The claims

The whole record, in the order it is written up: each chapter derives its results in sections, and a claim sits in the section that proves it. The open questions follow, grouped by the theme they belong to, and the verification records last.

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Universal information structuresource

U1Quantized Fisher-information identity

A hard label's information is the covariance of cell mean scores

Cell masses and conditional score means determine the retained Fisher information of a fixed rule in a regular model.

Established

FI-QUANT-IDENTITY

U2Exact information-loss decomposition

Binning loss is within-cell score scatter

The difference between full and retained Fisher information equals the within-cell second-moment loss.

Derived from the literature

FI-LOSS-DECOMPOSITION

U3Local Fisher-losslessness criterion

When is a hard label locally lossless?

Fisher information is preserved exactly when the score is measurable from the label under the stated model assumptions.

Derived from the literature

FI-LOSSLESS-IFF-SCORE-MEASURABLE

U4Rank ceiling

A bin budget limits information rank

A regular score model with K categories has retained-information rank at most K minus one.

Derived from the literature

FI-RANK-CEILING

U5Refinement monotonicity

Refining cells cannot lose information

Splitting cells increases retained information in the positive-semidefinite order.

Derived from the literature

FI-REFINEMENT-MONOTONICITY

Refinement bounds hold for every monotone criterion

A refinement upper-bounds coarser assignments for any criterion monotone in the information matrix.

Derived from the literature

GENERAL-REFINEMENT-BB

U6Reparameterization invariance of D

D partitions survive reparameterization

An invertible change of parameter coordinates preserves the D-optimal partition.

Derived from the literature

D-REPARAM-INVARIANCE

U7Normalized retained-information spectrum

Every retention eigenvalue lies between zero and one

Whitening against the full information reveals the retained fraction in each information direction.

Derived from the literature

INFO-RETENTION-SPECTRUM

The trace criterionsource

T1Fisher-whitened normalized trace equals weighted k-means

Trace retention is whitened k-means

Maximizing normalized trace is equivalent to minimizing within-cell scatter in Fisher-whitened score space.

Derived from the literature

TRACE-WHITENED-KMEANS

First-order and finite screeningsource

G1Infinitesimal cell-transfer identity

A cell transfer has an explicit first variation

The first variation connects changes in cell moments to the information criterion’s local geometry.

Proved here

GENERAL-FIRST-VARIATION

G2General concavity/supergradient screening

A tangent bound safely screens D moves

A tangent upper bound can exclude unprofitable moves before exact gain evaluation.

Proved here

D-TANGENT-SCREENING

A tangent bound safely screens profiled moves

A first-order upper bound excludes moves unable to improve the exact profiled objective.

Proved here

DS-TANGENT-SCREENING

One supergradient principle screens moves safely

Concavity turns supergradients into upper bounds for finite objective changes.

Proved here

GENERAL-SUPERGRADIENT-SCREENING

Tangent stability can certify exchange stability

The stated weighted tangent conditions exclude profitable exact relocations.

Proved here

GENERAL-WEIGHTED-TANGENT-STABILITY

D-optimalitysource

D1Population stationary geometry

Stationary population D partitions are geometric

The population first-order condition gives nearest-centre geometry under its measure and regularity assumptions.

Proved here

D-POP-VORONOI

D2Exact weighted rank-two relocation

Relocation changes information by rank two

Moving one weighted observation changes the two affected cell contributions through an exact rank-two identity.

Proved here · machine-checked

D-RANK2-MOVE

D3Exact log-det relocation gain

Every D relocation has an exact closed-form gain

A classical determinant identity gives a closed-form gain for the coupled cell-centroid update.

Proved here · machine-checked

D-LOGDET-GAIN

D4Leverage inequality

A cell's D leverage is bounded

The leverage inequality constrains cell centres in the inverse retained-information metric.

Derived from the literature · machine-checked

D-LEVERAGE

D5Exchange stability implies strict D-Voronoi geometry

Stable D partitions have a geometric rule

Under the theorem’s hypotheses, exact exchange stability forces strict self-consistent nearest-centre geometry.

Proved here · machine-checked · audited

D-EXCHANGE-IMPLIES-VORONOI

A scalar inequality drives the exchange argument

A scalar inequality supplies the core step of the quantitative exchange-to-geometry argument.

Proved here · machine-checked

D-EXCHANGE-SCALAR-CORE

Geometry violations have a measurable gain

A wrongly assigned atom supplies a strictly improving relocation with an explicit lower bound under the stated hypotheses.

Proved here · machine-checked

D-EXCHANGE-VIOLATION-LOWER-BOUND

D6Exact finite inductive closure / compiler

Repeated events inherit their group's label

Rows sharing a score are grouped before fitting; the compiled rule gives each of them the label its group received.

Proved here · machine-checked

D-CLOSURE-DUPLICATE-INHERITANCE

A solver that stops early promises less

Stopping at a positive tolerance weakens the compile guarantee to one stamped with that tolerance, so ties are broken by convention rather than ruled out.

Proved here · audited

D-COMPILE-TOLERANCE-GUARANTEE

A stable D assignment compiles into a rule

The terminal centres and information metric define a rule reproducing the assignment under the exact theorem’s conditions.

Proved here · machine-checked

D-FINITE-INDUCTIVE-CLOSURE

D7Every finite global D optimum is geometrically realizable

Global D optima are geometric

Global optimality implies exchange stability, and hence geometric realizability under the structural theorem’s hypotheses.

Proved here

D-GLOBAL-GEOMETRIC-REALIZABILITY

D8Monotone exact one-point exchange

Finite D exchange terminates

Accepting only strictly positive exact gains yields monotone search on finitely many assignments.

Proved here · machine-checked

D-EXCHANGE-TERMINATES

D9Adaptive Mahalanobis Lloyd is not monotone

Exact gains guard Lloyd proposals

Accepting a Lloyd proposal only after an exact objective check preserves monotonicity.

Proved here

D-GUARDED-LLOYD

Lloyd reassignment can reduce D

Updating the Mahalanobis metric and reassigning all observations need not increase the determinant objective.

Counterexample

D-LLOYD-NONMONOTONE

D10Voronoi fixed point does not imply exchange stability

Nearest-centre consistency is not enough

A self-consistent D-Voronoi partition can still admit a profitable exact relocation.

Counterexample

D-VORONOI-NOT-EXCHANGE

D11Exact global enumeration for fixed (d,K)

Exact D search is polynomial at fixed dimension and bin count

The geometric structure yields an XP complexity bound; the polynomial exponent depends on dimension and bin count.

Proved here

D-GLOBAL-XP

D12Singleton-refinement branch-and-bound bound

Singleton refinement bounds unfinished D assignments

Singleton refinement supplies an upper bound for branch-and-bound certification on small finite problems.

Proved here

D-BB-SINGLETON-BOUND

Profiled Ds-optimalitysource

DS0Profiled objective and Schur notation

Profiled information is a Schur complement

The profiled determinant measures efficient information for the parameters of interest after nuisance adjustment.

Established

DS-SCHUR

DS1Classical D_s design theory

Classical design theory already covers nuisance parameters

Established optimal-design theory supplies the background for profiled determinant criteria.

Established

DS-CLASSICAL-DESIGN-THEORY

DS2Gradient / efficient semimetric

The profiled gradient defines a semimetric

First-order profiled geometry acts through an efficient projection rather than a full positive-definite score metric.

Proved here

DS-GRADIENT-EFFICIENT-SEMIMETRIC

DS3Exact finite one-point objective oracle

Profiled moves have an exact gain

An exact one-point objective oracle compares feasible finite assignments with nuisance profiling.

Proved here

DS-EXACT-MOVE-ORACLE

Finite profiled exchange terminates

Strictly improving feasible profiled moves terminate on a finite assignment space.

Proved here

DS-EXCHANGE-TERMINATES

DS4D-style finite geometry theorem fails

Profiled stability does not force geometry

Finite profiled exchange stability does not imply the strict geometric conclusion available for D.

Counterexample

DS-FINITE-GEOMETRY-FAILS

DS5A global finite D_s optimum can be non-geometric

A profiled global optimum can violate geometry

An exhaustive exact witness has a global optimum that violates its own first-order rule.

Counterexample

DS-GLOBAL-NONGEOMETRIC

DS6Approximate finite efficient-Voronoi bound

Profiled stability implies approximate geometry

The first-order violation bound scales with observation weight relative to the affected cell masses.

Proved here

DS-OKN-BOUND

DS7Full-data efficient-score domination

Efficient scores bound profiled information

Between-cell information of the full-data efficient score dominates the profiled information for the same rule.

Proved here

DS-EFFICIENT-SCORE-DOMINATION

An efficient-score problem bounds profiled optimization from above

Optimizing efficient-score information over the same bin budget bounds the global profiled optimum.

Proved here

DS-EFFICIENT-SCORE-GLOBAL-UPPER

DS8Scalar efficient-score upper problem is exactly solvable by DP

Dynamic programming solves the scalar upper problem exactly

Interval dynamic programming optimizes the scalar efficient-score upper problem within the finite capacity limit.

Proved here

DS-SCALAR-EFFICIENT-DP

DS9The K≤ d feasibility split

Too few cells make full profiling singular

The rank ceiling obstructs full in-bin profiling when the cell budget is too small for the full score dimension.

Proved here

DS-FULL-PROFILE-K-LE-D-SINGULAR

The projected problem needs fewer cells

Efficient-score projection reduces the relevant rank requirement to the number of parameters of interest.

Proved here

DS-PROJECTED-K-REQUIREMENT

Profiled Ds: from a sample to the populationsource

DS11Variational form of the profiled objective and Φ-neutral splits

Global profiled optima can have degenerate ties

An exact witness exposes a tie class at a global optimum, limiting geometric conclusions.

Counterexample

DS-GLOBAL-TIE-DEGENERACY

Profiling has a variational form

Minimization over nuisance tilts expresses the profiled objective through a classical variational identity.

Proved here · audited

DS-PROFILED-VARIATIONAL

Efficient-score domination is tight exactly when nuisance projections agree

The exact gap and equality conditions identify when the efficient-score upper bound is attained.

Proved here

OPEN-DS-DOMINATION-EQUALITY

DS12Population stationary geometry for profiled D_s

Population stationarity can defeat a geometric rule

Degenerate projected centres allow stationary profiled partitions that no efficient-semimetric rule reproduces.

Counterexample

DS-POP-WASTED-CELLS

Stationary profiled population quantizers are efficient-Voronoi

An efficient projection describes stationarity under regularity assumptions, but stationarity alone does not force distinct projected centres.

Proved here, no precedent found · audited

OPEN-DS-POP-COMMON-METRIC

DS13Exact profiled leverage stability bound

Profiled stability bounds cell leverage

An exact bound quantifies what one-point profiled exchange stability says about first-order geometry.

Proved here, no precedent found · audited

DS-EXCHANGE-LEVERAGE-BOUND

DS14Conditional finite→population D_s bridge

Margined profiled stable labellings become geometric in the limit

The proved bridge requires the stated regularity and margin hypotheses; it is not unrestricted consistency.

Proved here, no precedent found · audited

OPEN-DS-FINITE-POP-BRIDGE

DS15Margins dichotomy at global finite D_s optima

Global profiled optima can lose conditioning

In the audited scalar conditionally centred class, optimal value converges while binned nuisance information vanishes.

Proved here, no precedent found · audited

OPEN-DS-MARGINS-AT-OPTIMA

DS16Margin price at arbitrary labelings, the value funnel, and the profiled compile verdict

Regularity margins have an information price

For the stated scalar conditionally centred class, maintaining nuisance information separates achievable value from the unrestricted ceiling.

Proved here, no precedent found · audited

DS-STABLE-MARGINS-PRICE

Selection of profiled stable states was measured

Experiments find both margin regimes and examine which states ascent selects at increasing sample sizes.

Measured · audited

DS-STABLE-STATE-SELECTION

DS17Inhabitation of the margin-certified branch: the conditional-centering obstruction and the fixed-point gate

Conditional centering obstructs stable margins

In the stated class of laws, the margin-certified stable branch is empty under the audited sequence assumptions.

Proved here, no precedent found · audited

DS-STABLE-BASINS-CENTERED-OBSTRUCTION

A root equation gates regular self-consistency

A scalar tilt equation is necessary for the stated empirical inhabitation problem; no converse or basin-transfer result follows.

Proved here, no precedent found · audited

DS-STABLE-BASINS-FIXED-POINT-GATE

The margin gate was probed numerically

Finite searches compare empty and inhabited regimes on specified laws; they are measured probes, not existence decisions in general.

Measured · audited

DS-STABLE-BASINS-GATE-SCANS

Wasted cells carry the remaining margins

Under linear conditional means, merging coincident projected centres collapses the relevant rank; constrained-value attainment remains unresolved.

Proved here · audited

DS-STABLE-BASINS-LCM-CLASSIFICATION

DS18Exact off-class global basin and empirical stable transfer

One explicit non-centered law has convergent global profiled optima

An explicit atomless law has a unique isolated population optimum approached by exact finite global optima; practical ascent selection is not claimed.

Proved here, no precedent found · audited

DS-NONCENTERED-GLOBAL-BASIN-TRANSFER

DS19The tilt-DP bracket: weak certification, an order-one gap, and a value-consistent DS18 primal

Matrix tilts need not be quasiconvex

With multiple parameters of interest, the outer tilt ceiling can fail quasiconvexity.

Counterexample · audited

DS-MATRIX-TILT-NONQUASICONVEX

A certificate decides profiled compilation

Observable certificate conditions separate justified profiled conclusions from cases requiring refusal.

Proved here, no precedent found · audited

DS-PROFILED-COMPILE-CERTIFICATE

An interval rule is consistent on an explicit law

The zero-tilt interval dynamic program approaches the profiled value on the audited law without implying exchange stability or compilation authority.

Proved here, no precedent found · audited

DS-STRIP-DP-DELTA-CONSISTENCY

A tilted dual brackets the profiled optimum

Scalar-interest tilt dynamic programming gives two-sided bounds, with exact closure when an exhibited labelling satisfies the saddle condition.

Proved here, no precedent found · audited

DS-TILT-DUAL-CERTIFICATE

The profiled dual gap can be strictly positive

Exact finite witnesses show a genuine primal–dual gap; a nonclosing reported bracket alone does not prove such a gap.

Counterexample · audited

DS-TILT-DUAL-STRONG-DUALITY-FAILS

The certified profiled solver question closes by reduction

The settled reduction distinguishes an exhibited saddle, an open bracket, conditional geometry, and refusal; broader exact complexity remains open.

Proved here, no precedent found · audited

OPEN-DS-PRACTICAL-CERTIFIED-SOLVER

E-optimalitysource

E1Supergradient structure

The E criterion has explicit supergradients

Classical spectral analysis describes the supergradient when the smallest eigenvalue is simple or repeated.

Established

E-SUPERGRADIENT

E2Repeated-eigenvalue one-transfer degeneracy

Repeated eigenvalues weaken E geometry

A repeated minimum eigenvalue makes one-transfer first-order geometry non-identifying.

Proved here

E-REPEATED-EIGEN-DEGENERACY

E3Finite global E geometry fails even with simple minimum eigenvalue

Global E optima can violate their own rule

An exact finite optimum need not satisfy its rank-one nearest-cell rule.

Counterexample

E-GLOBAL-GEOMETRY-FAILS

E4Positive first-order E margin need not imply exact improvement

A positive E margin need not improve a finite move

An exact counterexample separates infinitesimal screening information from the gain of a whole-observation relocation.

Counterexample

E-FIRSTORDER-NOT-FINITE

E5Safe E screening and B&B

Refinement bounds also apply to E

Information monotonicity under refinement supplies a branch-and-bound upper bound for the smallest-eigenvalue criterion.

Proved here

E-BB-APPLIES

Supergradients safely screen E moves

A valid supergradient gives a safe bound for rejecting finite moves.

Proved here

E-TANGENT-SCREENING

E6Common-supergradient population geometry

Can one supergradient describe population E geometry?

Seek a common geometric description under the smallest-eigenvalue criterion.

Open

OPEN-E-COMMON-SUPERGRADIENT

A-optimalitysource

A1Exact finite move oracle

Finite A moves have an exact gain

Relocation updates give an exact objective comparison for the A criterion under its feasibility conditions.

Proved here

A-EXACT-MOVE-ORACLE

Finite A exchange terminates

Strictly improving feasible moves cannot cycle through the finite set of assignments.

Proved here

A-EXCHANGE-TERMINATES

A2D-style finite geometry theorem fails

A stability does not force geometry

A counterexample prevents transferring the finite D exchange-to-geometry theorem to A.

Counterexample

A-FINITE-GEOMETRY-FAILS

A3Concavity screening remains valid

A tangent bound safely screens A moves

A tangent bound can reject moves that cannot improve the exact A objective.

Proved here

A-TANGENT-SCREENING

Randomized and soft quantizerssource

S1Soft assignments are an actual randomized quantizer

Soft assignments define a randomized quantizer

The soft information objective corresponds to an actual randomized labelling mechanism.

Proved here

SOFT-RANDOMIZED-FIM

S2Exact soft-assignment gradient

Soft assignments have an exact gradient

An exact gradient supports optimization of the randomized-assignment information objective.

Proved here

SOFT-ASSIGNMENT-GRADIENT

S3Hard empirical geometric objective is piecewise constant

Hard empirical objectives have flat regions

Moving a boundary without crossing a sample point leaves its labels and hard objective unchanged.

Proved here

HARD-GEOMETRIC-EMPIRICAL-PIECEWISE-CONSTANT

S4Fixed-temperature soft optimization has ordinary stationary-point guarantees

Fixed-temperature optimization reaches only ordinary stationary points

Standard nonconvex optimization theory supplies stationary guarantees at a fixed temperature, not global hard optimality.

Derived from the literature

SOFT-FIXED-TEMP-STATIONARY

S5Atomless purification

Randomized assignments purify on atomless laws

Classical purification preserves the cell masses and first moments that determine the information objective.

Established

DWW-PURIFICATION-MOMENTS

Randomized and hard population values can coincide

Atomless purification equates optimal values for criteria depending on the preserved cell moments.

Derived from the literature

SOFT-HARD-ATOMLESS-EQUIVALENCE

Sample-to-population consistencysource

C1Restricted affine-class consistency

Consistency within a restricted rule family

Uniform convergence supports value consistency in a compact finite-capacity family with mass and conditioning margins.

Proved here

CONSISTENCY-RESTRICTED-AFFINE

Score, ratio and classifier accesssource

O1Density ratios suffice for local scores

Density ratios determine the local score

Differentiating a local model density ratio yields the local score under the stated regularity conditions.

Derived from the literature

RATIO-LOCAL-SCORE

O2Linear-mixture component ratios suffice

Density ratios determine the mixture score

Component density ratios suffice for explicit score construction in a linear-mixture model.

Derived from the literature

MIXTURE-RATIO-SCORE

O3Calibrated classifier posteriors provide ratios

Class probabilities determine the mixture score

Classifier posteriors yield explicit mixture score coordinates under the stated model and calibration assumptions.

Derived from the literature

CLASSIFIER-MIXTURE-SCORE-FORMULA

Calibrated posterior odds recover density ratios

Calibrated posterior odds recover component density ratios after correcting for class priors.

Established

CLASSIFIER-RATIO-ORACLE

O4True retained FI under an estimated score

Proxy-built labels retain the true score's information, not the proxy's

The true score, conditioned on proxy-derived labels, determines achieved Fisher information; proxy moments measure a different quantity.

Derived from the literature

PROXY-TRUE-RETAINED-FI

O5Representation loss and quantization loss separate

Information loss splits into representation and binning parts

Conditional projection decomposes information lost in the representation and in its subsequent hard labels.

Derived from the literature

REPRESENTATION-QUANTIZATION-LOSS

O6Frozen-rule scalar retention: plug-in asymptotics and a consistent variance

A frozen scalar rule's retention has a limiting distribution

A plug-in central limit theorem gives a consistent influence-function variance for independent evaluation under the stated nondegeneracy assumptions.

Derived from the literature · audited

RETENTION-PLUGIN-CLT-FROZEN-SCALAR

Coverage of scalar retention intervals was measured

A classifier-based experiment tests the frozen-rule interval’s coverage; the evidence is numerical.

Measured · audited

RETENTION-PLUGIN-COVERAGE-DOOR3

O7Frozen-rule vector retention: plug-in asymptotics of the geometric-mean retention

A frozen vector rule's retention has a limiting distribution

A plug-in central limit theorem describes geometric-mean determinant retention under its regularity and nonsingularity conditions.

Derived from the literature · audited

RETENTION-PLUGIN-CLT-FROZEN-VECTOR

Coverage of vector retention intervals was measured

Closed-form laws test the vector interval’s coverage and its limitations numerically.

Measured · audited

RETENTION-PLUGIN-COVERAGE-VECTOR

Singularity changes the plug-in retention rate

At singular retained information, geometric-mean retention has an upward plug-in bias with a rank-dependent rate.

Proved here, no precedent found · audited

RETENTION-PLUGIN-SINGULAR-ENDPOINT-RATE

O8Score-error budget for a frozen rule: what an estimated score costs, to first order

Calibration error bounds the score error from below, the excess Brier score from above

The mixture score is Lipschitz in the classifier posterior, so the score error that drives the reporting budget is controlled by the excess Brier score, only half of which is observable without truth; AUC controls nothing.

Derived from the literature · audited

CLASSIFIER-CALIBRATION-SCORE-LIPSCHITZ

The score-error budget was measured on the classifier example

On the logistic-regression rungs and a two-dimensional mixture the budget brackets every measured gap and shows which of its two terms dominates.

Measured · audited

SCORE-ERROR-DOOR3-MEASURED

A proxy score misreports retention by a computable two-term budget

The gap between the retention reported for an estimated score and the true retention of the same labels is an alignment term scaling with the rule's own information loss plus a spurious-information term set by the weakest retained direction, with explicit constants.

Proved here, no precedent found · audited

SCORE-ERROR-RETENTION-BUDGET

Applying a rule to the proxy instead of the truth costs a margin-controlled amount

The mass of events the rule labels differently on the estimated score is bounded through a margin condition at the cell boundaries, and the retained information changes by at most an explicit matrix of the mislabelled mass.

Proved here · audited

SCORE-ERROR-RULE-TRANSFER

Information-efficiency outputssource

I1D-efficiency

D-efficiency measures determinant retention

D-efficiency compares binned and unbinned information through a dimension-normalized determinant ratio.

Derived from the literature

INFO-D-EFFICIENCY

I2D_s-efficiency

Profiled D-efficiency measures profiled determinant retention

The profiled efficiency compares information for parameters of interest after nuisance adjustment.

Derived from the literature

INFO-DS-EFFICIENCY

I3Directional diagnostics

Retention can be inspected direction by direction

Directional diagnostics reveal information losses hidden by a single aggregate efficiency.

Derived from the literature

INFO-DIRECTIONAL-DIAGNOSTICS

Open questions

Estimated scores and classifiers

How does calibration error become information loss?

A general propagation bound from classifier calibration error to retained Fisher information remains open.

Open

OPEN-CLASSIFIER-CALIBRATION-FI

Can representation and binning loss be estimated separately?

Determine how to estimate both contributions rather than only their combined effect.

Open

OPEN-REPRESENTATION-LOSS-ESTIMATION

How robust is the rule to estimated scores?

Relate score estimation error to changes in cell moments, boundaries, and retained information.

Open

OPEN-SCORE-PERTURBATION

How many bins a target needs

What does determinant efficiency guarantee by direction?

Seek directional retention guarantees from a determinant-based aggregate.

Open

OPEN-D-DIRECTIONAL-BOUND

How many bins meet an information target?

Sharp efficiency bounds as a function of bin count remain unresolved.

Open

OPEN-D-EFFICIENCY-VS-K

What happens with many D-optimal bins?

Develop high-rate asymptotics for determinant-preserving quantization.

Open

OPEN-D-HIGH-RATE

Away from the reference point

Can one binning serve several reference models?

Study robust quantization across multiple parameter reference points.

Open

OPEN-MULTIREFERENCE-ROBUST

What is lost away from the reference point?

Bound the degradation when the operating parameter differs from the design reference.

Open

OPEN-PARAMETER-MISMATCH

How uncertain is a finite-sample retention estimate?

Beyond frozen-rule asymptotics, finite-sample guarantees for retention estimates remain open.

Open

OPEN-RETENTION-UNCERTAINTY

Template fits made explicit

Can count and shape information be combined?

Make the quantization problem explicit for extended template fits using both event count and event shape.

Open

OPEN-EXTENDED-COUNT-SHAPE

How do template fits scale with nuisance parameters?

Understand systematic template morphing and the cost of larger nuisance models.

Open

OPEN-HEP-NUISANCE-SCALING

Which mixture score coordinates should be used?

A canonical parameterization for linear-mixture score construction remains to be developed.

Open

OPEN-MIXTURE-PARAMETERIZATION

Completing the D core

Do exchange-stable solutions converge?

Study population limits for local exchange-stable D solutions, rather than only global optima.

Open

OPEN-D-EXCHANGE-CONSISTENCY

Does finite optimality survive the population limit?

Establish unrestricted global D consistency beyond compact rule families with imposed margins.

Open

OPEN-D-UNRESTRICTED-CONSISTENCY

Do unrestricted profiled D and E optima have population limits?

Determine when finite optimal assignments converge without the restricted theory’s assumptions.

Open

OPEN-DS-E-UNRESTRICTED-CONSISTENCY

Does the scalar margins dichotomy extend beyond centred laws?

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

Open

OPEN-DS-MARGINS-NONCENTERED

Foundations

How far can stable A geometry deviate?

Seek a quantitative necessity bound despite the failure of an exact exchange-to-geometry theorem.

Open

OPEN-A-QUANTITATIVE-BOUND

Can randomization help on atomic laws?

Determine the possible advantage of randomized labels when atomless purification is unavailable.

Open

OPEN-ATOMIC-RANDOMIZATION-GAP

Which criteria force geometry?

Characterize matrix criteria for which finite exchange stability implies first-order geometric consistency.

Open

OPEN-CRITERION-CHARACTERIZATION

Can exact D search improve beyond XP?

Resolve the parameterized complexity of global finite quantization in dimension and bin count.

Open

OPEN-D-PARAMETERIZED-COMPLEXITY

Can constrained profiled values be attained?

Wasted-cell tracking and constrained-value attainment remain unresolved beyond the established obstructions.

Open

OPEN-DS-STABLE-BASINS

What is the exact bit complexity of tilt optimization?

Resolve remaining exact-complexity questions as nuisance dimension and cell count vary.

Open · parked

OPEN-DS-TILT-DUAL-EXACT-COMPLEXITY

Does a spectral gap control E geometry?

Seek a quantitative finite necessity bound under eigenvalue separation.

Open

OPEN-E-SPECTRAL-GAP-BOUND

Does soft optimization approach the hard problem?

Understand the zero-temperature limit and what survives hardening.

Open

OPEN-SOFT-HARD-ZEROTEMP

Are there tighter bounds for exact search?

Improve branch-and-bound ceilings beyond singleton refinement.

Open

OPEN-STRONGER-BB

Can larger local moves improve the search?

Study stronger neighbourhoods beyond single-observation relocation.

Open

OPEN-STRONGER-LOCAL-NEIGHBORHOODS

Not yet written into a chapter

Splitting duplicates breaks strict geometry

Coincident score rows assigned separately can defeat strict Voronoi closure, explaining the merged-atom hypothesis.

Counterexample

D-UNMERGED-DUPLICATES-FAIL

Verification records

An independent re-derivation of a result already in the record. A verification record carries no result of its own; it says that someone went through the argument again and what they found.