Threshold-Subtracted Clock Observable

• Threshold-subtracted clock observable is defined by removing a universal threshold term (such as a 1/√E divergence or a critical exponent barrier) to reveal the mechanism of interest.
• In stationary quantum scattering, the subtraction isolates resonant delay by eliminating the continuum-edge background, thereby sharpening the Breit–Wigner resonance peaks.
• In Euler dynamics, isolating the α=1/3 threshold in the matrix-clock inequality prevents finite-time collapse and refines the analysis of the deformation gradient.

Searching arXiv for the cited works and closely related context. Searching arXiv for "threshold-subtracted clock observable", "Resonant delay in a stationary quantum clock", and "conditional Lagrangian clock barrier". I’ll look up the arXiv records for the two papers by title and id. Searching arXiv by identifier: (Davies et al., 1 Jun 2026) and (Avadanei, 29 May 2026). Threshold-subtracted clock observable denotes a class of clock-type quantities in which a distinguished threshold contribution is removed, or explicitly separated, so that the residual behavior reflects the mechanism of interest rather than a universal background. In the supplied arXiv literature, the term appears in two technically distinct settings. In stationary one-dimensional quantum scattering, it refers to a modified Peres clock time obtained by subtracting the universal low-energy $1/\sqrt{E}$ continuum-edge term from the raw stationary clock, thereby isolating resonant delay (Davies et al., 1 Jun 2026). In axisymmetric Euler without swirl, it refers to a threshold comparison at the Hölder exponent $\alpha=1/3$, where the matrix-clock or scalar clock is compared against the critical power law, yielding a barrier against finite-time clock collapse for $\alpha\ge 1/3$ (Avadanei, 29 May 2026).

1. Terminological scope and common structure

The phrase has a narrow technical meaning in each source, but the two constructions share a common architecture: a clock observable is dominated near a threshold by a term that is not itself the phenomenon one wants to detect. In the quantum setting, the threshold is the continuum edge $E\to 0^+$, and the contaminating term is an explicit $1/\sqrt{E}$ divergence. In the Euler setting, the threshold is the regularity exponent $\alpha=1/3$, and the relevant operation is not literal subtraction of a function of time, but isolation of the critical power law in the clock inequality (Davies et al., 1 Jun 2026).

Setting	Clock observable	Threshold operation
Stationary quantum scattering	$\tau_P(E)=\frac{d}{dE}\Arg[t(E)e^{ikL}]$	subtract $\ell_{\rm thr}/\sqrt{E}$
Axisymmetric Euler without swirl	$\nu(t)=\sigma_{\min}(F(t))$, on-axis $J(t)$ with $\alpha=1/3$0	compare with the critical exponent $\alpha=1/3$1

This suggests that “threshold-subtracted clock observable” is best understood as a structural label rather than a single universal formula. In one case the construction is a renormalized scattering time; in the other it is a threshold-isolated Lagrangian clock inequality.

2. Stationary quantum clock and the origin of threshold subtraction

For one-dimensional scattering by a real, compactly supported potential $\alpha=1/3$2 with $\alpha=1/3$3, the raw stationary Peres clock is defined from the transmission amplitude $\alpha=1/3$4 by

$\alpha=1/3$5

with $\alpha=1/3$6. If $\alpha=1/3$7 is the usual Wigner phase, then

$\alpha=1/3$8

where $\alpha=1/3$9 (Davies et al., 1 Jun 2026).

The threshold problem arises from the low-energy expansion of the Cauchy-data transfer matrix

$\alpha\ge 1/3$0

which is even in $\alpha\ge 1/3$1. The existence of a bounded zero-energy solution, described as a “half-bound” state, is equivalent to $\alpha\ge 1/3$2. In the generic sector $\alpha\ge 1/3$3,

$\alpha\ge 1/3$4

with

$\alpha\ge 1/3$5

The coefficient $\alpha\ge 1/3$6 is therefore fixed entirely by low-energy scattering data. The paper identifies this $\alpha\ge 1/3$7 term as universal in the sense that it is inherited from the vanishing exterior momentum and the associated scattering matching, rather than from resonant delay itself (Davies et al., 1 Jun 2026).

The threshold-subtracted stationary clock is then defined by

$\alpha\ge 1/3$8

By construction, $\alpha\ge 1/3$9 as $E\to 0^+$0. In the exceptional half-bound sector $E\to 0^+$1, one instead has

$E\to 0^+$2

and the corresponding coefficient is subtracted instead.

3. Square-well realization and resonant Lorentzian structure

For the attractive square well

$E\to 0^+$3

with $E\to 0^+$4, $E\to 0^+$5, and $E\to 0^+$6, matching at $E\to 0^+$7 and $E\to 0^+$8 gives the exact transmission amplitude

$E\to 0^+$9

The resulting exact stationary clock time is

$1/\sqrt{E}$0

In the generic case $1/\sqrt{E}$1,

$1/\sqrt{E}$2

so that

$1/\sqrt{E}$3

In the half-bound tuning $1/\sqrt{E}$4,

$1/\sqrt{E}$5

and the exceptional subtraction coefficient is $1/\sqrt{E}$6 (Davies et al., 1 Jun 2026).

The subtraction becomes especially consequential near isolated transmission resonances $1/\sqrt{E}$7, defined by $1/\sqrt{E}$8. The local phase expansion yields

$1/\sqrt{E}$9

with

$\alpha=1/3$0

Equivalently,

$\alpha=1/3$1

The paper states that the threshold-subtracted clock acquires the expected local Lorentzian form near isolated transmission resonances. This is the central reason for the subtraction: it exposes the pole-sensitive resonant structure that is masked in the raw clock by the universal threshold background (Davies et al., 1 Jun 2026).

4. Comparison with dwell time, Wigner delay, and near-threshold masking

The same study compares the threshold-subtracted clock with two standard scattering-time notions. The dwell time in the interior well is

$\alpha=1/3$2

and, as $\alpha=1/3$3 with $\alpha=1/3$4, one has $\alpha=1/3$5. The usual Wigner phase delay satisfies

$\alpha=1/3$6

Thus both $\alpha=1/3$7 and $\alpha=1/3$8 inherit the $\alpha=1/3$9 divergence, whereas $\tau_P(E)=\frac{d}{dE}\Arg[t(E)e^{ikL}]$0 is threshold-benign (Davies et al., 1 Jun 2026).

At resonance, the dwell-time peak has the same height

$\tau_P(E)=\frac{d}{dE}\Arg[t(E)e^{ikL}]$1

and width

$\tau_P(E)=\frac{d}{dE}\Arg[t(E)e^{ikL}]$2

with

$\tau_P(E)=\frac{d}{dE}\Arg[t(E)e^{ikL}]$3

In particular, for near-threshold resonances $\tau_P(E)=\frac{d}{dE}\Arg[t(E)e^{ikL}]$4, the peak heights scale as $\tau_P(E)=\frac{d}{dE}\Arg[t(E)e^{ikL}]$5 and the widths coincide asymptotically. This is the precise sense in which the subtracted clock aligns with the resonant content of the dwell time and transmission Wigner phase delay (Davies et al., 1 Jun 2026).

The continuum-edge masking effect is quantified by letting the well depth satisfy $\tau_P(E)=\frac{d}{dE}\Arg[t(E)e^{ikL}]$6, $\tau_P(E)=\frac{d}{dE}\Arg[t(E)e^{ikL}]$7, so that

$\tau_P(E)=\frac{d}{dE}\Arg[t(E)e^{ikL}]$8

Then both $\tau_P(E)=\frac{d}{dE}\Arg[t(E)e^{ikL}]$9 and $\ell_{\rm thr}/\sqrt{E}$0 scale as $\ell_{\rm thr}/\sqrt{E}$1, whereas the unsubtracted threshold background at $\ell_{\rm thr}/\sqrt{E}$2 scales as

$\ell_{\rm thr}/\sqrt{E}$3

Hence the ratio of the visible resonance to the background in the raw clock vanishes like $\ell_{\rm thr}/\sqrt{E}$4. The paper describes this as a threshold “mask”: without subtraction, any near-threshold resonance is hidden by the continuum-edge term (Davies et al., 1 Jun 2026).

The same conclusion is tested in two control examples. For a symmetric barrier–well–barrier cavity, subtraction of the coefficient $\ell_{\rm thr}/\sqrt{E}$5 obtained from the zero-energy Cauchy-data transfer matrix removes the universal threshold term without touching the narrow Breit–Wigner peaks. For an asymmetric two-step attractive well,

$\ell_{\rm thr}/\sqrt{E}$6

a least-squares fit over a low-$\ell_{\rm thr}/\sqrt{E}$7 window gives, for the quoted numerical example, $\ell_{\rm thr}/\sqrt{E}$8, and subtracting $\ell_{\rm thr}/\sqrt{E}$9 yields mild near-threshold behavior in which the same resonant peaks, identical to those in $\nu(t)=\sigma_{\min}(F(t))$0, stand out clearly (Davies et al., 1 Jun 2026).

5. Lagrangian matrix-clock, scalar reduction, and threshold subtraction at $\nu(t)=\sigma_{\min}(F(t))$1

A distinct use of the same phrase appears in the study of axisymmetric no-swirl solutions to the three-dimensional incompressible Euler equations with initial velocity in $\nu(t)=\sigma_{\min}(F(t))$2, $\nu(t)=\sigma_{\min}(F(t))$3. Along a fixed Lagrangian label $\nu(t)=\sigma_{\min}(F(t))$4, the deformation gradient is

$\nu(t)=\sigma_{\min}(F(t))$5

and the singular-value decomposition is written

$\nu(t)=\sigma_{\min}(F(t))$6

with

$\nu(t)=\sigma_{\min}(F(t))$7

The smallest singular value

$\nu(t)=\sigma_{\min}(F(t))$8

is called the matrix-clock observable (Avadanei, 29 May 2026).

In the on-axis setting, symmetry forces $\nu(t)=\sigma_{\min}(F(t))$9 and a diagonal $J(t)$0. The reduced meridional Jacobian

$J(t)$1

satisfies

$J(t)$2

Hence the matrix-clock $J(t)$3 and the scalar clock $J(t)$4 are equivalent via $J(t)$5. The threshold subtraction in this setting consists of isolating the Hölder exponent threshold $J(t)$6 in the driver estimate. Specifically, there exist

$J(t)$7

such that

$J(t)$8

and, on-axis,

$J(t)$9

where $\alpha=1/3$00 (Avadanei, 29 May 2026).

The right-Dini derivative of the smallest singular value satisfies

$\alpha=1/3$01

and the resulting master matrix-clock inequality is

$\alpha=1/3$02

Using $\alpha=1/3$03 and the exact kinematic law

$\alpha=1/3$04

one obtains the reduced scalar clock inequality

$\alpha=1/3$05

This is the precise content of the threshold subtraction at $\alpha=1/3$06: the critical power $\alpha=1/3$07 is isolated, and the sign structure of the clock ODE changes from the subcritical mechanism studied in Shkoller’s framework to a barrier regime for $\alpha=1/3$08 (Avadanei, 29 May 2026).

6. Consequences, hypotheses, and conceptual limits

The Euler paper compares the reduced inequality with the auxiliary ODE

$\alpha=1/3$09

In the critical case $\alpha=1/3$10, one has $\alpha=1/3$11, so

$\alpha=1/3$12

and Grönwall yields

$\alpha=1/3$13

which excludes finite-time collapse. In the supercritical case $\alpha=1/3$14, one sets $\alpha=1/3$15, obtaining

$\alpha=1/3$16

and Grönwall again shows that $\alpha=1/3$17 stays finite and bounded away from $\alpha=1/3$18, so $\alpha=1/3$19 for all $\alpha=1/3$20. Consequently, $\alpha=1/3$21 for all $\alpha=1/3$22, so neither $\alpha=1/3$23 nor $\alpha=1/3$24 can collapse. By incompressibility and the Cauchy formula $\alpha=1/3$25, one then rules out $\alpha=1/3$26 and uses Beale–Kato–Majda to continue the solution past $\alpha=1/3$27 (Avadanei, 29 May 2026).

These conclusions are conditional. The hypotheses are stated as cusp-tail, Dini coherence, near-field compatibility, and bounded transverse distortion. More specifically, the exposition lists a cusp-tail reservoir, Dini coherence as averaged principal-value control with $\alpha=1/3$28, near-field compatibility in the form

$\alpha=1/3$29

and bounded transverse distortion as uniform control of the ratio $\alpha=1/3$30 on $\alpha=1/3$31. The paper also states a limitation that is important for interpretation: these results do not enlarge the known Lorentz-space global regularity classes. Rather, they identify the supercritical Lagrangian obstruction dual to Shkoller’s subcritical blow-up mechanism in the case $\alpha=1/3$32 (Avadanei, 29 May 2026).

A common misconception would be to treat the two threshold-subtracted clock constructions as the same operation. They are not. In the quantum problem, threshold subtraction is an explicit additive removal of a universal continuum-edge divergence from a stationary time delay. In the Euler problem, it is a threshold isolation at the level of a differential inequality, tied to the exponent $\alpha=1/3$33 and to the non-collapse of the smallest singular value of the deformation gradient. What they share is more abstract: each separates a threshold-determined contribution from the mechanism under study. This suggests a unifying viewpoint in which a clock observable is useful only after the threshold contribution has been factored out, whether by explicit subtraction or by critical-exponent comparison (Davies et al., 1 Jun 2026).