Zero: Critical Boundaries in Math, Physics & ML

• Zero is defined as a marker for vanishing quantities, absent resources, and critical boundaries across mathematics, physics, and machine learning.
• It reveals singular regimes in analytic functions, partitions zeros indicating phase transitions, and zero supervision protocols that optimize learning models.
• Practical applications span probing Riemann zeros, diagnosing quantum-critical phenomena, and engineering metamaterials with zero-frequency and zero-wavenumber gaps.

Searching arXiv for papers related to “zero” across mathematics, physics, machine learning, and quantum information. Zero functions in contemporary research as a technically precise marker of vanishing quantity, absent resource, or critical boundary. In the arXiv literature represented here, the term denotes at least four distinct but structurally related notions: roots of analytic or partition functions in complex variables, regimes in which a conventional supervisory signal is removed, protocols constrained to use no classical communication, and wave systems engineered to exhibit forbidden behavior at vanishing frequency or vanishing wavenumber. Across these settings, zero is not merely a numeral; it defines singular regimes in which asymptotics, identifiability, universality, and algorithmic design become especially transparent or especially difficult (Reyna, 2024, Kim, 2017, Wang et al., 15 Jan 2026, George et al., 2023, Lemkalli et al., 6 May 2025).

1. Zero as vanishing value, absent resource, and critical boundary

A compact way to organize the research usage of zero is to distinguish three recurrent meanings. First, zero denotes a root condition such as $R(\rho)=0$ for the auxiliary Riemann function or $Z(h_i)=0$ for a partition function analytically continued into a complex parameter (Reyna, 2024, D'Emidio, 2023). Second, zero denotes the explicit removal of an ordinarily required resource, as in “zero annotation,” “zero-guidance,” and “zero communication,” where labels, prompts, or classical messages are disallowed by construction (Wang et al., 15 Jan 2026, Rewatbowornwong et al., 2023, George et al., 2023). Third, zero denotes a kinematic or spectral endpoint, such as $\omega=0$ or $k=0$, where conventional elastic or phononic behavior is usually gapless but can be altered by external control (Lemkalli et al., 6 May 2025).

Sense of zero	Representative formulation	Domain
Root	$Z(h_i)=0$, $R(\rho)=0$	Analytic number theory; critical phenomena
Resource absence	zero annotation; zero-guidance; zero communication	Multimodal learning; segmentation; entanglement transformation
Spectral endpoint	zero-frequency gap; zero-wavenumber gap	Elastic metamaterials

This taxonomy suggests that zero often marks a regime where standard perturbative intuition fails. In the partition-function setting, zeros do not occur on the real axis for finite systems but encode phase transitions through their approach to that axis in the thermodynamic limit (D'Emidio, 2023). In learning systems, zero supervision is not the absence of structure; rather, structure is reconstructed from internal consistency, pseudo-labeling, or pretrained priors (Wang et al., 15 Jan 2026, Rewatbowornwong et al., 2023). In zero-communication entanglement conversion, zero does not mean triviality, because nontrivial optimal fidelities remain achievable under local unitaries, local operations, and shared randomness (George et al., 2023).

2. Zero loci in analytic number theory

In analytic number theory, zero refers to points at which a complex function vanishes. One preprint stated that it “present[s] a proof of the Riemann hypothesis” and that zeros of the Riemann zeta function “should be on the line with the real value 1/2” in the region where the real part lies between $0$ and $1$ (Lee, 2013). That statement is a claim in the abstract of the preprint itself.

A more detailed and technically specific treatment appears for Riemann’s auxiliary function $R(s)$, introduced in Riemann’s Nachlass and treated via the Riemann–Siegel expansion. Writing $s=\sigma+it$, $Z(h_i)=0$0, $Z(h_i)=0$1, and $Z(h_i)=0$2, the paper gives an explicit asymptotic representation

$Z(h_i)=0$3

with $Z(h_i)=0$4 an explicit remainder term of order $Z(h_i)=0$5 for fixed $Z(h_i)=0$6 (Reyna, 2024). The zeros of the auxiliary function are the complex numbers $Z(h_i)=0$7 in the upper half-plane for which

$Z(h_i)=0$8

The principal theorem of that work establishes a right-half-plane exclusion region. If

$Z(h_i)=0$9

then every zero $\omega=0$0 of $\omega=0$1 with $\omega=0$2 satisfies

$\omega=0$3

Equivalently, $\omega=0$4 has no zeros in the half-strip

$\omega=0$5

The argument combines explicit asymptotic expansions of $\omega=0$6, lower bounds for $\omega=0$7 in $\omega=0$8, explicit Dirichlet-series and van der Corput-type estimates, and Rouché’s theorem applied on overlapping rectangles (Reyna, 2024).

Several explicit inequalities organize the proof. For example, for $\omega=0$9,

$k=0$0

and for $k=0$1, $k=0$2,

$k=0$3

The paper also proves that for $k=0$4 and $k=0$5,

$k=0$6

hence $k=0$7 in that region (Reyna, 2024).

Numerically, the same work reports that all computed zeros of $k=0$8 with $k=0$9 had real part strictly below $Z(h_i)=0$0, and formulates the conjecture that every zero $Z(h_i)=0$1 of $Z(h_i)=0$2 with $Z(h_i)=0$3 satisfies $Z(h_i)=0$4 (Reyna, 2024). The paper further states that, by Siegel’s correspondence, if every zero of $Z(h_i)=0$5 lies strictly to the left of $Z(h_i)=0$6, then each such zero generates two zeros of $Z(h_i)=0$7 on the line $Z(h_i)=0$8. This suggests that zero sets of auxiliary functions can be used as indirect probes of the geometry of zeta zeros.

3. Partition-function zeros and quantum-critical diagnostics

In statistical mechanics and quantum many-body theory, zeros of the partition function in complexified parameters provide a complementary description of phase transitions. For a spin system with Hamiltonian $Z(h_i)=0$9 and external field $R(\rho)=0$0 coupling to an order parameter $R(\rho)=0$1, one writes

$R(\rho)=0$2

For real $R(\rho)=0$3, the free energy is analytic on any finite system, but after analytic continuation into the complex $R(\rho)=0$4 plane, the zeros $R(\rho)=0$5 defined by $R(\rho)=0$6 can accumulate and pinch the real axis as system size tends to infinity (D'Emidio, 2023).

That framework is used in large-scale quantum Monte Carlo for two-dimensional quantum antiferromagnets. In the stochastic series expansion formalism, one obtains the free-energy-ratio identity

$R(\rho)=0$7

and analogous formulas for modified couplings. For a complex staggered field embedded in bond operators, the paper derives

$R(\rho)=0$8

so zeros of $R(\rho)=0$9 are found by locating $0$0 such that $0$1 numerically, or where $0$2 (D'Emidio, 2023).

Finite-size scaling of the leading Lee–Yang zero $0$3 yields critical exponents. General scaling implies

$0$4

and one may equivalently write

$0$5

with $0$6 (D'Emidio, 2023). In the Heisenberg bilayer at the O(3) quantum critical point, zeros in a staggered imaginary field lie exactly on the imaginary axis, and fits of the first three zeros give $0$7, $0$8, and $0$9, in agreement with the best-known classical 3D Heisenberg values (D'Emidio, 2023). In the square-lattice $1$0-$1$1 model, leading zeros for staggered and VBS fields lie purely on the imaginary axis and yield exponents below O(3) values, with moderate drift as $1$2 grows, reflecting the still-debated nature of the transition (D'Emidio, 2023).

The same work studies simultaneous complex Néel and VBS fields through the joint ratio

$1$3

At the critical $1$4-$1$5 coupling, zeros in the $1$6 plane coalesce into nearly circular rings around the origin; away from criticality, the rings elongate in the $1$7 or $1$8 direction (D'Emidio, 2023). The paper interprets those rings as a signature of order-parameter competition and approximate emergent SO(5) symmetry at the deconfined quantum critical point.

Fisher zeros in the complex temperature plane provide the thermal analogue. For the two-dimensional $1$9-state clock model, the partition function is

$R(s)$0

To avoid binning artifacts in models with irregular energy spectra, the Hamiltonian

$R(s)$1

is rewritten as

$R(s)$2

with integer-valued $R(s)$3 and $R(s)$4, and integer partial energies

$R(s)$5

so that $R(s)$6 (Kim, 2017). A two-dimensional Wang–Landau random walk in the $R(s)$7 plane estimates the joint density of states $R(s)$8 without ad hoc energy binning.

The normalized partition function at complex $R(s)$9 is

$s=\sigma+it$0

with

$s=\sigma+it$1

Leading zeros are found by locating the intersection of $s=\sigma+it$2 and $s=\sigma+it$3, then minimizing $s=\sigma+it$4 near that intersection (Kim, 2017).

For a generic second-order transition one expects

$s=\sigma+it$5

For a BKT transition, where $s=\sigma+it$6, the leading zero follows

$s=\sigma+it$7

with $s=\sigma+it$8 for the 2D XY-type BKT transition (Kim, 2017). The paper reports that for $s=\sigma+it$9, and the XY limit via HOTRG data, the upper-transition zeros lie on a universal curve

$Z(h_i)=0$00

using $Z(h_i)=0$01 and $Z(h_i)=0$02. By contrast, $Z(h_i)=0$03 shows a distinct finite-size trajectory that within accessible sizes mimics an effective exponent $Z(h_i)=0$04–$Z(h_i)=0$05 (Kim, 2017). The mutual collapse of $Z(h_i)=0$06 onto $Z(h_i)=0$07, $Z(h_i)=0$08, and the XY limit is presented as strong direct evidence that the upper transition of the six-state clock model belongs to the same BKT universality class.

A technical limitation arises from the nondivergent specific heat at the BKT transition. The Gaussian envelope of oscillations in $Z(h_i)=0$09 satisfies

$Z(h_i)=0$10

With $Z(h_i)=0$11 and $Z(h_i)=0$12 only logarithmically in $Z(h_i)=0$13, the overall factor decays as $Z(h_i)=0$14, so oscillations around the leading zero become exponentially small relative to statistical fluctuations in the estimated density of states (Kim, 2017). This shows that zero-finding can be limited not only by physics but also by estimator noise.

4. Zero supervision and zero guidance in multimodal learning

In machine learning, zero often marks the deliberate removal of human-provided supervision or guidance. “V-Zero” is a post-training framework for vision-LLMs that uses exclusively unlabeled images and no human annotation (Wang et al., 15 Jan 2026). The framework establishes a co-evolutionary loop between two role-specialized agents initialized from the same base VLM, such as Qwen2.5-VL-7B-Instruct: a Questioner $Z(h_i)=0$15 and a Solver $Z(h_i)=0$16.

Given an image $Z(h_i)=0$17, the Questioner generates a multiple-choice question $Z(h_i)=0$18 with exactly four options, an intuitive answer $Z(h_i)=0$19, and a concise visual description $Z(h_i)=0$20. The Solver samples $Z(h_i)=0$21 chain-of-thought answers $Z(h_i)=0$22, forms a pseudo-label $Z(h_i)=0$23 by majority voting, and computes confidence

$Z(h_i)=0$24

The Questioner is then updated using a dual-track reasoning reward

$Z(h_i)=0$25

combined with strict format checking: $Z(h_i)=0$26 Both roles are optimized by Group Relative Policy Optimization, whose standardized advantage is

$Z(h_i)=0$27

and whose clipped objective is

$Z(h_i)=0$28

For Solver training, samples are filtered by the majority-vote confidence score

$Z(h_i)=0$29

keeping only questions with $Z(h_i)=0$30 (Wang et al., 15 Jan 2026).

The reported setup uses Qwen2.5-VL-3B-Instruct and Qwen2.5-VL-7B-Instruct, an unlabeled image pool of about $Z(h_i)=0$31K images from OpenVLThinker’s GRPO-medium/hard splits, $Z(h_i)=0$32 NVIDIA A800 GPUs (80 GB) plus $Z(h_i)=0$33 GPUs for feedback loops, batch size $Z(h_i)=0$34, learning rate $Z(h_i)=0$35, sampling temperature $Z(h_i)=0$36, $Z(h_i)=0$37, $Z(h_i)=0$38, $Z(h_i)=0$39, $Z(h_i)=0$40, and token limits $Z(h_i)=0$41 for the Questioner and $Z(h_i)=0$42 for the Solver. Training time is about $Z(h_i)=0$43 hours per iteration, and the schedule alternates Questioner and Solver updates for two full iterations (Wang et al., 15 Jan 2026).

On Qwen2.5-VL-7B-Instruct, average multiple-choice accuracy over MMMU, MMStar, MathVision, MathVerse, MathVista, and LogicVista increases from $Z(h_i)=0$44 for the base model to $Z(h_i)=0$45 for supervised GRPO, $Z(h_i)=0$46 for V-Zero Iter 1, and $Z(h_i)=0$47 for V-Zero Iter 2, an overall gain of $Z(h_i)=0$48. The paper also reports gains of $Z(h_i)=0$49 on visual mathematical reasoning and $Z(h_i)=0$50 on general vision-centric tasks (Wang et al., 15 Jan 2026). Ablations show degradation when freezing the Questioner, removing the dual-track reward, or disabling data filtering. Output validity rises from $Z(h_i)=0$51 to $Z(h_i)=0$52, and average question difficulty increases from $Z(h_i)=0$53 to $Z(h_i)=0$54 by Iter 2 (Wang et al., 15 Jan 2026). This suggests that zero annotation does not eliminate curriculum design; instead, the curriculum is generated endogenously.

A related but distinct formulation is “zero-guidance segmentation,” where an image is automatically partitioned into semantic regions and each region is labeled with free-form natural-language text, without any user-provided class list, text prompt, or “what-to-find” query (Rewatbowornwong et al., 2023). The baseline uses only DINO, CLIP, and ZeroCap, with no further training on any segmentation dataset.

The pipeline has four stages. First, DINO-ViT last-attention key projections are clustered agglomeratively into $Z(h_i)=0$55 over-segments, optionally refined with DenseCRF. Second, each candidate mask $Z(h_i)=0$56 is embedded in CLIP space by applying masked self-attention in the last CLIP layers to the full image. With flattened mask $Z(h_i)=0$57, the masked attention output is defined through

$Z(h_i)=0$58

$Z(h_i)=0$59

A saliency score

$Z(h_i)=0$60

controls a “global subtraction” step,

$Z(h_i)=0$61

with masking applied to CLIP layers $Z(h_i)=0$62–$Z(h_i)=0$63 and $Z(h_i)=0$64 (Rewatbowornwong et al., 2023).

Third, ZeroCap steers a GPT-2 captioner initialized with “Image of a ….” so that the text embedding approaches the region embedding in CLIP space. Fourth, semantically similar segments are merged according to

$Z(h_i)=0$65

considering only sibling segments in the original DINO tree and requiring similarity above $Z(h_i)=0$66, for example $Z(h_i)=0$67 (Rewatbowornwong et al., 2023).

Evaluation is performed after reassignment of free-form predicted labels to ground-truth classes, either by text-to-text matching with SBERT or segment-to-text matching with CLIP. On Pascal Context PC-59 under segment-to-text reassignment with $Z(h_i)=0$68, the reported scores are Segmentation IoU$Z(h_i)=0$69 of $Z(h_i)=0$70, Segment Recall$Z(h_i)=0$71 of $Z(h_i)=0$72, and Text Generation Quality of $Z(h_i)=0$73 with oracle masks (Rewatbowornwong et al., 2023). The Crop-and-Mask baseline gives IoU about $Z(h_i)=0$74, masking only gives about $Z(h_i)=0$75, no merging gives about $Z(h_i)=0$76, and the full method gives about $Z(h_i)=0$77 (Rewatbowornwong et al., 2023). The method also reports qualitative labels such as “Mona Lisa,” “crowd observing,” “red barn,” and “circular fountain.” A plausible implication is that zero-guidance here means the removal of externally specified semantics at inference time, not the removal of semantic priors altogether, since the method relies entirely on pretrained representations.

5. Zero communication in entanglement transformation

In quantum information theory, zero communication refers to bipartite state-conversion protocols that forbid classical message exchange between the two parties. The relevant problem is approximate pure-state conversion under local unitaries or under local operations and shared randomness (George et al., 2023).

For two density operators $Z(h_i)=0$78 and $Z(h_i)=0$79, the fidelity is

$Z(h_i)=0$80

For pure states, this reduces to $Z(h_i)=0$81. Every bipartite pure state $Z(h_i)=0$82 admits a Schmidt decomposition

$Z(h_i)=0$83

with nonincreasing Schmidt spectrum $Z(h_i)=0$84 (George et al., 2023).

The zero-communication local-unitary benchmark is

$Z(h_i)=0$85

The paper proves the exact formula

$Z(h_i)=0$86

where $Z(h_i)=0$87 and $Z(h_i)=0$88 are the Schmidt spectra of $Z(h_i)=0$89 and $Z(h_i)=0$90, padded with zeros if necessary (George et al., 2023). Thus optimal local-unitary conversion is obtained by aligned comparison of the ordered Schmidt coefficients.

Allowing local operations and shared randomness leads to

$Z(h_i)=0$91

and the optimal fidelity

$Z(h_i)=0$92

where $Z(h_i)=0$93 and $Z(h_i)=0$94 denotes descending reordering (George et al., 2023). Exact zero-error conversion under LOSR is possible if and only if there exists a distribution $Z(h_i)=0$95 such that

$Z(h_i)=0$96

The paper interprets this as recovering catalytic majorization in the classical probability setting.

The same reduction yields a catalytic, or embezzling, formulation under local unitaries. For a catalyst $Z(h_i)=0$97 of Schmidt rank $Z(h_i)=0$98, the maximum fidelity is

$Z(h_i)=0$99

As a corollary, if

$\omega=0$00

then for any $\omega=0$01 and any target $\omega=0$02 of rank $\omega=0$03, choosing $\omega=0$04 guarantees $\omega=0$05 (George et al., 2023).

Several qualitative trade-offs are reported. In two-qubit cases, $\omega=0$06; for higher rank, LOSR can strictly improve fidelity over LU. In the many-copy i.i.d. regime, zero-communication fidelity under LU decays exponentially with the number of copies whenever the Schmidt spectra differ (George et al., 2023). This places zero communication in a nontrivial intermediate regime: stronger than unrestricted LOCC, but richer than purely formal impossibility.

6. Zero-frequency and zero-wavenumber bandgaps

In elastodynamics, zero can denote a spectral endpoint at which conventional bosonic phonon systems are expected to remain gapless. A 2025 paper constructs space-time elastic metamaterials using optical trapping forces to generate both zero-frequency and zero-wavenumber bandgaps in mass-spring chains (Lemkalli et al., 6 May 2025).

For an infinite one-dimensional monoatomic chain with mass $\omega=0$07, nearest-neighbor spring stiffness $\omega=0$08, and an optical trapping force acting as an on-site spring of stiffness $\omega=0$09, the displacement $\omega=0$10 obeys

$\omega=0$11

Under the Bloch ansatz $\omega=0$12, the dynamical matrix is

$\omega=0$13

and the trap force is

$\omega=0$14

Without optical trapping,

$\omega=0$15

which yields the usual acoustic band from $\omega=0$16 at $\omega=0$17 to $\omega=0$18 at the zone edge. With positive on-site optical stiffness $\omega=0$19,

$\omega=0$20

Hence at the $\omega=0$21 point,

$\omega=0$22

and the interval $\omega=0$23 becomes forbidden: a zero-frequency bandgap (Lemkalli et al., 6 May 2025).

Using the normalized acoustic scale $\omega=0$24 and $\omega=0$25, the dispersion becomes

$\omega=0$26

The gap-opening condition is simply

$\omega=0$27

The paper notes experimental parameters $\omega=0$28 and $\omega=0$29, giving $\omega=0$30 (Lemkalli et al., 6 May 2025).

To generate a zero-wavenumber gap, the system must have two degrees of freedom per cell. For sublattices $\omega=0$31 and $\omega=0$32 with displacements $\omega=0$33 and $\omega=0$34 and on-site optical terms $\omega=0$35 and $\omega=0$36, the Bloch-reduced dynamical matrix is

$\omega=0$37

with dispersion from

$\omega=0$38

At $\omega=0$39,

$\omega=0$40

Choosing $\omega=0$41 and $\omega=0$42 yields

$\omega=0$43

Because there is no zero eigenvalue at $\omega=0$44, the paper concludes that no propagating mode can have arbitrarily small $\omega=0$45, so a zero-wavenumber gap opens (Lemkalli et al., 6 May 2025).

The authors interpret the positive optical stiffness as a stable supplement to elastic restoring forces, whereas an effectively negative on-site stiffness borders on instability and therefore must be realized through temporal modulation and phase locking rather than as a truly static negative spring (Lemkalli et al., 6 May 2025). This indicates that “zero” in band engineering is not merely descriptive: it identifies spectral constraints that require external time-dependent control to circumvent.

7. Cross-disciplinary patterns and recurring misconceptions

Across these literatures, zero is repeatedly associated with edge cases that are more informative than generic operating points. Zeros of analytic functions encode singular behavior not visible on the real axis of a finite system (D'Emidio, 2023, Kim, 2017). Zero-resource protocols expose which parts of performance can be recovered from internal structure alone, whether by majority-vote pseudo-labels, pretrained vision-language priors, or catalytic spectra (Wang et al., 15 Jan 2026, Rewatbowornwong et al., 2023, George et al., 2023). Zero-frequency and zero-wavenumber gaps identify precisely those long-wavelength or static limits that ordinary elastic systems do not suppress (Lemkalli et al., 6 May 2025).

Several misconceptions are corrected by the cited works. One is that zero supervision implies no informative signal: V-Zero instead relies on internally generated rewards, confidence filtering, and self-consistency (Wang et al., 15 Jan 2026). Another is that zero-guidance segmentation means fully unconstrained vision; in practice, the method depends on DINO, CLIP, and ZeroCap as frozen generalist priors (Rewatbowornwong et al., 2023). A further misconception is that zero communication entails negligible conversion power; the fidelity formulas for LU, LOSR, and catalytic settings show otherwise (George et al., 2023). In critical phenomena, a common simplification is to equate all finite-size zero trajectories with ordinary power-law scaling; the clock-model study emphasizes that BKT behavior requires a different trajectory,

$\omega=0$46

and that estimator noise becomes exponentially problematic near leading zeros (Kim, 2017).

A broader interpretation, marked here as inference, is that zero operates as a boundary concept linking absence and structure. In every case summarized above, setting a quantity or resource to zero does not trivialize the problem. Instead, it often exposes latent geometry: the geometry of zero loci in the complex plane, the geometry of pretrained embedding spaces, the ordering geometry of Schmidt spectra, or the dispersion geometry of space-time metamaterials.