
+
+This is a limit on the fixed words, capacities and paid-profile comparison,
+not a new kappa or supplier claim. Reaching even this small headroom still
+requires a legal chronological distinct-family fill (or a changed interface
+with its own fully charged costs). New words, capacities, stock or transfer
+interfaces fall outside the bound.
+
+## Verification level and reproduction
+
+`verify.py` uses only the Python standard library and the unchanged source
+rational enclosure engine. No floating optimizer is part of the certificate.
+It independently rescans binary frame dimensions in all four raw words,
+checks source widths, declared local profiles and template multiplicities, checks exact
+primal balance, enumerates all 841 patterns for three exact duals, certifies
+the uniform real exponent interval, checks the root signs, and rejects altered
+mixture, primal, pattern-coverage, dual and paid-profile controls.
+
+```
+python3.12 -I -B verify.py --source source4088 --controls
+```
+
+By default recomputation is compared with the saved `RESULT.json`; it does not
+rewrite it. `--output PATH` explicitly writes a regenerated certificate.
+
+The 841-pattern coverage and rational enclosure computation are finite exhaustive
+checks. The accompanying affine/weak-duality argument and monotonicity proof
+cover every real mixture in `[0,3240]` and every exponent in the stated
+interval; the monotonicity argument gives the claimed global root ceiling.
+This is a written proof backed by exact finite arithmetic, not a formal kernel
+proof and not an all-size multiplication theorem. Source native and operator
+claims remain attributed conditional premises; the source's complete owner and
+operator replay has not been repeated by this relaxation checker.

research/staged-fourth-mixture-ceiling/README.md

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+# Sharp pooled ceiling for a fourth staged entry
+
+For the exact THREE/FOUR words in [PR4088](https://github.com/CrocSwap/integer-mult-bounds/pull/4088)
+at `76088d49e3ed4142a958659aa996b0ca4da5e88e`, the best pooled complete moment
+uses **1,080 fourth-entry calls out of 3,240**. This is the same one-third
+fraction as its literal construction. Every real mixture is allowed in this
+relaxation, and all whole-helper patterns and residual children are charged.
+
+The sharp pooled root lies in
+
+```
+[0.001730980255989036959962,
+ 0.001730980255989036959963].
+```
+
+Thus any improvement to the pinned literal fill within this fixed two-provider
+invoice is **less than 1.951e-9 in the complete-moment root**. This is an upper
+bound on remaining headroom, not an achieved supplier or a new kappa.
+
+The proof certifies all 841 bin patterns, three duals and a piecewise affine
+optimum for every real staging count `0 <= x <= 3240` and every exponent
+`0.00173 <= a <= 0.00174`. An eight-pattern **integer pooled** witness attains
+the optimum at `x=1080`; it is not a chronological owner map. Its idle profile
+is `{3:22788,5:1620,48:138492}`. The literal fill exceeds it by
+`-3 H3 + 5 H5 + H16 + 2 H32 - 2 H48`, a strictly positive power cost for
+`H_r=r^(1-a)` and `0<a<1`.
+
+## Reproduction and evidence
+
+From the repository root with standard-library Python 3.12:
+
+```sh
+python3.12 -B scripts/research_package.py --package research/staged-fourth-mixture-ceiling
+python3.12 -B scripts/check_research_commit.py --package research/staged-fourth-mixture-ceiling --check-source-history
+```
+
+`source_binding.py` re-executes the credited predecessor binary-frame cost
+ledger on all four words and extracts the current 360-replica bins.
+`verify.py` checks the exact primal/dual and complete invoice.
+`independent/check.py` uses a separately implemented fixed-point rational enclosure
+engine and pattern enumeration; `independent/review_broad.py` separately checks the whole exponent interval and affine fills. The normal and optimized commands retain
+explicit rejecting controls. `RESULT.json` and `PROOF.md` give the exact
+certificate and derivation. No optimizer is needed for replay. All four standalone commands in `VERIFY.json` also run from the official committed export, with `{package}` replaced by the exported package path; add `-O` for optimized replay. The metadata commands above require Git.
+
+Fourteen predecessor files are retained byte-for-byte. `SOURCE-EXPORT.json`
+records their original head, path, Git blob and SHA256; `SOURCES.json` binds
+their committed local freeze so a cold checkout needs no pending-PR objects.
+The original template contains historical 37-replica material: only its
+original child-port capacities are used, with the current `ORDER.json`
+supplying all 360 replicas. No historical replica factor is silently reused.
+
+## Scope and limits
+
+The model fixes the two local invoices, width-5 through width-9 whole helpers,
+all original capacity-8/16/32/48 bins, one paid residual per bin, width 73 and
+stock `896*360`. It drops chronology, helper identity, family exclusions,
+offsets and integral pattern multiplicities to obtain a lower cost bound.
+Additional implementation costs cannot improve that bound. Changed words,
+capacities, cross-slot operator fusions or stock conventions require a new model.
+
+The mathematical result combines a written linear-program dual argument with
+finite exhaustive pattern coverage and exact rational interval certificates.
+There is no formal-kernel or independent human review. The predecessor ledger
+is reused and freshly executed for local costs; dirty-state identities, native
+phases, physical hosting, mixed-provider full release, precision, tape, BIT and
+outer transfer are not freshly verified here. The selected record is unchanged.
+
+## Open next questions
+
+Can legal exchanges attain the pooled eight-pattern inventory? Inspect the
+literal fill's one rank-16 and two rank-32 residual bins, preserve all source
+windows and physical-family constraints, and require complete owner/phase and
+release validation. The maximum possible gain is now explicit. A larger change
+must alter a named provider or interface premise and reprice its complete cost.
+Structured questions and blockers are in `TASK.json`.
+
+Prepared for Ross Richardson with substantial GPT-6.1 Sol assistance, including
+two mathematical specialists. Credit Cinna Davis/PR4088; Rohan Arun/PR4076;
+Dugongue and David Ganz/PR4085/3945; Nick Petalas; dataisfire/PR3734;
+ChaloupeMusic and Erwan Secousses/PR3949/3815; the staged-release, solver,
+native, owner and transfer lineage. See `NOTICE.md` and the retained source
+notice. Apache-2.0.