Research: independently audit PR #4091's pooled duals with rational binomial bounds

research/staged-fourth-dual-binomial-audit/PROOF.md

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+# Independent certificate for the fixed PR4091 pooled dual
+
+This is a scoped verification of Ross Richardson's PR4091 theorem, not a new
+packing theorem or construction. Its source is
+`a29bdd8f56cd18b805917775cdec09ff935eddc4`; the package is unchanged at the
+scientific base `32890bc5de034ab8fc93ac88ead7279a6a416d04`. Its frozen PR4088
+inputs come from Cinna Davis's `76088d49e3ed4142a958659aa996b0ca4da5e88e`.
+The eight input files and their exact bytes are listed in `SOURCES.json`.
+
+## 1. Declared model and source boundary
+
+Write `H_r(a)=r^(1-a)`, with `H_0=0`. There are 3,240 calls and a real variable
+`0 <= x <= 3240` for FOUR calls. The reported local invoices are
+
+```
+L3 = 2487 H1 + 713 H2 + 224 H3 + 102 H4 + 20 H5 + 8 H8,
+d = L4-L3 = 2 H1 + 3 H3 - 3 H4.
+```
+
+This audit **inherits those local reports**. It does not recompute the scalar
+word's paid transitions or operator correctness. It separately enumerates
+the complete binary spans of the entry and exit frames of all 408 helpers in
+both orientations. Entry is contained in exit, and only helper 601 changes,
+from residual width 8 to 7. This gives the item inventory
+
+```
+(N5,N6,N7,N8,N9) = (90720,58320,181440+x,301320-x,690120).
+```
+
+The template has 112 ports, each with six capacity-8 and four each of
+capacity-16/32/48 children. The current order is exactly nine stages times
+360 replicas, so the available bins are
+
+```
+(B8,B16,B32,B48) = (241920,161280,161280,161280).
+```
+
+The historical template's `replicas:37` fields are not multipliers. The stock
+coefficient `(896*360)H73` is a declared comparison premise of PR4091's proof,
+not inferred from the historical template's physical-field count.
+
+A pattern consists of a capacity `c`, five nonnegative integral helper counts
+`n_w`, and a residual `r=c-sum(w*n_w)>=0`. Each bin pays exactly `H_r`.
+The relaxed objective is
+
+```
+3240 L3 + x d + sum_pattern y_pattern H_residual,
+```
+
+where `y>=0` obeys all four bin balances and all five item balances. This
+retains every empty bin and allows real multiplicities. Chronology, identities,
+physical-family exclusions and offsets are omitted. A legal source fill maps
+to this relaxation only under the declared whole-helper/residual accounting
+contract. This audit supplies no new physical justification of that contract.
+
+## 2. Complete patterns by two finite methods
+
+The checker enumerates the Cartesian product
+`0 <= n_w <= floor(c/w)` and keeps precisely the capacity-feasible tuples.
+Independently, let `A(s)` be the coefficient of `z^s` in
+
+```
+product_(w=5..9) 1/(1-z^w).
+```
+
+The ordinary unbounded-coin recurrence computes these coefficients through
+degree 48. For each capacity and every residual, the first enumeration must
+have exactly `A(c-r)` distinct patterns. It also verifies nonnegative integral
+counts and the exact capacity equation for every tuple. Thus there are
+5, 21, 158 and 657 patterns respectively, 841 in total. Missing empty or
+nonempty patterns, duplicates at unchanged total count, and wrong residuals
+are rejected. The complete ordered census is hashed in `RESULT.json`.
+
+## 3. A different exact power enclosure
+
+The numerical part imports no predecessor Python and uses no logarithms,
+exponentials, Decimal powers, floating-point decisions or optimizer. For an
+integer `r>=1`, put `z=1-1/r`. If `0<a<1`, the binomial series gives
+
+```
+(H_r(a)-r)/a = -r sum_(k>=1) t_k(a),
+t_1(a)=z,
+t_(k+1)(a)=t_k(a) * (k-a) * z/(k+1).
+```
+
+Equivalently,
+`t_k(a)=z^k/k * product_(j=1..k-1)(1-a/j)`. Every term is nonnegative and
+nonincreasing in `a`. The convergent sum therefore shows that the normalized
+power `g_r(a)=(H_r(a)-r)/a` is nondecreasing in `a` (including `r=1`, where
+it is zero). This is enough for an interval result, not merely a grid test.
+
+After `N` included terms, consecutive omitted terms have ratio at most `z`.
+Their sum is at most `t_(N+1)/(1-z)=r*t_(N+1)`. At scale `S=10^40`, the
+implementation starts with floor/ceiling bounds for `S*z`, rounds each positive
+rational recurrence outward using integer division, and sums those bounds.
+If `L_k <= S*t_k <= U_k`, positivity of the multiplier preserves this relation.
+Consequently its final bounds are
+
+```
+-r * (sum_(k=1..N) U_k + r*U_(N+1))/S
+    <= g_r(a) <=
+-r * sum_(k=1..N) L_k/S.
+```
+
+It uses `N=2048`. The tail is included even if it is much larger than rounding
+error. Independent exact rational-power fixtures and short Fraction sums check
+the arithmetic, direction of rounding and first-omitted-term index. A one-term
+enclosure is deliberately too wide for the dual and must fail closed.
+
+Every nonzero dual slack below has zero rank mass: `sum c_r*r=0`. Hence
+
+```
+(sum c_r H_r(a))/a = sum c_r g_r(a).
+```
+
+For each positive coefficient, use the lower `g_r` bound at `a0=173/100000`
+and the upper bound at `a1=174/100000`; reverse them for negative coefficients.
+This encloses the expression **for all real a in [a0,a1]**. The checker refuses
+to cancel the linear part when the rank mass is nonzero.
+
+## 4. Recovering the duals and their faces
+
+There are nine dual variables: four capacity prices `u_c` and five item prices
+`v_w`. The desired inequalities are
+
+```
+u_c + sum n_w v_w <= H_(c-sum w*n_w).
+```
+
+The source certificate's eight central primal patterns are declarative data,
+not executable evidence. We solve their eight tightness equations symbolically
+over rational coefficients of `H_r`. The ninth equation is one of:
+
+- middle: `v8-v7=d` (free-mixture stationarity);
+- left: tightness of capacity-32 pattern `7+8+8+9`;
+- right: tightness of capacity-48 pattern `6+6+9+9+9+9`.
+
+Each exact linear system is nonsingular, and its solution must equal the
+published potential expressions. Every recovered dual is then evaluated on
+all 841 patterns. Zero expressions are recognized algebraically; all others
+have strictly positive interval lower bounds. The following displayed bounds
+are conservative rational truncations of `RESULT.json`:
+
+| Dual | Identically tight | Strict | Lower bound on every nonzero slack/a |
+|---|---:|---:|---:|
+| Left | 11 | 830 | > 4.38806 |
+| Middle | 8 | 833 | > 0.69698 |
+| Right | 9 | 832 | > 2.37161 |
+
+The left face has eleven tight patterns, even though the chosen affine witness
+uses nine. No uniqueness claim about every left-face allocation is needed.
+The middle face has exactly the eight source patterns.
+
+## 5. Primal balance, uniqueness and the whole interval of mixtures
+
+For the middle support, solve all nine balance equations with its eight
+multiplicities and `x` as unknowns. The determinant is exactly `-10` and the
+unique solution is
+
+```
+x=1080,
+y=(241920,161280,1620,10620,58320,90720,138492,22788)
+```
+
+in the source table's order. Each multiplicity is positive. Its idle invoice
+is `22788 H3+1620 H5+138492 H48`. All other patterns have strict dual slack,
+so any global optimum must use this support. The exact balance system then
+forces the displayed multiplicities and mixture. This proves uniqueness in
+the relaxed pattern program for every exponent in the certified interval.
+
+For each side, add its ninth tight pattern and solve the nine balances with
+right-hand side affine in `x`. The result gives every multiplicity as
+`b_j+d_j*x`; no sampled interpolation is used to infer this form. Exact
+nonnegativity at each segment's endpoints proves it throughout that segment.
+All patterns in the affine witnesses are symbolically complementary to the
+corresponding dual. Thus both witnesses attain the dual bound for every real
+`x` in `[0,1080]` or `[1080,3240]` respectively.
+
+The complete objective slopes are independently recovered as
+
+```
+s_left  = d-H3/20-H5/4+H48/20,
+s_right = d+(H5-H3)/2.
+```
+
+The binomial certificate encloses `s_left/a` in
+`[-0.348858,-0.348494]` and `s_right/a` in `[4.365719,4.365912]`.
+The complete bill therefore strictly decreases to 1080 and strictly increases
+after 1080. These facts concern the complete invoice, not scalar ADD counts.
+
+At `x=1090`, the derived right-face witness has idle
+`22783 H3+1625 H5+138492 H48` and exactly five capacity-48 bins containing
+`6+6+9+9+9+9`. This verifies the pooled feasibility question raised by PR5280.
+The eight central patterns alone cannot serve `x=1090`: the central balance
+system forces `x=1080`. The original central idle multiset itself also has ten
+too few residual rank units at `x=1090`. Thus PR4170's request for that same
+idle multiset at 1090 must change the mixture or the invoice; it is not an
+unresolved chronological rearrangement. All these formulas already follow
+from PR4091. Their derivation and audit, not their discovery, are claimed here.
+
+## 6. Root implication and limits
+
+The derived complete profile is
+
+```
+8060040 H1 + 2310120 H2 + 751788 H3 + 327240 H4
+ + 66420 H5 + 25920 H8 + 138492 H48.
+```
+
+The checker evaluates only the coarse endpoints `a0,a1`, obtaining negative
+and positive signed moments respectively against `(896*360)H73`.
+Every paid rank is positive and below 73. After division by `H73`, every term
+is strictly increasing in `a`, so this profile has exactly one root in the
+certified interval. At that root the middle dual bounds every feasible
+complete profile below by the same stock bill. Each such normalized profile
+is also strictly increasing, so none has a larger root, including outside
+the interval where these dual formulas were certified.
+
+This independently checks the pooled optimality implication. The published
+24-place root bracket is not recomputed, refined or used as an acceptance
+input. No κ improvement, chronological construction, native/operator replay,
+stock implementation, compiler, precision/tape, BIT/outer admission or
+all-size multiplication theorem is claimed. The current selected result is
+a changed mechanism and is not subject to this historical fixed-model cap.
+
+Prepared for commdata2338 (@commdata2338), with substantial OpenAI Codex assistance.
+Ross Richardson, Cinna Davis and the predecessor lineage credited in README.md
+retain their construction and theorem credit. Apache-2.0.

research/staged-fourth-dual-binomial-audit/README.md

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+# Independent binomial-series audit of the PR4091 pooled dual
+
+**No discrepancy found in the scoped pooled proof. No new κ.** A separate
+standard-library implementation verifies all three duals of
+[Ross Richardson's PR4091](https://github.com/CrocSwap/integer-mult-bounds/pull/4091)
+on every whole-helper bin pattern, uniformly for
+`0.00173 <= a <= 0.00174`. It reconstructs the affine primal witnesses for
+every real mixture `0 <= x <= 3240` and the unique global optimum `x=1080`.
+
+This closes a specific independent-verification obligation on the revision-7
+`independent-moment` card. It does not recompute more digits of a root or
+search for another packing. The mathematical optimum and its witnesses remain
+PR4091's contribution.
+
+| Checked object | Result |
+|---|---|
+| Complete pattern census, independently counted by a generating-function recurrence | 5 / 21 / 158 / 657 at capacities 8 / 16 / 32 / 48; 841 total |
+| Left / middle / right dual faces | 11 / 8 / 9 identically tight patterns; all others strictly positive throughout the interval |
+| Nonzero slack divided by `a`, conservative lower bounds | > 4.38806 / 0.69698 / 2.37161 |
+| Central balance matrix | determinant −10; unique `x=1080` and all eight multiplicities recovered |
+| Complete objective slope divided by `a` | left in [−0.348858,−0.348494]; right in [4.365719,4.365912] |
+| `x=1090` consequence | right-face optimum has five `48:(6,6,9,9,9,9)` bins and idle `{3:22783,5:1625,48:138492}` |
+
+The last row answers the pooled-feasibility question in
+[PR5280](https://github.com/CrocSwap/integer-mult-bounds/pull/5280).
+The central eight-pattern support alone forces 1080 calls. The central idle
+multiset also has ten too few residual rank units for 1090. Thus
+[PR4170](https://github.com/CrocSwap/integer-mult-bounds/pull/4170)'s open
+question about reaching that same multiset **at 1090** requires changing the
+mixture or invoice. These are independently checked consequences of PR4091's
+existing formulas, not newly discovered constructions.
+
+## Independence and evidence
+
+The checker imports no predecessor Python. It recovers dual prices from exact
+linear equations instead of executing the published formulas. It enumerates
+patterns by a Cartesian product and checks every residual count against a
+separate coin-generating-function recurrence. Its power enclosures use the
+positive binomial series for `(r^(1-a)-r)/a`, integer outward rounding and an
+explicit geometric tail. They use no logarithm/exponential engine, floating
+acceptance, optimizer or external numerical library.
+
+[PROOF.md](PROOF.md) proves the interval enclosure, weak-duality implication,
+unique central optimum and global root-ceiling implication. [RESULT.json](RESULT.json)
+records exact rational evidence, the census hash, affine solutions and input
+hashes. Only the coarse endpoints `.00173` and `.00174` are evaluated to put
+the root inside the dual-certified interval. The published 24-place root
+bracket is not freshly verified or refined.
+
+Twenty-one arithmetic/control tests include exact rational-root fixtures, a
+Fraction partial-sum oracle, omitted-tail discrimination, missing/duplicate
+patterns, a zero-mass perturbation that violates dual feasibility, inadequate
+precision, wrong mixture/width/replica factor, an unpaid local charge and
+corrupt source bytes. Explicit checks remain active under Python `-O`.
+
+This is an independently implemented audit, author-checked in this package.
+There was no separate human, external-agent or formal-kernel review.
+
+## Source and accounting boundaries
+
+Base: `32890bc5de034ab8fc93ac88ead7279a6a416d04`. Audited PR4091:
+`a29bdd8f56cd18b805917775cdec09ff935eddc4`; its package is unchanged at the base.
+Frozen PR4088: `76088d49e3ed4142a958659aa996b0ca4da5e88e`.
+
+The eight exact input files are pinned in [SOURCES.json](SOURCES.json) and
+declared in [VERIFY.json](VERIFY.json). The audit reconstructs helper widths
+by explicit binary spans in all four word exports and derives bin counts
+from the template and current order. It **inherits the local paid-profile
+reports and comparison stock coefficient**. It does not rescan every paid
+frame transition or establish native/operator correctness.
+
+The complete bill includes `3240 L3 + x(L4-L3)` and every residual child,
+including empty bins, against `(896*360)H73`. Words, whole-helper widths,
+capacities, stock and this accounting boundary are fixed. Changing them
+requires a new model. The selected PR5413 mechanism is outside this historical
+ceiling's scope.
+
+Not checked: chronological/family/offset feasibility, native and dirty-state
+identities, source local-profile correctness, actual stock provision, compiler,
+all-size, precision/tape, BIT or outer admission. No supplier promotion or
+complete multiplication theorem follows from this audit. PR4181's reported
+chronological attainment is credited but not replayed here.
+
+## Reproduce
+
+From the repository root, Python 3.12 and its standard library:
+
+```sh
+python3 -B scripts/research_package.py --package research/staged-fourth-dual-binomial-audit --check-only
+python3 -B scripts/research_package.py --package research/staged-fourth-dual-binomial-audit
+```
+
+The registered commands run the checker and all 21 tests in normal and
+optimized modes. A direct checker run takes about one second on the measured
+Linux/Python 3.12 environment; timing is host-dependent. Expected checker line:
+
+```text
+PASS_STAGED_FOURTH_DUAL_BINOMIAL_AUDIT patterns=841 duals=3 tight=11/8/9 unique_x=1080
+```
+
+The direct scripts also work from an official export containing the declared
+inputs, without Git or network access:
+
+```sh
+python3 -I -W error -B research/staged-fourth-dual-binomial-audit/verify.py
+python3 -I -W error -B research/staged-fourth-dual-binomial-audit/test_verify.py
+```
+
+Add `-O` to check optimized execution. Normal replay compares the complete
+canonical JSON result and never rewrites it. Regeneration is explicit:
+`verify.py --output PATH`.
+
+## Open next questions
+
+No further fixed-inventory packing search is justified by this audit:
+[PR4181](https://github.com/CrocSwap/integer-mult-bounds/pull/4181) already reports
+attaining PR4091's cap. A consequential follow-up must name a changed word,
+inventory or consumer boundary and derive its complete signed invoice before
+reusing a ceiling. The first step is to identify which of the model's item,
+capacity, residual or stock equations changes; absent such a source-bound
+change, this audit supplies no reason to expand the search. Physical admission
+and independent source-transition coverage remain separate obligations, not
+closed by these LP checks.
+
+## Credit
+
+Prepared for commdata2338 (@commdata2338), with substantial OpenAI Codex assistance.
+
+Ross Richardson and the credited PR4091 specialists supplied the pooled
+theorem, all three duals and the affine witnesses. Cinna Davis/PR4088 supplied
+the fixed words and fill. Nick Petalas/PR4170 and unpaidmoon/PR5280 supplied
+the source-transfer checks and follow-up questions; their checks explicitly
+exclude the pooled dual proof. Rohan Arun/PR4181 supplied the chronological
+attainment. Retain Rohan Arun/PR4076; David Ganz (Dugongue)/PR3945 and
+zxv/PR4085 with their lineage; dataisfire/PR3734; Erwan Secousses
+(ChaloupeMusic)/PR3949/3815; and the original geometry, native, owner, compiler and transfer
+contributors credited in PR4091's notices. This account's PR3237 concerned a
+different PR2979 final-fill model and is not repeated here.
+
+The new work is the binomial enclosure implementation, independently solved
+dual/primal certificate, finite audit and controls. Substantial OpenAI Codex
+(GPT-6 Astra) assistance was used for derivation, implementation, checking and
+writing. No independent human review is claimed. Apache-2.0.