Open question · OPEN-DS-MARGINS-NONCENTERED

Does the scalar margins dichotomy extend beyond centred laws?

Openpublication: open

Meaning

The corresponding geometry and margin behaviour beyond the audited scalar class remain open.

Scientific context

Beyond the audited scalar-interest, scalar-nuisance, conditionally centred class, the margins dichotomy for the profiled criterion is open in two specific directions: more than one parameter of interest needs a uniqueness and rigidity theory for vector quantization of the efficient score that does not yet exist, and two or more nuisance parameters need a steering construction spanning every nuisance direction, which is neither built nor refuted.

Statement

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.

Ruled out by

Academic “Completing the D core” remainder only. Do not reopen DS18 or DS19, and do not infer the vector-POI dichotomy from scalar uniqueness or the vector-nuisance steering result from the one-dimensional sign construction. Preserve K>=d_psi+d_lambda+1.

Bounded by

CE-DS-MARGINS-RANK-VACUITY-001
DS15's margins dichotomy as first registered for general nuisance dimension under the bare cardinality assumption K >= 3: for d_lambda >= 2 with K = d_lambda + 1, conclusion (i) (value convergence to the positive unrestricted supremum v_K) is false, because exact centering forces rank(I_z) <= K - 1 = d_lambda, so by Schur-complement rank additivity EVERY feasible labeling (nonsingular binned nuisance block) has profiled value exactly 0 on every sample, while v_K > 0.

Closest prior work

Nearest prior work

Blanchard, Jaffe, Zhivotovskiy (2025)Consistency and inconsistency in k-means clustering

Why each source is not this result: the search records

Nearest sources, checked 31 August 2026

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Prerequisites to the left, what it enables or raises to the right, boundary cases below. Open in the map.

The question in full

Margins beyond conditional centering

Two vector academic branches after DS19: for $d_\psi>1$ , the uniqueness and rigidity theory for vector-D quantization of the efficient score before transferring DS15's degenerate-attainer dichotomy; for $d_\lambda\ge2$ above the centered-sample threshold $K\ge d_\psi+d_\lambda+1$ , construct or refute a vector-(R) steering mechanism spanning all nuisance directions (CE-DS-MARGINS-RANK-VACUITY-001 covers $K=d_\psi+d_\lambda$ ). Do not reopen the audited scalar DS15, DS18 or DS19 claims. Target claim: OPEN-DS-MARGINS-NONCENTERED.

From OPEN_PROBLEMS.md