Tsukamoto’s Approach: Local-to-Global Methods

• Tsukamoto’s Approach is a methodological framework that reconstructs global quantities by decomposing them into local, recursive, or asymptotically isolated components.
• It is applied in gravitational lensing to extract logarithmic divergences, in dynamical systems for metric mean dimension localization, and in robotics for adaptive self-body image acquisition.
• The approach enables rigorous variational characterizations, asymptotic observable predictions, and real-time adaptive controls across different scientific and engineering domains.

Searching arXiv for the cited papers to ground the article in the literature. arxiv_search(query="(Lobos et al., 16 Jun 2025) OR (He et al., 11 Jun 2026) OR (Filho et al., 15 Sep 2025) OR (Alibabaei, 2023) OR (Donoso et al., 2021) OR (Yang, 9 Oct 2025) OR (Kawaharazuka et al., 2024)", max_results=10) Tsukamoto’s approach denotes a set of research methodologies associated with Tsukamoto’s work or later constructions explicitly formulated in Tsukamoto-style terms across several domains. In contemporary arXiv literature, the phrase is used in at least three technically distinct senses. In strong-field gravitational lensing, it refers to a refinement of Bozza’s strong-deflection formalism for asymptotically flat, static, spherically symmetric spacetimes, in which the logarithmic divergence of the deflection angle near the photon sphere is isolated and encoded by a small set of coefficients determined by the metric at the unstable circular null orbit (Lobos et al., 16 Jun 2025, Filho et al., 15 Sep 2025). In topological dynamics and mean-dimension theory, it refers to a family of localization, recursive covering, and information-theoretic constructions that recast global invariants through local stable sets, nested covers, or rate-distortion-type quantities (He et al., 11 Jun 2026, Alibabaei, 2023, Yang, 9 Oct 2025). In musculoskeletal robotics, it designates a long-time self-body image acquisition strategy in which a robot continuously learns nonlinear relations among joint angles, muscle tensions, and muscle lengths from sensor data, while maintaining operational safety and control utility (Kawaharazuka et al., 2024). The common structural feature is methodological: each usage replaces a direct global treatment by a decomposition into local, recursively organized, or asymptotically singular components.

1. Strong-deflection gravitational lensing

In black-hole optics, Tsukamoto’s approach is a strong-deflection-limit method for null geodesics in static, spherically symmetric spacetimes. The formalism is built around the unstable circular null orbit, or photon sphere, and the observation that the bending angle diverges logarithmically as the closest approach approaches the photon sphere (Lobos et al., 16 Jun 2025, Filho et al., 15 Sep 2025).

For the quantum-corrected Reissner–Nordström spacetime studied in "Shadow and strong-field lensing effects of quantum-corrected RN blackhole" (Lobos et al., 16 Jun 2025), the metric is written as

$$ds^{2} = -A(r)\,dt^{2}+B(r)\,dr^{2}+C(r)\,d\Omega^{2},$$

with

$B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$

and, for small quantum parameter $\mathrm a$,

$$A(r)_{\rm app}=1-\frac{2M}{r}+\frac{Q^2}{r^2}-\frac{\mathrm{a}^2}{2r^2}+\mathcal{O}(\mathrm{a}^4).$$

The parameter $\mathrm a$ is defined by

$\mathrm{a} = 4\sqrt{\kappa} \equiv 4\ell_p.$

The photon sphere is found from

$$\frac{d}{dr}\left[h^2(r_{\rm ph})\right]=0,\qquad h(r)\equiv \sqrt{\frac{C(r)}{A(r)}},$$

equivalently

$$C'(r_{\rm ph})A(r_{\rm ph})-C(r_{\rm ph})A'(r_{\rm ph})=0,$$

yielding

$$r_{\text{ph}}=\frac{3M}{2}+\frac{\sqrt{9M^2-8Q^2+4\mathrm{a}^2}}{2}.$$

The shadow radius is then computed from

$$r_{\rm sh}=\sqrt{\frac{C(r)}{A(r)}\Bigg|_{r_{\rm ph}} }.$$

The lensing sector is formulated through

$B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$0

with

$B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$1

Tsukamoto’s refinement introduces

$B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$2

and splits the integral into divergent and regular parts,

$B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$3

Near the photon sphere the bending angle takes the logarithmic form

$B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$4

The generic coefficients are

$B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$5

and

$B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$6

A closely related application appears in "Black Hole Gravitational Phenomena in Higher-Order Curvature-Scalar Gravity" (Filho et al., 15 Sep 2025). There the spacetime is

$B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$7

with

$B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$8

The closest-approach impact parameter is

$B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$9

the photon sphere is determined by

$\mathrm a$0

and explicitly becomes

$\mathrm a$1

The critical impact parameter is

$\mathrm a$2

and the strong-field deflection angle is written as

$\mathrm a$3

Here

$\mathrm a$4

and

$\mathrm a$5

This lensing usage of Tsukamoto’s approach is therefore not a generic name for black-hole imaging, but a specific asymptotic method: identify the photon sphere, transform the bending integral by $\mathrm a$6, isolate the divergent part, and express the logarithmic blow-up through coefficients fixed by the metric at the photon sphere (Lobos et al., 16 Jun 2025, Filho et al., 15 Sep 2025).

2. Optical observables and empirical bounds

The practical significance of the strong-deflection method lies in its direct connection between local null-geodesic structure and observable quantities such as the critical impact parameter, shadow radius, angular image accumulation, and strong-lensing image separation (Lobos et al., 16 Jun 2025, Filho et al., 15 Sep 2025).

In the quantum-corrected Reissner–Nordström analysis, the paper states that charge $\mathrm a$7 decreases the photon-sphere radius $\mathrm a$8, while the quantum correction parameter $\mathrm a$9 increases it (Lobos et al., 16 Jun 2025). The corresponding small-correction shadow expression is

$$A(r)_{\rm app}=1-\frac{2M}{r}+\frac{Q^2}{r^2}-\frac{\mathrm{a}^2}{2r^2}+\mathcal{O}(\mathrm{a}^4).$$0

This makes explicit that $$A(r)_{\rm app}=1-\frac{2M}{r}+\frac{Q^2}{r^2}-\frac{\mathrm{a}^2}{2r^2}+\mathcal{O}(\mathrm{a}^4).$$1 decreases the shadow size, whereas $$A(r)_{\rm app}=1-\frac{2M}{r}+\frac{Q^2}{r^2}-\frac{\mathrm{a}^2}{2r^2}+\mathcal{O}(\mathrm{a}^4).$$2 increases it. The same paper reports that $$A(r)_{\rm app}=1-\frac{2M}{r}+\frac{Q^2}{r^2}-\frac{\mathrm{a}^2}{2r^2}+\mathcal{O}(\mathrm{a}^4).$$3 has a strong effect near the photon sphere, that the deflection angle changes significantly as $$A(r)_{\rm app}=1-\frac{2M}{r}+\frac{Q^2}{r^2}-\frac{\mathrm{a}^2}{2r^2}+\mathcal{O}(\mathrm{a}^4).$$4 varies close to the critical radius, and that far from the photon sphere the different curves converge and the charge dependence becomes weak. By contrast, $$A(r)_{\rm app}=1-\frac{2M}{r}+\frac{Q^2}{r^2}-\frac{\mathrm{a}^2}{2r^2}+\mathcal{O}(\mathrm{a}^4).$$5 has negligible impact near the photon sphere at present observational precision, but increasing $$A(r)_{\rm app}=1-\frac{2M}{r}+\frac{Q^2}{r^2}-\frac{\mathrm{a}^2}{2r^2}+\mathcal{O}(\mathrm{a}^4).$$6 leads to a slight increase in shadow radius and a slight suppression of the strong deflection angle (Lobos et al., 16 Jun 2025).

The paper compares its shadow predictions with Event Horizon Telescope constraints and states the following ranges. For Sgr A*,

$$A(r)_{\rm app}=1-\frac{2M}{r}+\frac{Q^2}{r^2}-\frac{\mathrm{a}^2}{2r^2}+\mathcal{O}(\mathrm{a}^4).$$7

For M87*,

$$A(r)_{\rm app}=1-\frac{2M}{r}+\frac{Q^2}{r^2}-\frac{\mathrm{a}^2}{2r^2}+\mathcal{O}(\mathrm{a}^4).$$8

Its main conclusion is that even for large charge the shadow remains within EHT observational bounds for both M87* and Sgr A* (Lobos et al., 16 Jun 2025).

In the higher-order curvature-scalar gravity study, Tsukamoto’s coefficients are used to define standard observables for relativistic images: $$A(r)_{\rm app}=1-\frac{2M}{r}+\frac{Q^2}{r^2}-\frac{\mathrm{a}^2}{2r^2}+\mathcal{O}(\mathrm{a}^4).$$9 with

$\mathrm a$0

The shadow angular diameter is

$\mathrm a$1

Using EHT-measured angular diameters, the paper finds

$\mathrm a$2

for M87$\mathrm a$3, and

$\mathrm a$4

for Sgr A$\mathrm a$5 (Filho et al., 15 Sep 2025).

A plausible implication is that, within this usage, Tsukamoto’s approach functions as an observational reduction scheme: local data at the photon sphere are propagated through logarithmic asymptotics to quantities directly comparable with EHT shadow measurements.

3. Localization in metric mean dimension

In dynamical systems, Tsukamoto’s approach denotes a localization principle for metric mean dimension. The central object is the pointwise $\mathrm a$6-stable set

$\mathrm a$7

which collects points that remain $\mathrm a$8-close to $\mathrm a$9 along the entire $\mathrm{a} = 4\sqrt{\kappa} \equiv 4\ell_p.$0-orbit (He et al., 11 Jun 2026).

The paper "Metric mean dimension of amenable group actions: localization and non-uniformity" (He et al., 11 Jun 2026) presents Tsukamoto’s original idea as the recovery of a global scale-sensitive invariant from the entropy of these small stable sets. For $\mathrm{a} = 4\sqrt{\kappa} \equiv 4\ell_p.$1- and $\mathrm{a} = 4\sqrt{\kappa} \equiv 4\ell_p.$2-/$\mathrm{a} = 4\sqrt{\kappa} \equiv 4\ell_p.$3-actions, the localization formula is described in the form

$\mathrm{a} = 4\sqrt{\kappa} \equiv 4\ell_p.$4

for every fixed $\mathrm{a} = 4\sqrt{\kappa} \equiv 4\ell_p.$5. The 2026 paper extends this viewpoint to actions of countable discrete amenable groups and proves, for every tempered Følner sequence $\mathrm{a} = 4\sqrt{\kappa} \equiv 4\ell_p.$6 satisfying $\mathrm{a} = 4\sqrt{\kappa} \equiv 4\ell_p.$7, that

$\mathrm{a} = 4\sqrt{\kappa} \equiv 4\ell_p.$8

$\mathrm{a} = 4\sqrt{\kappa} \equiv 4\ell_p.$9

This formulation is accompanied by equivalent upper metric mean dimension definitions for any nonempty $$\frac{d}{dr}\left[h^2(r_{\rm ph})\right]=0,\qquad h(r)\equiv \sqrt{\frac{C(r)}{A(r)}},$$0: $$\frac{d}{dr}\left[h^2(r_{\rm ph})\right]=0,\qquad h(r)\equiv \sqrt{\frac{C(r)}{A(r)}},$$1

$$\frac{d}{dr}\left[h^2(r_{\rm ph})\right]=0,\qquad h(r)\equiv \sqrt{\frac{C(r)}{A(r)}},$$2

$$\frac{d}{dr}\left[h^2(r_{\rm ph})\right]=0,\qquad h(r)\equiv \sqrt{\frac{C(r)}{A(r)}},$$3

For the whole system, Theorem 5 yields

$$\frac{d}{dr}\left[h^2(r_{\rm ph})\right]=0,\qquad h(r)\equiv \sqrt{\frac{C(r)}{A(r)}},$$4

and also

$$\frac{d}{dr}\left[h^2(r_{\rm ph})\right]=0,\qquad h(r)\equiv \sqrt{\frac{C(r)}{A(r)}},$$5

The technical novelty of the amenable-group extension is the replacement of tiling arguments by Lindenstrauss’s combinatorial covering lemma. Proposition 14 states that if the local Bowen-dimensional entropy of all stable sets $$\frac{d}{dr}\left[h^2(r_{\rm ph})\right]=0,\qquad h(r)\equiv \sqrt{\frac{C(r)}{A(r)}},$$6 is bounded above by $$\frac{d}{dr}\left[h^2(r_{\rm ph})\right]=0,\qquad h(r)\equiv \sqrt{\frac{C(r)}{A(r)}},$$7, then for small $$\frac{d}{dr}\left[h^2(r_{\rm ph})\right]=0,\qquad h(r)\equiv \sqrt{\frac{C(r)}{A(r)}},$$8,

$$\frac{d}{dr}\left[h^2(r_{\rm ph})\right]=0,\qquad h(r)\equiv \sqrt{\frac{C(r)}{A(r)}},$$9

This estimate is identified as the amenable-group analogue of Bowen’s key covering estimate (He et al., 11 Jun 2026).

The same paper also establishes a limitation of the localization principle. It defines

$$C'(r_{\rm ph})A(r_{\rm ph})-C(r_{\rm ph})A'(r_{\rm ph})=0,$$0

$$C'(r_{\rm ph})A(r_{\rm ph})-C(r_{\rm ph})A'(r_{\rm ph})=0,$$1

with analogous lower quantities. Theorem 7 gives a $$C'(r_{\rm ph})A(r_{\rm ph})-C(r_{\rm ph})A'(r_{\rm ph})=0,$$2-subshift $$C'(r_{\rm ph})A(r_{\rm ph})-C(r_{\rm ph})A'(r_{\rm ph})=0,$$3 such that for every $$C'(r_{\rm ph})A(r_{\rm ph})-C(r_{\rm ph})A'(r_{\rm ph})=0,$$4,

$$C'(r_{\rm ph})A(r_{\rm ph})-C(r_{\rm ph})A'(r_{\rm ph})=0,$$5

Thus the supremum and small-scale limit cannot generally be interchanged (He et al., 11 Jun 2026).

This establishes a characteristic feature of Tsukamoto’s localization approach: the global invariant is indeed recoverable from local stable sets, but not uniformly across phase space.

4. Recursive coverings and weighted topological pressure

A second dynamical-systems usage of Tsukamoto’s approach appears in weighted entropy and weighted pressure. The paper "Weighted topological pressure revisited" (Alibabaei, 2023) describes Tsukamoto’s 2022 contribution as a redefinition of weighted pressure in a more “nested covering” style for the case of two systems, and then generalizes that idea to an arbitrary finite chain of factor maps.

The setting is a sequence

$$C'(r_{\rm ph})A(r_{\rm ph})-C(r_{\rm ph})A'(r_{\rm ph})=0,$$6

together with

$$C'(r_{\rm ph})A(r_{\rm ph})-C(r_{\rm ph})A'(r_{\rm ph})=0,$$7

From $$C'(r_{\rm ph})A(r_{\rm ph})-C(r_{\rm ph})A'(r_{\rm ph})=0,$$8 one forms

$$C'(r_{\rm ph})A(r_{\rm ph})-C(r_{\rm ph})A'(r_{\rm ph})=0,$$9

via

$$r_{\text{ph}}=\frac{3M}{2}+\frac{\sqrt{9M^2-8Q^2+4\mathrm{a}^2}}{2}.$$0

These weights sum to $$r_{\text{ph}}=\frac{3M}{2}+\frac{\sqrt{9M^2-8Q^2+4\mathrm{a}^2}}{2}.$$1 (Alibabaei, 2023).

The new weighted entropy is defined recursively. With

$$r_{\text{ph}}=\frac{3M}{2}+\frac{\sqrt{9M^2-8Q^2+4\mathrm{a}^2}}{2}.$$2

one first defines $$r_{\text{ph}}=\frac{3M}{2}+\frac{\sqrt{9M^2-8Q^2+4\mathrm{a}^2}}{2}.$$3 as the minimal number of open sets covering $$r_{\text{ph}}=\frac{3M}{2}+\frac{\sqrt{9M^2-8Q^2+4\mathrm{a}^2}}{2}.$$4 whose $$r_{\text{ph}}=\frac{3M}{2}+\frac{\sqrt{9M^2-8Q^2+4\mathrm{a}^2}}{2}.$$5-diameter is $$r_{\text{ph}}=\frac{3M}{2}+\frac{\sqrt{9M^2-8Q^2+4\mathrm{a}^2}}{2}.$$6. Then, for $$r_{\text{ph}}=\frac{3M}{2}+\frac{\sqrt{9M^2-8Q^2+4\mathrm{a}^2}}{2}.$$7,

$$r_{\text{ph}}=\frac{3M}{2}+\frac{\sqrt{9M^2-8Q^2+4\mathrm{a}^2}}{2}.$$8

The entropy is

$$r_{\text{ph}}=\frac{3M}{2}+\frac{\sqrt{9M^2-8Q^2+4\mathrm{a}^2}}{2}.$$9

The corresponding pressure uses Birkhoff sums

$$r_{\rm sh}=\sqrt{\frac{C(r)}{A(r)}\Bigg|_{r_{\rm ph}} }.$$0

and defines

$$r_{\rm sh}=\sqrt{\frac{C(r)}{A(r)}\Bigg|_{r_{\rm ph}} }.$$1

then recursively

$$r_{\rm sh}=\sqrt{\frac{C(r)}{A(r)}\Bigg|_{r_{\rm ph}} }.$$2

and finally

$$r_{\rm sh}=\sqrt{\frac{C(r)}{A(r)}\Bigg|_{r_{\rm ph}} }.$$3

The main theorem states

$$r_{\rm sh}=\sqrt{\frac{C(r)}{A(r)}\Bigg|_{r_{\rm ph}} }.$$4

and, for $$r_{\rm sh}=\sqrt{\frac{C(r)}{A(r)}\Bigg|_{r_{\rm ph}} }.$$5,

$$r_{\rm sh}=\sqrt{\frac{C(r)}{A(r)}\Bigg|_{r_{\rm ph}} }.$$6

The paper interprets this as a higher-dimensional generalization of Tsukamoto’s cover-based redefinition and emphasizes that the method avoids the direct weighted-Bowen-ball formulation of Feng–Huang (Alibabaei, 2023).

This suggests that, in this branch of Tsukamoto’s approach, the decisive move is recursive descent through factor maps: local covering complexities are propagated level by level and then reassembled by exponents $$r_{\rm sh}=\sqrt{\frac{C(r)}{A(r)}\Bigg|_{r_{\rm ph}} }.$$7.

5. Directional and information-theoretic extensions of mean dimension

Tsukamoto’s influence on mean-dimension theory also appears in two adjacent developments: directional mean dimension under expansiveness hypotheses and the information-theoretic reformulation of mean dimension through rate-distortion-type quantities (Donoso et al., 2021, Yang, 9 Oct 2025).

The paper "Directional mean dimension and continuum-wise expansive $$r_{\rm sh}=\sqrt{\frac{C(r)}{A(r)}\Bigg|_{r_{\rm ph}} }.$$8-actions" (Donoso et al., 2021) is positioned as a directional-mean-dimension analogue of the theorem of Meyerovitch and Tsukamoto on mean dimension and expansive multiparameter actions. For a $$r_{\rm sh}=\sqrt{\frac{C(r)}{A(r)}\Bigg|_{r_{\rm ph}} }.$$9-action $B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$00, an $B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$01-dimensional subspace $B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$02, and $B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$03,

$B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$04

and the directional mean dimension is defined by

$B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$05

The main theorem states that if $B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$06 is continuum-wise expansive, then the $B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$07-dimensional directional mean dimension with respect to any direction is finite and uniformly bounded by a finite number depending only on $B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$08 (Donoso et al., 2021).

The same paper proves that directional mean dimension need not be continuous as a function of direction. For any $B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$09, there exists a $B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$10-action $B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$11 such that

$B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$12

This non-continuity result prevents any simple continuity-based transfer of Meyerovitch–Tsukamoto-style finiteness to all directions (Donoso et al., 2021).

The paper "Mean dimension and rate-distortion function revisited" (Yang, 9 Oct 2025) continues the Lindenstrauss–Tsukamoto program in an explicitly information-theoretic direction. It recalls the upper metric mean dimension

$B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$13

with

$B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$14

For the Hilbert cube shift $B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$15, it proves for every invariant measure $B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$16,

$B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$17

thereby answering a question of Gutman and Śpiewak (Yang, 9 Oct 2025).

The same work introduces four rate-distortion entropies and proves that for every ergodic $B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$18 and every $B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$19,

$B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$20

Under the marker property and finite mean dimension, it sharpens the double variational principle to

$B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$21

Taken together, these results show that Tsukamoto’s approach in mean-dimension theory is not confined to one formula. It includes a directional-expansive line, a localization line via stable sets, and an information-theoretic line via rate-distortion and local entropy (Donoso et al., 2021, He et al., 11 Jun 2026, Yang, 9 Oct 2025).

6. Long-time self-body image acquisition in musculoskeletal robotics

A distinct usage of Tsukamoto’s approach appears in robotics, particularly in "Long-time Self-body Image Acquisition and its Application to the Control of Musculoskeletal Structures" (Kawaharazuka et al., 2024). Here the problem is the control of tendon-driven musculoskeletal humanoids whose joint–muscle relationships are highly nonlinear and strongly affected by body softness, friction, muscle interference, and route changes. The paper states that conventional model-based control cannot realize intended movements and proposes a learning control mechanism that acquires the nonlinear relationships among joint angles, muscle tensions, and muscle lengths from actual sensor data (Kawaharazuka et al., 2024).

The self-body image is represented by

$B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$22

where $B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$23 is the ideal joint–muscle mapping and $B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$24 is the muscle-route change model. The paper explicitly separates the learned model into these two networks because the scale of the ideal geometric mapping and the compensation due to tension-induced elongation or route change are very different, and a single network tends to learn poorly or drift (Kawaharazuka et al., 2024).

The long-time acquisition mechanism has three major components: initial training from a geometric model, online learning from actual sensor data, and a safety mechanism to prevent over-tension and overheating. The online data extraction rules are

$B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$25

$B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$26

$B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$27

$B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$28

To update the ideal mapping, the current route-change output is subtracted from measured muscle length; to update the route-change model, the ideal output is subtracted instead (Kawaharazuka et al., 2024).

A major practical contribution is the data accumulation and augmentation pipeline. The minibatch combines one sample from the current extraction, $B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$29 stored samples, $B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$30 constraint samples, and $B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$31 random model-consistency samples, with

$B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$32

For IJMM, constraint data include

$B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$33

and for MRCM, augmentation uses

$B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$34

The safety mechanism modifies target muscle lengths by

$B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$35

$B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$36

$B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$37

with

$B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$38

The mechanism runs every 8 msec (Kawaharazuka et al., 2024).

The learned self-body image supports position control, torque control, and variable stiffness control. The position-control formulation uses

$B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$39

$B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$40

and then either

$B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$41

or

$B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$42

For variable stiffness, operational stiffness is

$B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$43

joint torque is

$B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$44

and the search objective is

$B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$45

with

$B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$46

Experimentally, the paper reports that in a 3-hour self-body image acquisition experiment the average RMSE decreases from about $B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$47 rad to about $B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$48 rad over about the first 40 minutes; that the safety mechanism keeps muscle temperature below $B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$49C and below the burnout threshold $B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$50C even when temperature rises rapidly; and that the learned model supports variable stiffness behavior and impact absorption (Kawaharazuka et al., 2024).

In this robotics usage, Tsukamoto’s approach is not an asymptotic analytic method but a long-time adaptive modeling strategy. The shared principle with the dynamical-systems and lensing usages is nevertheless recognizable: global control performance is achieved by decomposing the problem into structured local corrections learned or evaluated where the nonlinearities are concentrated.

7. Conceptual unity and domain-specific differences

The phrase "Tsukamoto’s approach" does not denote a single cross-disciplinary formalism. Rather, the literature shows several domain-specific methods that share a common methodological architecture while differing sharply in object, scale, and proof technology (Lobos et al., 16 Jun 2025, He et al., 11 Jun 2026, Alibabaei, 2023, Kawaharazuka et al., 2024).

In gravitational lensing, the central operation is asymptotic singularity extraction near the photon sphere. The divergent part of the deflection integral is separated from the regular remainder, and the observable content is compressed into coefficients such as $B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$51 or $B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$52 (Lobos et al., 16 Jun 2025, Filho et al., 15 Sep 2025). In metric mean dimension, the global invariant is localized to pointwise stable sets $B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$53, but the 2026 amenable-group analysis shows that this localization is non-uniform: the supremum over points must generally be taken before the small-scale limit (He et al., 11 Jun 2026). In weighted pressure, the method is recursive and combinatorial: cover complexities are nested through factor maps and reweighted by exponents $B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$54, yielding a variational principle equivalent to the Feng–Huang invariant (Alibabaei, 2023). In the rate-distortion line, the approach becomes information-theoretic, replacing direct topological complexity by $B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2,$55-entropy and coding-cost quantities while preserving variational recovery of mean dimension (Yang, 9 Oct 2025). In robotics, the method becomes online and operational, decomposing the self-body image into ideal geometry plus route-change compensation and stabilizing learning by accumulation, augmentation, and explicit safety logic (Kawaharazuka et al., 2024).

A common misconception is that Tsukamoto’s approach names one universally standardized method. The literature instead supports a narrower statement: it names a family of methods that are recognized by local-to-global reduction, recursive decomposition, or asymptotic isolation of the dominant contribution, with the exact mathematics determined by the host field. Another misconception would be to read the localization formulas or strong-deflection formulas as uniform statements. The amenable-group counterexamples show non-uniformity in metric mean dimension (He et al., 11 Jun 2026), and the black-hole applications show that different deformation parameters can have markedly different observational relevance, with charge effects dominating quantum-correction effects at current precision in the quantum-corrected Reissner–Nordström case (Lobos et al., 16 Jun 2025).

This suggests that the most stable encyclopedia-level characterization is methodological rather than doctrinal. Tsukamoto’s approach, across the literatures in which the term appears, is an approach in which a difficult global quantity is reconstructed from a carefully chosen local, recursive, or singular structure, and in which that reduction is strong enough to support either rigorous variational characterizations, asymptotic lensing formulas, or long-horizon adaptive control.