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When Can We Trust the Matching Principle? Robust Deployment Geometry Under Finite-Sample and Model Uncertainty
Vishal Rajput
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Abstract:Match only geometry you can identify; otherwise spread the penalty. We quantify that decision by the trust ratio tau = epsilon / gamma (estimation uncertainty over spectral separation). Under the linear-quadratic Matching response, oracle-relative drift between estimated and oracle projector matching scales as tau^2 for probes in the chosen top-r deployment subspace -- O(tau^2) in the Davis-Kahan separation region tau < 1/2, with practical usefulness depending on constants. Confidence-Calibrated Matching (CCM) turns tau into a policy -- directional when tau is small, progressively isotropic when not -- with thresholds from calibration, not from the theorem (match sits in the separation region; soft is mostly heuristic). Experiments show both regimes, including UCI HAR embeddings where always-match is worse than abstain on every cell.
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- Source: https://arxiv.org/abs/2610.02894v1
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Vishal Rajput When Can We Trust the Matching Principle? Robust Deployment Geometry Under Finite-Sample and Model Uncertainty Vishal Rajput vishal.stark42@gmail.com Affiliation: Independent researcher Affiliation: Leuven, Belgium Abstract Match only geometry you can identify; otherwise spread the penalty. We quantify that decision by the trust ratio τ=ε/γτ= /γ (estimation uncertainty over spectral separation). Under the linear–quadratic Matching response, oracle-relative drift between estimated and oracle projector matching scales as τ2τ^2 for probes in the chosen top-r deployment subspace—O(τ2)O(τ^2) in the Davis–Kahan separation region τ<1/2τ<1/2, with practical usefulness depending on constants. Confidence-Calibrated Matching (CCM) turns τ into a policy—directional when τ is small, progressively isotropic when not—with thresholds from calibration, not from the theorem (match sits in the separation region; soft is mostly heuristic). Experiments show both regimes, including UCI HAR embeddings where always-match is worse than abstain on every cell. †heading: 0 2026 1-1 8/26 †shortheadings: When Can We Trust the Matching Principle? / Rajput†firstpage: 1 keywords matching principle, trust ratio, deployment geometry, Davis–Kahan, confidence-calibrated matching 1 Introduction Suppose Matching would be optimal if the deployment direction were known. We estimate it, regularise aggressively—and do worse than isotropic. The problem is not whether directional matching is sharp; it is whether the direction is identifiable. Match only geometry you can identify; otherwise spread the penalty. That is the whole decision. Finite samples scramble subspaces when the eigengap is small; a wrong directional penalty can hurt more than spreading capacity evenly. Picture the flagship failure: on UCI HAR embeddings, always-matching the estimated direction is worse than abstaining on 6/66/6 cells—not because Matching is false, but because the estimate was not trustworthy. The trust ratio. Write ε for a high-probability operator-norm bound on ‖Σ^−Σ‖op\| - \|_op and γ=λr−λr+1γ= _r- _r+1 for the eigengap of the top deployment subspace. The scalar that organises the decision is τ:=εγ=estimation uncertaintyspectral separation.τ\;:=\; γ\;=\; estimation uncertaintyspectral separation. Small τ: the estimated direction is identifiable; matching it is close to oracle Matching. Large τ: do not trust the direction—even-spread remains the default. Davis–Kahan enters later as a proof tool that converts τ into a subspace-error bound; it is not the idea. What the math says vs. what we do. Under the LQ Matching response, for probes in the chosen top-r deployment subspace the oracle-relative absolute drift gap scales as |D^−D⋆|≲C⋅τ2| D-D | C·τ^2 (Theorem 3)—O(τ2)O(τ^2) in the Davis–Kahan separation region τ<1/2τ<1/2, with practical usefulness depending on constants. Confidence-Calibrated Matching (CCM) is a policy that uses τ: directional match, soft blend, or even-spread abstention. Its thresholds are frozen by a calibration protocol (§7)—not a corollary of the theorem—but the match threshold sits inside that separation region; most of the soft band does not. Synthetic grids and identifiable positives (Fashion rotations) show the complementary regime where trust is warranted. Scope. Oracle-relative regret of estimated projector Matching under the LQ response (§2)—not a new covariance estimator, not a new Davis–Kahan theorem, and not global optimality among regularisers. CCM thresholds are policy, not theorem. Deep / real blocks are experiments under a specified protocol, not a nonlinear Matching extension. Optional companions (Rajput (2026b), Rajput (2026a)) motivate why geometry matters and what coverage means; this paper restates the LQ objects it needs and is readable alone. 2 LQ objects (self-contained preliminaries) A cold reader needs only the objects below. Notation follows Rajput (2026a); every formula used later is stated here. Displacements and second moments. Let Δ be a label-preserving input displacement with Σ=[ΔΔ⊤]⪰0 =E[ ] 0. Large eigenvalues of Σ are directions where deployment moves a lot. Linear–quadratic response. For a training penalty Σ′⪰0 0, probe v, and weight λ>0λ>0, wλ(Σ′,v)=(I+2λΣ′)−1v,DΣ(w)=w⊤Σw.w_λ( ;v)=(I+2λ )^-1v, D_ (w)=w w. As λ→∞λ→∞, drift DΣ(wλ)D_ (w_λ) vanishes for every v∈range(Σ)v ( ) if and only if range(Σ′)⊇range(Σ)range( ) ( ) (cover-or-floor). Miss any deployment direction and no larger λ removes the residual floor. Even-spread vs. match. Even-spread takes Σ′∝I I: no axis estimate; in this LQ response it strictly improves on task-only training when axes are unknown. Matched training takes Σ′ aligned with Σ (or its top subspace): sharper when axes are known. Wrong-axis penalties leave a permanent floor. Operational rule used below: prefer even-spread unless the estimated geometry is trustworthy enough to match (§7). What this paper adds. The geometry to match is treated as given in the cover question above. We study Σ′ as a function of a finite-sample estimate Σ and ask when that choice stays close to oracle matched drift DΣ(wλ(ΠW,v))D_ (w_λ( _W;v) ). The answer is organised by τ=ε/γτ= /γ (§5–6), not by a new cover theorem. 3 Related work We situate by distinction: same tools, different question. Coverage design. Rajput (2026b) and Rajput (2026a) supply the cover question and even-spread default. Distinction: we ask when an estimate of that geometry is trustworthy enough to match. Classical matrix tools. Operator-norm covariance bounds (Vershynin, 2018), spiked models (Johnstone, 2001), Davis–Kahan / Wedin (Davis and Kahan, 1970; Yu, Wang, and Samworth, 2015), and Ledoit–Wolf shrinkage (Ledoit and Wolf, 2004) are ingredients or baselines. Distinction: we package ε/γ /γ as τ and prove that Matching drift gaps scale as τ2τ^2 on the chosen top-r subspace under the LQ response—not a new matrix inequality. Domain shift and DRO. CORAL (Sun and Saenko, 2016), DANN (Ganin et al., 2016), DRO / group-robust training (Ben-Tal et al., 2009; Rahimian and Mehrotra, 2019; Sagawa et al., 2020), and IRM / DG benchmarks (Arjovsky et al., 2019; Gulrajani and Lopez-Paz, 2021) address shift with different objectives and usually without a Matching coverage certificate. Distinction: CCM is a finite-sample gate on Matching, not a competitor to those programmes. Closest concurrent. Landers (2026) regularise along an estimated drift subspace and report tolerance to moderate subspace misspecification. Distinction: their object is pathwise volatility under a frozen predictor; ours is oracle-relative Matching drift regret as a function of τ=ε/γτ= /γ, with CCM as a calibrated match/soft/abstain map. Selective prediction. Selective classification (Geifman and El-Yaniv, 2017; El-Yaniv and Wiener, 2010) abstains on uncertain labels. Distinction: CCM abstains from directional geometry matching, falling back to even-spread rather than refusing a prediction. 4 Deployment geometry estimation From displacements Δ1,…,Δn _1,…, _n form Σ^=n−1∑iΔiΔi⊤ =n^-1 _i _i _i . Under standard sub-Gaussian assumptions, ‖Σ^−Σ‖op≤ε(n,δ)\| - \|_op≤ (n,δ) with probability at least 1−δ1-δ (Vershynin, 2018). We treat ε as a cited ingredient and use the proxy ε≈c‖Σ‖op(dn+log(1/δ)n+d+log(1/δ)n), \;≈\;c\,\| \|_op ( dn+ (1/δ)n+ d+ (1/δ)n ), (1) with δ=0.05δ=0.05. Exp. 1 calibrates c=0.5c=0.5 on the synthetic spike grid (full coverage; ≈1.5×≈ 1.5× slack). On high-dimensional real Δ , ambient-d ε over-abstains; Exp. 6 reports a practical numerator alongside ambient-τ as a limitation (§8.1). 5 The trust ratio Estimation error alone does not decide whether matching is trustworthy. A large eigengap can absorb moderate covariance error; a tiny gap makes small ε fatal. The decision object is the ratio of those two scales. Definition 1 (Trust ratio). Let ε(n,δ) (n,δ) be a high-probability bound on ‖Σ^−Σ‖op\| - \|_op and γ:=λr−λr+1>0γ:= _r- _r+1>0 the eigengap cutting the top-r deployment subspace. The trust ratio is τ:=ε(n,δ)γ.τ\;:=\; (n,δ)γ. (2) Numerator: how uncertain the estimate is. Denominator: how identifiable the subspace is. Operational reading—before any perturbation lemma: τ≪1τ 1 means the direction is identifiable enough to match; τ≫1τ 1 means spread the penalty; τ=O(1)τ=O(1) is the soft band in between. Population vs. plug-in. Theorem 3 and Exp. 2 use the population gap γ. CCM must run from data alone, so it substitutes the plug-in γ^=λ^r−λ^r+1 γ= λ_r- λ_r+1 (§7). When γ γ is unstable the resulting τ is large and CCM abstains—the intended behaviour. 6 What the theorem says Proofs: Appendix A. The intellectual object is τ=ε/γτ= /γ. Large estimation error relative to the eigengap means: do not trust the subspace. Davis–Kahan converts that ratio into a projector gap; the LQ Matching response then converts the projector gap into an oracle-relative drift gap. Neither conversion is the idea—τ is. 6.1 Perturbation lemma Lemma 2 (Subspace error via Davis–Kahan). Assume a spiked spectrum with gap γ=λr−λr+1>0γ= _r- _r+1>0 and ‖Σ^−Σ‖op≤ε(n,δ)\| - \|_op≤ (n,δ) with probability at least 1−δ1-δ. On the event 2ε<γ2 <γ (equivalently τ<1/2τ<1/2), if W W spans the top-r eigenspace of Σ and W that of Σ , then with probability at least 1−δ1-δ, ‖ΠW^−ΠW‖op≤CDKτ,\| _ W- _W\|_op\;≤\;C_DK\,τ, for an absolute constant CDKC_DK as in Yu, Wang, and Samworth (2015) (cf. Davis and Kahan (1970)). When τ≥1/2τ≥ 1/2 the bound is uninformative: the direction is not identifiable at this precision. 6.2 Matching consequence Theorem 3 (Drift gap scales as τ2τ^2 on the deployment subspace). Assume the LQ response of §2, the spiked spectrum of Lemma 2, and ‖Σ^−Σ‖op≤ε\| - \|_op≤ with probability at least 1−δ1-δ. Write α:=2λ/(1+2λ)∈(0,1)α:=2λ/(1+2λ)∈(0,1). On the event τ<1/2τ<1/2, for every unit probe v∈range(ΠW)v ( _W) (the chosen top-r deployment subspace) let w⋆=wλ(ΠW,v)w_ =w_λ( _W;v) and wn=wλ(ΠW^,v)w_n=w_λ( _ W;v). Then, with probability at least 1−δ1-δ, |DΣ(wn)−DΣ(w⋆)|≤(2α+2α2)CDK2‖Σ‖opτ2≲C‖Σ‖opτ2. |D_ (w_n)-D_ (w_ ) |\;≤\; (2α+2α^2 )\,C_DK^2\,\| \|_op\,τ^2\; \;C\,\| \|_op\,τ^2. (3) In shorthand: |D^−D⋆|≲C⋅τ2| D-D | C·τ^2 (absolute gap; Appendix A). Proof sketch. Lemma 2 gives s:=‖ΠW^−ΠW‖op≤CDKτs:=\| _ W- _W\|_op≤ C_DKτ. For orthogonal projectors the compressed gap satisfies ‖ΠW(I−ΠW^)ΠW‖op≤s2\| _W(I- _ W) _W\|_op≤ s^2. Expanding the LQ projector response on v∈range(ΠW)v ( _W) then yields an O(s2)=O(τ2)O(s^2)=O(τ^2) drift gap (Appendix A). Ambient probes outside range(ΠW)range( _W) retain only the weaker O(τ)O(τ) estimate (Remark 10). ∎ What this does—and does not—say. As τ→0τ→ 0, estimated projector matching recovers oracle matched drift at rate τ2τ^2 for probes in range(ΠW)range( _W). Large τ does not justify trusting W W; even-spread (Σ′∝I I) remains the no-axis default (Rajput, 2026a). The bound is oracle-relative regret under correct second-moment specification—not absolute optimality among regularisers, and not a numerical recipe for CCM thresholds. Those thresholds are a calibrated policy (§7). The theorem gives O(τ2)O(τ^2) scaling in the Davis–Kahan separation region τ<1/2τ<1/2; practical usefulness depends on constants (Remark 11). CCM’s match threshold sits inside that separation region; most of the soft band does not. We state the theorem for projectors to isolate subspace error through τ. Full Σ matching adds bulk eigenvalues that can shrink task directions (Appendix 12); experiments therefore treat full Σ as the dangerous naive baseline and the projector as CCM’s match arm. 7 How CCM uses it Theorem 3 motivates a decision rule: trust ΠW _ W when τ is small; distrust it when τ is large. Confidence-Calibrated Matching (CCM) turns that motivation into one penalty with three arms—match, soft blend, even-spread abstention. Policy, not corollary. CCM is an operational map from τ to a penalty. Its thresholds are fixed by the calibration protocol below and then frozen. They are not implied by Theorem 3. The soft band is a calibrated heuristic; most of it sits outside the proved Davis–Kahan separation region τ<1/2τ<1/2, where Theorem 3 gives O(τ2)O(τ^2) scaling (constants decide numeric usefulness). 7.1 Inputs and three penalties From Σ and n, compute ε , the plug-in gap γ^=λ^r−λ^r+1 γ= λ_r- λ_r+1, and τ=ε/γ^τ= / γ (τ=∞τ=∞ if γ^≤0 γ≤ 0). This is the operational counterpart of Definition 1; the theorem uses population γ, the method uses γ γ. Write μ:=tr(Σ^)/dμ:=tr( )/d and let ΠW _ W be the top-r eigenprojector. • Full Σ : uses every eigenvalue of the estimate. Bulk directions can shrink task signal even when the top subspace is accurate; experiments treat this as the dangerous naive baseline. • Top-r projector ΠW _ W: the object controlled by Theorem 3; CCM’s match arm. • Trace-scaled even-spread μIμ I: no axis estimate; CCM’s abstention arm. 7.2 Decision rule Fix 0<τlow<τhigh0< _low< _high and α(τ)=clip(τ−τlowτhigh−τlow, 0, 1).α(τ)=clip\! ( τ- _low _high- _low,\,0,\,1 ). Then ΣCCM′=ΠW^,τ≤τlow(match),μI,τ≥τhigh(even-spread),(1−α(τ))ΠW^+α(τ)μI,otherwise(soft). _CCM= cases _ W,&τ≤ _low (match),\\[3.0pt] μ I,&τ≥ _high (even-spread),\\[3.0pt] (1-α(τ) ) _ W+α(τ)\,μ I,&otherwise (soft). cases (4) 7.3 Threshold-selection protocol Thresholds are not free per-evaluation knobs and are not implied by Theorem 3. They are chosen once by a predetermined procedure, then frozen. Synthetic (Exp. 3–5). On a development gap×n× n grid with the same spike family as Exp. 3 but seed 1111 (disjoint from the reported Exp. 3 seed), we declare a pair (τlow,τhigh)( _low, _high) feasible if (i) match, soft, and even-spread all appear, and (i) on mid-n cells (n≥400n≥ 400) CCM beats both always-full-Σ and always-iso on mean dual cost in at least half the cells. A coarse search over 0.2,0.35,0.5×1.5,2.5,4.0\0.2,0.35,0.5\×\1.5,2.5,4.0\ yields six feasible pairs (code: experiments/calibrate_thresholds.py; artifact: results/calibration/threshold_protocol.json). We freeze (τlow,τhigh)=(0.35,2.5)( _low, _high)=(0.35,2.5) —feasible, with balanced regime mass—for all reported synthetic tables. The dual-cost minimiser on that coarse grid is (0.5,4.0)(0.5,4.0); it does not change the qualitative Exp. 3–5 conclusions (CCM wins mid/high n; caution premium at n=80n=80). Theorem 3 gives O(τ2)O(τ^2) scaling in the separation region τ<1/2τ<1/2 (Remark 11); the frozen match threshold τlow=0.35 _low=0.35 sits inside that region as a calibrated choice, not because CDK≲1C_DK 1 makes every τ<1/2τ<1/2 numerically live. Most of the soft interval (τlow,τhigh]( _low, _high] does not lie in τ<1/2τ<1/2: the blend there is a calibrated heuristic, not a certified interpolation of the theorem. Real data (Exp. 6). Ambient-d ε forces permanent even-spread, so Exp. 6 switches to the practical numerator of §8.1 and therefore needs a separate freeze. On a Fashion development split (half the loaded images; not the reported test split), we require an n-ladder that progresses out of even-spread toward soft/match. Among (0.5,4),(1,8),(1.5,10),(2,12)\(0.5,4),(1,8),(1.5,10),(2,12)\, the pair (1,8)(1,8) is feasible and is frozen for Office-31 and UCI HAR. Ambient-τ is reported alongside to show what the unadjusted proxy would have done. Ledoit–Wolf (Ledoit and Wolf, 2004) remains a covariance baseline, not the claim. Oracle ΠW _W is a synthetic upper reference only. 8 Experiments The flagship claim is failure avoidance: when τ says do not trust the direction, blindly matching hurts. Supporting blocks check that τ tracks subspace error, that CCM wins when the direction is identifiable, and that a practical ε opens those regimes on real features. Theorem 3 is linear–quadratic and oracle-relative; the blocks below are experiments under a specified protocol—not a proof that CCM is optimal for every architecture. Code: experiments/; frozen numbers: results/index.json and results/exp0*/summary.json. 8.1 Flagship: always-match hurts when τ forbids trust UCI HAR embeddings (Exp. 6, abstention value). TCN pooled hidden states (d=96d=96). Deltas Δ=haug−hclean =h_aug-h_clean use label-preserving sensor nuisance (channel dropout / time-warp style) on the same TCN; geometry Σ is estimated from those Δ , then CCM chooses the penalty for a downstream activity probe. (Checkpoint provenance: Matching release (Rajput, 2026a), reproducibility only.) Practical τ correlates with subspace error (corr(τ,sinΘ)=0.87corr(τ, )=0.87), but always-match hurts on 6/66/6 cells—nuisance and task directions are entangled in the embedding. CCM stays soft/even-spread, beats always-match on 6/66/6, and beats B0 on 3/63/6 (Figure 1, bottom). That is the discovery in operational form: match only geometry you can identify; otherwise spread the penalty. A trust gate that only helps when matching helps is incomplete; the HAR cell shows the complementary failure avoided. Fashion-MNIST rotation (identifiable positive control). Same Exp. 6 protocol on pixel deltas Δ=rotateθ(x)−x =rotate_θ(x)-x with |θ|≤15∘|θ|≤ 15 . As n grows, CCM regimes progress even-spread→ → ; corr(τ,sinΘ)=0.92corr(τ, )=0.92; always-match and CCM beat B0 on 6/66/6 n-cells (Figure 1, middle). This is the clean “match when identifiable” complement to HAR. Office-31 Amazon→ (mid-confidence). Frozen ResNet-18 penultimate features (d=512d=512); same-class cross-domain deltas. corr(τ,sinΘ)=0.69corr(τ, )=0.69; always-match beats B0 on 3/43/4 cells; mid-n soft CCM is best among deployable arms (0.6890.689 vs. always-match 0.6740.674). Ambient-d τ is 55–10×10× larger and would never leave even-spread: the practical numerator is doing real work here. Figure 1: Exp. 6 flagship panel: practical vs. ambient trust ratio (left) and accuracy arms (right) for Office-31, Fashion rotation, and UCI HAR. HAR is the failure-avoidance cell—always-match hurts; CCM abstains. Fashion is the identifiable positive control. Not a domain-adaptation SOTA table. Shared real-data protocol. • Geometry pool. Form label-preserving deltas Δ ; estimate Σ from a pool of size n; choose rank r by largest absolute eigengap among 4,8,16\4,8,16\. • Practical ε . Ambient-d concentration keeps τ huge on d∈96,512,784d∈\96,512,784\ and forces permanent even-spread even when always-match helps. Exp. 6 uses ε=min(εeff(n,deff=4r),‖Σ^1−Σ^2‖op), = ( _eff(n,d_eff=4r),\;\| _1- _2\|_op ), the split-sample term being an empirical reproducibility radius. Ambient-τ is reported alongside as a limitation, not used for decisions. • CCM thresholds. Freeze (1,8)(1,8) from the Fashion development split (§7); held fixed for Office and HAR—not retuned per dataset. • Downstream probe. Standardised logistic regression after B0 = no projection; always-match = project orthogonal to W W; CCM = match / soft / identity per (4). What Exp. 6 does not show. It does not certify ambient-d concentration on real features; it does not prove CCM thresholds transfer beyond the Fashion freeze; it does not claim Matching beats every DA baseline on Office. It shows that a trust-gated match/abstain rule avoids the always-match failure mode on HAR and matches when identifiable on Fashion. 8.2 Supporting synthetic programme (Exp. 1–5) Default spike: d=32d=32, rank r=2r=2, λ=5λ=5, concentration constant c=0.5c=0.5 (Exp. 1 tests the proxy (1): coverage 1.01.0; average ε/err0.95≈1.5× /err_0.95≈ 1.5×). CCM thresholds: frozen synthetic pair (0.35,2.5)(0.35,2.5) from §7. Dual cost. Raw mean LQ drift favours isotropic over-shrinkage. Exp. 3 and 5 use J=DWDWB0+(1−retu⟂W),J\;=\; D_WD_W^B0\;+\; (1-ret_u W ), with residual nuisance drift DWD_W on the true subspace W and task retention on u⟂Wu W. Naive baseline: full Σ . CCM match: top-r projector. Even-spread: (trΣ^/d)I(tr\, /d)\,I. Exp. 1–2 (trust ingredients). Exp. 1 confirms that the proxy ε covers ‖Σ^−Σ‖op\| - \|_op. Exp. 2 finds corr(τ,sinΘ)≈0.83corr(τ, )≈ 0.83 and corr(τ,drift-gap)≈0.91corr(τ,drift -gap)≈ 0.91 (Figure 2): τ tracks the quantities in Theorem 3. Packaging validation—secondary to the HAR failure-avoidance cell. Figure 2: Exp. 2 trust curve on the synthetic spike grid: τ versus subspace error (left) and LQ drift gap to oracle projector match (right). Correlations: 0.830.83 and 0.910.91. Supporting evidence that τ organises the theorem quantities—not the flagship claim. Exp. 3 (CCM decision on spikes). On the 3×33× 3 gap×n× n grid (Table 1, Figure 3), regime frequencies are match 33%33\% / soft 59%59\% / even-spread 8%8\%. CCM beats isotropic on dual cost in 9/99/9 cells and beats both pure policies in 6/96/9. At n≥400n≥ 400, CCM dual cost drops to 0.010.01–0.110.11 versus 0.300.30–0.440.44 for full Σ . At n=80n=80, CCM pays a caution premium versus noisy Σ —the abstention tradeoff foreshadowing HAR. Table 1: Exp. 3 dual cost J (lower better; d=32d=32, 3030 reps). Bold: best among naive / iso / CCM. Regime == modal per-rep CCM decision; cell-mean CCM need not equal the pure policy named by the mode (e.g. γ=0.08γ=0.08, n=80n=80: 19/3019/30 even-spread and 11/3011/30 soft, hence CCM 0.653≠0.653≠ iso 0.7790.779). γ n regime naive Σ iso μIμ I CCM LW 0.080.08 8080 even 0.4250.425 0.7790.779 0.6530.653 0.5370.537 0.080.08 400400 soft 0.4410.441 0.7740.774 0.1120.112 0.4710.471 0.080.08 16001600 match 0.4390.439 0.7670.767 0.0130.013 0.4440.444 0.250.25 8080 soft 0.3180.318 0.7590.759 0.3740.374 0.3590.359 0.250.25 400400 soft 0.3350.335 0.7550.755 0.0700.070 0.3410.341 0.250.25 16001600 match 0.3310.331 0.7680.768 0.0090.009 0.3330.333 0.700.70 8080 soft 0.2780.278 0.7500.750 0.3870.387 0.3070.307 0.700.70 400400 soft 0.3020.302 0.7490.749 0.0790.079 0.3070.307 0.700.70 16001600 match 0.3080.308 0.7450.745 0.0090.009 0.3090.309 Figure 3: Exp. 3: regime frequencies (left) and dual cost versus fixed policies (right) on the gap×n× n grid. Exp. 5 (stress). The same dual-cost story holds across d∈16,32,64,128d∈\16,32,64,128\ at fixed n/d∈2.5,12.5,50n/d∈\2.5,12.5,50\ (Table 2): CCM beats both pure policies in 9/129/12 cells and stays within 5%5\% of hindsight on 84%84\% of reps. Table 2: Exp. 5 dual cost at gap γ=0.25γ=0.25 (means over 2020 reps). CCM wins whenever n/dn/d is not tiny. d n/dn/d regime naive CCM iso 1616 2.52.5 soft 0.3740.374 0.5620.562 0.7510.751 1616 12.512.5 soft 0.3470.347 0.1220.122 0.7540.754 1616 5050 match 0.3500.350 0.0090.009 0.7570.757 3232 2.52.5 soft 0.3120.312 0.4660.466 0.7650.765 3232 12.512.5 soft 0.3280.328 0.0760.076 0.7520.752 3232 5050 match 0.3350.335 0.0090.009 0.7500.750 6464 2.52.5 soft 0.2840.284 0.3040.304 0.7770.777 6464 12.512.5 soft 0.2950.295 0.0450.045 0.7810.781 6464 5050 match 0.2960.296 0.0090.009 0.7700.770 128128 2.52.5 soft 0.2560.256 0.2070.207 0.7960.796 128128 12.512.5 soft 0.2640.264 0.0310.031 0.7880.788 128128 5050 match 0.2670.267 0.0090.009 0.8310.831 Exp. 4 (scope; secondary). Same-Σ / different-P shifts are invisible to τ; wrong-family geometry breaks naive match while CCM’s soft/iso fallback remains safer. Statistical confidence is not model correctness—controlled misspecification, not the main product. 9 Failure boundary Theorem 3 and CCM assume the displacements used for Σ share the deployment Σ that matters at test time. When that assumption fails, confidence is not correctness: a small τ does not repair a misspecified geometry family. Same Σ , different P. Exp. 4 (Case A): τ can look fine while laws differ in higher-order structure. Second-moment confidence is not model correctness (a natural next question outside LQ risk). Wrong geometry family. Matching ΣA _A under deployment ΣB _B fails; CCM soft/even-spread is the safer fallback (Exp. 4, Case B)—abstention, not a repair of the wrong estimand. Weak gaps and tiny n. Then τ is large and CCM abstains by design. Ambient-d concentration on real Δ . Proxy (1) can over-abstain on high-dimensional features. Exp. 6 states the practical numerator; sharper real-data bounds remain open. Out of scope. Nonlinear Matching extensions; Dirichlet / heavy nonlinear Matching; online / LLM Matching; new shrinkage estimators; ambient-d certificates for Office/Fashion/HAR. 10 Conclusion Match only geometry you can identify; otherwise spread the penalty. The trust ratio τ=ε/γτ= /γ organises that rule: under the LQ response, oracle-relative projector-matching drift scales as τ2τ^2 for probes in the chosen top-r deployment subspace (Theorem 3)—O(τ2)O(τ^2) in the separation region τ<1/2τ<1/2, with practical usefulness depending on constants. CCM is a separate calibrated policy from τ to match / soft / abstain—not a theorem corollary—with match inside that separation region and soft mostly outside it. The flagship evidence is failure avoidance: on UCI HAR embeddings, always-match hurts on 6/66/6 cells while CCM abstains; Fashion shows the identifiable complement. Open. Sharper real-data ε with guarantees; adaptive rank with certificates; trust gates beyond LQ. acknowledgments-disclosure-of-funding.No third-party funding. Optional citations (not required to read this paper): arXiv:2604.21395 (PMH) and arXiv:2605.22800 (Matching Principle). Appendix A Proof of the deployment-drift gap We prove Lemma 2 and Theorem 3. The LQ response is inherited from Rajput (2026a); matrix concentration and Davis–Kahan are cited tools. The intellectual object remains τ=ε/γτ= /γ (Definition 1). A.1 Assumptions (restated) Assumption 4 (Label-preserving second moment). Displacements Δ are label-preserving with finite second moment Σ:=[ΔΔ⊤]⪰0 :=E[ ] 0. Theory assumes correct second-moment specification (misspecification is Exp. 4). Assumption 5 (Linear–quadratic response). Fix a unit probe v∈ℝdv ^d and penalty weight λ>0λ>0. For Σ′⪰0 0, wλ(Σ′,v):=(I+2λΣ′)−1v,DΣ(w):=w⊤Σw.w_λ( ;v):=(I+2λ )^-1v, D_ (w):=w w. Assumption 6 (Spiked deployment spectrum). Σ has eigenvalues λ1≥⋯≥λd≥0 _1≥·s≥ _d≥ 0 with gap γ:=λr−λr+1>0γ:= _r- _r+1>0. Write W for an orthonormal top-r eigenbasis and ΠW:=WW⊤ _W:=W . Assumption 7 (Finite-sample geometry). I.i.d. sub-Gaussian (or bounded) samples yield Σ^=n−1∑iΔiΔi⊤ =n^-1 _i _i _i with ‖Σ^−Σ‖op≤ε(n,δ)\| - \|_op≤ (n,δ) w.p. at least 1−δ1-δ under a standard matrix-concentration bound (Vershynin, 2018). Experiment 1 calibrates a transparent proxy radius; we do not claim a new concentration theorem. A.2 Proof of Lemma 2 Proof of Lemma 2. Under Assumptions 6–7, write E:=Σ^−ΣE:= - and assume the concentration event ‖E‖op≤ε(n,δ)\|E\|_op≤ (n,δ). If moreover 2ε(n,δ)<γ2 (n,δ)<γ, the top-r eigenspace remains separated under the perturbation. The Davis–Kahan / Wedin form of Yu, Wang, and Samworth (2015) then yields ‖ΠW^−ΠW‖op≤CDKε(n,δ)γ=CDKτ\| _ W- _W\|_op≤ C_DK\, (n,δ)γ=C_DK\,τ with probability at least 1−δ1-δ, for an absolute constant CDKC_DK (cf. Davis and Kahan (1970)). When 2ε≥γ2 ≥γ one has τ≥1/2τ≥ 1/2 and the bound is uninformative; CCM treats that regime as non-identifiable and abstains (§7). ∎ A.3 Projector response identity For the Matching consequence we only need the closed-form projector response (I+2λΠ)−1=I−αΠ(I+2λ )^-1=I-α with α:=2λ/(1+2λ)α:=2λ/(1+2λ). (A general resolvent Lipschitz bound exists for non-projector penalties and recovers a looser factor 4λ4λ; we isolate projectors precisely to remove that λ-growth. The resolvent is a calculation, not the trust idea.) Lemma 8 (Projector response gap). If A,BA,B are orthogonal projectors and α:=2λ/(1+2λ)α:=2λ/(1+2λ), then (I+2λA)−1−(I+2λB)−1=α(B−A),(I+2λ A)^-1-(I+2λ B)^-1=α\,(B-A), so ‖(I+2λA)−1−(I+2λB)−1‖op=α‖A−B‖op \|(I+2λ A)^-1-(I+2λ B)^-1 \|_op=α\|A-B\|_op. Proof. Eigenvalues of I+2λΠI+2λ lie in 1,1+2λ\1,1+2λ\, hence (I+2λΠ)−1=I−αΠ(I+2λ )^-1=I-α . Subtracting the two closed forms gives the claim. ∎ A.4 Proof of Theorem 3 Lemma 9 (Compressed projector gap). If P,QP,Q are orthogonal projectors and s:=‖P−Q‖op<1s:=\|P-Q\|_op<1, then ‖P(I−Q)P‖op≤s2.\|P(I-Q)P\|_op\;≤\;s^2. Proof. In principal angles, the nonzero singular values of P−QP-Q are sinθi _i and those of P(I−Q)P(I-Q)P are sin2θi ^2 _i (Davis and Kahan, 1970), so ‖P(I−Q)P‖op=maxisin2θi≤s2\|P(I-Q)P\|_op= _i ^2 _i≤ s^2. ∎ Proof of Theorem 3. On the 1−δ1-δ event of Assumption 7, Lemma 2 gives s:=‖ΠW^−ΠW‖op≤CDKτs:=\| _ W- _W\|_op≤ C_DKτ. Write P:=ΠWP:= _W, Q:=ΠW^Q:= _ W, and fix a unit v∈range(P)v (P). Then w⋆=(I−αP)v=(1−α)vw_ =(I-α P)v=(1-α)v and DΣ(w⋆)=(1−α)2v⊤ΣvD_ (w_ )=(1-α)^2\,v v. Also wn=(I−αQ)vw_n=(I-α Q)v, so DΣ(wn) D_ (w_n) =v⊤Σv−2αv⊤QΣv+α2v⊤QΣQv, =v v-2α\,v Q v+α^2\,v Q Qv, DΣ(wn)−DΣ(w⋆) D_ (w_n)-D_ (w_ ) =2αv⊤(I−Q)Σv−α2v⊤(Σ−QΣQ)v. =2α\,v (I-Q) v-α^2\,v ( -Q Q)v. Because v∈range(P)v (P) and ΣP=PΣ P=P , both Σv v and v lie in range(P)range(P). For a,b∈range(P)a,b (P), a⊤(I−Q)b=a⊤P(I−Q)Pba (I-Q)b=a P(I-Q)Pb, so Lemma 9 yields |v⊤(I−Q)Σv|≤s2‖Σ‖op‖v‖22. |v (I-Q) v |\;≤\;s^2\| \|_op\|v\|_2^2. Moreover Σ−QΣQ=(I−Q)Σ+QΣ(I−Q) -Q Q=(I-Q) +Q (I-Q), hence |v⊤(Σ−QΣQ)v| |v ( -Q Q)v | ≤|v⊤(I−Q)Σv|+|(Qv)⊤Σ(I−Q)v| ≤ |v (I-Q) v |+ |(Qv) (I-Q)v | ≤s2‖Σ‖op‖v‖22+‖Σ‖op‖(Q−P)v‖2‖(I−Q)v‖2 ≤ s^2\| \|_op\|v\|_2^2+\| \|_op\|(Q-P)v\|_2\|(I-Q)v\|_2 ≤2s2‖Σ‖op‖v‖22, ≤ 2s^2\| \|_op\|v\|_2^2, where ‖(I−Q)v‖2=‖(I−Q)Pv‖2≤s‖v‖2\|(I-Q)v\|_2=\|(I-Q)Pv\|_2≤ s\|v\|_2 and Qv=v+(Q−P)vQv=v+(Q-P)v. The triangle inequality then yields the absolute gap |DΣ(wn)−DΣ(w⋆)| |D_ (w_n)-D_ (w_ ) | ≤2α|v⊤(I−Q)Σv|+α2|v⊤(Σ−QΣQ)v| ≤ 2α\, |v (I-Q) v |+α^2\, |v ( -Q Q)v | ≤(2α+2α2)‖Σ‖ops2≤(2α+2α2)CDK2‖Σ‖opτ2, ≤ (2α+2α^2 )\| \|_op\,s^2\;≤\; (2α+2α^2 )C_DK^2\| \|_op\,τ^2, which is (3). ∎ Remark 10 (Ambient probes: O(τ)O(τ) only). If v∉range(ΠW)v ( _W), the bulk term v⊤Q(I−ΠW)Σ(I−ΠW)v Q(I- _W) (I- _W)v is only O(s)O(s) in general. The crude estimate |DΣ(wn)−DΣ(w⋆)|≤2αCDK‖Σ‖opτ‖v‖22|D_ (w_n)-D_ (w_ )|≤ 2α C_DK\| \|_opτ\|v\|_2^2 still holds for every unit v (expand |(Δw)⊤Σ(wn+w⋆)||( w) (w_n+w_ )| with ‖Δw‖2≤αs‖v‖2\| w\|_2≤α s\|v\|_2). Theorem 3 concerns the chosen top-r deployment subspace range(ΠW)range( _W)—CCM’s projector match arm. Equality range(Σ)=range(ΠW)range( )=range( _W) holds only under exact rank-r (λr+1=0 _r+1=0); if λr+1>0 _r+1>0 the Matching cover game is larger, and probes outside range(ΠW)range( _W) retain only the O(τ)O(τ) ambient estimate. Remark 11 (Numeric liveness). Trivially |DΣ(wn)−DΣ(w⋆)|≤‖Σ‖op‖v‖22|D_ (w_n)-D_ (w_ )|≤\| \|_op\|v\|_2^2. The τ2τ^2 bound beats that when (2α+2α2)CDK2τ2<1(2α+2α^2)C_DK^2τ^2<1. 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