CERO: Adaptive RL, Collider, and Zero Isotherm

• CERO is a multi-domain term defining an RL rollout optimization method, a circular energy-recovery collider design, and the zero isotherm concept in thermodynamics.
• In reinforcement learning, it employs a Fenchel-dual online algorithm for adaptive rollout budgeting that improves training signals and accuracy over fixed-budget approaches.
• In accelerator and thermodynamic contexts, it conceptualizes innovative particle recycling for high luminosity colliders and derives the Nernst heat theorem from second-law principles.

In the cited works, CERO denotes three distinct technical objects: Cross-Epoch Adaptive Rollout Optimization, a rollout-budgeting method for reinforcement-learning post-training of LLMs; the Circular Energy-Recovery Option/Collider, a generic concept for an $e^+e^-$ collider based on energy-recovery linacs in a large circular tunnel; and, in a thermodynamic context, absolute zero, the zero isotherm $T=0$. The term therefore spans machine learning, accelerator physics, and thermodynamics, with each usage carrying a different formal structure, set of assumptions, and research agenda.

1. Scope and principal meanings

The cited literature uses the same label for an RL resource-allocation framework, a collider architecture, and the thermodynamic zero isotherm. These meanings are defined respectively in (Zong et al., 4 Jun 2026, Litvinenko et al., 2022), and (Martín-Olalla, 2024).

Meaning of CERO	Research area	Core definition
Cross-Epoch Adaptive Rollout Optimization	RL post-training	A rollout budgeting framework that adaptively allocates rollouts across prompts and epochs under a fixed global budget
Circular Energy-Recovery Option/Collider	Accelerator physics	A generic concept of an $e^+e^-$ collider in a large circular tunnel using energy-recovery linacs instead of storage rings, with both energy and particle recycling
Absolute zero / zero isotherm	Thermodynamics	The isotherm $T=0$, used in a Carnot-based proof of the Nernst heat theorem

The first usage is algorithmic and concerns adaptive data collection for GRPO-style training. The second is architectural and concerns luminosity, synchrotron-radiation power, and polarized beams in a $\sim 100$ km machine. The third is conceptual and concerns the low-temperature structure of entropy derived from the second law.

2. CERO as Cross-Epoch Adaptive Rollout Optimization

In RL post-training, CERO is formulated as an online resource-allocation problem over prompts $i \in \{1,\dots,M\}$ and epochs $k \in \{1,\dots,K\}$. Each prompt receives an integer number of rollouts $N_{i,k}$, with cumulative allocation

$N_i=\sum_{k=1}^K N_{i,k},$

subject to the global budget constraint

$\sum_{i=1}^M \sum_{k=1}^K N_{i,k}\le B.$

Its motivation is that GRPO and similar methods often use a fixed rollout budget per prompt even though prompt difficulty is heterogeneous: some prompts are too easy or too hard and therefore provide little learning signal, whereas medium-difficulty prompts produce both successes and failures and thus stronger gradients. CERO addresses this by explicitly modeling prompt-level diminishing returns through a concave increasing utility $T=0$0 over cumulative allocation (Zong et al., 4 Jun 2026).

For verifiable reasoning tasks, each rollout produces a binary success indicator $T=0$1 with unknown pass probability $T=0$2. CERO places a Beta prior on $T=0$3,

$T=0$4

and after $T=0$5 rollouts with $T=0$6 successes updates

$T=0$7

The posterior mean and variance are

$T=0$8

The key informativeness score is the posterior expected Bernoulli variance,

$T=0$9

Because $e^+e^-$0 is maximized at $e^+e^-$1 and vanishes at $e^+e^-$2, this score prioritizes medium-difficulty prompts; as evidence accumulates, $e^+e^-$3 decreases, thereby encoding diminishing returns.

With temperature $e^+e^-$4, CERO constructs the prompt utility

$e^+e^-$5

whose marginal utility is

$e^+e^-$6

In implementation, $e^+e^-$7 is replaced by an epoch-wise posterior estimate $e^+e^-$8. The resulting global problem is

$e^+e^-$9

subject to

$T=0$0

Because utility depends on cumulative allocations across epochs, the objective is temporally nonseparable.

CERO handles this nonseparability through a Fenchel-dual reformulation. For

$T=0$1

the concave conjugate is

$T=0$2

with closed form, for $T=0$3,

$T=0$4

Introducing prompt-level dual variables $T=0$5 and a budget dual variable $T=0$6, the dual objective decomposes into identical per-epoch subproblems,

$T=0$7

The allocation rule is a threshold policy: $T=0$8 clipped by the remaining budget. CERO thus acts as a drop-in data-collection layer on top of GRPO-style policy optimization rather than changing the GRPO objective itself.

3. Theory, implementation, and empirical behavior in RL post-training

Under fixed prompt utilities, CERO proves regret against the offline allocation benchmark. If $T=0$9 denotes the offline optimum and $\sim 100$0 the achieved utility, the regret is

$\sim 100$1

With projected online gradient descent on compact domains and bounded gradients,

$\sim 100$2

The prompt-level update is

$\sim 100$3

and the budget-level update is

$\sim 100$4

The analysis assumes fixed $\sim 100$5, closed and concave $\sim 100$6, compact projection domains, and bounded supergradients.

The implementation uses Beta$\sim 100$7 priors, maintains $\sim 100$8 and $\sim 100$9, and triggers a GRPO policy update when the rollout buffer reaches size $i \in \{1,\dots,M\}$0. Per epoch, allocation and dual updates cost $i \in \{1,\dots,M\}$1, posterior maintenance costs $i \in \{1,\dots,M\}$2 sufficient statistics per prompt, and total memory is $i \in \{1,\dots,M\}$3. The reported practical overhead is less than $i \in \{1,\dots,M\}$4 second per allocation update.

The empirical study uses DeepSeek-R1-Distill-Qwen-1.5B, Qwen3-4B-Base, Qwen3-4B-Instruct, and Qwen2.5-Math-7B, trained on DAPO-Math-17K with $i \in \{1,\dots,M\}$5 training prompts in the verl framework for $i \in \{1,\dots,M\}$6 optimization steps. The average rollout count is $i \in \{1,\dots,M\}$7, the per-epoch cap is $i \in \{1,\dots,M\}$8, and hardware is H800 GPUs: the 1.5B and 4B models use 4 GPUs, the 7B model uses 8 GPUs, with approximate GPU hours of $i \in \{1,\dots,M\}$9, $k \in \{1,\dots,K\}$0, and $k \in \{1,\dots,K\}$1, respectively. Evaluation is on AIME24, AIME25, AIME26, and AMC23 using accuracy as pass rate. Against vanilla GRPO with fixed per-prompt rollout budget, CERO improves the average across all four benchmarks for every model–benchmark pair (Zong et al., 4 Jun 2026).

Model	Avg accuracy GRPO $k \in \{1,\dots,K\}$2 CERO	Improvement
R1-Distill-1.5B	$k \in \{1,\dots,K\}$3	$k \in \{1,\dots,K\}$4 points
Qwen3-4B-Base	$k \in \{1,\dots,K\}$5	$k \in \{1,\dots,K\}$6 points
Qwen3-4B-Instruct	$k \in \{1,\dots,K\}$7	$k \in \{1,\dots,K\}$8 points
Qwen2.5-Math-7B	$k \in \{1,\dots,K\}$9	$N_{i,k}$0 points

A central diagnostic is the effective prompt ratio, defined as the fraction of prompts whose rollout groups contain both successes and failures, yielding nonzero within-group reward variance and hence nonzero GRPO gradients. The reported observation is that GRPO’s effective prompt ratio collapses later in training, whereas CERO maintains a higher ratio by reallocating budget away from exhausted or hopeless prompts. This is presented as the mechanism sustaining useful training signal and explaining the performance gains.

4. CERO as the Circular Energy-Recovery Option/Collider

In accelerator physics, CERO is the generic concept of an $N_{i,k}$1 collider built in a large circular tunnel that uses energy-recovery linacs instead of storage rings. The paper on the Circular Energy Recovery Collider treats CERC as a concrete realization of this generic CERO idea in a $N_{i,k}$2 km tunnel. Its defining features are simultaneous energy recovery and particle recycling, very high luminosity at moderate wall-plug power, and extension of center-of-mass energy to $N_{i,k}$3–$N_{i,k}$4 GeV with polarized $N_{i,k}$5 beams (Litvinenko et al., 2022).

The topology places two superconducting ERLs in the same $N_{i,k}$6 km tunnel used for recirculation arcs. Beams make four passes through the linacs and arcs, accelerating to collision energy, colliding at the interaction region, then being phase-shifted by half an RF period and decelerated through the same linacs. After deceleration, both electrons and positrons are reinjected into damping rings for particle recycling and emittance cooling. Lost particles are replenished from compact injectors, so most particles are recycled cycle-to-cycle. To fit the damping rings’ momentum acceptance and minimize IBS, bunches are compressed during low-energy passes and decompressed at the end of the deceleration cycle before ring reinjection; decompression factors up to $N_{i,k}$7 at top energies reduce accumulated energy spread to the $N_{i,k}$8 damping-ring acceptance. Damping-ring energies are $N_{i,k}$9 GeV for top beam energies $N_i=\sum_{k=1}^K N_{i,k},$0 GeV, $N_i=\sum_{k=1}^K N_{i,k},$1 GeV at $N_i=\sum_{k=1}^K N_{i,k},$2 GeV, $N_i=\sum_{k=1}^K N_{i,k},$3 GeV at $N_i=\sum_{k=1}^K N_{i,k},$4 GeV, and $N_i=\sum_{k=1}^K N_{i,k},$5 GeV at $N_i=\sum_{k=1}^K N_{i,k},$6 GeV.

The synchrotron-radiation energy loss per turn is written as

$N_i=\sum_{k=1}^K N_{i,k},$7

and the corresponding storage-ring SR power is

$N_i=\sum_{k=1}^K N_{i,k},$8

In CERC, SR losses are summed over recirculation passes through the arcs. The design caps total SR at about $N_i=\sum_{k=1}^K N_{i,k},$9 MW for both beams, approximately $\sum_{i=1}^M \sum_{k=1}^K N_{i,k}\le B.$0–$\sum_{i=1}^M \sum_{k=1}^K N_{i,k}\le B.$1 MW per beam depending on energy; SR energy is not recovered, but acceleration energy is recovered in the linacs. The claimed consequence is a large reduction in site AC power relative to storage rings, with CERC AC power lower by about $\sum_{i=1}^M \sum_{k=1}^K N_{i,k}\le B.$2 MW across the energy range. The paper lists total RF-to-beam, magnet, and cryogenic AC powers at several operating points: at Z ($\sum_{i=1}^M \sum_{k=1}^K N_{i,k}\le B.$3 GeV per beam), approximately $\sum_{i=1}^M \sum_{k=1}^K N_{i,k}\le B.$4 MW RF-to-beam, $\sum_{i=1}^M \sum_{k=1}^K N_{i,k}\le B.$5 MW magnets, and $\sum_{i=1}^M \sum_{k=1}^K N_{i,k}\le B.$6 MW cryo AC; at $\sum_{i=1}^M \sum_{k=1}^K N_{i,k}\le B.$7 ($\sum_{i=1}^M \sum_{k=1}^K N_{i,k}\le B.$8 GeV), $\sum_{i=1}^M \sum_{k=1}^K N_{i,k}\le B.$9 MW RF-to-beam, $T=0$00 MW magnets, and $T=0$01 MW cryo AC; at HH ($T=0$02 GeV), $T=0$03 MW RF-to-beam, $T=0$04 MW magnets, and $T=0$05 MW cryo AC; and at $T=0$06 ($T=0$07 GeV), $T=0$08 MW RF-to-beam, $T=0$09 MW magnets, and $T=0$10 MW cryo AC. The RF system assumes $T=0$11 MHz BNL-3 five-cell cavities, $T=0$12, $T=0$13-m cryomodules with $T=0$14 cavities each, a $T=0$15 accelerating-field filling factor, and HOM loss factors of $T=0$16 V/pC for $T=0$17 mm and $T=0$18 V/pC for $T=0$19 mm. The total HOM power is reported as $T=0$20 kW and $T=0$21 W per absorber.

The $T=0$22-km arcs use a regular FODO lattice with combined-function magnets, zero-chromaticity cells, strong sextupole components, and small magnet gaps of about $T=0$23 mm. The cell period is about $T=0$24 m, with two $T=0$25 m combined-function magnets and $T=0$26 m drifts. At top energy, the dipole field is about $T=0$27 T, quadrupole gradients are $T=0$28 T/m, and sextupole components are $T=0$29 T/m and $T=0$30 T/m. The small gaps are used to reduce magnet power; for the $T=0$31 beam lines, the paper states that magnet consumption is about half that of storage-ring magnets in FCC-ee.

Luminosity is expressed as

$T=0$32

or, at fixed beam current, as $T=0$33. The machine design employs very small $T=0$34 values and micron-scale vertical beam sizes, together with large disruption parameters and beam pinch. Reported disruption parameters span approximately $T=0$35–$T=0$36 horizontally and $T=0$37–$T=0$38 vertically, with strong–strong collision simulations indicating acceptable emittance growth, limited to about $T=0$39 in the vertical plane for selected $T=0$40 values. Predicted luminosities, in units of $T=0$41, are approximately $T=0$42 at Z, $T=0$43 at W, $T=0$44 at HZ, $T=0$45 at $T=0$46, $T=0$47 at HH, and $T=0$48 at $T=0$49. Corresponding bunch charges are $T=0$50–$T=0$51 nC, bunch frequencies range from $T=0$52 kHz at Z to $T=0$53 kHz at HH, and beam current per beam decreases from about $T=0$54 mA at Z to $T=0$55 mA at $T=0$56.

A major design claim is the combination of energy reach and polarization. The machine is presented as extending practical operation to $T=0$57 GeV for $T=0$58 and to $T=0$59 GeV for $T=0$60. The paper states that at $T=0$61 GeV CERC can deliver integrated luminosity of about $T=0$62 even at about $T=0$63 of the power of a conventional SR-based collider. Polarization is tied to continuous production and recycling of particles through damping rings; ZGOUBI simulations on the proposed lattice report depolarization below $T=0$64 per full path. MadGraph estimates in the paper give cross-section enhancement factors relative to the unpolarized case: for $T=0$65, $T=0$66, $T=0$67 scales by $T=0$68, $T=0$69 by $T=0$70, and $T=0$71 by $T=0$72; for $T=0$73, $T=0$74, the factors are $T=0$75, $T=0$76, and $T=0$77; and for $T=0$78, $T=0$79, they are $T=0$80, $T=0$81, and $T=0$82.

5. CERO as absolute zero and the zero isotherm

In the thermodynamic paper, “CERO” is read as absolute zero, the zero isotherm $T=0$83. The central result is a purely thermodynamic proof of the Nernst heat theorem derived from the second law, without assuming vanishing specific heats or unattainability of $T=0$84 as a premise (Martín-Olalla, 2024).

The formal statement is that for any two equilibrium states $T=0$85 and $T=0$86 of a chemically homogeneous finite-density system that are isothermal at temperature $T=0$87,

$T=0$88

Equivalently, for mechanical variables $T=0$89,

$T=0$90

where $T=0$91 is a constant independent of $T=0$92. The proof starts from the second law in Planck form and uses the Carnot theorem as the constructive bridge to temperature: $T=0$93 The Clausius inequality is

$T=0$94

with equality for reversible cycles.

The key logical step is that $T=0$95 in the Carnot relation implies $T=0$96. If a reversible engine operating with a $T=0$97 cooler had any nonzero heat intake from the hot reservoir, then it would raise a weight while cooling only the hot reservoir, contradicting the second law. Consistency therefore requires

$T=0$98

and hence the isothermal entropy change at zero temperature must vanish: $T=0$99 By universality of the Carnot theorem, this conclusion does not depend on the working substance or on its mechanical or chemical configuration.

Standard thermodynamic continuity then yields

$e^+e^-$00

and therefore

$e^+e^-$01

At $e^+e^-$02, entropy is thus independent of mechanical coordinates,

$e^+e^-$03

Using Maxwell relations,

$e^+e^-$04

the proof implies, as $e^+e^-$05,

$e^+e^-$06

and similarly for other generalized variables. This is the usual differential form of the Nernst theorem.

The paper explicitly states that the argument does not assume $e^+e^-$07, and does not assume unattainability of $e^+e^-$08. Instead, unattainability is obtained as a corollary: since for any $e^+e^-$09,

$e^+e^-$10

adiabatic entropy cannot decrease, so adiabatic cooling cannot reach $e^+e^-$11. The remaining independent content of the third law is narrowed to the assertion that the entropy of a finite-density, chemically homogeneous body must not be negative. One may then choose the absolute entropy scale so that

$e^+e^-$12

which yields the common third-law convention.

The conditions of validity are explicit: the proof assumes internal equilibrium, chemical homogeneity, finite density, and existence of the usual thermodynamic derivatives in the low-temperature limit. It excludes cases such as phase coexistence, metastability, or frozen-in disorder at $e^+e^-$13, since these would invalidate reversibility or uniqueness of the equilibrium state. On this basis, residual entropy is not treated as a counterexample within the theorem’s scope; rather, it is associated with systems outside the proof’s assumptions.

6. Conditions, limitations, and domain-specific significance

For RL post-training, CERO’s theoretical guarantee is proved for fixed prompt utilities, meaning fixed $e^+e^-$14, whereas the implementation updates $e^+e^-$15 online. The paper therefore states that the regret bound does not fully capture the nonstationarity introduced by policy changes. It also notes that misspecified priors or rapidly drifting task difficulty can transiently misallocate budget, and that very large $e^+e^-$16 increases dual-memory linearly. At the same time, the framework is situated within a broader literature on rollout optimization, including sample selection, within-batch reallocation, and post-generation filtering, and is linked conceptually to bandits with knapsacks and online convex optimization with long-term constraints (Zong et al., 4 Jun 2026).

For collider design, the cited work states that no show-stoppers have been identified for a next-generation polarized $e^+e^-$17 collider based on ERLs and particle recycling, but it also makes the dependence on R&D explicit. The outstanding items include SRF cavity performance at $e^+e^-$18 MHz, realistic multi-pass BBU studies, damping-ring optics with $e^+e^-$19 momentum acceptance, polarized positron source and ring injection, full beam-beam and beamstrahlung simulations, small-gap combined-function magnet prototyping, and refined system-level power integration. The proposed staging path runs from Z/W/Higgs energies through $e^+e^-$20 and then to HH and $e^+e^-$21, while keeping total SR at about $e^+e^-$22 MW (Litvinenko et al., 2022).

For thermodynamics, the proof’s significance lies in relocating the Nernst heat theorem from a postulate about unattainability or low-temperature heat capacities to a consequence of second-law consistency plus the Carnot theorem. Its limitations are exactly those stated in the paper: internal equilibrium, chemical homogeneity, finite density, and regular low-temperature thermodynamic derivatives. Statistical and quantum proofs instead appeal to spectral properties such as ground-state structure; this thermodynamic proof does not. Residual entropy, where present, is interpreted as a sign of frozen-in disorder, metastability, or broken homogeneity rather than as a failure of the theorem within its domain of validity (Martín-Olalla, 2024).

Taken together, the cited literature uses CERO for three technically unrelated but individually precise constructs: an adaptive rollout-budgeting method with a Fenchel-dual online primal–dual algorithm and $e^+e^-$23 regret under fixed utilities; a circular ERL-based $e^+e^-$24 collider concept with simultaneous energy and particle recovery, polarized beams, and operation up to $e^+e^-$25–$e^+e^-$26 GeV; and the zero isotherm $e^+e^-$27, through which the Nernst heat theorem is derived from the second law.