A tilted dual brackets the profiled optimum

Independent audit (2 Sep 2026, AUDITS/AUDIT-DS-PRACTICAL-CERTIFIED-SOLVER-001.md). Verified with hardened assumptions (Tier A), verified (T5, Tier B), umbrella reduced with a narrower remainder. Independent exact falsification over 125,491 canonical partitions found zero weak-duality, DS11, contiguity or refinement violations; closure $\Leftrightarrow$ saddle held on 54/54 exact-$d$ tables. Five hardenings, folded into the text below: (H1) the tie lemma — at a fixed tilt an optimal labeling exists in which every group of equal tilted values is wholly in one cell or split only among pure-tied cells, so the interval-DP value is tie-order independent and the $O(KN^2)$ bound (in fact $O(KN)$ after sorting, Grønlund et al. 2017) holds with ties; the frozen text asserted this without proof. (H2) the saddle gate is set-valued: a deterministic DP tie policy can return a non-closing member of $\mathcal D(\beta^*)$ when the bracket closes (CE-DS-TILT-DUAL-TIE-MASK-001, $N=3,K=2$, pairwise-distinct tilted values), so a reported open interval is not a gap certificate. (H3) the certified-$\varepsilon$ bracket's bit model is explicit: singleton-versus-rest coercivity radius, subgradient oracle, exact cutting-plane LP lower certificate, ellipsoid/GLS rounding discipline. (H4) "support-minimal" holds for $K=3$ only; the overall minimum is CE-DS-TILT-DUAL-GAP-002 ($N=3,K=2$, $g^+=1/3$, $d=1/2$, gap $1/6$); the exact dual minimum of fixture 001 is $44729/4232$ at the rational crossing $\beta^*=-8/23$, exact gap $534361/781333$, so the registered mixture bound is a lower certificate, as stated. (H5) zero-weight rows are excluded and the collapsed-atom versus unrestricted-row domain is named. Scope widened (audit §7.5): at $d_\lambda=1$ the exact dual minimiser is computable in polynomial bit complexity for every $K$ — the breakpoints are roots of degree-$\le2$ polynomials of polynomial height, so a bisection on the exact one-sided DP derivatives certifies a rational or quadratic-irrational minimiser after polynomially many probes; for fixed $d_\lambda\ge2$ and variable $K$ exact minimisation is arithmetic-polynomial by Toledo (1993) parametric search. The (EXACT-PARAMETRIC-DP) question below is therefore narrowed to a polynomial bit bound for fixed $d_\lambda\ge2$ and to variable $d_\lambda$. The audit authorizes no src/ change.

Normalization (protocol A). Tier A has $d_\psi=1$, arbitrary $d_\lambda\ge1$, and finite positive rational weights. All empirical second moments are about the score-space origin: scores are not sample-centered. The decision variable on the comparison side is a nonempty $K$-cell labeling. The generalized comparison domain uses DS11's pseudo-inverse value $\Phi^+(z)$; the ordinary in-bin domain is its subset with a nonsingular nuisance block. For $\beta\in\mathbb R^{d_\lambda}$, put

$$T_{\beta i}=s_{\psi i}-\beta s_{\lambda i},\qquad V_z(\beta)=\sum_b\frac{\bigl(\sum_{i:z_i=b}w_iT_{\beta i}\bigr)^2} {\sum_{i:z_i=b}w_i}.$$

Let $v_K(\beta)=\max_zV_z(\beta)$. By scalar contiguity this value is returned by the exact interval DP after sorting $T_\beta$. Let

$$g^+=\max_z\Phi^+(z),\quad g_{\rm reg}=\max_{z:I_{\lambda\lambda}(z)\succ0}\Phi(z),\quad d=\min_\beta v_K(\beta).$$

For the primal, a tie policy is part of the object. If ${\cal D}(\beta)$ is the set of DP-optimal labelings at $\beta$, define

$$p^+=\max_{\beta,\,z\in{\cal D}(\beta)}\Phi^+(z),\qquad p_{\rm reg}=\max_{\beta,\,z\in{\cal D}(\beta),\, I_{\lambda\lambda}(z)\succ0}\Phi(z).$$

An implementation may use one deterministic member of ${\cal D}(\beta)$; that only weakens its lower bound. (Audit H2: it can also hide a closure — the member it holds need not be the saddle even when $g^+=d$; an open reported interval therefore certifies nothing about the gap.) The information-loss implication is a finite train-sample $D_s$ value interval. It supplies no held-out or population retention statement except in the named DS18 corollary below.

Theorem DS19.1 (the valid bracket and its exact closure gate)

On the generalized domain,

$$\boxed{p^+\le g^+\le d.}$$

On the ordinary in-bin domain,

$$\boxed{p_{\rm reg}\le g_{\rm reg}\le g^+\le d.}$$

Thus the same dual is a ceiling on the whole DS11 pseudo-inverse class and, a fortiori, on DS9. A singular DP state is a valid generalized lower bound but is not an in-bin lower bound; the regular filter in $p_{\rm reg}$ is load-bearing.

Moreover the generalized bracket closes, $p^+=g^+=d$, iff there are $(\beta^*,z^*)$ such that

$$z^*\in{\cal D}(\beta^*),\qquad \beta^*I_{\lambda\lambda}(z^*)=I_{\psi\lambda}(z^*).$$

Equivalently, $z^*$ maximizes $V_z(\beta^*)$ over labelings and $\beta^*$ minimizes $V_{z^*}(\beta)$: it is a saddle pair. If in addition $I_{\lambda\lambda}(z^*)\succ0$, the same equality certifies $z^*$ as an ordinary in-bin global optimum. At a singular block only the normal equation is meaningful; no quotient slope is asserted.

Proof. DS11 gives $\Phi^+(z)=\min_\gamma V_z(\gamma)$. Hence for every $z,\beta$,

$$\Phi^+(z)\le V_z(\beta)\le\max_{z'}V_{z'}(\beta)=v_K(\beta).$$

Taking the indicated maxima and minimum proves weak duality and both lower bounds. If a saddle pair exists, all inequalities are equalities. Conversely, if $g^+=d$, choose a primal maximizer $z^*$ and a dual minimizer $\beta^*$. Then

$$d=\min_\gamma V_{z^*}(\gamma) \le V_{z^*}(\beta^*) \le v_K(\beta^*)=d.$$

Thus both inequalities are equalities: $z^*$ is DP-optimal at $\beta^*$, and DS11's attainment set makes $\beta^*$ satisfy exactly the displayed normal equation. The same pair also gives an attained $p^+=d$. The finite maximum of nonnegative convex quadratics attains its infimum after quotienting directions that are null for every empirical nuisance moment. ∎

Ties and duplicate tilts. Scalar contiguity supplies the exact value $v_K(\beta)$ even when equal tilted values are pooled (audit H1, tie lemma: each mixed cell's term $(M+tg)^2/(W+g)$ is convex in the tied mass $g$ it receives, so an optimal labeling keeps every tied group in one cell or only in pure-tied cells, and every tie order yields the same DP value; the labeling active on $(\beta,\beta+\varepsilon)$ is contiguous in the order breaking ties by decreasing $s_\lambda$). At a crossing, different row-level refinements can have the same DP value and different derivatives. Any active derivative is a valid subgradient of $v_K$, but a closure certificate must exhibit the concrete labeling whose normal equation is checked. A deterministic score-only quantizer must be constant on an exact duplicate-score class; unrestricted finite row assignment is a larger comparison domain and must be named separately.

Theorem DS19.2 (what is polynomial, and the exact-computation reduction)

For a supplied rational $\beta$, $v_K(\beta)$, one active labeling, its exact rational gradient, and the regular/generalized primal values are computable in $O(KN^2)$ rational operations by sorted interval DP. The map $v_K$ is a finite maximum of convex quadratics, hence convex. With rational input, full empirical nuisance span, and a requested rational tolerance $\varepsilon>0$, a standard separation-oracle convex method therefore returns certified rational lower/upper bounds on $d$ of width $\varepsilon$ in time polynomial in the input bit length and $\log(1/\varepsilon)$. A search radius is observable: singleton-versus-rest nuisance partitions give a positive quadratic coercivity bound on the effective nuisance span; common-null directions are first quotiented out.

Exact algebraic computation was established in the frozen proof only when $K$ and $d_\lambda$ are fixed; the audit widens this (H-scope above): polynomial bit complexity at $d_\lambda=1$ for every $K$, and arithmetic-polynomial exact minimisation for fixed $d_\lambda\ge2$, variable $K$ (Toledo 1993). The fixed-$(K,d_\lambda)$ argument follows. The $O(N^{2d_\lambda})$ tilt-order arrangement, including its tie faces, has polynomial size then; each cell has $\binom{N-1}{K-1}$ contiguous candidates, and fixed-dimensional real-algebraic comparison of their quadratic envelopes returns the exact minimum. This is not polynomial jointly in variable $K,d_\lambda$.

The study's requested general exact polynomial-time claim is therefore reduced to one explicit assumption/question:

(EXACT-PARAMETRIC-DP) Does the oracle-defined convex envelope $\min_\beta v_K(\beta)$, with $K$ and $d_\lambda$ part of the input, admit exact algebraic optimization in polynomial bit complexity, including active tie refinements, without materializing its potentially large parametric-DP envelope?

No exact-computation claim is inferred from crossings-plus-midpoints or from an $\varepsilon$-accurate convex solve.

Counterexample DS19.3 (strong duality can fail by order one)

CE-DS-TILT-DUAL-GAP-001 is the equal-weight table

$$(-11/2,39/8),\quad(3/2,-65/8),\quad (7/2,31/8),\quad(9/2,-49/8),$$

with $K=3$. All six canonical nonempty partitions are regular, so DS9 and DS11 coincide. Exact enumeration gives

$$g=g^+=g_{\rm reg}=\frac{116805}{11816}.$$

For $z_1=(0,0,1,2)$ and $z_2=(0,1,2,2)$,

$$q_1(\beta)=\frac{925}{64}\beta^2+\frac{15}{4}\beta+\frac{81}{8}, \quad q_2(\beta)=\frac{1477}{64}\beta^2+24\beta+\frac{129}{8}.$$

With $\alpha=14/25$, $v_K(\beta)\ge \alpha q_1(\beta)+(1-\alpha)q_2(\beta)$. The mixed quadratic has vertex $-10128/29197$ and minimum $61717893/5839400$. Therefore

$$\boxed{ d-g\ge \frac{61717893}{5839400}-\frac{116805}{11816} =\frac{105329256}{154014175}>0.68 .}$$

Since $p^+\le g$, the proposed primal-dual bracket has at least the same gap. The witness is support-minimal for $K=3$: $N=3,K=3$ has only the singleton partition. (Audit H4: over all $K$ the minimum is $N=3,K=2$, CE-DS-TILT-DUAL-GAP-002: rows $(-1,0),(0,-1),(1,0)$, equal weights, $g^+=1/3$, $d=1/2$ exactly at $\beta^*=0$ by the mixture $\tfrac16\beta^2+\tfrac12$, gap $1/6$. The exact minimum of the $N=4$ envelope is $44729/4232$ at $\beta^*=-8/23$.)

This is genuinely $\Theta(1)$, not merely a fixed-size notation. For $r\to\infty$, give the four displayed atoms total mass $1-1/r$ and add $r$ distinct bounded rational atoms of total mass $1/r$. Cellwise Cauchy--Schwarz bounds show uniformly, for $\beta$ in a compact set, that every added-only cell contributes $O(1/r)$, while every cell containing a base atom perturbs its quadratic by $O(1/r)$. The mixed quadratic above is coercive, so all dual minimizers remain in one compact interval. The same cellwise argument applied to the scalar Schur formula (a base-singular cell has cross term $O(1/r)$ and nuisance term $O(1/r)$) gives $d_r\to d$ and $g_r^+\to g^+$. Hence $d_r-g_r^+$ stays bounded below by a positive constant. This construction uses positive weights and makes no split-duplicate assumption.

Theorem DS19.4 (the DS18 strip DP is $\Delta$-consistent)

On the exact DS18 law, let $\tilde z_N$ be the exact three-interval DP labeling of the uncentered scalar values $X_i=T_{0,i}$. It is computable in polynomial time. Almost surely, it is regular eventually and

$$0\le\Delta_N :=\hat v_{3,N}(X)-\hat\Phi_{D_s}(\tilde z_N) =\frac{\hat I_{\psi\lambda}(\tilde z_N)^2} {\hat I_{\lambda\lambda}(\tilde z_N)}\longrightarrow0.$$

Proof. DP optimality gives $\hat I_{\psi\psi}(\tilde z_N)=\hat v_{3,N}(X)$, so the equality is the ordinary scalar Schur formula, with no centering. DS18's selection-independent uniform codebook event and the unique equal-third scalar optimum imply that the DP cells converge, up to labels, to the fixed cuts $\pm1/3$. Bounded scores then transfer all cell moments. Consequently $\hat I_{\psi\lambda}(\tilde z_N)\to0$ and $\hat I_{\lambda\lambda}(\tilde z_N)\to32/81$, proving regularity and the limit. ∎

The audited DS18 disagreement inequality therefore applies to $\tilde z_N$. This is strictly a value statement. The interval seed can be one-point exchange-unstable (CE-DS-INTERVAL-SEED-UNSTABLE-001), and raw population-cut labels can be unstable (CE-DS-NONCENTERED-POPULATION-CUT-UNSTABLE-001). Nothing here proves local ascent selects the basin, exchange stability of $\tilde z_N$, perturbation robustness, or a compile authorization.

Tier B: the matrix-tilt outer problem is not quasiconvex

Weak duality still holds for $d_\psi>1$. Convexity and even quasiconvexity of the proposed outer log-determinant map fail. In CE-DS-MATRIX-TILT-NONQUASICONVEX-001, take the eight centered equal-weight rows $\{\pm2e_j:j=1,\ldots,4\}$, split the first two coordinates as POI and the last two as nuisance, and take $K=N=8$. The singleton partition is the only nonempty partition and its blocks are $I_2,0,I_2$. Hence

$$f(B)=\log\det(I_2+BB^\top).$$

For $B_0=\operatorname{diag}(4,0)$, $B_1=\operatorname{diag}(0,4)$, and $B_{1/2}=(B_0+B_1)/2=\operatorname{diag}(2,2)$, the exact determinants are $17,17,25$. Thus $f(B_{1/2})>\max(f(B_0),f(B_1))$, disproving quasiconvexity and therefore convexity. The certified ceiling remains valid, but minimizing it is a nonconvex outer problem in general. Tier B closes disproved; no library approximation follows.

Observable decision rule and deployment boundary

1. A regular exhibited saddle pair closes the bracket and certifies finite global optimality on both domains. A singular saddle certifies only the generalized DS11 problem.
2. A nonzero bracket reports the interval $[p_{\rm reg},d]$ (or $[p^+,d]$ when explicitly generalized). It does not authorize the word “optimal”. The counterexample proves the interval may stay macroscopically open.
3. The projected full-efficient-score interval rule remains the currently established unconditional projected-problem compiler. It is not silently relabeled as an in-bin profiled solution.
4. A DS14 companion rule requires the complete audited sequence hypotheses and margins (M1)--(M5). A measured nuisance floor, one finite root, or $\Delta$-consistency alone is diagnostic only.
5. In every other state the correct profiled compile action is refusal.

This result authorizes no src/ or public API change. The theorem and both counterexamples require a fresh independent audit before any compile surface can consume them.

Self-adversarial notes. The dual ceiling, primal lower bound, and closure condition name their comparison domains separately. Exact-K nonempty cells are used; empty-cell conventions are not hidden. The fixed-tilt value tolerates ties, but a concrete closure labeling is mandatory. The order-one family does not use duplicate splitting. Cardinality remains $K\ge d_\psi+d_\lambda+1$ for centered regular samples. The DS18 proof uses the raw rows and only a value limit. Estimated scores, score-estimation error, held-out performance, and law perturbations remain outside the claim.

Stop-condition verdict. TIER A REDUCED. Validity on both named domains, the exact saddle closure condition, an exact $\Theta(1)$ duality-gap counterexample, polynomial certified-accuracy computation, and DS18 $\Delta$-consistency are settled. General exact polynomial bit-complexity is reduced precisely to (EXACT-PARAMETRIC-DP). TIER B DISPROVED: the outer matrix-tilt map need not be quasiconvex. The deployment question is answered: finite brackets are honest diagnostics/certificates at closure, while compile authorization remains projected-rule, full DS14 companion hypotheses, or refusal.

Next dependency-blocking question. After the audit, (EXACT-PARAMETRIC-DP) is settled at $d_\lambda=1$ (bit-polynomial) and for fixed $d_\lambda$ (arithmetic-polynomial); does a polynomial bit bound hold for fixed $d_\lambda\ge2$ with variable $K$, and is variable $d_\lambda$ hard? The deployment blocker is a selection theorem: which certified or $\Delta$-consistent labeling does a practical solver return, and when does it inherit exchange stability.