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The Paired Thermal Channel Is Two Orthogonal Axes: A Controlled Attribution Study of Calibration Cost and Detection Gain in Infrared Person Detection

Tian Zhang, Yang Chen, Bing Bai, Weikai He, Xiang Gao, Zhirong Su

arXiv:2610.33118v1cs.CVcs.MM

TL;DR

Infrared detection pipelines often discard calibrated temperature in favor of grayscale renderings, while uncertainty-based data decisions commonly rely on separate criteria. This paper uses temperature-evidential learning and five controlled attribution arms to examine calibrated-channel effects. It reports distinct calibration and detection axes, including a +0.0221 mAP advantage for true ΔT over scale-matched grayscale pseudo-ΔT.

  • Problem

    Conventional thermal pipelines discard the calibrated temperature matrix, and existing frameworks use separate criteria for annotation, pseudo-labeling, and replay.

    Source: §1 · p. 1

  • Method

    Five controlled attribution arms in TEL vary pairing, calibrated scale, power-spectrum structure, and spatial phase to examine the calibrated channel’s effects.

    Source: §1 · p. 2

  • Results

    +0.0221 mAP: true ΔT outperforms scale-matched grayscale pseudo-ΔT, while calibration cost and detection gain arise along distinct structural axes.

    Source: §1 · p. 2

  • Takeaways & Limitations

    A calibrated physical channel can serve as an evidence variable for unified annotation, pseudo-labeling, and discard decisions in TEL.

    Source: §1 · p. 2

    The information-theoretic analysis adopts a local Gaussian shift model for the annular contrast.

    Source: §3.3 · p. 8

Abstract

from arXiv · show

The calibrated thermal channel in infrared person detection comprises two structurally orthogonal axes. True calibrated $\Delta T$ outperforms scale-matched grayscale pseudo-$\Delta T$ by $+0.0221$ mAP (paired Wilcoxon $p=0.0098$, Cohen's $d=+1.08$), confirming that the calibrated channel carries information the grayscale channel lacks. The calibration axis is driven by the joint pairing-scale condition. Preserving both paired correspondence and absolute calibrated scale raises expected calibration error (ECE) from $\approx 0.05$ to $0.142$. The $2\times 2$ interaction ($p=0.0059$) localizes the rise to this condition. Breaking either factor alone returns ECE to the broken-factor level (true-vs-const $p=0.002$; true-vs-shuffled $p=0.002$). Under post-hoc temperature scaling, the pairing cost survives recalibration ($p=0.002$), while the scale cost is largely absorbed ($p=0.16$). As regards the detection axis, the gain stems from structural de-structuring. Phase randomization, which destroys spatial structure while preserving per-sample variance, yields $+0.0185$ mAP over true $\Delta T$ on FLIR ADAS ($p=0.014$, $d=+0.96$). The FLAME~3 gain is $+0.0064$ mAP ($p=0.078$, n=8). On the calibration axis, the same intervention is insignificant on FLIR ADAS ($p=0.47$) but significant on FLAME~3 ($p=0.0078$). These findings rest on a five-arm controlled attribution with temperature-evidential learning (TEL) as the testbed. TEL obtains a single evidence-derived routing scalar together with a complementary diagnostic entropy, and establishes a finite-evidence degeneracy theorem proving that the acquisition product degenerates to this scalar with a closed-form bound. A physical $0.5^\circ$C threshold admits cross-dataset physical readings, whereas a gray threshold of $20$ inverts without physical interpretation.

1 Introduction

The paper treats calibrated thermal data as a physical evidence channel that can address blind spots in rendered grayscale and unify decisions across the data lifecycle. Its controlled attribution identifies separate calibration and detection axes, with gains tied to calibrated information and structural de-structuring.

1 Introduction

  • ECE rises from ≈0.05 to 0.142 when paired correspondence and absolute calibrated scale are both preserved; breaking either factor alone returns it to the broken-factor level.The interaction is significant (p = 0.0059); after temperature scaling, the pairing gap persists (p = 0.002), while the scale gap is absorbed (p = 0.16).Source: §1 · p. 2
  • +0.0221 mAP: true ΔT outperforms scale-matched grayscale pseudo-ΔT, indicating that the calibrated channel carries information grayscale lacks.The paired Wilcoxon test reports p = 0.0098 and d = +1.08.Source: §1 · p. 2
  • Detection gain tracks structural de-structuring: the phase-randomized arm exceeds true ΔT by +0.0185 mAP while preserving per-sample variance and removing spatial phase.The paired Wilcoxon test reports p = 0.014 and d = +0.96; flattening the power spectrum accounts for a further +0.0156 (p = 0.004).Source: §1 · p. 2
  • The cross-dataset contrast shows that the calibration and detection axes vary independently, with each significant on the dataset where its structural counterpart is most pronounced.The findings are presented as a five-arm controlled attribution using true, const-ΔT, shuffled-ΔT, var-phase, and random arms.Source: §1 · p. 2
  • Physical thresholds expressed in °C remain semantically consistent across datasets, whereas a gray threshold of 20 inverts and has no physical reading.The attribution protocol is defined for generic calibrated auxiliary channels and validated on two radiometric benchmarks spanning different calibrated regimes.Source: §1 · p. 2
  • TEL uses one evidence-derived scalar u to govern annotation, pseudo-labeling, and discard, and proves that the acquisition product degenerates to u in the finite-evidence regime.The routing partition is invariant under strictly increasing affine reparametrizations.Source: §1 · p. 2

2 Related Work

Prior work develops evidential uncertainty, calibration, sample selection, replay, infrared detection, and multimodal methods, but typically treats temperature as preprocessing and uses separate decision criteria. TEL instead uses a calibrated physical channel as an evidence variable within a unified decision framework.

2.1 Uncertainty Estimation in Deep Learning

  • EDL places a Dirichlet prior over class probabilities and has been applied to detection, but cited work raises concerns about regularization sensitivity, underestimated epistemic uncertainty, and calibration under shift.TEL keeps the EDL loss and regularization common across attribution arms, so their paired contrasts share those effects.Source: §2.1 · pp. 2–3

2.2 Confidence Calibration in Detection

  • Detection calibration work studies localization-aware ECE and post-hoc calibration, while TEL examines whether a calibrated physical input changes an evidential detector’s intrinsic confidence distribution.The paper notes that ECE is definition-sensitive and that post-hoc calibration can be strong.Source: §2.2 · p. 4, §2.7 · p. 5

2.3 Active Learning and Pseudo-Labeling

  • Active learning and pseudo-labeling often use separate scores; TEL uses one temperature-modulated scalar to unify annotation, pseudo-labeling, and discard.Existing evidential approaches cited include selection scores and filtering strategies, but the paper characterizes them as separate criteria for these decisions.Source: §2.3 · p. 4

2.4 Continual Learning and Replay

  • Replay methods select historical samples using gradient or visual criteria, whereas TEL weights replay by |ΔT| and temperature-modulated uncertainty.The related work frames replay as a response to catastrophic forgetting and the choice of stored historical samples.Source: §2.4 · p. 4

2.5 Infrared Pedestrian Detection and Cross-Modal Distillation

  • Infrared detection and cross-modal distillation address fusion, misalignment, and data-scale constraints; TEL makes supervision strength a function of physical contrast rather than globally fixed band weights.FreqKD uses asymmetric frequency supervision, while other cited methods use globally fixed band weights.Source: §2.5 · pp. 4–5

2.6 Temperature Information in Radiometric Imagery

  • Prior radiometric-image work uses temperature in preprocessing or feature extraction, while TEL makes it a structural variable in uncertainty modeling.The paper characterizes the resulting calibration cost as a joint pairing-scale condition.Source: §2.6 · p. 5

2.7 Multi-Modal Uncertainty Fusion and Modality Reliability

  • Multimodal reliability and misalignment studies treat pairing and reliability as robustness variables; TEL intervenes on paired correspondence to test calibration cost as an attribution question.The stated research gap is the absence, to the authors’ knowledge, of a framework using a calibrated physical channel to determine annotation, pseudo-labeling, replay, and distillation decisions.Source: §2.7 · p. 5

3 Method

The method retains calibrated temperature alongside grayscale, computes local multiscale temperature contrast, and uses it to modulate evidential uncertainty and route samples. It also formalizes the physical and information-theoretic basis for this calibrated channel.

3.2 Physical Layer: Separability of the Paired Thermal Channel

  • The method retains the calibrated temperature matrix T alongside grayscale G and computes local annular contrast at three dilation scales to keep the reference background local.The scales are r ∈ {0.1, 0.2, 0.4}; a learned attention module combines them, and signed contrast preserves warm-target and shade regimes.Source: §3.1 · p. 5, §3.2 · p. 6

3.1 Problem Setup

  • Under a fixed annotation budget, the setup allocates decisions among detection, annotation, pseudo-labeling, and replay; samples are ranked by u for annotation.The budget constraint limits total annotations, and Proposition 3(i) establishes monotonicity of u in the leading expected-information-gain term in the large-evidence limit.Source: §3.1 · pp. 5–6

3.2 Physical Layer: Separability of the Paired Thermal Channel

  • A sigmoid modulation is selected by a maximal-growth constraint; temperature-conditioned priors and a modulation factor then adjust detector evidence.The prior can depend on ΔT, extending beyond purely multiplicative modulation of base evidence.Source: §3.2 · pp. 6–7
  • Uncertainty reduction depends jointly on physical contrast through m(ΔT) and evidence concentration a, so the pairing factor determines whether a is meaningful and the scale factor whether ΔT is informative.The proposition gives a two-sided bound and an error-propagation result for estimated concentration.Source: §3.2 · p. 7
  • A strictly increasing affine reparameterization of temperature preserves the direction of uncertainty reduction and the ordering of routing scores, while changing their physical interpretation and reference scale.The stated relation is T′ = aT + b with a > 0.Source: §3.2 · p. 7
  • The attribution probes isolate different inputs: const-ΔT removes absolute scale, shuffled-ΔT breaks within-mask pairing, and var-phase removes spatial phase while preserving variance and power-spectrum magnitude.The grayscale pseudo-ΔT arm uses the same predicted boxes and annular geometry as true ΔT, with grayscale as the value source; structured arms share a fixed target scale.Source: §3.2 · pp. 7–8

3.3 Information-Theoretic Layer: KL Separability of the Temperature Channel

  • The information-theoretic layer uses a KL separability index under a local Gaussian shift model and treats that model as a diagnostic for the temperature channel.The index is described as closed-form and unbounded in signal-to-noise ratio under the stated model.Source: §3.3 · p. 8

T and SNRG

This section bounds temperature-channel KL separability and develops a unified uncertainty-driven routing layer, examining the routing scalar’s relation to acquisition-product rankings, evidence regimes, and optimization under controlled routing switches. It also distinguishes diagnostic entropy’s role and presents replay and distillation as unevaluated extensions.

T and SNRG

  • The KL separability index has a closed-form sandwich that is tight as grayscale information collapses but loosens as grayscale information grows, when separability is governed by grayscale.The upper bound remains fixed while the lower bound decays as 1/(1+x) as x → ∞.Source: T and SNRG · p. 8

3.4 Decision Layer: Unified Routing via Evidential Scoring

  • The predicted value is R2 = 0.9936 and measured value is R2 = 0.9958, with slope Ψ̄ = 0.4982 and residual standard deviation 0.007054; the acquisition product therefore degenerates to u in this regime.η = 6.5 × 10−3.Source: §3.4 · p. 10
  • Across 45 seed pairs, the pseudo-label error bound exceeded observed accepted error for u, margin, and confidence, with a mean gap of +0.030, and for entropy, with a gap of +0.046.The bound is tighter when model errors are less correlated; at ρ = 0 it is ε^2/[ε^2 + (1 − ε)^2], while at ρ = 1 it equals ε.Source: §3.4 · pp. 10–11
  • In actual TEL, routing uses u alone while entropy is diagnostic: u and H are anti-correlated (Pearson −0.077; Spearman −0.266), and the acquisition product is nearly collinear with u (Pearson +0.998).The acquisition product is essentially uncorrelated with H (Pearson −0.015).Source: §3.4 · pp. 11–12
  • All four tested strictly increasing affine temperature reparameterizations preserved the quantile-router partition, with routing consistency κroute = 1.0000.The theoretical invariance applies to the annotation, pseudo-label, and discard branches.Source: §3.4 · p. 12

3.5 Scale Structure and Two-Ended Separation of the Routing Scalar

  • u and diagnostic entropy Ψ relate oppositely to correctness (Pearson +0.4763 ± 0.0291 and −0.4301, respectively); quantile routing is scale-free, but Ψ is not affine-invariant.The two quantities therefore serve structurally different roles: u is operational, whereas Ψ is diagnostic and scale-dependent.Source: §3.5 · p. 12
  • The fit slope 0.4982 matches Ψ̄, and R2 = 0.9958 confirms the bound, showing that Δacq is a rescaled copy of u with no independent ranking information from Ψ.Corollary 2 predicts R2 ≥ 0.9936.Source: §3.5 · p. 12
  • The scalar u is a two-ended separator rather than a monotone ranking: the top-10% annotate branch captures saturated bin 9, while the bottom-30% discard branch has mean error 0.9941 ± 0.0063.The annotate branch has mean error 0.6773 ± 0.0583; controlling for box area preserves 98% of u’s and 99.6% of Ψ’s raw correlations with correctness.Source: §3.5 · pp. 12–13, Table 6 · p. 13

3.6 Design Extensions: Replay and Distillation

  • Replay and frequency-decoupled distillation are proposed as extensions, but their empirical evaluation is left to future work.Distillation adapts high-frequency supervision to physical contrast ΔT, unlike FreqKD’s globally fixed weight [45].Source: §3.6 · p. 13

3.7 Optimization Layer: Closed-Loop Convergence

  • Under the Lipschitz-router case, the switching term is O(1) when routing switches are controlled.The optimization analysis states convergence conditions in terms of routing switches and bounds the switching contribution.Source: §3.7 · p. 14

4 Experiments

Across FLIR ADAS and FLAME 3, the section uses controlled attribution to examine calibrated temperature’s contribution and distinguish calibration effects from detection accuracy. It also discusses routing probes that characterize u as the acquisition signal and a pool-based selection probe that does not improve over warmup.

4.1 Attribution Design

  • The evaluation uses radiometric FLIR ADAS and FLAME 3 data, a two-stage oracle-free protocol, and a native grayscale channel pixel-aligned with calibrated thermal data.FLIR ADAS supports the main attribution experiments; FLAME 3 provides auxiliary validation. Ground-truth boxes are used for scoring, not for constructing ∆T maps.Source: §4.1 · p. 14, Table 7 · p. 14

4.2 Statistical Methodology and Confirmatory Family

  • The primary confirmatory contrasts are significant: true-∆T versus pseudo-∆T on mAP (p ≈ 0.0098) and true-∆T versus shuffled-∆T on ECE (p = 0.002), under Holm correction.The secondary true-versus-constant-∆T ECE contrast is also significant (p = 0.002); remaining true-versus-pseudo contrasts on ECE, Recall, and LCR are not.Source: §4.2 · p. 15, Table 8 · p. 16

4.3 Step 1: Is the Temperature Channel Consumed?

  • True ∆T raises modulated evidence magnitude by +38.42% and top detection score by +2.48% over zero modulation, while random modulation raises evidence by only +8.78% and leaves score nearly unchanged (+0.05%).The probe attributes channel consumption primarily to physical contrast rather than generic modulation.Source: §4.3 · p. 15, Table 9 · p. 16

4.4 Step 2: Does the Gain Come from the Physical Channel?

  • True ∆T significantly outperforms scale-matched grayscale pseudo-∆T on FLIR ADAS mAP (p = 0.0098, d = +1.08), whereas the corresponding FLAME 3 mAP contrast is not significant.On FLAME 3, ∆G/∆T redundancy is high (Pearson r = 0.76, Spearman ρ = 0.85); true ∆T is significantly worse than random modulation on mAP and ECE.Source: §4.4 · p. 16, Table 8 · p. 16, Table 11 · p. 17

4.5 Step 3: Attribution of the Calibration and Detection Axes

  • The pairing-scale interaction raises ECE (+0.0756, p = 0.0059), while phase randomization raises FLIR ADAS mAP to 0.5034 from 0.4849 (p = 0.014, d = +0.96).Breaking either pairing or calibrated scale alone returns ECE to the broken-factor level (both p = 0.002); phase randomization preserves per-sample variance while removing spatial phase.Source: §4.5 · p. 16, Table 8 · p. 16, Table 14 · p. 18

4.6 Step 4: Cross-Dataset Reading of the Two Axes

  • On FLAME 3, phase randomization yields a +0.0064 mAP contrast over true ∆T (p = 0.078), while its ECE contrast is significant there (−0.2456, p = 0.0078) but not on FLIR ADAS (p = 0.47).The detection-axis effect has the same sign and comparable magnitude across datasets; the calibration-axis effect appears on FLAME 3 but not FLIR ADAS. A 20-level gray threshold inverts across datasets (73.6% vs. 36.0%) and lacks physical interpretation.Source: §4.6 · pp. 17–18, Table 11 · p. 17, Table 16 · p. 19

4.7 Qualitative Analysis

  • Across FLIR ADAS ∆T buckets, mAP rises through B4 and declines at B5; the var-phase advantage over true ∆T changes sign across buckets and is largest at B4.This qualitative pattern shows the detection-axis gain is not concentrated in a single ∆T regime.Source: §4.7 · p. 19

4.8 Discussion

  • The retained absolute scale supports physical threshold interpretation across datasets, while the study notes radiometric-calibration assumptions and that no other public benchmark currently provides calibrated temperature data.The authors also leave annotation-budget sweeps for future work; with relative counts, physical interpretation is lost even though quantile routing remains invariant.Source: §4.8 · pp. 19–20
  • Preserving both pairing and calibrated scale raises ECE by about 3× over broken-factor arms, while detection accuracy is unaffected.De-structuring lowers ECE without being required for the mAP gain. FLIR ADAS ECE variance is localized to true-ΔT, where seed s2 has a non-rescalable component; shuffled and constant ΔT are stable, and main-text results use α = 1.0.Source: §4.8 · p. 20
  • The routing scalar u has a +0.4677 partial correlation with correctness controlling for box area, while diagnostic entropy Ψ correlates oppositely at −0.4301.Ψ remains robust to area control at −0.4284.Source: §4.8 · p. 20
  • η = 6.5×10^-3 and the fit Δacq = 0.4982 u−0.0003 with R2 = 0.9958 support finite-evidence degeneracy: the acquisition product is a rescaled copy of u.In this regime, Ψ contributes no independent ranking information.Source: §4.8 · p. 20
  • +0.0185 mAP is the paired var-phase gain over true ΔT on FLIR ADAS; its largest positive bucket gap is at B4.The var-phase curve is above true at B1, B2, and B4, and below it at B3 and B5.Source: Figure 6 · p. 20
  • On FLIR ADAS, top-10% pool selection does not improve over warmup: 0.5194 ± 0.0058 versus utop10 0.4409 ± 0.0434 and Htop10 0.4586 ± 0.0244.The pool probe measures end-to-end annotation efficiency, unlike routing-consistency results that characterize u’s local ordering behavior; KL-regularization effects are common across attribution arms and cancel in paired contrasts.Source: §4.8 · p. 20

5 Conclusion

The calibrated thermal channel has two structurally orthogonal axes: calibration tracks a joint pairing-scale condition, while detection responds to structural de-structuring. Across FLIR ADAS and FLAME 3, the detection-axis effect holds in sign and comparable magnitude, while the calibration-axis effect varies by dataset.

5 Conclusion

  • The phase-randomized arm raises mAP over true ∆T, while across datasets the detection-axis effect holds in sign and comparable magnitude.The calibration-axis effect registers on one dataset and recedes on the other.Source: §5 · pp. 20–21
  • On the calibration axis, ECE tracks the joint pairing-scale condition, and breaking either factor alone returns ECE to the broken-factor level.Post-hoc recalibration absorbs more of the scale cost than of the pairing cost.Source: §5 · p. 20
  • TEL produces an evidence-derived routing scalar u and complementary diagnostic entropy Ψ, with a finite-evidence degeneracy theorem giving the bound R2 ≥ 1 − η.Absolute scale provides deployment-time value through cross-dataset semantic consistency.Source: §5 · p. 21

A Proofs for Lemma 1, Proposition 1, and Corollary 1

The proofs characterize the admissible modulation function and show how modulation changes uncertainty. Positive rescaling of ∆T preserves the direction of the uncertainty reduction and its sample ordering.

A Proofs for Lemma 1, Proposition 1, and Corollary 1

  • Under the growth constraint, equality yields m(t) = σ(t + c), while input rescaling gives the admissible family σ(κt + β) for 0 < κ ≤ 1.The proof establishes uniqueness by applying Picard–Lindelöf to the logistic ODE.Source: A Proofs for Lemma 1, Proposition 1,… · p. 23
  • The modulated evidence gives an uncertainty reduction bounded by the endpoint values of γma/(1 + γma), with error propagation bounded by γm · ϵa.The proof derives the reduction from Smod = Saug[1 + γm(∆T)a].Source: A Proofs for Lemma 1, Proposition 1,… · p. 23
  • For a > 0, rescaling ∆T to ∆T′ = a∆T preserves the direction of uncertainty reduction and the ordering of u across samples.The proof uses the strictly increasing modulation map and the comonotonicity of m and m′.Source: A Proofs for Lemma 1, Proposition 1,… · p. 23

B Proof of Proposition 2

The proof bounds a Gaussian divergence expression and couples its large- and small-signal behavior to the routing scalar u. The resulting approximation is stated for weak and strong regimes.

B Proof of Proposition 2

  • For equal-variance univariate Gaussians with means 0 and µ, the proof gives KL(N(µ, σ²)∥N(0, σ²)) = µ²…The supplied proof passage continues with lower and upper bounds involving SNR terms; the bounds coincide as x → 0 and the sandwich loosens as x → ∞.Source: B Proof of Proposition 2 · p. 24
  • In the weak-modulation regime, γma/(1 + γma) expands as γma + O((γma)²), and the proof relates γma to 1 − e^−Λ(b) to leading order.In the strong regime, the latter term tends to 1 as Λ(b) → ∞.Source: B Proof of Proposition 2 · p. 24

C Proof of Proposition 3 and Corollary 3

These proofs connect evidence-based acquisition and routing to information gain, pseudo-label error under correlated predictions, and bounds on discarded examples. They also invoke concentration and online-regret results for their respective guarantees.

C Proof of Proposition 3 and Corollary 3

  • At large evidence strength, the exact information gain has leading term (K − 1)/(2K)u(x), so ∆acq(x) = u(x)Ψ(x) is used as a monotone acquisition proxy.The passage distinguishes the exact information gain from the proxy.Source: C Proof of Proposition 3 and Corollary 3 · p. 24
  • Under correlated prediction errors, the proof expresses the accepted pseudo-label error as a function of marginal error ε and correlation ρ.It derives the joint-error range from the marginal and joint probability constraints.Source: C Proof of Proposition 3 and Corollary 3 · p. 24
  • Hoeffding supplies the concentration step, a covariance-corrected mixing bound extends it, and standard online gradient descent regret applies to convex G-Lipschitz losses.The supplied passages identify these proof tools but do not state the full resulting bounds.Source: C Proof of Proposition 3 and Corollary 3 · p. 24
  • Within the Dirichlet evidence model, the Bayes error bound yields εdiscard(τlow) ≤ (K − 1)/(K/τlow + 1), increasing with τlow.The proof obtains this by substituting S = K/u and conditioning on u < τlow.Source: C Proof of Proposition 3 and Corollary 3 · p. 24

D Proofs of Proposition 4 and Corollary 2

The proof decomposes the acquisition product into a term proportional to u and a residual, then bounds the residual and its effects on ranking and correlation.

D Proofs of Proposition 4 and Corollary 2

  • The acquisition product decomposes as Δacq = u · Ψ̄ + u · δ, with a variance bound on the residual yielding an R² bound.The proof also bounds the pointwise deviation |Δacq − Ψ̄u| by |uδ|.Source: D Proofs of Proposition 4 and… · p. 24
  • Under independence of δ and u, the relationship between Δacq and u has R² = 1/(1 + η).For the reported values, the predicted R² is 0.9936 versus a measured 0.9958; the text says the gap is consistent with weak negative dependence.Source: D Proofs of Proposition 4 and… · pp. 24–25
  • A rank flip requires the relative u-gap between samples to be at most 2ε/(1 − ε) when the residual satisfies |δ|/|Ψ̄| ≤ ε.Thus, sufficiently separated u values retain their ordering under the bounded perturbation.Source: D Proofs of Proposition 4 and… · p. 24

E Proof of Proposition 5, Remark 4, and Corollary 4

This section establishes properties of u-based routing, relates uncertainty signals empirically, and validates bounds and routing behavior across seeds.

E Proof of Proposition 5, Remark 4, and Corollary 4

  • Under the fixed-shape idealization, u is the only quantity that varies with evidence strength S and provides ranking information across samples.In the actual TEL evidence distribution, u and entropy are negatively associated, while the information-gain product is nearly collinear with u and is not used for routing.Source: E Proof of Proposition 5, Remark 4,… · p. 25
  • The affine-router proof relies on strictly increasing transformations preserving quantiles and order; empirical consistency is κroute = 1.0000 across four tested affine reparameterizations.The same consistency applies to the pseudo-label and discard branches.Source: E Proof of Proposition 5, Remark 4,… · p. 25

F Proof of Remark 6

  • Entropy-regularized replay assigns weights increasing in |ΔTi|, while reliability weighting has non-negative covariance with reliability under p(u).The proof derives these properties from the exponential weight and the Chebyshev association inequality, respectively.Source: F Proof of Remark 6 · p. 25

G Proofs of Lemmas 2–3 and Proposition 6

  • With bounded density near routing thresholds, expected switches are bounded by summing per-sample crossing probabilities; under contraction, the expected number of switches is O(log T).The convergence proof separates non-switching iterations, where the descent lemma applies, from switching iterations, where loss changes are bounded.Source: G Proofs of Lemmas 2–3 and Proposition 6 · p. 25

H Oracle-based validation details

  • The pseudo-label error bound exceeds observed accepted error in all 180 records across four signals and 45 seed pairs; Table 18 disaggregates per-seed discard error.The tightest reported case is u at seed pair (7, 8), with a positive gap of +0.0019.Source: H Oracle-based validation details · p. 26, Table 18 · p. 26

I Datasets and Hyperparameters

The experiments use calibrated thermal data from FLIR ADAS and FLAME 3, with the supplement describing a limitation of the FLAME 3 low-luminance splits.

I Datasets and Hyperparameters

  • FLIR ADAS provides 14-bit calibrated thermal imagery, with a standard split of 4,129 training and 1,013 test pairs.The dataset has 26,442 annotated frames and more than 520,000 boxes across 15 categories.Source: I Datasets and Hyperparameters · p. 26
  • FLAME 3 contains 738 image quartets from six prescribed burns, each pairing visible and radiometric thermal imagery, with fire or no-fire labels.Each quartet includes raw RGB, raw thermal, corrected field-of-view RGB, and a thermal TIFF with per-pixel temperatures.Source: I Datasets and Hyperparameters · p. 26

J Supplemental Tables on the Joint Pairing–Scale Effect

K Post-hoc temperature scaling: interaction and leave-one-out

Post-hoc temperature scaling separates the pairing and scale contributions to calibration cost, while distributional distances do not distinguish those factors.

K Post-hoc temperature scaling: interaction and leave-one-out

  • After temperature scaling, the true-vs-shuffled contrast remains significant (p = 0.002), whereas true-vs-const is non-significant (p = 0.16).The post-hoc interaction is positive but non-significant (c̄TS = +0.0488, p = 0.13), attenuated from the raw-ECE interaction (c̄raw = +0.0756, p = 0.0059).Source: K Post-hoc temperature scaling:… · pp. 26, 28
  • Excluding seed s2, true-vs-shuffled remains significant (+0.0259, p = 0.0039), while true-vs-const remains non-significant (+0.0124, p = 0.30).Seed s2 has fitted temperature T = 3.89 and its ECE barely changes under recalibration.Source: K Post-hoc temperature scaling:… · p. 28

L Distributional distance across arms

  • Distributional distances do not separate pairing and scale factors: neither total-variation distance nor the Kolmogorov-Smirnov statistic orders the arms like post-hoc ECE.Distance to random is larger under both metrics, consistent with the reported global confidence collapse.Source: L Distributional distance across arms · p. 28

M Modulation-amplitude sensitivity and the log-smoothing offset

The α = 0 mAP increase is an artifact of the evidence-log smoothing offset, while the offset stabilizes training. Routing-switch measurements use per-image fractions because aggregate counts saturate at the fixed evaluation-set size.

M Modulation-amplitude sensitivity and the log-smoothing offset

  • At α = 0, mAP is 0.5005 for the true arm versus 0.4720 at α = 1, but the OOD-region mAP declines monotonically across α ∈[0.5, 2].The α = 0 gain comes from confidence-dependent re-ranking introduced by the 0.1 smoothing offset, not from a ∆T contribution.Source: M Modulation-amplitude sensitivity and… · p. 28
  • Clamping the logarithm destabilizes training, collapsing mAP to 0.161 at α = 1.0; at α = 0, mAP is 0.446, near the unmodulated baseline of 0.4664.The comparison supports the conclusion that the 0.1 smoothing offset causes the original α = 0 gain and is necessary for training stability.Source: M Modulation-amplitude sensitivity and… · p. 28
  • At α = 1.0, the log-smoothing offset is common across attribution arms and subtracts out in paired contrasts, leaving the physical-channel effect.Source: M Modulation-amplitude sensitivity and… · p. 28

N Routing switch statistics

  • Per-image switch fraction avoids the aggregate-count ceiling from evaluating 100 images across 152 checkpoint pairs, though it does not resolve νT scaling with T.The reported statistic is the fraction of checkpoint pairs in which an individual image changes branch.Source: N Routing switch statistics · pp. 28–29
  • Mean per-image switch fractions are 0.031 on training and 0.069 on test for the fixed threshold, versus 0.092 and 0.152 for the quantile router.The quantile router has higher fractions on both sets, consistent with thresholds that drift with the evidence distribution.Source: N Routing switch statistics · pp. 28–29
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