KAN-IDIR: Multi-Disciplinary Insights

• KAN-IDIR is a multifaceted term used to denote unrelated methods in deformable image registration, intrusion detection, and 2-category theory.
• It integrates techniques like Kolmogorov–Arnold Networks in implicit neural representations and DKD-based knowledge distillation for lightweight edge inference.
• The term’s interpretation must be anchored to its specific disciplinary context, emphasizing local usage over a universally fixed definition.

KAN-IDIR is a domain-dependent research label rather than a single standardized concept. One explicit formal usage names KAN-IDIR and RandKAN-IDIR as KAN-based implicit neural representations for deformable image registration (Drozdov et al., 26 Sep 2025). The same string also appears as a shorthand for “KAN-based Intrusion Detection with Resource constraints via DKD” in industrial control system security (Alikhani, 3 Mar 2026), as an abbreviation for Kan injectivity and induced KZ-pseudomonads in 2-category theory (Liberti et al., 2022), and as a mnemonic organizing several KAN-centered summaries in equation discovery, phase-transition identification, quantum-inspired KAN design, and kaon identification (Pal et al., 2024, Xu et al., 5 Mar 2025, Sharma et al., 9 Oct 2025, Massri, 2016). The term therefore designates a family of unrelated constructions unified only by local disciplinary usage.

1. Terminological scope

The label is used across several technically distinct literatures. In medical image analysis, it denotes an instance-specific DIR method built from KAN-based INRs. In industrial cybersecurity, it denotes a teacher–student IDS pipeline in which a high-capacity KAN transfers knowledge to a tiny MLP via DKD. In higher category theory, it denotes a theory of Kan injectivity and the KZ-pseudomonads induced by a set of 1-cells. Other appearances are mnemonic rather than canonical, but they preserve the same pattern of attaching “KAN” to an application-specific expansion.

Usage	Core meaning	Representative paper
Medical image registration	KAN-IDIR and RandKAN-IDIR for deformable registration	(Drozdov et al., 26 Sep 2025)
ICS intrusion detection	“KAN-based Intrusion Detection with Resource constraints via DKD”	(Alikhani, 3 Mar 2026)
2-category theory	Kan injectivity and induced KZ-pseudomonads	(Liberti et al., 2022)
Other mnemonic uses	KAN summaries in equation discovery, phase transitions, QuIRK, kaon ID	(Pal et al., 2024, Xu et al., 5 Mar 2025, Sharma et al., 9 Oct 2025, Massri, 2016)

This multiplicity matters because the same string spans two different meanings of “KAN”: Kolmogorov–Arnold networks in machine learning and Kan extensions or Kan injectivity in category theory. A plausible implication is that any technical reading of KAN-IDIR must be anchored to its source paper rather than inferred from the acronym alone.

2. KAN-IDIR in deformable image registration

In deformable image registration, KAN-IDIR and RandKAN-IDIR are introduced as the first integration of Kolmogorov-Arnold Networks into DIR with implicit neural representations (Drozdov et al., 26 Sep 2025). The problem is posed by seeking a continuous displacement field $u(x)\in\mathbb{R}^d$ on a domain $\Omega\subset\mathbb{R}^d$, with transformation

$\phi(x)=x+u(x).$

The instance-specific objective combines image similarity, regularization, and a Jacobian-based anti-folding penalty:

$$L(\phi)=L_{\mathrm{sim}}(I_f,I_m\circ\phi)+\lambda R(u)+\beta P(J_\phi),$$

with $J_\phi(x)=\nabla\phi(x)=I+\nabla u(x)$.

The default similarity term is the negative normalized cross-correlation computed over minibatches of coordinates. Regularization is total variation on the displacement gradient,

$$R_{\mathrm{TV}}(u)=\frac{1}{|\Omega_p|}\sum_{p\in\Omega_p}\|\nabla u(p)\|_1,$$

and folding reduction is handled by

$$P_{J\det}=\frac{1}{|\Omega_p|}\sum_{p\in\Omega_p}\max(0,-|J_\phi(p)|+\epsilon),$$

with $\epsilon=0.1$. The reported loss is

$$L=L_{\mathrm{sim}}+\lambda L_{\mathrm{smooth}}+\gamma L_{J\det},$$

with $\lambda=0.4$ and $\Omega\subset\mathbb{R}^d$0.

The network is a Chebyshev-KAN INR with two hidden layers of width $\Omega\subset\mathbb{R}^d$1. A single layer computes

$\Omega\subset\mathbb{R}^d$2

where $\Omega\subset\mathbb{R}^d$3 are Chebyshev polynomials and inputs are normalized to $\Omega\subset\mathbb{R}^d$4 via $\Omega\subset\mathbb{R}^d$5. Each layer also includes a learnable linear skip connection,

$\Omega\subset\mathbb{R}^d$6

with $\Omega\subset\mathbb{R}^d$7 given as SiLU. The full INR is a composition of these layers, and its output head maps coordinates to $\Omega\subset\mathbb{R}^d$8.

KAN-IDIR uses a sequential basis set $\Omega\subset\mathbb{R}^d$9 shared across layers. RandKAN-IDIR instead samples

$\phi(x)=x+u(x).$0

which reduces the number of evaluated basis functions while preserving diversity across layers. The main experiments use $\phi(x)=x+u(x).$1 and $\phi(x)=x+u(x).$2.

The optimization is instance-specific rather than dataset-trained. The model is initialized randomly and trained with Adam at learning rate $\phi(x)=x+u(x).$3 for $\phi(x)=x+u(x).$4 iterations, keeping the learning rate constant for the first $\phi(x)=x+u(x).$5 of iterations and then using cosine annealing. Each iteration samples $\phi(x)=x+u(x).$6 points from an organ mask or application-specific spatial subset. This places KAN-IDIR between classical iterative registration and dataset-trained CNN or transformer pipelines: it avoids training on large cohorts, but still requires per-pair optimization.

3. Empirical behavior of KAN-IDIR and RandKAN-IDIR

The reported evaluation spans lung CT, brain MRI, and cardiac MRI, with comparisons to other INR methods, dataset-trained deep learning models, and classical registration (Drozdov et al., 26 Sep 2025). On DIR-Lab, performance is measured by TRE; on OASIS-1 and ACDC, by DSC, HD95, and NJD.

Dataset	KAN-IDIR	RandKAN-IDIR
DIR-Lab	TRE $\phi(x)=x+u(x).$7 mm; $\phi(x)=x+u(x).$8 s; $\phi(x)=x+u(x).$9 GB	TRE $$L(\phi)=L_{\mathrm{sim}}(I_f,I_m\circ\phi)+\lambda R(u)+\beta P(J_\phi),$$0 mm; $$L(\phi)=L_{\mathrm{sim}}(I_f,I_m\circ\phi)+\lambda R(u)+\beta P(J_\phi),$$1 s; $$L(\phi)=L_{\mathrm{sim}}(I_f,I_m\circ\phi)+\lambda R(u)+\beta P(J_\phi),$$2 GB
OASIS-1	DSC $$L(\phi)=L_{\mathrm{sim}}(I_f,I_m\circ\phi)+\lambda R(u)+\beta P(J_\phi),$$3; HD95 $$L(\phi)=L_{\mathrm{sim}}(I_f,I_m\circ\phi)+\lambda R(u)+\beta P(J_\phi),$$4 mm; NJD $$L(\phi)=L_{\mathrm{sim}}(I_f,I_m\circ\phi)+\lambda R(u)+\beta P(J_\phi),$$5	DSC $$L(\phi)=L_{\mathrm{sim}}(I_f,I_m\circ\phi)+\lambda R(u)+\beta P(J_\phi),$$6; HD95 $$L(\phi)=L_{\mathrm{sim}}(I_f,I_m\circ\phi)+\lambda R(u)+\beta P(J_\phi),$$7 mm; NJD $$L(\phi)=L_{\mathrm{sim}}(I_f,I_m\circ\phi)+\lambda R(u)+\beta P(J_\phi),$$8
ACDC	DSC $$L(\phi)=L_{\mathrm{sim}}(I_f,I_m\circ\phi)+\lambda R(u)+\beta P(J_\phi),$$9; HD95 $J_\phi(x)=\nabla\phi(x)=I+\nabla u(x)$0 mm; NJD $J_\phi(x)=\nabla\phi(x)=I+\nabla u(x)$1	DSC $J_\phi(x)=\nabla\phi(x)=I+\nabla u(x)$2; HD95 $J_\phi(x)=\nabla\phi(x)=I+\nabla u(x)$3 mm; NJD $J_\phi(x)=\nabla\phi(x)=I+\nabla u(x)$4

On DIR-Lab, the outlier distribution for KAN-IDIR is reported as $J_\phi(x)=\nabla\phi(x)=I+\nabla u(x)$5, $J_\phi(x)=\nabla\phi(x)=I+\nabla u(x)$6, $J_\phi(x)=\nabla\phi(x)=I+\nabla u(x)$7, and $J_\phi(x)=\nabla\phi(x)=I+\nabla u(x)$8 for thresholds $J_\phi(x)=\nabla\phi(x)=I+\nabla u(x)$9 mm, $$R_{\mathrm{TV}}(u)=\frac{1}{|\Omega_p|}\sum_{p\in\Omega_p}\|\nabla u(p)\|_1,$$0 mm, $$R_{\mathrm{TV}}(u)=\frac{1}{|\Omega_p|}\sum_{p\in\Omega_p}\|\nabla u(p)\|_1,$$1 mm, and $$R_{\mathrm{TV}}(u)=\frac{1}{|\Omega_p|}\sum_{p\in\Omega_p}\|\nabla u(p)\|_1,$$2 mm; RandKAN-IDIR reports $$R_{\mathrm{TV}}(u)=\frac{1}{|\Omega_p|}\sum_{p\in\Omega_p}\|\nabla u(p)\|_1,$$3, $$R_{\mathrm{TV}}(u)=\frac{1}{|\Omega_p|}\sum_{p\in\Omega_p}\|\nabla u(p)\|_1,$$4, $$R_{\mathrm{TV}}(u)=\frac{1}{|\Omega_p|}\sum_{p\in\Omega_p}\|\nabla u(p)\|_1,$$5, and $$R_{\mathrm{TV}}(u)=\frac{1}{|\Omega_p|}\sum_{p\in\Omega_p}\|\nabla u(p)\|_1,$$6. Classical pTV attains $$R_{\mathrm{TV}}(u)=\frac{1}{|\Omega_p|}\sum_{p\in\Omega_p}\|\nabla u(p)\|_1,$$7 mm TRE but requires approximately $$R_{\mathrm{TV}}(u)=\frac{1}{|\Omega_p|}\sum_{p\in\Omega_p}\|\nabla u(p)\|_1,$$8 s, while IDIR requires $$R_{\mathrm{TV}}(u)=\frac{1}{|\Omega_p|}\sum_{p\in\Omega_p}\|\nabla u(p)\|_1,$$9 s and ccIDIR $$P_{J\det}=\frac{1}{|\Omega_p|}\sum_{p\in\Omega_p}\max(0,-|J_\phi(p)|+\epsilon),$$0 s. This makes the KAN variants competitive among INR methods while remaining substantially faster than pTV.

The paper emphasizes seed stability. Across $$P_{J\det}=\frac{1}{|\Omega_p|}\sum_{p\in\Omega_p}\max(0,-|J_\phi(p)|+\epsilon),$$1 random seeds on DIR-Lab, the difference in $$P_{J\det}=\frac{1}{|\Omega_p|}\sum_{p\in\Omega_p}\max(0,-|J_\phi(p)|+\epsilon),$$2 mm outlier counts between best and worst seed is $$P_{J\det}=\frac{1}{|\Omega_p|}\sum_{p\in\Omega_p}\max(0,-|J_\phi(p)|+\epsilon),$$3 for KAN-IDIR versus $$P_{J\det}=\frac{1}{|\Omega_p|}\sum_{p\in\Omega_p}\max(0,-|J_\phi(p)|+\epsilon),$$4 for the SIREN-based baseline. The reported interpretation is that KAN-based coordinate-to-displacement maps are less seed-sensitive than MLP or SIREN INRs.

RandKAN-IDIR is also compared with an adaptive index-learning variant, A-KAN-IDIR. RandKAN-IDIR is reported to slightly outperform A-KAN-IDIR while eliminating its additional training-time complexity, with approximately $$P_{J\det}=\frac{1}{|\Omega_p|}\sum_{p\in\Omega_p}\max(0,-|J_\phi(p)|+\epsilon),$$5 runtime reduction and approximately $$P_{J\det}=\frac{1}{|\Omega_p|}\sum_{p\in\Omega_p}\max(0,-|J_\phi(p)|+\epsilon),$$6 less VRAM in the reported tests. This makes randomized basis sampling the preferred sparse-basis variant in the study.

The accuracy–efficiency trade-off is described as nearly linear in basis count. Increasing $$P_{J\det}=\frac{1}{|\Omega_p|}\sum_{p\in\Omega_p}\max(0,-|J_\phi(p)|+\epsilon),$$7 in KAN-IDIR or $$P_{J\det}=\frac{1}{|\Omega_p|}\sum_{p\in\Omega_p}\max(0,-|J_\phi(p)|+\epsilon),$$8 in RandKAN-IDIR improves accuracy but scales runtime and memory with $$P_{J\det}=\frac{1}{|\Omega_p|}\sum_{p\in\Omega_p}\max(0,-|J_\phi(p)|+\epsilon),$$9. On DIR-Lab, TRE improves up to about $\epsilon=0.1$0 and then plateaus. For RandKAN-IDIR, randomized sampling with $\epsilon=0.1$1, $\epsilon=0.1$2 is reported to exceed the sequential choice $\epsilon=0.1$3, $\epsilon=0.1$4 by $\epsilon=0.1$5 DSC and to match $\epsilon=0.1$6, $\epsilon=0.1$7 while running faster and using less memory.

4. KAN-IDIR as intrusion detection under resource constraints

In industrial control system security, KAN-IDIR denotes a lightweight intrusion-detection framework that uses a high-capacity KAN teacher and a tiny MLP student trained by decoupled knowledge distillation (Alikhani, 3 Mar 2026). The target setting is binary cyber-attack detection for SWaT and WADI, both operating at about $\epsilon=0.1$8 Hz sampling and involving multivariate sensor and actuator streams. The method is explicitly designed for edge and real-time CPS environments in which model size, inference latency, and memory use are critical.

The teacher is a KAN modeled through the Kolmogorov–Arnold representation

$\epsilon=0.1$9

with univariate edge activations defined as

$$L=L_{\mathrm{sim}}+\lambda L_{\mathrm{smooth}}+\gamma L_{J\det},$$0

The reported KAN parameter count is

$$L=L_{\mathrm{sim}}+\lambda L_{\mathrm{smooth}}+\gamma L_{J\det},$$1

Teacher settings are $$L=L_{\mathrm{sim}}+\lambda L_{\mathrm{smooth}}+\gamma L_{J\det},$$2, $$L=L_{\mathrm{sim}}+\lambda L_{\mathrm{smooth}}+\gamma L_{J\det},$$3 for WADI and $$L=L_{\mathrm{sim}}+\lambda L_{\mathrm{smooth}}+\gamma L_{J\det},$$4, $$L=L_{\mathrm{sim}}+\lambda L_{\mathrm{smooth}}+\gamma L_{J\det},$$5 for SWaT.

The student is a compact MLP trained from hard labels and a DKD objective. The decomposition

$$L=L_{\mathrm{sim}}+\lambda L_{\mathrm{smooth}}+\gamma L_{J\det},$$6

separates target-class and non-target-class transfer. The total loss uses a warm-up schedule blending hard-label loss with DKD. Reported hyperparameters are $$L=L_{\mathrm{sim}}+\lambda L_{\mathrm{smooth}}+\gamma L_{J\det},$$7, $$L=L_{\mathrm{sim}}+\lambda L_{\mathrm{smooth}}+\gamma L_{J\det},$$8, $$L=L_{\mathrm{sim}}+\lambda L_{\mathrm{smooth}}+\gamma L_{J\det},$$9, $\lambda=0.4$0 for SWaT, and $\lambda=0.4$1, $\lambda=0.4$2, $\lambda=0.4$3, $\lambda=0.4$4 for WADI. Inputs are standardized by

$\lambda=0.4$5

The resulting student models are extremely small: $\lambda=0.4$6 parameters for SWaT and $\lambda=0.4$7 for WADI. On SWaT, the bare student reports Acc $\lambda=0.4$8, Prec $\lambda=0.4$9, Rec $\Omega\subset\mathbb{R}^d$00, F1 $\Omega\subset\mathbb{R}^d$01, whereas DKD-MLP reports Acc $\Omega\subset\mathbb{R}^d$02, Prec $\Omega\subset\mathbb{R}^d$03, Rec $\Omega\subset\mathbb{R}^d$04, F1 $\Omega\subset\mathbb{R}^d$05, a gain of $\Omega\subset\mathbb{R}^d$06 F1 points. On WADI, the bare student reports Acc $\Omega\subset\mathbb{R}^d$07, Prec $\Omega\subset\mathbb{R}^d$08, Rec $\Omega\subset\mathbb{R}^d$09, F1 $\Omega\subset\mathbb{R}^d$10, whereas DKD-MLP reports Acc $\Omega\subset\mathbb{R}^d$11, Prec $\Omega\subset\mathbb{R}^d$12, Rec $\Omega\subset\mathbb{R}^d$13, F1 $\Omega\subset\mathbb{R}^d$14, a gain of $\Omega\subset\mathbb{R}^d$15 F1 points. The WADI KAN teacher has $\Omega\subset\mathbb{R}^d$16 parameters and reports Acc $\Omega\subset\mathbb{R}^d$17, Prec $\Omega\subset\mathbb{R}^d$18, Rec $\Omega\subset\mathbb{R}^d$19, F1 $\Omega\subset\mathbb{R}^d$20.

The framework uses per-instance inference with no sliding windows and only standardization preprocessing. The reported memory footprint is approximately $\Omega\subset\mathbb{R}^d$21 KB for the SWaT student and approximately $\Omega\subset\mathbb{R}^d$22 KB for the WADI student, assuming float32 weights. Estimated compute per instance is approximately $\Omega\subset\mathbb{R}^d$23k MACs for SWaT and approximately $\Omega\subset\mathbb{R}^d$24k MACs for WADI. Throughput, latency, and energy are not measured in the paper, but the student is explicitly positioned for PLC or edge-gateway deployment. The paper also notes a reporting inconsistency for SWaT: a results table labels the teacher as an “MLP teacher,” while the narrative consistently describes a KAN teacher.

5. KAN-IDIR in 2-category theory

In category theory, KAN-IDIR denotes Kan injectivity and induced KZ-pseudomonads (Liberti et al., 2022). The setting is an adequate 2-category $\Omega\subset\mathbb{R}^d$25 that is locally small, has the required small weighted bicolimits, and satisfies a smallness hypothesis ensuring convergence of transfinite pseudochains. The central object is left Kan injectivity with respect to a set $\Omega\subset\mathbb{R}^d$26 of 1-cells.

For $\Omega\subset\mathbb{R}^d$27 and $\Omega\subset\mathbb{R}^d$28, a left Kan extension of $\Omega\subset\mathbb{R}^d$29 along $\Omega\subset\mathbb{R}^d$30 is a pair $\Omega\subset\mathbb{R}^d$31 with $\Omega\subset\mathbb{R}^d$32 and $\Omega\subset\mathbb{R}^d$33 satisfying the usual universal property. Equivalently, precomposition with $\Omega\subset\mathbb{R}^d$34 induces an adjunction on hom-categories, with left adjoint $\Omega\subset\mathbb{R}^d$35. The universal property is expressed by the natural isomorphism

$\Omega\subset\mathbb{R}^d$36

Two forms of injectivity are distinguished. Weak left Kan injectivity of $\Omega\subset\mathbb{R}^d$37 with respect to $\Omega\subset\mathbb{R}^d$38 requires existence of the relevant left adjoints $\Omega\subset\mathbb{R}^d$39, or equivalently that $\Omega\subset\mathbb{R}^d$40 be a right adjoint. Strong left Kan injectivity further requires the canonical Kan 2-cell $\Omega\subset\mathbb{R}^d$41 to be invertible. In the presence of bicocomma objects, weak and strong notions coincide after saturation, written $\Omega\subset\mathbb{R}^d$42 for a canonical closure $\Omega\subset\mathbb{R}^d$43.

The induced KZ-pseudomonad arises from a transfinite pseudochain. For each object $\Omega\subset\mathbb{R}^d$44, one builds $\Omega\subset\mathbb{R}^d$45 by inserting 1-dimensional Kan-extension data via wide bipushouts and 2-dimensional universality constraints via bicoequinserters or bicoequifiers, taking bicolimits at limit ordinals. Choosing a sufficiently large regular cardinal $\Omega\subset\mathbb{R}^d$46, one sets

$\Omega\subset\mathbb{R}^d$47

By Marmolejo–Wood’s left Kan pseudomonad construction, this yields a KZ-pseudomonad $\Omega\subset\mathbb{R}^d$48.

The main theorem identifies the pseudoalgebras of $\Omega\subset\mathbb{R}^d$49 with the objects left Kan injective with respect to $\Omega\subset\mathbb{R}^d$50, and the pseudoalgebra morphisms with the morphisms preserving Kan extensions along $\Omega\subset\mathbb{R}^d$51. A pseudoalgebra structure map $\Omega\subset\mathbb{R}^d$52 satisfies

$\Omega\subset\mathbb{R}^d$53

Examples in $\Omega\subset\mathbb{R}^d$54 recover familiar cocompletion doctrines. With appropriate choices of $\Omega\subset\mathbb{R}^d$55, one obtains finite-colimit completion and links to Garner–Lack’s $\Omega\subset\mathbb{R}^d$56-lex-colimit KZ-pseudomonads. In this literature, KAN-IDIR has no connection to Kolmogorov–Arnold networks; it is entirely about Kan extensions, injectivity, and lax-idempotent 2-monadic structure.

6. Related KAN-IDIR-style usages in adjacent literatures

Several other papers use the string as a mnemonic rather than as the official title of a method. These uses are technically separate but are informative because they emphasize recurring themes of interpretability, sparsity, and structure discovery.

In nonlinear equation discovery, KAN/MultKAN combined with Sequentially Regularized Derivatives for Denoising and Physics-Informed Spline Fitting forms a pipeline for identifying governing ODEs and PDEs from noisy data (Pal et al., 2024). The framework proceeds through SRDD denoising, KAN-guided library construction, and PISF-based term elimination. It was tested on the forced Duffing oscillator, Van der Pol oscillator, Burgers’ equation, and the Bouc–Wen model; it converged to the true equation for the first three and produced an approximate hysteresis-capable model for Bouc–Wen. The summary explicitly attributes the framework to Pal et al. (2024). A plausible implication is that the mnemonic use of KAN-IDIR here foregrounds interpretable discovery rather than a fixed acronym expansion.

In statistical physics, a KAN-based classifier is trained directly on raw site-percolation and Ising configurations (Xu et al., 5 Mar 2025). The reported architecture uses input size $\Omega\subset\mathbb{R}^d$57, a hidden layer of $\Omega\subset\mathbb{R}^d$58 nodes, and an output layer of $\Omega\subset\mathbb{R}^d$59 nodes, with edge activations $\Omega\subset\mathbb{R}^d$60 and regularization based on activation $\Omega\subset\mathbb{R}^d$61 norm and entropy. At $\Omega\subset\mathbb{R}^d$62 training steps, reported accuracies reach $\Omega\subset\mathbb{R}^d$63, $\Omega\subset\mathbb{R}^d$64, $\Omega\subset\mathbb{R}^d$65, and $\Omega\subset\mathbb{R}^d$66 for Ising at $\Omega\subset\mathbb{R}^d$67, and $\Omega\subset\mathbb{R}^d$68, $\Omega\subset\mathbb{R}^d$69, $\Omega\subset\mathbb{R}^d$70, and $\Omega\subset\mathbb{R}^d$71 for percolation. Finite-size scaling of the learned output yields $\Omega\subset\mathbb{R}^d$72 for the Ising model and $\Omega\subset\mathbb{R}^d$73 for percolation. Here the string is used to organize a KAN-based identification program rather than to name a separate architecture family.

In "QuIRK: Quantum-Inspired Re-uploading KAN," the explanatory label “KAN-IDIR” is expanded as Kolmogorov–Arnold Networks, Data Re-uploading, Interpretability, and closed-form Derivations (Sharma et al., 9 Oct 2025). QuIRK replaces B-spline edge functions with single-qubit data re-uploading models, so each edge function becomes a trigonometric polynomial obtained from a single-qubit expectation. A representative expression is

$\Omega\subset\mathbb{R}^d$74

and the paper states that QuIRK retains interpretability and closed-form solutions while often using fewer parameters than classical KANs, especially for periodic targets.

A further mnemonic usage appears in the NA62 kaon-identification literature, where the supplied summary maps KAN-IDIR to “kaon identification in-rate differential Cherenkov ring tagging” (Massri, 2016). The underlying system is the CEDAR counter upgraded to KTAG, operating in a $\Omega\subset\mathbb{R}^d$75 MHz unseparated hadron beam with a kaon component of about $\Omega\subset\mathbb{R}^d$76. The reported performance includes kaon-tagging efficiency exceeding $\Omega\subset\mathbb{R}^d$77 with at least $\Omega\subset\mathbb{R}^d$78 sectors in coincidence, pion misidentification of $\Omega\subset\mathbb{R}^d$79, and overall event-time resolution of about $\Omega\subset\mathbb{R}^d$80 ps. This usage is terminologically incidental, but it shows that KAN-IDIR can also be repurposed outside the Kolmogorov–Arnold and categorical contexts.

Across these variants, the string functions less as a stable acronym than as a movable label attached to structure-aware inference or identification. This suggests that “KAN-IDIR” is best treated bibliographically as a context-sensitive term whose meaning is determined entirely by the paper in which it appears.