Reparametrization Invariant Symmetry Scaling (RISS)

• RISS is a framework where scaling behavior is fixed by invariant reparametrization structures, exemplified by the AdS₂ string’s Diff(S¹)/SL(2,ℝ) formulation.
• It unifies diverse applications—from neutrino oscillations to multi-scalar QFT—by organizing physical quantities in a basis-independent, symmetry-controlled manner.
• In neural networks and gravitational models, RISS guides invariant optimization and dynamics, ensuring predictions remain unaffected by arbitrary parameter choices.

Searching arXiv for the cited papers to ground the article in current arXiv records. Reparametrization Invariant Symmetry Scaling (RISS) denotes a class of constructions in which scaling behavior is controlled by reparametrization-invariant structures rather than by a fixed coordinate, basis, or parameter choice. In its most explicit formulation, for the AdS$_2$ string, RISS is the statement that a non-local reparametrization action with conformal kernels and physical $SL(2,\mathbb{R})$ invariance fixes the scaling of defect-CFT correlators and the exact double-scaling form of the out-of-time-order correlator (OTOC) (Giombi et al., 2022). In other areas the terminology is less uniform. A natural broader usage, consistent with later work on neutrino amplitudes, renormalisation-group invariants, neural-network parameter spaces, thermodynamic geometry, and cosmology, identifies RISS with symmetry relations that remain invariant under admissible reparametrizations together with associated scaling operations (Minakata, 2023).

1. Conceptual scope and recurring structure

The designation is not used in a completely uniform way across the cited literature. In the AdS$_2$ string analysis, RISS is introduced directly as an overview of reparametrization invariance, conformal kernels, and symmetry-controlled correlator scaling. In neutrino oscillations and multi-scalar renormalisation-group analyses, the term is presented as a natural definition or as a framing of an already existing symmetry-and-scaling construction. This suggests that RISS is best regarded not as a single formalism, but as a recurring pattern: a quotient or gauge-fixed description is identified, symmetry-compatible scaling data are isolated, and physical quantities are organized so that they do not depend on arbitrary reparametrization choices (Minakata, 2023).

Across the cited works, the invariant object changes with context. For the AdS$_2$ string it is the boundary reparametrization path integral over $\mathrm{Diff}(S^1)/SL(2,\mathbb{R})$. In neutrino oscillations it is the equivalence class of rephased amplitudes $S_{\alpha\beta}^{\text{Reph-1}}$ and $S_{\alpha\beta}^{\text{Reph-2}}$. In multi-scalar field theory it is the basis-independent spurion or invariant-ring description of couplings. In deep learning it is the quotient of parameter space by scaling symmetries, together with a metric or manifold constraint that removes parameterization dependence. In minisuperspace cosmology and thermodynamics it is a gauge-fixed remnant of time or scale reparametrization, often encoded by a first-class constraint or a global scale variable (Pilaftsis, 18 May 2026).

Domain	Invariant structure	Characteristic scaling data
AdS$_2$ string	$\mathrm{Diff}(S^1)/SL(2,\mathbb{R})$ and bilocal kernel	$K_{\text{circle}}=[2\sin(\Delta\theta/2)]^{-2}$, $SL(2,\mathbb{R})$0 (Giombi et al., 2022)
Neutrino oscillations	Rep symmetry acting on rephased amplitudes	$SL(2,\mathbb{R})$1, $SL(2,\mathbb{R})$2 (Minakata, 2023)
Multi-scalar QFT	Spurion charges, SI rays, basis invariants	$SL(2,\mathbb{R})$3 on symmetry-protected directions (Pilaftsis, 18 May 2026)
2HDM symmetry analysis	Basis-invariant ring and symmetry map	relations among invariants or vanishing covariants (Bento et al., 2020)
Deep networks	Quotient by scaling symmetries; metric/manifold methods	SM, UN, $SL(2,\mathbb{R})$4 (Badrinarayanan et al., 2015)
FRW/BBN	Time reparametrization, Möbius residual symmetry, or $SL(2,\mathbb{R})$5	$SL(2,\mathbb{R})$6, CF-type restriction, fitted $SL(2,\mathbb{R})$7 (Achour, 2021)

2. AdS$SL(2,\mathbb{R})$8 string realization

The most developed realization of RISS appears in the AdS$SL(2,\mathbb{R})$9 open-string worldsheet dual to the half-BPS Wilson line. In conformal gauge, the bosonic Polyakov action is written with dynamical longitudinal coordinates $_2$0 and a boundary reparametrization $_2$1, with boundary condition $_2$2. Two distinct $_2$3 symmetries act: a physical $_2$4 corresponding to AdS$_2$5 isometries, and a gauge $_2$6 corresponding to residual worldsheet coordinate transformations. The conformal-gauge analysis leads to a path integral over $_2$7 governed not by the Schwarzian, but by a non-local quadratic action for the fluctuation $_2$8, with Fourier-space dispersion $_2$9 on the circle and $_2$0 on the line (Giombi et al., 2022).

The kernel is the universal one-dimensional conformal kernel,

$_2$1

acting bilocally on $_2$2. Variational derivatives of the transverse action define the dressed bilocal

$_2$3

and correlators are computed as reparametrization averages of powers of $_2$4. Because $_2$5 transforms covariantly under the physical $_2$6 and the reparametrization action is invariant under $_2$7, boundary two-point and normalized four-point functions depend only on the one-dimensional conformal cross-ratio $_2$8. This is one of the defining features of RISS in this setting: scaling laws are fixed by the interplay between the bilocal kernel, the reparametrization measure, and physical $_2$9 covariance (Giombi et al., 2022).

The same framework controls chaos. In the thermal OTOC with $\mathrm{Diff}(S^1)/SL(2,\mathbb{R})$0, the cross-ratio along the out-of-time-order contour is $\mathrm{Diff}(S^1)/SL(2,\mathbb{R})$1. In the double-scaling limit $\mathrm{Diff}(S^1)/SL(2,\mathbb{R})$2, $\mathrm{Diff}(S^1)/SL(2,\mathbb{R})$3 with

$\mathrm{Diff}(S^1)/SL(2,\mathbb{R})$4

held fixed, the exact OTOC is

$\mathrm{Diff}(S^1)/SL(2,\mathbb{R})$5

with early-time growth

$\mathrm{Diff}(S^1)/SL(2,\mathbb{R})$6

and Lyapunov exponent $\mathrm{Diff}(S^1)/SL(2,\mathbb{R})$7, saturating the chaos bound. Although the string reparametrization action is non-local and differs from the Schwarzian of JT gravity, the double-scaled OTOC has the same functional form because both are governed by the same near-horizon eikonal scattering. In the Wilson-line defect CFT, this exact result agrees with the analytic bootstrap prediction to three-loop order at strong coupling (Giombi et al., 2022).

3. Rephasing symmetry and neutrino-amplitude realizations

In neutrino oscillations, the relevant reparametrization is not a boundary diffeomorphism but a discrete exchange of matter eigenstates combined with amplitude rephasing. For a flavor-basis S-matrix $\mathrm{Diff}(S^1)/SL(2,\mathbb{R})$8, the paper defines two physically equivalent rephased amplitudes,

$\mathrm{Diff}(S^1)/SL(2,\mathbb{R})$9

A set of discrete reparametrization symmetries found by the Symmetry Finder method, collectively denoted Symmetry X-DMP with $S_{\alpha\beta}^{\text{Reph-1}}$0 IA, IB, IIA, IIB, IIIA, IIIB, IVA, IVB, exchange the first two matter eigenstates and act on the flavor-basis S-matrix by conjugation with diagonal matrices $S_{\alpha\beta}^{\text{Reph-1}}$1. The central result is that these transformations map $S_{\alpha\beta}^{\text{Reph-1}}$2 to $S_{\alpha\beta}^{\text{Reph-1}}$3 and vice versa, up to a channel-dependent sign, thereby realizing S-matrix rephasing invariance locally and manifestly at the amplitude level (Minakata, 2023).

This construction is described as strongly indicative of the quantum-mechanical nature of the reparametrization symmetry. It also has structural consequences. In DMP perturbation theory, the Rep–rephasing link forbids a pure $S_{\alpha\beta}^{\text{Reph-1}}$4 exchange symmetry: the atmospheric resonance is not perfectly isolated from solar-scale effects in that framework. The paper further analyzes eigenvalue exchange $S_{\alpha\beta}^{\text{Reph-1}}$5 to fourth order, derives eigenvalue sum rules to fourth order in DMP, SRP, and helio perturbation theory, and reports numerical evidence of convergence up to twelfth order in DMP (Minakata, 2023).

The most explicit RISS-like statement in this setting is the one-parameter family of rephased amplitudes

$S_{\alpha\beta}^{\text{Reph-1}}$6

for any real $S_{\alpha\beta}^{\text{Reph-1}}$7. Because the Rep symmetry maps $S_{\alpha\beta}^{\text{Reph-1}}$8 to $S_{\alpha\beta}^{\text{Reph-1}}$9 for all $S_{\alpha\beta}^{\text{Reph-2}}$0, the paper proposes a natural definition of RISS as the invariance of symmetry relations under continuous reparametrizations and rephasing scalings. Here the scaling variable is not a geometrical coordinate but the rephasing parameter $S_{\alpha\beta}^{\text{Reph-2}}$1, and the invariant content is the physical equivalence of amplitudes under changes of phase origin (Minakata, 2023).

4. Basis invariance, invariant rings, and renormalisation-group protection

In multi-scalar quantum field theory, RISS-like structures appear when scaling symmetry is combined with non-overlapping global symmetries and formulated in a basis-independent language. One construction uses scale-invariant field directions together with spurion-charge assignments to derive renormalisation-group invariants (RGIs). Couplings $S_{\alpha\beta}^{\text{Reph-2}}$2 satisfy $S_{\alpha\beta}^{\text{Reph-2}}$3, and an invariant $S_{\alpha\beta}^{\text{Reph-2}}$4 obeys $S_{\alpha\beta}^{\text{Reph-2}}$5. The key claim is that the synergy of scale invariance along selected field directions and non-overlapping global symmetries constrains the allowed monomials in the $S_{\alpha\beta}^{\text{Reph-2}}$6-functions strongly enough to protect specific combinations to all loops in dimensional regularization. In the two-Higgs-doublet model, under $S_{\alpha\beta}^{\text{Reph-2}}$7, $S_{\alpha\beta}^{\text{Reph-2}}$8, and $S_{\alpha\beta}^{\text{Reph-2}}$9 with a CP2-symmetric quartic and Yukawa sector, the formalism yields

$_2$0

to all loops, and the paper explicitly states that its “scaling + non-overlapping symmetry + spurion” methodology is a realization of RISS (Pilaftsis, 18 May 2026).

A complementary basis-invariant program is developed for the general 2HDM scalar potential. There the central objects are the basis-covariant building blocks $_2$1, $_2$2, and $_2$3, together with fully contracted basis invariants $_2$4 and CP-odd invariants $_2$5. The resulting “Symmetry Map” classifies all global symmetries of the model in terms of reparametrization-independent objects. The paper’s main insight is that symmetries manifest in two algebraically distinct ways: either by non-trivial relations among basis invariants, or by vanishing of basis-covariant building blocks. These two possibilities lead to different reductions of the invariant ring and different counts of remaining physical parameters. For example, CP2 is characterized by $_2$6 and $_2$7, while CP3 adds the cubic relation $_2$8 (Bento et al., 2020).

Taken together, these analyses show that RISS in field theory is not limited to coordinate redefinitions. It also includes basis changes in field space, provided physical statements are recast in invariant form. A plausible implication is that the common content of these constructions lies in replacing parameter-dependent descriptions by symmetry-compatible invariants: spurion charges and SI rays in the renormalisation-group setting, or invariant rings and syzygies in the 2HDM symmetry map (Pilaftsis, 18 May 2026).

5. Neural-network parameter spaces and symmetry-invariant optimization

Deep-network parameter spaces supply a different realization of RISS. In common feedforward, convolutional, and batch-normalized architectures, positive homogeneity of linear layers, ReLU, max-pooling, and subsampling produces continuous scaling-based reparameterization symmetries. Typical examples are neuron-wise or filter-wise rescalings of incoming weights compensated by reciprocal rescalings of outgoing weights, or arbitrary row-wise scalings in batch-normalized layers. These transformations leave the network function unchanged, but Euclidean gradients do not transform covariantly. The basic consequence is that standard stochastic gradient descent depends on the chosen parameterization even when the represented function is identical (Badrinarayanan et al., 2015).

Two practical symmetry-invariant updates were proposed for this setting. The first is the Scaled Metric (SM) update, which equips each weight matrix with a diagonal Riemannian metric based on row norms and rescales the gradient accordingly. The second is the Unit-Norm (UN) update, which constrains each filter to lie on the oblique manifold and performs stochastic gradient descent by tangent projection followed by row normalization. In a four-layer batch-normalized architecture on MNIST with bold-driver learning-rate annealing, the reported mean test errors were $_2$9 for B-SGD, $\mathrm{Diff}(S^1)/SL(2,\mathbb{R})$0 for SM, and $\mathrm{Diff}(S^1)/SL(2,\mathbb{R})$1 for UN. A related study focused on the same symmetry class in convolution–BN–ReLU–pool pipelines and argued that constraining filters to unit norm removes the continuous scaling redundancy while preserving computational efficiency (Badrinarayanan et al., 2015).

A more general geometric treatment reformulates the issue on a parameter manifold $\mathrm{Diff}(S^1)/SL(2,\mathbb{R})$2 endowed with a metric $\mathrm{Diff}(S^1)/SL(2,\mathbb{R})$3. Under a smooth reparametrization $\mathrm{Diff}(S^1)/SL(2,\mathbb{R})$4, the covariant Hessian $\mathrm{Diff}(S^1)/SL(2,\mathbb{R})$5 and the metric transform tensorially, so the endomorphism

$\mathrm{Diff}(S^1)/SL(2,\mathbb{R})$6

transforms by similarity, $\mathrm{Diff}(S^1)/SL(2,\mathbb{R})$7. Its spectrum, trace, and determinant are therefore reparametrization invariant. The same paper defines a quantity or procedure as RISS-invariant when it is invariant both under smooth coordinate changes and under symmetry scalings $\mathrm{Diff}(S^1)/SL(2,\mathbb{R})$8 that leave the network function unchanged. In practice this is implemented either by quotienting $\mathrm{Diff}(S^1)/SL(2,\mathbb{R})$9 by symmetry orbits or by projecting onto the horizontal subspace orthogonal to those orbits. Natural-gradient flow with respect to $K_{\text{circle}}=[2\sin(\Delta\theta/2)]^{-2}$0 is then coordinate invariant, whereas Euclidean gradient flow is not (Kristiadi et al., 2023).

A common misconception in this area is that invariance is absent unless the network itself is modified. The geometric account argues instead that invariance is inherent once the metric is kept explicit and transformed correctly. The earlier SM and UN methods can be read as concrete realizations of that principle: SM uses a symmetry-adapted metric, and UN fixes a representative in each scaling-equivalence class by imposing unit norms (Kristiadi et al., 2023).

6. Reparametrization symmetry in gravity, thermodynamics, and cosmology

In reparametrization-invariant mechanics, the basic structure already has the form later abstracted as RISS. Replacing $K_{\text{circle}}=[2\sin(\Delta\theta/2)]^{-2}$1 by $K_{\text{circle}}=[2\sin(\Delta\theta/2)]^{-2}$2 and $K_{\text{circle}}=[2\sin(\Delta\theta/2)]^{-2}$3 yields an RI action whose canonical Hamiltonian is proportional to

$K_{\text{circle}}=[2\sin(\Delta\theta/2)]^{-2}$4

with $K_{\text{circle}}=[2\sin(\Delta\theta/2)]^{-2}$5 on every solution. Quantization imposes $K_{\text{circle}}=[2\sin(\Delta\theta/2)]^{-2}$6, which is precisely the Schrödinger equation $K_{\text{circle}}=[2\sin(\Delta\theta/2)]^{-2}$7. In this formulation, reparametrization invariance is a gauge symmetry generated by a