When Is the Sharp Covariance Envelope Tight? Feature-Only Geometry for Volume-Sampled Least Squares

Motivation

• Frozen-feature refitting can vary substantially across randomized support subsets, so the paper targets the most variable coefficient direction rather than aggregate same-pool loss.The analysis isolates least-squares readouts and does not address classification loss or representation quality.Source: §1 · p. 1

Problem formulation

• For a fixed regression pool, ordinary indexed fixed-size volume sampling and selected unweighted least squares yield centered, full-Gram-whitened coefficient covariance whose worst residual margin depends only on features.The response is fixed while subset choice supplies randomness, and the covariance preserves directional information hidden by a trace.Source: §1 · p. 1

Global benchmark

• Theorem 1 establishes a globally sharp Loewner coefficient-covariance envelope for every fixed pool, response, and legal budget, extending beyond the previously documented rank-size endpoint s = d [11, 12].Earlier work established all-size identities and endpoint formulas, while expected-loss bounds for s > d remained open [11, 12].Source: §1 · p. 2

Fixed-design phase

• Theorem 3 shows that the feature-only margin νA exactly separates uniform strictness for every compatible residual from existential spectral tightness on the realized pool.A zero-margin pool has a compatible positive-loss residual attaining the normalized envelope, whereas a positive-margin pool has a uniform positive gap for every compatible residual.Source: §1 · p. 2

Mechanism and tightness

• Residual augmentation supplies the response-aware contraction and boundary-valid one-sided resolvent, while zero margin forces support saturation and a common tight residual across strict-interior subset sizes.The quantitative slack is only lower-bounded by a positive multiple of νA, not identified exactly by it.Source: §1 · p. 2

Critical interpretation and consequences

• Theorem 4 identifies the zero-margin boundary with the classical projective repeated-pair boundary in the critical equal-leverage class, and Corollary 5 yields conservative cardinality decisions from sound lower certificates.Frozen public feature pools produce nonvacuous certificate decisions and measure their fixed-pool cost.Source: §1 · p. 3

2 Setting, metric, and scope

• The setting uses X ∈ R^{m×d} with full column rank and an arbitrary deterministic fixed response y ∈ R^m, conditioning every expectation on (X, y).Source: §2 · p. 3, §2.1 · p. 4
• Ordinary unrescaled fixed-size volume sampling selects indexed row subsets of size d ≤ s ≤ m, treating repeated rows at different indices as distinct observations.Only positive-volume subsets support the estimator; zero-volume sets receive no pseudoinverse, ridge term, rescaling, weighting, replacement rule, or selected fit.Source: §2 · pp. 3–4
• The metric is the centered second moment of selected-fit coefficients, expressed invariantly and normalized as q_s = λmax(M_s)/(αL*) when αL* > 0.This is a conditional fixed-pool quantity, not a raw Euclidean eigenvalue; no directional quotient is formed at a zero denominator.Source: §2 · p. 4

2.1 Endpoint and support conventions

• Endpoint conventions remove undefined ratios: m = d gives zero loss and covariance, s = m gives M_s = 0 and α = 0, L* = 0 gives M_s = 0, and s = d < m uses interpolation without the later rank-(d + 1) augmentation.At s = d < m, supported square systems interpolate; the full-pool loss identity is used separately.Source: §2.1 · p. 4

• The proof combines indexed Cauchy–Binet normalization, rank-size basis moments, uniform padding, selected-residual first moments, and total covariance; its inverse-Gram volume-sampling ingredients are standard [22].The resulting trace bound follows through the full-pool Pythagoras identity and is conditional on the same pool.Source: §3 · pp. 4–5
• Theorem 1 bounds centered whitened coefficient covariance in Loewner order for every full-column-rank pool, fixed response, and legal budget d ≤ s ≤ m.The bound is design-independent and therefore controls every coefficient contrast simultaneously before any trace is taken.Source: §3 · pp. 4–5
• On the regular positive-loss interior, α remains the perturbative supremum, whereas arbitrary full-rank boundary designs require a determinant-weighted directional limit rather than the interior slack decomposition.Endpoint and zero-volume support conventions apply before forming ratios or residual directions.Source: §3 · pp. 4–5
• The envelope coefficient α = (m−s)/(m−d) is globally sharp, attained by the full-column-rank core-plus-zero family, while positive-loss row-general-position designs with d < s < m are strictly below it.The attainment claim is coefficient-level, not an equality classification.Source: §3 · pp. 4–5

• The resulting linear consequence provides a response-uniform slack input: a feature-only lower bound on R rules out the universal envelope for every compatible residual.This is the mechanism used to convert the response-aware analysis into a feature-only certificate.Source: §4 · pp. 5–6
• Theorem 2 uses the normalized residual as an analysis-only augmentation, preserving the original ordinary volume law and selected OLS sampling-and-fitting primitive.The construction applies when L∗ > 0 and d < s < m under row general position.Source: §4 · p. 5
• An exact loss-weighted change of measure from the ordinary law to the residual-dependent auxiliary law PB yields the response-aware resolvent representation.The relation begins with PX(S)LS = βL∗PB(S), after which classical operator Jensen supplies the interior resolvent bound.Source: §4 · pp. 5–6
• At arbitrary full-column-rank boundaries, only a congruent one-sided resolvent inequality remains because the exact inverse-moment covariance identity does not survive rank-changing support.The auxiliary support may be smaller than the ordinary support, and no inverse is taken on zero augmented volume.Source: §4 · p. 6
• Under the same augmented law, positive-contraction structure gives an exact resolvent identity and a computable second-centered-Gram-moment tightening of the Jensen ceiling.Proposition 16 states the resulting interior Loewner hierarchy and its one-sided full-column-rank extension.Source: §4 · p. 6

• The margin is computed from feature geometry alone, while the response-uniform target maximizes spectral covariance over compatible normalized residual directions.Source: §5 · pp. 6–7
• The feature-only margin ν_A exactly classifies attainability: ν_A = 0 admits a compatible residual attaining the normalized spectral envelope, whereas ν_A > 0 keeps every compatible residual strictly below it.The classification applies for d < s < m under the stated full-rank, no-coloop, ordinary volume-sampling setup.Source: §5 · p. 7
• When ν_A = 0, one residual direction is spectrally tight simultaneously at every strict-interior budget s′ ∈ {d + 1, …, m − 1}.Source: §5 · p. 7
• The lower-bound expression quantifies guaranteed spectral slack, but the theorem establishes phase classification rather than an exact formula for positive slack magnitude.Source: §5 · p. 7
• The proof combines a residual-augmented response-aware covariance bound for positive uniform slack with support saturation and determinant calculations for zero-margin equality.Source: §5 · pp. 7–8

6 Critical equal-leverage geometry

• The critical residual weights are coupled fractional weights generated by an admissible residual direction, not freely chosen frame weights or binary erasures [5].They arise from a fixed Naimark complement after removing one unit direction, linking the construction to Parseval frame and frame-erasure geometry.Source: §6 · p. 8, Figure 2 · p. 9
• At m = 2d with equal leverage, ν_A = 0 exactly at the projectively repeated-pair boundary with an orthogonal remainder.Theorem 4 characterizes the equivalent critical geometry, while the mechanism uses Parsevalness and unit-norm constraints to force the repeated pair.Source: §6 · pp. 8–9
• Coherence bounds ν_A from below and above but generally does not determine it exactly; the continuous margin measures distance from the repeated-pair boundary.The lower bound follows from capped-simplex residual squares and directional frame-energy control, while the upper bound uses a coherence-maximizing pair.Source: §6 · pp. 8–9
• When ν_A = 0, one compatible residual fixed before sampling attains the universal spectral envelope at every strict-interior budget s ∈ {d + 1, …, 2d − 1}.The witness is independent of the budget and has an exact rank-one covariance described in Appendix F.Source: §6 · p. 9

Sound computable pre-response certificates

• A zero certificate, failed sufficient test, or abstention is inconclusive and does not establish νA = 0, envelope attainment, or the absence of another sound certificate.The certificate is intentionally one-sided rather than a complete test for margin zero.Source: §7 · p. 9
• Computable certificates cX and tρ(A) satisfy 0 ≤ cX, tρ(A) ≤ νA, providing sound lower bounds on the exact feature margin.The general spectral certificate applies for full-column-rank X with m > d; the critical-coherence certificate applies under Theorem 4’s assumptions.Source: §7 · p. 9
• Any verified feature-only certificate t(A) > 0 certifies strictness uniformly over every compatible residual direction on the stated no-coloop, positive-loss, strict-interior domain.Computing the exact residual-sphere minimum is therefore unnecessary for certifying the positive branch.Source: §7 · p. 9

Same-primitive action

• For the predeclared (m, d, ξ) = (512, 64, 1/3) profile, a verified pass authorizes s = 338; otherwise, the same primitive falls back to s = 363.Both branches retain the same one-sided covariance ceiling, with the predicate, guards, integer arithmetic, and nonoptimality qualification specified in Appendix G.Source: §7 · p. 10

Pre-response certificate decisions

• Frozen features yielded action/fallback counts of 20/4, 4/20, and 5/19 across the three released blocks, showing both authorized reductions and abstentions.The counts cover 72 fixed frozen-encoding cells under the sound sufficient rule.Source: §8 · p. 10

Measured cost

• In all nine released action cells, the post-action lower-budget excess-risk difference was positive, quantifying a fixed-pool cost for the authorized reduction.These descriptive results do not estimate TRUspec, test Theorem 3, or establish population performance or selector dominance.Source: §8 · p. 10

Scope and limitations

• Fallbacks are inconclusive rather than evidence of the zero phase, and the fresh Fashion–MNIST rows are disjoint from the initial row block but share its frozen encoder inventory.Exact costs, covariance and accuracy summaries, matched-budget selector context, draws, and selection notes appear in Appendix B.Source: §8 · p. 10

Nearest theorem-level comparisons

• Prior results cover all-size determinant laws, selected-OLS unbiasedness, inverse moments, and rank-size arbitrary-fixed-response loss and prediction-covariance formulas.Dereziński and Warmuth’s Theorems 5–6 average matched selected-Gram inverse or pseudoinverse objects, while the residual-augmented calculation introduces a response-induced law–target mismatch confined to the supplementary contraction correction; operational sampling remains under P_X.Source: §9 · pp. 10, 12
• Other audited neighbors study different primitives or statistics, including volume-rescaled and random-design regression, generic-sketch covariance, with-replacement debiased fits, and active or dependent-leverage designs [24].The section makes no worldwide absence claim about these neighboring approaches.Source: §9 · p. 10

Critical frame and erasure geometry

• Parseval-frame, Naimark-complement, projection-diagonal, and weighted-frame results provide geometric ancestry [1] [5], but the overlap is partial because the residual-coupled functional and bounds differ.The paper translates the classical repeated-pair binary-erasure boundary of Bodmann and Paulsen [5] into a continuous constrained margin, using Naimark-complement and equal-norm projection machinery from Casazza et al..Source: §9 · p. 10

Pre-response selection and certificates

• Related fixed-pool subset-design work changes the fitted objective, sampling law, estimator, or statistic, whereas the present sufficient test changes only cardinality when it passes [24].Chen and Price use weighted sparsification and weighted empirical risk minimization; Gittens and Magdon-Ismail study residual/leverage deletion for scalar full-pool loss; Shimizu et al. use dependent-leverage sampling with an inverse-inclusion rescaled fit [24].Source: §9 · p. 10

Nearest theorem-level comparisons

• Table 1 organizes nearest theorem-level comparisons by sampling law, estimator, budget-specific object, response model, covariance target, and equality quantifier.The comparison is explicitly not a systematic literature census, and its relations concern the present fixed-pool coefficient-covariance envelope and strict/tight phase.Source: §9 · p. 10

10 Limitations and conclusion

• The analysis concerns conditional centered, full-Gram-whitened coefficient covariance under ordinary fixed-size volume sampling and selected unweighted OLS, not population generalization or selector dominance.Feature-only screens are sufficient but inconclusive on failure or abstention; exact ν_A complexity is not established, and the residual-augmented identity additionally requires row general position.Source: §10 · p. 12
• Under positive full-fit loss, strict-interior budgets, and no coloops, ν_A > 0 gives uniform strictness, whereas ν_A = 0 yields a compatible residual attaining the envelope at every strict-interior budget.Critical geometry clarifies the boundary, and sound positive lower certificates support conservative cardinality decisions without changing the sampler or fit.Source: §10 · p. 12

Use of generative AI

• Generative-AI tools assisted drafting, source organization, compilation diagnostics, and verification workflows, but their outputs were not evidence for correctness, novelty, or empirical claims.The author retains responsibility for the paper’s mathematical statements, proofs, citations, calculations, experiments, and final artifacts; no AI system is listed as an author.Source: Use of generative AI · p. 12

Proof support and supplementary material

• The appendix derives the ordinary law and covariance envelope, residual augmentation, pre-response and critical geometry, and response-uniform feature certificate in dependency order.It also records unordered indexed subsets, excludes zero-volume sets from sampling and fitting, and retains balanced and finite diagnostics as supplementary context.Source: Proof support and supplementary material · p. 16

Supplementary fixed-profile instantiation

• For the profile (m, d, ξ) = (512, 64, 1/3), n = 448, strict guards and the fixed-threshold predicate certify ν_A ≥ 13/60.The certificate uses the derivation in Appendix G and requires 0 < ℓ_i < 1 for every i together with Φ13(X) ⪰ 0.Source: Supplementary fixed-profile… · p. 16
• The sufficient route yields s13 = 338 for every fixed positive-loss response when its strict guards and predicate pass.This is a certified count, not a claim of minimality or optimality.Source: Supplementary fixed-profile… · p. 16
• If the sufficient route abstains, the fallback q_H = 149 yields s_H = 363 with the same one-sided tolerance certificate.Both branches retain ordinary indexed fixed-size volume sampling and selected unweighted OLS; the rule claims neither label savings nor predictive utility.Source: Supplementary fixed-profile… · p. 16

Theorem-to-proof map and scope

• The theorem chain establishes universal normalization, unbiasedness, covariance-envelope bounds, attainment, strict-interior slack, residual augmentation, and boundary resolvent inequalities under their stated domains.The exact augmented identity is restricted to its interior domain, while arbitrary full-rank boundaries receive only the one-sided resolvent inequality.Source: Theorem-to-proof map and scope · pp. 16–17

Scope summary

• The robust phase theorem gives an exact zero-versus-positive classification for the feature-only margin on no-coloop whitened designs, while its displayed slack magnitude is only one-sided.The classification applies for m ≥ d + 2, d < s < m, L∗ > 0, and the maximum over all unit compatible residuals; certificate failure or abstention is inconclusive.Source: Theorem-to-proof map and scope · p. 17, Scope summary · p. 18, Table 2 · p. 19
• Critical equal-leverage geometry identifies the boundary regime and supplies an existential sharp covariance witness, including the repeated-projective-pair boundary’s classical binary-erasure ancestry.The critical setting is m = 2d with d ≥ 2 and one fixed-complement residual sphere.Source: Theorem-to-proof map and scope · p. 17, Scope summary · p. 18

A Detailed claim and assumption ledger

• The response-uniform feature certificate is sufficient for cardinality decisions, but it is silent for d < m ≤ 2d and positivity is not guaranteed merely by m > 2d.Balanced lower-side results and class-infimum witnesses provide conservative lower certificates rather than universal positivity claims.Source: Theorem-to-proof map and scope · p. 17, Scope summary · p. 18
• The claim ledger distinguishes exact phase equivalence from one-sided slack bounds, records conditional quantifiers and assumptions, and excludes supplementary diagnostics from evidence for the robust phase theorem.The supplementary asymptotic class quantity is retained only in its stated r = o(d) regime, with the witness having h1 = 1.Source: Theorem-to-proof map and scope · p. 17, Scope summary · p. 18, Table 2 · p. 19

B Supplementary evidence ledger

• Supplementary fixed-pool experiments report feature-only actions and descriptive covariance or excess-risk summaries, not population generalization, worst-residual phase quantities, or selector dominance.The fresh-row block uses five action-selected cells and 100 domain-separated draws per arm; its selection is not random-cell or task-level replication.Source: Table 3 · p. 20, Table 4 · p. 21, B Supplementary evidence ledger · p. 21

C The universal fixed-pool covariance envelope

• The core primitive conditions on full-rank fixed X and fixed y, samples indexed subsets without replacement under ordinary unrescaled volume sampling, and fits unweighted least squares only on positive-volume sets.Endpoint conventions make covariance zero when the pool is exhausted, the loss is zero, or the selected system interpolates at s = d.Source: C The universal fixed-pool covariance… · p. 22

C.1 Fixed-cardinality normalizer and theorem statement

• Theorem 8 gives a universal Loewner covariance envelope for every full-column-rank X, fixed y, and budget d ≤ s ≤ m.The normalized directional ratio is defined only when αL∗ > 0; no normalized ratio is formed at s = m or L∗ = 0.Source: §C.1 · p. 23

C.2 Rank-size basis moments and uniform padding

• Rank-size volume sampling admits a basis-plus-uniform-padding representation that yields an exact covariance decomposition in row general position.The selected size-s set can be generated by sampling a volume-weighted d-row basis and then choosing a uniformly padded size-s superset.Source: §C.2 · p. 24

C.3 The selected residual first moment

• The selected residual has an exact first-moment identity for every full-column-rank X, including cases with zero full-pool loss.The derivation treats singular selected subsets as zero determinant-weighted contributions and covers the interpolating endpoint s = d.Source: §C.3 · pp. 24–25

C.4 The Loewner envelope in row general position

• In row general position, the residual identity and inverse-Gram order combine to prove both covariance-envelope inequalities and the associated full-pool loss bound.The argument uses GS ⪯ G and congruence by G1/2 to obtain the matrix inequalities.Source: §C.4 · p. 25

C.5 Arbitrary full-rank boundary designs

• Polynomial continuation preserves the residual mean, while one-sided perturbation extends the covariance envelope and loss bound to every full-column-rank boundary design.The boundary proof retains limiting positive-volume terms and drops only newly supported nonnegative terms, without extending the exact row-general-position covariance decomposition.Source: §C.5 · pp. 25–26

C.6 Exact matrix attainment, strict interior slack, and supremum

• The core-plus-zero family attains the envelope exactly, whereas for d ≥ 2 and d < s < m row-general-position designs have strict slack but approach the coefficient as an operator-norm supremum.Equality holds at s = d for every positive-loss row-general-position design; at s = m no normalized ratio is formed.Source: §C.6 · pp. 27–28

D Residual-augmented representation and response-aware resolvent envelope

• The response-aware analysis is restricted to L∗ > 0 and d < s < m, using ordinary indexed fixed-size volume sampling with supported least-squares fits and separate endpoint branches.Zero-volume sets receive no pseudoinverse or assigned fit; s = d, s = m, L∗ = 0, and m = d are handled separately.Source: D Residual-augmented representation… · pp. 28–29

D.1 Whitening and the exact change of measure

• After whitening, the augmented rank-(d + 1) volume law defines a response-dependent analysis measure, distinct from the ordinary feature-only sampling law used by the estimator.The whitened matrices satisfy A⊤A = I_d, A⊤z = 0, ∥z∥2 = 1, and B⊤B = I_{d+1}.Source: §D.1 · p. 29
• The augmented change of measure matches ordinary selected-fit probabilities on ordinary-supported sets and extends to a finite measure on all sets, but its zero branch does not define an inverse fit.Augmented support can be strictly smaller than ordinary support, and all inverses are restricted to sets with positive augmented volume.Source: §D.1 · p. 30

D.2 Exact row-general-position covariance transform

• For row-general-position designs with positive loss and strict-interior budgets, basis coupling and total covariance give an exact selected-OLS covariance transform, with the restriction forming part of the theorem statement.The coupling uses a size-d volume-sampled basis contained in the size-s sample; related basis-moment and basis-padding representations appear in.Source: §D.2 · p. 30

D.3 Augmented exclusion marginal and the augmented Gram first moment

• Fixed-cardinality Cauchy–Binet applied to the augmented design gives an augmented exclusion marginal with factor γ(1−h_i), and the augmented Gram first moment is strictly positive.The strict positivity follows from positive-definite augmented Grams on augmented-supported samples and nonempty augmented-law support.Source: §D.3 · p. 31

D.4 Classical operator Jensen and two resolvent bounds

• Classical operator Jensen inversion, combined with the volume-specific change of measure and augmented exclusion moment, gives two resolvent bounds, including the linear response-uniform slack input.The contraction inequality also yields a positive-semidefinite resolvent remainder, but no high-dimensional sharpness claim is made for the resulting bound.Source: §D.4 · p. 31

D.5 Second-centered-Gram refinement of the response-aware resolvent

• The contraction-specific refinement yields an ordered inverse-remainder hierarchy whose correction is computable from singleton and pair exclusions, while arbitrary designs retain only a one-sided comparison.The coefficient of the second-moment term is exactly one, not a Taylor approximation; outside row-general position, the result is neither an exact covariance identity nor an exact covariance-gap decomposition.Source: §D.5 · pp. 31–32

D.6 One-sided extension to arbitrary full-column-rank boundary designs

• For arbitrary full-column-rank boundary designs, the response-aware upper inequalities survive by perturbation and directional limits, but the exact inverse-moment covariance identity can fail.The boundary argument drops newly supported nonnegative terms; Proposition 17 exhibits strict resolvent slack despite failure of the exact identity.Source: §D.6 · pp. 33–35

E Proof of the robust pre-response phase boundary

• Because βsγs > 0, νA > 0 implies strict normalized spectral slack, spec(A, s) < αs, while a feature-geometry witness attains the envelope simultaneously for all s′ ∈ {d + 1, . . . , m − 1}.Source: §E.2 · p. 36, §E.3 · p. 37

E.4 Envelope equality forces zero margin

• Envelope equality conversely forces ν_A = 0, completing the phase equivalences under positive loss, strict-interior budgets, and the stated no-coloop scope.The converse follows from a maximizing residual and top eigenvector forcing the minimum eigenvalue of the residual matrix to vanish; no-coloop is needed for the forward support argument, not this converse.Source: §E.4 · p. 37, §E.5 · p. 38

F Critical geometry: proofs and source boundary

• Critical geometry identifies ν_A = 0 through exactly two saturated coordinates with signed duplicate rows, and bounds the margin using the coherence sandwich.The fixed-complement residual representation gives positive semidefiniteness, while the zero-locus equivalence and coherence calculation characterize and bound the boundary.Source: §F.1 · p. 38, §F.2 · p. 39, §F.3 · p. 40

F.4 Ordinary-volume sharp witness

• A duplicated-pair residual is an ordinary-volume sharp witness for every strict-interior budget in the critical regime, but its rank-one covariance does not saturate the trace-derived scalar-loss envelope.The witness uses direct support enumeration without extending an interior inverse-moment identity to the boundary; the related scalar-loss result of Derezi´nski et al. [13] is distinguished from this fixed-residual covariance equality.Source: §F.4 · pp. 40–41

G Sound pre-response certificates: proofs and boundaries

• The certificate section separates the exact margin ν_A from sufficient lower bounds, including a general spectral certificate computable by compressing the residual-space geometry.These certificates preserve the ordinary indexed fixed-size volume law and selected unweighted OLS rather than changing the sampling rule or estimator.Source: G Sound pre-response certificates:… · p. 41, §G.1 · p. 41

G.1 The general spectral certificate

• Feature-coordinate invariance lets the general spectral certificate avoid forming a dense m × m projector while preserving its geometric value.Whitened matrices change by a right orthogonal factor, leaving the certificate quantities unchanged up to orthogonal congruence.Source: §G.1 · p. 41

G.2 Capped, top-K, coherence, and fixed-threshold routes

• At m = 2d with equal leverage, tρ = (1 −ρA)/4 ≤νA gives a sharper coherence certificate, while completed fixed-threshold screening certifies 13/60 ≤νA.Zero rows and leverage-one rows fail route eligibility rather than proving νA = 0.Source: §G.2 · p. 42

G.3 Propagation to robust slack

• A proved lower certificate t(A) yields a sufficient robust-slack bound through the one-sided residual resolvent, without providing the exact phase theorem’s converse.The argument applies to arbitrary full-rank support boundaries and conditions on one fixed indexed pool and response, with randomness only in the ordinary subset draw.Source: §G.3 · p. 43

G.4 Fixed-profile cardinality arithmetic

• For the frozen profile n = 448, the scalar certificate admits s13 = 338 versus sH = 363, changing only the certified budget and not weights, fits, or the covariance target.The larger count is specific to inequality (191), not a minimal-data or optimal-cardinality claim.Source: §G.4 · pp. 43–44

G.5 Structural calibration and abstention boundaries

• The inexpensive cX route is intentionally conservative: the full-rank Parseval core-plus-zero design A = [Id; 0(m−d)×d] has cX = 0 for arbitrarily large m, and cX is silent at m = 2d.These zeros demonstrate abstention, not impossibility; cX = 0 does not show νA = 0. The positive-loss and strict-interior formulas are not evaluated at L∗ = 0, s = d, s = m, or m = d.Source: §G.5 · p. 44

H Supplementary balanced result

• The supplementary balanced result shows that cX = 0 at m = 2d does not prevent a response-aware gap below the universal covariance ceiling.It studies whitened A with A⊤A = Id, m = 2d, and s = d + r through the relative shortfall Δ(A, y; r).Source: §H.1 · p. 44

H.1 Balanced ordinary geometry and the worst direction

• A single deterministic pair exists for d = 4n and 1 ≤r < d/2 with Δ(Ad, yd; r) < r d −r, yielding a Θ(r/d) class-infimum squeeze only in that restricted regime.The normalized worst-direction ratio approaches the envelope only for r = o(d) along d = 4n; the witness has an augmented coloop, so no augmented-margin claim follows.Source: §H.1 · p. 45

H.2 Balanced projective separation and the worst direction

• For pairwise projective separation, every balanced design and positive-loss response satisfies a uniform gap Δ(A, y; r) ≥ r d −1ψ(σ), while the universal lower clause has a separate deterministic witness.The response-aware inequality remains valid beyond row general position by a one-sided boundary argument, and its centered second moment is the covariance.Source: §H.2 · p. 48
• For d = 4n and 1 ≤ r < d/2, balanced projective separation gives a uniform lower gap and a strictly larger worst-direction gap, both proportional to r/d.The two-sided class quantity lies between (3/38)(r/d) and 2r/d; an explicit full-spark witness supplies the strict upper inequality.Source: §H.2 · p. 52
• The explicit witness approaches the universal normalized covariance envelope when r = o(d), but this near-attainment claim excludes positive limiting budget fractions and trace statements.The witness has an augmented coloop, so the construction does not establish a common d-minor or augmented-leverage margin.Source: §H.2 · p. 52

I Supplementary finite diagnostics

• The supplementary finite diagnostics are descriptive records that do not test the robust phase theorem, extend its domain, or provide theorem evidence.Their computations concern fixed configurations and do not establish prevalence, typicality, population effects, or response-uniform certificate behavior.Source: I Supplementary finite diagnostics · p. 52, §I.3 · p. 54, §I.3.1 · p. 56

I.1 Controlled residual localization

• Across all 11 controlled residual-localization trajectories, τres increased at every displayed grid step, with exact endpoints r/(3d + r) and r/(d + r).Dimension four used exact enumeration; dimensions 16 and 64 used 16,384 seeded draws per cell, and sampled diagnostic endpoints remained below bs at the reported resolution.Source: §I.1 · p. 53

I.2 Public fixed-pool diagnostic

• For the three complete public pools, the response-aware ceiling contracts the universal ceiling, but the Monte-Carlo plug-in occupies only 11.3%–15.3% of it.The corresponding ceiling-to-plug-in slack is 6.52–8.82-fold; these ratios are descriptive and are not exact qs values, confidence intervals, or covariance reductions.Source: §I.2 · pp. 53–54, Figure 4 · p. 54

I.3.1 Controlled residual-localization record

• The controlled residual-localization record fixes design, sampling law, estimator, and budget while varying only the residual, using exact d = 4 cells and seeded Monte Carlo otherwise.The 55-cell record reports monotonicity only on the displayed grid; the smallest sampled upper-endpoint gap below bs was 0.00555.Source: §I.3.1 · p. 55, Table E2 · p. 56

I.3.2 Companion fixed-design response-direction diagnostic

• In the companion fixed-design diagnostic, no listed budget qualified because the range across four generic residual directions was never at least 0.05.Four targeted stress directions were excluded by definition, and the result is a configuration-level diagnostic failure rather than theorem falsification.Source: §I.3.2 · p. 56, Table E3 · p. 56

J Fixed-pool diagnostic record

• The fixed-pool record covers Airfoil, Concrete, and Energy pools with five budgets each and reports poolwise median contractions τs of 0.98138, 0.95482, and 0.94033.Their bqs/bs ranges are 0.1263–0.1363, 0.1133–0.1530, and 0.1316–0.1534, respectively; all medians exceed the uncalibrated 0.10 display cutoff.Source: J Fixed-pool diagnostic record · p. 58