The frontier

First-order reporting and rule-transfer bounds for frozen score rules are audited with corrections (SCORE-ERROR-RETENTION-BUDGET, SCORE-ERROR-RULE-TRANSFER; AUDIT-SCORE-ERROR-BUDGET-001). Remaining: uniform error control plus optimization-gap certification, D_s, refitted rules and sharp second-order constants. No uniform relative/log reporting bound follows from an L2 score-error upper bound alone without retention conditioning.

Let s be the true score with V = E[s s^T] > 0, s_hat = s + e a proxy, Z = q(s_hat) a frozen label map with p_b > 0, c_b = E[s | Z=b], e_b = E[e | Z=b], I_Z = sum_b p_b c_b c_b^T (true retained information), I_tilde_Z = sum_b p_b (c_b + e_b)(c_b + e_b)^T and V_tilde = E[s_hat s_hat^T] (what the library computes from the proxy), eps^2 = E[e^T V^{-1} e], eps_Z^2 = sum_b p_b e_b^T V^{-1} e_b <= eps^2, eps_R^2 = sum_b p_b e_b^T I_Z^{-1} e_b <= eps_Z^2 / rho_min, R = V^{-1/2} I_Z V^{-1/2} with eigenvalues rho_i in [0, 1]. (B1) For every direction a: |a^T (I_tilde_Z - I_Z) a - a^T E_Z a| <= 2 sqrt((a^T I_Z a)(a^T E_Z a)) with E_Z = sum_b p_b e_b e_b^T <= E[e e^T], and the same for V_tilde - V with E[e e^T]. (B2) If I_Z > 0, I_tilde_Z > 0 and V_tilde > 0, the reported geometric-mean retention eta_tilde_D = (det I_tilde_Z / det V_tilde)^(1/d) and the true eta_D = (det I_Z / det V)^(1/d) satisfy log(eta_tilde_D / eta_D) = T_1 + T_2 + r_3 with T_1 = (2/d)(E[e_Z^T I_Z^{-1} c_Z] - E[e^T V^{-1} s]) (alignment term), |T_1| <= (2/d) sqrt(d - tr R) (eps_Z sqrt((1 - rho_min)/rho_min) + sqrt(eps^2 - eps_Z^2)); separately |T_1| <= (2/sqrt d)(eps_R + eps); T_2 = (eps_R^2 - eps^2)/d (spurious-information term); and the curvature remainder r_3 in (1/d)[-U_d(eps_R)^2/(2(1-eps_R)^2), U_d(eps)^2/(2(1-eps)^2)], U_d(x) = 2 sqrt(d) x + x^2, for eps, eps_R < 1. Without expansion, (1/d)[L_d(eps_R) - U_d(eps)] <= log(eta_tilde_D/eta_D) <= (1/d)[U_d(eps_R) - L_d(eps)] with L_d(x) = max(2 d log(1-x), x^2 - 2 sqrt(d) x - U_d(x)^2/(2(1-x)^2)). The numerator log-det first-order constant 2 sqrt(d) is attained (e_b = kappa c_b); this does not assert sharpness for the reporting ratio or an ordering of its two alignment bounds. No finite uniform relative/log reporting bound follows from an upper bound on eps alone (CE-SCORE-ERROR-RHO-MIN-NECESSARY-001): the retention-weighted eps_R is the scale, with eps_Z^2/rho_min an upper bound on eps_R^2, not an identity. For fixed labels, the reported number is invariant under nonsingular linear transformations A of the proxy, not translations; a rule must be transported as q_A(u)=q(A^{-1}u) to preserve labels. Every reporting bound applies with e=A s_hat-s and recomputed errors. The unconstrained least-squares A*=E[s s_hat^T] V_tilde^{-1} has eps_lin^2=d-sum_i r_i^2 for uncentred canonical correlations r_i, but is admissible for retention invariance only if E[s s_hat^T] is nonsingular. Otherwise this value is an infimum over invertible maps, not a usable singular minimizer. The roles of s and s_hat are symmetric, so the truth is bracketed from the reported eta_tilde_D, the reported smallest retention eigenvalue rho_tilde_min and the error measured in the proxy metric. Endpoints: at eta_D = 0 (I_Z singular) the reported number is uncontrolled; at eps_R >= 1 the lower bracket is void and the reported information can vanish.

Frozen rule q on score space, Z = q(s), Z_hat = q(s_hat), s_hat = s + e, both retained informations computed from the true s; M = {Z != Z_hat}, pi = P(M). (i) If q(s_hat) != q(s) then dist(s, boundary of the cell of q(s)) <= ||e|| in any norm (the segment [s, s_hat] is connected); for a nearest-centre rule with positive-definite metric G, distinct centres and distance measured in G the squared distance is min_{b' != b} (||s - mu_b'||_G^2 - ||s - mu_b||G^2)^2 / (4 ||mu_b - mu_b'||G^2), rational on rational data. Hence with the margin function M_q(t) = P(dist_V(s, boundary q) <= t) and eps^2 = E[e^T V^{-1} e]: pi <= inf_t {M_q(t) + eps^2 / t^2}; if M_q(t) <= C t^alpha then pi <= (1 + alpha/2)(2/alpha)^(alpha/(alpha+2)) C^(2/(alpha+2)) eps^(2 alpha/(alpha+2)), i.e. pi <= 1.89 C^(2/3) eps^(2/3) at alpha = 1. (ii) With c_b the Z-cell means and c_hat_b the Z_hat-cell means of s: I_Zhat >= I_Z - Gamma and I_Z >= I_Zhat - Gamma_hat in the Loewner order, Gamma = 2 E[(s s^T + c{Z_hat} c{Z_hat}^T) 1_M], Gamma_hat = 2 E[(s s^T + c_hat_Z c_hat_Z^T) 1_M]. Corollaries: |tr R_Z - tr R_Zhat| <= 2 E[||w||^2 1_M] + 2 pi max_b max(||c'_b||^2, ||c_hat'b||^2) (whitened); log eta_D(Z_hat) - log eta_D(Z) >= log(1 - gamma_R) with gamma_R = tr(I_Z^{-1} Gamma) <= tr(V^{-1} Gamma)/rho_min and tr(V^{-1} Gamma) = 2 E[(||w||^2 + ||c'{Z_hat}||^2) 1_M], symmetrically above; order O(pi) for uniformly bounded whitened scores; O(sqrt pi) under a uniform fourth moment and stable positive base-cell masses as specified in the assumptions. Determinant rates also require a retained eigenvalue floor. Composed with (i) at alpha = 1: O(eps^(2/3)) bounded, O(eps^(1/3)) with a fourth moment. Without a margin condition nothing holds: CE-SCORE-ERROR-BOUNDARY-ATOM-001 has an atom of mass 1/2 on a cell boundary, pi = 1/2 for every eps > 0 and the retention jumps 4/5 -> 1/2.

For the mixture score of CLASSIFIER-MIXTURE-SCORE-FORMULA, Phi_a(eta) = (eta_a/pi_a) / sum_b theta0_b eta_b/pi_b on the simplex with theta0 interior: D(eta) >= D_min = min_b theta0_b/pi_b, 0 <= Phi_a <= 1/theta0_a, and |Phi_a(eta') - Phi_a(eta)| <= (|eta'_a - eta_a|/pi_a + (1/theta0_a) sum_b (theta0_b/pi_b) |eta'_b - eta_b|) / D_min, hence ||Phi(eta') - Phi(eta)||_2 <= L ||eta' - eta||2 with L = (1/pi_min + Q sqrt(m) ||1/theta0||2) / D_min, Q = max_b theta0_b/pi_b. With s = T Phi for the fixed linear chart T and s_hat = T Phi(eta_hat): the Fisher-whitened score error of SCORE-ERROR-RETENTION-BUDGET satisfies eps^2 <= ||V^{-1}|| ||T||^2 L^2 E{P_theta0} ||eta_hat - eta||^2 <= ||V^{-1}|| ||T||^2 L^2 Q E{P_pi} ||eta_hat - eta||^2, and under the training mixture P_pi with one-hot component label Y, E ||eta_hat - eta||^2 = BS(eta_hat) - BS(eta) is the excess Brier score, which splits orthogonally as reliability (calibration error, E||eta_hat - E[eta | eta_hat]||^2, estimable from labelled data) plus resolution gap (E||eta - E[eta | eta_hat]||^2, the posterior-level representation loss of O5, needing the truth). The inverse map eta_a=pi_a Phi_a/sum_b pi_b Phi_b has Lipschitz constant L_inv=Q(pi_max+||pi||_2) on its image. Thus posterior and full component-ratio errors are equivalent. For score error the reverse implication additionally requires sigma_T=inf{||Th||_2: theta0^T h=0, ||h||_2=1}>0. Then eps^2 >= D_min sigma_T^2/(L_inv^2 ||V||_op) E_Ppi||eta_hat-eta||^2 >= the same constant times reliability. Without this chart condition positive reliability can coexist with zero score error. Even with it, positive score/calibration error need not distort retention (invertible score rescalings cancel). Excess Brier error supplies a sufficient upper score-error budget, not a necessary calibration requirement for correct reporting; AUC does not determine the reporting gap.

Construct cross-fitted estimators and uncertainty bars for I_R=Var(E[s|R]) and I_q=Var(E[s|q(R)]) when truth scores are available only on simulation.

The audited CLASSIFIER-CALIBRATION-SCORE-LIPSCHITZ bridge supplies an upper score-error budget from excess Brier error. Reliability and resolution gap split that posterior error; a reverse score-error bound needs an injective chart and explicit constants. Calibration error does not certify retention distortion, and AUC does not determine proxy reporting error. Remaining: certify the resolution gap without truth scores (OPEN-REPRESENTATION-LOSS-ESTIMATION), sharper constants, and D_s efficiency.

Measured evidence only (py/score_error_budget.py door3 and synthetic2d, 8 Sep 2026; cut-aligned QUADPACK). door3 (examples/door3_classifier.py, logistic-regression rungs 15/60/300 events per class, D-exchange rule with K = 4, rebuilt through the O6 audit's closed forms): true retention of the deployed labels 0.8937 / 0.9557 / 0.9716 against reported 0.9671 / 0.9710 / 0.9738 (log ratio 0.079 / 0.016 / 0.0023); whitened score error eps = 0.500 / 0.289 / 0.096, after the least-squares linear reduction 0.416 / 0.235 / 0.067 (uncentred correlation between true and proxy score 0.909 / 0.972 / 0.998); eps_R = 0.447 / 0.262 / 0.091; exact first-order term T_1 = 0.084 / 0.025 / 0.0028; the sharper alignment bound 0.270 / 0.079 / 0.016 covers T_1 by 3-6x, the crude first-order budget 1.89 / 1.10 / 0.37 by 24x; rule applied to the true score retains 0.947 / 0.971 / 0.973; mislabel mass pi = 0.275 / 0.186 / 0.051; both Loewner sandwiches hold (Gamma = 0.80 / 0.41 / 0.10); the margin bound is vacuous at the two larger errors and 0.45 at rung 300. synthetic2d (the O7 bounded three-component mixture, d = 2, K = 4, eta_D = 0.3303, retention eigenvalues 0.131 and 0.830): a template with all component means shifted by delta = 0.05 / 0.1 / 0.2 / 0.4 has eps = 0.072 / 0.143 / 0.285 / 0.563 and reports log ratios -0.007 / +0.003 / +0.064 / +0.263; an error of the same eps aligned with the weakest retained direction and sign-correlated with the cells reports +0.021 / +0.068 / +0.204 / +0.468 (eps_R = 0.20 / 0.39 / 0.78 / 1.54) against +0.004 / +0.009 / +0.022 / +0.052 for the strongest direction (eps_R = 0.08 / 0.17 / 0.33 / 0.66); every row lies inside the B2 bracket; T_1 <= 0.04 on every row and (eps_R^2 - eps^2)/2 reproduces the weak-direction gap to 0.01 for eps <= 0.29.

Find distribution-dependent/distribution-free bounds on eta_D(K) and inversion rules for required bin count at target efficiency.

Bound the worst normalized retained eigenvalue from eta_D, and conversely, with useful additional assumptions.

Derive K->infinity asymptotics; test the hypothesis that first order is Fisher-whitened quadratic quantization and D-specific shape effects enter at second order.

Study average or minimax objectives over parameter regions and determine whether common-metric/affine geometry survives.

Develop influence-function or bootstrap confidence intervals for reported retention functionals (e.g. geometric-mean retention), handling the non-smoothness of hard assignment at cell boundaries, so retention is reported with error bars rather than as a point estimate. Settled special cases under a frozen rule with an iid equally weighted oracle-score evaluation sample: scalar true score (5 Sep 2026, RETENTION-PLUGIN-CLT-FROZEN-SCALAR, independently audited) and d-dimensional true score with the geometric-mean D-retention (det I_Z / det V)^(1/d) (6 Sep 2026, RETENTION-PLUGIN-CLT-FROZEN-VECTOR, independently audited the same day) - the plug-in is asymptotically normal with an explicit matrix influence function, a consistent plug-in variance and a valid Wald interval iff sigma^2 > 0, where sigma^2 = 0 iff every cell's conditional law sits on an ellipsoid (CE-O7-ELLIPSOID-ZERO-VARIANCE-001 shows atomless such laws exist for d >= 2), and at singular I_Z the plug-in is biased upward at rate n^{-(d-r)/d} (RETENTION-PLUGIN-SINGULAR-ENDPOINT-RATE). Still open: rules refitted on the evaluation sample, where the boundary non-smoothness enters; weighted samples; any uncertainty statement without an oracle score; the profiled D_s retention (a Schur-complement functional); the degenerate limits of n(eta_hat - eta) when sigma^2 = 0 and the joint limit of (n^{(d-r)/d} eta_hat_D, n^{(d-r)/d} sigma_hat/sqrt(n)) at the singular endpoint (the endpoint coverage is measured to stabilise, not proved to converge); and a second-order-corrected or transformed interval for heavy-tailed scores, where the first-order interval is measured to be liberal at n <= 3000.

Bound D/Ds efficiency loss when a quantizer optimized at theta0 is evaluated at theta0+delta; seek local perturbation results.

Formalize I_total=I_count+I_shape for the relevant extended likelihood and specify exactly what event hard quantization changes.

Develop scalable score/efficient-score construction for calibration, shape, normalization, MC-statistical, and correlated nuisance parameters.

Derive numerically stable score coordinates for simplex fractions, reference-component coordinates, unconstrained local coordinates, and extended yields; clarify D invariance and Ds POI/nuisance transformations.

Determine whether unrestricted empirical global D optima converge in value/decision to population global hard quantizers.

Determine whether empirical one-point-exchange-stable D solutions converge to the population stationary set and what stability margins prevent persistent spurious branches.

Determine whether non-geometric finite discrepancies vanish asymptotically and whether global finite objective values converge to population hard-quantizer optima.

Complete the academic margins dichotomy beyond the audited d_psi=d_lambda=1 DS15 class. For d_psi>1, establish the required uniqueness and rigidity theory for vector-D quantization of the efficient score before transferring the degenerate-attainer argument. For d_lambda>=2 and K>=d_psi+d_lambda+1, construct or refute a vector-(R) steering mechanism spanning all nuisance directions. The rank-vacuous boundary K=d_psi+d_lambda is already excluded by CE-DS-MARGINS-RANK-VACUITY-001. The deployment-facing non-centered DS18 remainder is closed by DS19's value-consistent interval-DP primal and is not part of this “Completing the D core” question.

Find necessary/sufficient conditions on concave F for one-point exchange stability to imply pointwise assignment under a common first-order metric/supergradient.

Improve singleton-refinement upper bounds using moment relaxations, SDP/convex envelopes, affine-realizability pruning, or mass constraints.

Resolve the two academic remainders left by DS17 on the scalar conditionally centered class: whether ordinary exchange-stable empirical sequences can track (M5)-free coincident-centroid wasted-cell configurations, and whether the constrained values v*(kappa) and v*+(kappa) are attained and one-sided continuous under their distinct DS16 conventions. The Gaussian sign-split construction proves only nonemptiness of the closed population feasible class for kappa<=1/pi. Deployment and non-centered Delta-selection are closed out of an earlier line of work by DS19 and are not part of this “Foundations” question.

Determine NP-hardness/FPT/W[1]/ETH boundaries and tightness of N^{O(Kd)}; valid reductions are required.

Analyze two-point swaps, move-two, merge-split, boundary perturbations, and rank-r updates; seek stronger guarantees and reduced multistart dependence.

Determine whether every suitable population E optimum admits one common supergradient supporting all cell-assignment inequalities almost everywhere.

For atomic/finite score laws, determine whether splitting an atom among labels can strictly improve the objective over every deterministic hard score quantizer.

Under a simple minimum eigenvalue with gap gamma, determine whether exchange stability implies an approximate rank-one Voronoi rule with explicit w/gamma scaling.

Determine when stationary points or optima of softened randomized affine/Voronoi families converge to hard stationary/optimal partitions as temperature tends to zero.

Derive or disprove an A analogue of the Ds O(w) first-order violation bound.

Determine whether min_beta v_K(S_psi-beta S_lambda), with K and d_lambda part of the rational input, admits exact algebraic optimization in polynomial bit complexity using the fixed-tilt interval-DP oracle, including active tie refinements, without materializing a potentially large parametric-DP envelope. Settled by DS-TILT-DUAL-CERTIFICATE and its audit: fixed-tilt exact evaluation, polynomial certified-epsilon minimization, exact polynomial-bit minimization at d_lambda=1 for every K (root-separation bisection), and exact minimization in polynomially many arithmetic operations for fixed d_lambda>=2 with variable K (Toledo 1993 fixed-dimension parametric search with degree-2 comparisons). Remaining: a polynomial bit bound for fixed d_lambda>=2 with variable K (a root-separation argument for the conic breakpoint arrangement carried through the parametric search), and any exact statement, or hardness obstruction, for variable d_lambda.