FFzero: Cross-Disciplinary Zero-Order Innovations

• FFzero is a multi-domain concept that removes higher-order corrective mechanisms, such as NLO splitting in QCD or backward gradients in neural learning.
• It is applied in fields like perturbative QCD, large-language-model optimization, federated learning, and few-/zero-shot segmentation to streamline computational processes.
• Practical implementations reveal trade-offs between speedup and stability, demonstrating that eliminating higher-order structures can yield efficiency gains while retaining domain-specific challenges.

Searching arXiv for “FFzero” and closely related entries to ground the article in current papers. Searching for exact title and term variants: “FFzero”, “FZOO”, “ZERO factorization scheme”, and “forward-only learning framework”. FFzero is a polysemous research label whose meaning depends strongly on disciplinary context. In perturbative QCD it denotes the ZERO factorization-scheme ideal, or an “approximately FFzero” scheme in which the NLO DGLAP splitting functions are negligible in effect; in large-language-model optimization it names a fast forward-only zeroth-order optimizer aligned with FZOO; in physical learning it is the explicit name of a forward-only, backpropagation-free training framework; and in several adjacent literatures it appears as a contextual shorthand for federated zero-order optimization, few-/zero-shot learning, or zero-sequence phenomena in Fock spaces (Kolar, 2012, Dang et al., 10 Jun 2025, Guo et al., 25 Mar 2026, Neto et al., 2024, Mhanna et al., 2024, Shu et al., 2023, Wang et al., 12 Nov 2025, Zhu, 2011).

1. Terminological scope

Domain	Meaning of FFzero	Representative paper
Perturbative QCD	ZERO or approximately ZERO factorization scheme	(Kolar, 2012)
LLM optimization	Fast forward-only zeroth-order optimizer	(Dang et al., 10 Jun 2025)
Physical learning	Forward-only local-learning framework without backpropagation	(Guo et al., 25 Mar 2026)
Federated optimization	Contextual shorthand for federated zero-order methods	(Neto et al., 2024, Mhanna et al., 2024, Shu et al., 2023)
3D segmentation	“FZ” denotes few- and zero-shot capability	(Wang et al., 12 Nov 2025)
Complex analysis	Zero sequences and maximality in Fock spaces	(Zhu, 2011)

The term is therefore not a single standardized object across arXiv. Two usages are especially prominent. One is historical and field-specific: the QCD notion of a factorization scheme in which all NLO splitting functions vanish. The other is algorithmic and contemporary: a forward-only zeroth-order learning procedure, either for LLM fine-tuning or for physically realizable neural-network training. Several neighboring papers do not use the literal name FFzero, but explicitly map their contributions to that conceptual space.

A useful organizing principle is that all usages revolve around some notion of eliminating or bypassing a higher-order corrective structure: NLO splitting kernels in QCD, backward gradients in neural-network training, gradient-vector communication in federated learning, or visual pre-training in few-/zero-shot segmentation. This suggests a family resemblance rather than a single unified theory.

2. FFzero in perturbative QCD: the ZERO and approximately ZERO factorization schemes

In the QCD literature, FFzero denotes the conceptual target of a factorization scheme in which all next-to-leading-order splitting functions vanish. The defining condition of the ZERO factorization scheme is

$P^{(1)}_{ij}(x) = 0 \quad \forall\, i,j.$

Within DGLAP evolution,

$$\frac{\mathrm{d} f_i(x, M^2)}{\mathrm{d}\ln M^2} = \sum_j \left[ P^{(0)}_{ij}(x) + \frac{\alpha_s(M^2)}{2\pi} P^{(1)}_{ij}(x) + \cdots \right] \otimes f_j(x, M^2),$$

so the ZERO scheme shifts all NLO effects from evolution kernels into the hard-scattering coefficient functions (Kolar, 2012).

The attraction of this construction is specific to NLO Monte Carlo event generators. Initial-state parton showers are typically LO and probabilistic, whereas NLO splitting functions in $\overline{\mathrm{MS}}$ are nontrivial, can be negative in some $x$ regions, and are more complicated than LO ones. The ZERO scheme removes that mismatch by leaving shower evolution purely LO while retaining NLO hard-scattering cross-sections. In this sense it is the theoretically optimal scheme for combining LO parton showers with NLO matrix elements (Kolar, 2012).

The central result, however, is negative. The paper shows that the exact ZERO scheme has a limited range of practical applicability and cannot be used at NLO over the full $x$-range relevant for QCD phenomenology. To search for an “approximately FFzero” alternative, the admissible singlet NLO splitting functions are parameterized as

$$\mathbf{P}^{(1)}(x) = \mathbf{P}^{(1)}_0(x) + \sum_{k=1}^{N} \lambda_k\, \mathbf{P}^{(1)}_k(x),$$

with non-singlet NLO splitting functions set to zero, while enforcing the NLO momentum sum-rule constraints

$$\int_0^1 x\left( P^{(1)}_{QQ}(x) + P^{(1)}_{GQ}(x) \right) \mathrm{d}x = 0, \quad \int_0^1 x\left( P^{(1)}_{QG}(x) + P^{(1)}_{GG}(x) \right) \mathrm{d}x = 0.$$

Practical applicability imposes an additional Mellin-space constraint in the singlet sector. For $n_f=5$, the relevant solutions of

$$b^2 - \left( P^{(0)}_{QQ}(n) - P^{(0)}_{GG}(n) \right)^2 - 4\,P^{(0)}_{QG}(n) P^{(0)}_{GQ}(n) = 0$$

are approximately $n \approx 1.9001$ and $$\frac{\mathrm{d} f_i(x, M^2)}{\mathrm{d}\ln M^2} = \sum_j \left[ P^{(0)}_{ij}(x) + \frac{\alpha_s(M^2)}{2\pi} P^{(1)}_{ij}(x) + \cdots \right] \otimes f_j(x, M^2),$$0, and the NLO splitting functions must satisfy a further compatibility relation at those values to avoid hidden singularities and loss of applicability, especially at small $$\frac{\mathrm{d} f_i(x, M^2)}{\mathrm{d}\ln M^2} = \sum_j \left[ P^{(0)}_{ij}(x) + \frac{\alpha_s(M^2)}{2\pi} P^{(1)}_{ij}(x) + \cdots \right] \otimes f_j(x, M^2),$$1 (Kolar, 2012).

The concrete candidate produced by the optimization is EP0. In EP0, non-singlet NLO splitting functions vanish, the singlet NLO functions satisfy the applicability constraint and momentum sum rules, and the effect of NLO evolution on proton PDFs becomes extremely small. Yet EP0 is not close to ZERO in the literal sense $$\frac{\mathrm{d} f_i(x, M^2)}{\mathrm{d}\ln M^2} = \sum_j \left[ P^{(0)}_{ij}(x) + \frac{\alpha_s(M^2)}{2\pi} P^{(1)}_{ij}(x) + \cdots \right] \otimes f_j(x, M^2),$$2: in the low-$$\frac{\mathrm{d} f_i(x, M^2)}{\mathrm{d}\ln M^2} = \sum_j \left[ P^{(0)}_{ij}(x) + \frac{\alpha_s(M^2)}{2\pi} P^{(1)}_{ij}(x) + \cdots \right] \otimes f_j(x, M^2),$$3 region its singlet NLO splitting functions are large in magnitude and can strongly dominate over both the $$\frac{\mathrm{d} f_i(x, M^2)}{\mathrm{d}\ln M^2} = \sum_j \left[ P^{(0)}_{ij}(x) + \frac{\alpha_s(M^2)}{2\pi} P^{(1)}_{ij}(x) + \cdots \right] \otimes f_j(x, M^2),$$4 NLO splitting functions and the LO ones. The conclusion is that no full-range NLO-applicable factorization scheme can be both practically usable and genuinely close to the ZERO ideal. Consequently, factorization-scheme freedom cannot significantly mitigate the LO-shower/NLO-matrix-element mismatch across the full phenomenological $$\frac{\mathrm{d} f_i(x, M^2)}{\mathrm{d}\ln M^2} = \sum_j \left[ P^{(0)}_{ij}(x) + \frac{\alpha_s(M^2)}{2\pi} P^{(1)}_{ij}(x) + \cdots \right] \otimes f_j(x, M^2),$$5-range; the ZERO scheme remains useful only in its limited region of applicability, such as heavy-object production (Kolar, 2012).

3. FFzero as a fast forward-only zeroth-order optimizer for LLM fine-tuning

In contemporary optimization for LLMs, FFzero is the fast zeroth-order optimizer described in “FZOO: Fast Zeroth-Order Optimizer for Fine-Tuning LLMs towards Adam-Scale Speed” (Dang et al., 10 Jun 2025). Its starting point is the memory asymmetry between first-order and zeroth-order methods: on MultiRC with approximately 400 input tokens on average, OPT-30B requires about $$\frac{\mathrm{d} f_i(x, M^2)}{\mathrm{d}\ln M^2} = \sum_j \left[ P^{(0)}_{ij}(x) + \frac{\alpha_s(M^2)}{2\pi} P^{(1)}_{ij}(x) + \cdots \right] \otimes f_j(x, M^2),$$6 under full-parameter Adam but about $$\frac{\mathrm{d} f_i(x, M^2)}{\mathrm{d}\ln M^2} = \sum_j \left[ P^{(0)}_{ij}(x) + \frac{\alpha_s(M^2)}{2\pi} P^{(1)}_{ij}(x) + \cdots \right] \otimes f_j(x, M^2),$$7 under FZOO/FFzero, and OPT-66B fits in $$\frac{\mathrm{d} f_i(x, M^2)}{\mathrm{d}\ln M^2} = \sum_j \left[ P^{(0)}_{ij}(x) + \frac{\alpha_s(M^2)}{2\pi} P^{(1)}_{ij}(x) + \cdots \right] \otimes f_j(x, M^2),$$8A100 under FFzero while Adam full fine-tuning requires $$\frac{\mathrm{d} f_i(x, M^2)}{\mathrm{d}\ln M^2} = \sum_j \left[ P^{(0)}_{ij}(x) + \frac{\alpha_s(M^2)}{2\pi} P^{(1)}_{ij}(x) + \cdots \right] \otimes f_j(x, M^2),$$9A100 (Dang et al., 10 Jun 2025).

FFzero combines four ingredients: batched one-sided zeroth-order gradient estimation, Rademacher perturbations, CUDA-parallel batched forward execution, and step normalization by the standard deviation of perturbed losses. With parameters $\overline{\mathrm{MS}}$0, perturbation radius $\overline{\mathrm{MS}}$1, and i.i.d. Rademacher directions $\overline{\mathrm{MS}}$2, the optimizer computes

$\overline{\mathrm{MS}}$3

where $\overline{\mathrm{MS}}$4 and $\overline{\mathrm{MS}}$5. It then forms

$\overline{\mathrm{MS}}$6

The theoretical claim is that this update is formally equivalent, up to constants and higher-order terms, to a normalized-SGD step. Specifically, for small $\overline{\mathrm{MS}}$7,

$\overline{\mathrm{MS}}$8

with $\overline{\mathrm{MS}}$9 and $x$0, so the normalization by $x$1 mimics scaling by $x$2 (Dang et al., 10 Jun 2025).

The engineering speedup comes from fused batched forwards and sign-flip arithmetic. With $x$3, FFzero executes $x$4 forwards per step, counting the shared baseline $x$5, yet costs only approximately $x$6 the wall clock of MeZO’s 2-forward step. Reported per-step times on an 80 GB A100 are $x$7 for OPT-125M under FFzero versus $x$8 for MeZO and $x$9 for Adam; for RoBERTa-large they are $x$0, $x$1, and $x$2, respectively. In aggregate convergence cost, FFzero exceeds MeZO by $x$3 accuracy with $x$4 fewer forward passes on average across 11 tasks and multiple models, and for RoBERTa-large achieves $x$5 average accuracy with an $x$6 reduction in forward passes, reaching convergence speeds comparable to Adam (Dang et al., 10 Jun 2025).

The method is explicitly forward-only, integrates with PEFT, and is also applicable to non-differentiable objectives such as SQuAD F1, where it outperforms MeZO by $x$7 F1 on average across OPT-125M to 13B. Its main limitations are the usual zeroth-order trade-offs: sensitivity to the perturbation radius $x$8, variance growth in high dimension, and the need for fused implementations to realize the advertised wall-clock gains (Dang et al., 10 Jun 2025).

4. FFzero as a forward-only learning framework for physical neural networks

A distinct and explicit usage appears in “Local learning for stable backpropagation-free neural network training towards physical learning,” where FFzero is a forward-only learning framework that combines layer-wise local learning, prototype-based representations, and directional-derivative-based optimization through forward evaluations only (Guo et al., 25 Mar 2026). The motivating setting is physical neural networks, where exact gradients, reverse-mode autodiff, and bidirectional signal propagation are often unavailable or physically implausible.

FFzero trains each layer independently using a local goodness objective. For layer $x$9, with pre-activation $$\mathbf{P}^{(1)}(x) = \mathbf{P}^{(1)}_0(x) + \sum_{k=1}^{N} \lambda_k\, \mathbf{P}^{(1)}_k(x),$$0, classification uses fixed unit-norm class prototypes $$\mathbf{P}^{(1)}(x) = \mathbf{P}^{(1)}_0(x) + \sum_{k=1}^{N} \lambda_k\, \mathbf{P}^{(1)}_k(x),$$1 on the hypersphere and defines the local goodness

$$\mathbf{P}^{(1)}(x) = \mathbf{P}^{(1)}_0(x) + \sum_{k=1}^{N} \lambda_k\, \mathbf{P}^{(1)}_k(x),$$2

The local classification loss is

$$\mathbf{P}^{(1)}(x) = \mathbf{P}^{(1)}_0(x) + \sum_{k=1}^{N} \lambda_k\, \mathbf{P}^{(1)}_k(x),$$3

where $$\mathbf{P}^{(1)}(x) = \mathbf{P}^{(1)}_0(x) + \sum_{k=1}^{N} \lambda_k\, \mathbf{P}^{(1)}_k(x),$$4 is a margin. Regression uses antipodal prototypes and

$$\mathbf{P}^{(1)}(x) = \mathbf{P}^{(1)}_0(x) + \sum_{k=1}^{N} \lambda_k\, \mathbf{P}^{(1)}_k(x),$$5

Optimization is purely zeroth-order. For a random direction $$\mathbf{P}^{(1)}(x) = \mathbf{P}^{(1)}_0(x) + \sum_{k=1}^{N} \lambda_k\, \mathbf{P}^{(1)}_k(x),$$6, FFzero estimates a directional derivative by central differences,

$$\mathbf{P}^{(1)}(x) = \mathbf{P}^{(1)}_0(x) + \sum_{k=1}^{N} \lambda_k\, \mathbf{P}^{(1)}_k(x),$$7

and updates

$$\mathbf{P}^{(1)}(x) = \mathbf{P}^{(1)}_0(x) + \sum_{k=1}^{N} \lambda_k\, \mathbf{P}^{(1)}_k(x),$$8

where $$\mathbf{P}^{(1)}(x) = \mathbf{P}^{(1)}_0(x) + \sum_{k=1}^{N} \lambda_k\, \mathbf{P}^{(1)}_k(x),$$9 is the number of trainable parameters in the layer and $$\int_0^1 x\left( P^{(1)}_{QQ}(x) + P^{(1)}_{GQ}(x) \right) \mathrm{d}x = 0, \quad \int_0^1 x\left( P^{(1)}_{QG}(x) + P^{(1)}_{GG}(x) \right) \mathrm{d}x = 0.$$0 is the number of directions averaged. Supplementary Information 2 proves the expectation identity

$$\int_0^1 x\left( P^{(1)}_{QQ}(x) + P^{(1)}_{GQ}(x) \right) \mathrm{d}x = 0, \quad \int_0^1 x\left( P^{(1)}_{QG}(x) + P^{(1)}_{GG}(x) \right) \mathrm{d}x = 0.$$1

which is the key unbiasedness statement for the local estimator (Guo et al., 25 Mar 2026).

The paper’s main empirical claim is comparative and qualitative: global directional-derivative training with a backprop-like objective, denoted BP+DD, degrades sharply with depth and can approach chance, whereas FFzero, denoted FF+DD, maintains stable accuracy across depths and widths in MLP classification, remains stable as channels increase in CNN classification, and consistently outperforms BP+DD on synthetic regression and MNIST-as-regression tasks. In a simulated photonic neural network based on a two-layer programmable Clements MZI mesh with electro-optic nonlinearity, FFzero significantly outperforms BP+DD on MNIST classification and produces embeddings that cluster around class prototypes (Guo et al., 25 Mar 2026).

The paper’s central interpretive claim is that localizing the objective prevents the accumulation of zeroth-order gradient-estimation errors across layers. This suggests a viable route to truly in-situ physical learning: the learning system requires only forward probes, local signal taps, and local measurements, not a digital twin or a backward computational graph (Guo et al., 25 Mar 2026).

5. Federated zero-order interpretations of FFzero

In federated optimization, FFzero is not a standard algorithm name, but several papers explicitly map their methods to the same conceptual space: federated training without backpropagated gradients, using only function-value information, compressed communication, and forward-only local computation (Neto et al., 2024, Mhanna et al., 2024, Shu et al., 2023).

CyBeR-0 is the first zero-order optimization algorithm for memory-and-communication efficient federated learning that is resilient to Byzantine faults. Each round uses $$\int_0^1 x\left( P^{(1)}_{QQ}(x) + P^{(1)}_{GQ}(x) \right) \mathrm{d}x = 0, \quad \int_0^1 x\left( P^{(1)}_{QG}(x) + P^{(1)}_{GG}(x) \right) \mathrm{d}x = 0.$$2 shared random directions, and each client sends only $$\int_0^1 x\left( P^{(1)}_{QQ}(x) + P^{(1)}_{GQ}(x) \right) \mathrm{d}x = 0, \quad \int_0^1 x\left( P^{(1)}_{QG}(x) + P^{(1)}_{GG}(x) \right) \mathrm{d}x = 0.$$3 scalar directional-derivative messages rather than $$\int_0^1 x\left( P^{(1)}_{QQ}(x) + P^{(1)}_{GQ}(x) \right) \mathrm{d}x = 0, \quad \int_0^1 x\left( P^{(1)}_{QG}(x) + P^{(1)}_{GG}(x) \right) \mathrm{d}x = 0.$$4-dimensional gradients. The server applies a scalar trimmed mean

$$\int_0^1 x\left( P^{(1)}_{QQ}(x) + P^{(1)}_{GQ}(x) \right) \mathrm{d}x = 0, \quad \int_0^1 x\left( P^{(1)}_{QG}(x) + P^{(1)}_{GG}(x) \right) \mathrm{d}x = 0.$$5

per direction, tolerating an attack fraction $$\int_0^1 x\left( P^{(1)}_{QQ}(x) + P^{(1)}_{GQ}(x) \right) \mathrm{d}x = 0, \quad \int_0^1 x\left( P^{(1)}_{QG}(x) + P^{(1)}_{GG}(x) \right) \mathrm{d}x = 0.$$6, and reconstructs

$$\int_0^1 x\left( P^{(1)}_{QQ}(x) + P^{(1)}_{GQ}(x) \right) \mathrm{d}x = 0, \quad \int_0^1 x\left( P^{(1)}_{QG}(x) + P^{(1)}_{GG}(x) \right) \mathrm{d}x = 0.$$7

The paper reports roughly $$\int_0^1 x\left( P^{(1)}_{QQ}(x) + P^{(1)}_{GQ}(x) \right) \mathrm{d}x = 0, \quad \int_0^1 x\left( P^{(1)}_{QG}(x) + P^{(1)}_{GG}(x) \right) \mathrm{d}x = 0.$$8 less communication than robust baselines for MNIST logistic regression, and “million-fold” communication savings for RoBERTa-Large fine-tuning compared to uncompressed gradient transmission, while maintaining competitive accuracy under Byzantine clients (Neto et al., 2024).

DZOFL pushes the compression further. It uses a single shared random direction $$\int_0^1 x\left( P^{(1)}_{QQ}(x) + P^{(1)}_{GQ}(x) \right) \mathrm{d}x = 0, \quad \int_0^1 x\left( P^{(1)}_{QG}(x) + P^{(1)}_{GG}(x) \right) \mathrm{d}x = 0.$$9 per round, and each client uploads exactly one quantized scalar

$n_f=5$0

which the server aggregates under packet drops as

$n_f=5$1

Clients then update locally with

$n_f=5$2

The paper proves nonconvex convergence with

$n_f=5$3

and a weighted average rate of order $n_f=5$4 when $n_f=5$5. Communication is reduced from $n_f=5$6 numbers per round to exactly one quantized scalar per client, and the energy model shows an $n_f=5$7 per-round communication-energy advantage over gradient-based FL (Mhanna et al., 2024).

FZooS addresses a different bottleneck: query inefficiency and client heterogeneity. Instead of finite-difference estimates requiring fresh queries at each step, it uses trajectory-informed Gaussian-process surrogate gradients built from previous function evaluations. The local surrogate is

$n_f=5$8

and a transferable global surrogate is built by random Fourier features and corrected adaptively: $n_f=5$9 The claimed advantage is that each gradient estimate requires zero extra function queries beyond the trajectory, and experimentally the method substantially outperforms FedZO, FedProx(FD), and SCAFFOLD(FD) in communication rounds and query counts across synthetic optimization, federated black-box adversarial attack, and non-differentiable metric optimization (Shu et al., 2023).

Taken together, these papers define a federated FFzero family in the conceptual sense: forward-only, function-query-based, communication-compressed training procedures that replace uploaded gradient vectors with a small number of scalars, shared seeds, or low-dimensional surrogate parameters.

6. Few-/zero-shot and language-guided uses of the “FZ” label

A related but distinct usage appears in EPSegFZ, where “FZ” denotes few- and zero-shot point-cloud semantic segmentation without visual pre-training (Wang et al., 12 Nov 2025). The method is not named FFzero, but it is directly relevant to the broader label because its zero-shot capability is central and because the paper explicitly frames the system as few- and zero-shot.

EPSegFZ uses three components. Multi-Prototype Sampling produces classwise raw prototypes

$$b^2 - \left( P^{(0)}_{QQ}(n) - P^{(0)}_{GG}(n) \right)^2 - 4\,P^{(0)}_{QG}(n) P^{(0)}_{GQ}(n) = 0$$0

ProERA adds learnable registers and mean-subtraction high-pass enhancement to self-attention in order to preserve high-frequency detail. LGPE injects CLIP-based text embeddings through

$$b^2 - \left( P^{(0)}_{QQ}(n) - P^{(0)}_{GG}(n) \right)^2 - 4\,P^{(0)}_{QG}(n) P^{(0)}_{GQ}(n) = 0$$1

and fuses them with visual prototypes according to

$$b^2 - \left( P^{(0)}_{QQ}(n) - P^{(0)}_{GG}(n) \right)^2 - 4\,P^{(0)}_{QG}(n) P^{(0)}_{GQ}(n) = 0$$2

DRPE then injects Euclidean and cosine relative positional encodings into query–prototype cross-attention.

The paper reports state-of-the-art mIoU on both S3DIS and ScanNet without any visual pre-training. On S3DIS, EPSegFZ + DGCNN reaches $$b^2 - \left( P^{(0)}_{QQ}(n) - P^{(0)}_{GG}(n) \right)^2 - 4\,P^{(0)}_{QG}(n) P^{(0)}_{GQ}(n) = 0$$3 mIoU in 2-way 1-shot and $$b^2 - \left( P^{(0)}_{QQ}(n) - P^{(0)}_{GG}(n) \right)^2 - 4\,P^{(0)}_{QG}(n) P^{(0)}_{GQ}(n) = 0$$4 in 2-way 5-shot, with a stated improvement of $$b^2 - \left( P^{(0)}_{QQ}(n) - P^{(0)}_{GG}(n) \right)^2 - 4\,P^{(0)}_{QG}(n) P^{(0)}_{GQ}(n) = 0$$5 over the previous SOTA. On ScanNet, it reaches $$b^2 - \left( P^{(0)}_{QQ}(n) - P^{(0)}_{GG}(n) \right)^2 - 4\,P^{(0)}_{QG}(n) P^{(0)}_{GQ}(n) = 0$$6 and $$b^2 - \left( P^{(0)}_{QQ}(n) - P^{(0)}_{GG}(n) \right)^2 - 4\,P^{(0)}_{QG}(n) P^{(0)}_{GQ}(n) = 0$$7, with a stated improvement of $$b^2 - \left( P^{(0)}_{QQ}(n) - P^{(0)}_{GG}(n) \right)^2 - 4\,P^{(0)}_{QG}(n) P^{(0)}_{GQ}(n) = 0$$8. In zero-shot evaluation on S3DIS, EPSegFZ with CLIP reaches $$b^2 - \left( P^{(0)}_{QQ}(n) - P^{(0)}_{GG}(n) \right)^2 - 4\,P^{(0)}_{QG}(n) P^{(0)}_{GQ}(n) = 0$$9 mIoU in 2-way 1-shot and $n \approx 1.9001$0 in 2-way 5-shot, outperforming prior zero-shot baselines. The zero-shot mechanism is prototype-based: if no visual support exists for a class, the class prototype is initialized from text alone and used with the same normalized dot-product decision rule (Wang et al., 12 Nov 2025).

This usage is terminologically adjacent rather than identical. It shows that “FZ” in current ML nomenclature often denotes a few-/zero-shot regime rather than a zeroth-order optimizer or a factorization-scheme ideal.

7. Mathematical and analytic uses: zero sequences in Bargmann–Fock spaces

In complex analysis, FFzero has also been used as a contextual shorthand for zero-sequence phenomena in Bargmann–Fock spaces, especially maximal zero sequences (Zhu, 2011). For $n \approx 1.9001$1 and $n \approx 1.9001$2, the weighted Fock space is

$n \approx 1.9001$3

with

$n \approx 1.9001$4

A sequence $n \approx 1.9001$5 is a zero sequence if some nonzero $n \approx 1.9001$6 has zero set exactly $n \approx 1.9001$7, counting multiplicities. The associated vanishing ideal is

$n \approx 1.9001$8

The paper establishes the existence of maximal zero sequences, defined by two simultaneous properties: $n \approx 1.9001$9 is one-dimensional, and for any $$\frac{\mathrm{d} f_i(x, M^2)}{\mathrm{d}\ln M^2} = \sum_j \left[ P^{(0)}_{ij}(x) + \frac{\alpha_s(M^2)}{2\pi} P^{(1)}_{ij}(x) + \cdots \right] \otimes f_j(x, M^2),$$00, the enlarged set $$\frac{\mathrm{d} f_i(x, M^2)}{\mathrm{d}\ln M^2} = \sum_j \left[ P^{(0)}_{ij}(x) + \frac{\alpha_s(M^2)}{2\pi} P^{(1)}_{ij}(x) + \cdots \right] \otimes f_j(x, M^2),$$01 is no longer a zero sequence. More generally, for any positive integer or $$\frac{\mathrm{d} f_i(x, M^2)}{\mathrm{d}\ln M^2} = \sum_j \left[ P^{(0)}_{ij}(x) + \frac{\alpha_s(M^2)}{2\pi} P^{(1)}_{ij}(x) + \cdots \right] \otimes f_j(x, M^2),$$02, there exists a zero sequence with $$\frac{\mathrm{d} f_i(x, M^2)}{\mathrm{d}\ln M^2} = \sum_j \left[ P^{(0)}_{ij}(x) + \frac{\alpha_s(M^2)}{2\pi} P^{(1)}_{ij}(x) + \cdots \right] \otimes f_j(x, M^2),$$03 (Zhu, 2011).

The construction uses the critical square lattice

$$\frac{\mathrm{d} f_i(x, M^2)}{\mathrm{d}\ln M^2} = \sum_j \left[ P^{(0)}_{ij}(x) + \frac{\alpha_s(M^2)}{2\pi} P^{(1)}_{ij}(x) + \cdots \right] \otimes f_j(x, M^2),$$04

and the associated Weierstrass sigma function $$\frac{\mathrm{d} f_i(x, M^2)}{\mathrm{d}\ln M^2} = \sum_j \left[ P^{(0)}_{ij}(x) + \frac{\alpha_s(M^2)}{2\pi} P^{(1)}_{ij}(x) + \cdots \right] \otimes f_j(x, M^2),$$05, whose zeros are exactly the lattice points. A key threshold is

$$\frac{\mathrm{d} f_i(x, M^2)}{\mathrm{d}\ln M^2} = \sum_j \left[ P^{(0)}_{ij}(x) + \frac{\alpha_s(M^2)}{2\pi} P^{(1)}_{ij}(x) + \cdots \right] \otimes f_j(x, M^2),$$06

For $$\frac{\mathrm{d} f_i(x, M^2)}{\mathrm{d}\ln M^2} = \sum_j \left[ P^{(0)}_{ij}(x) + \frac{\alpha_s(M^2)}{2\pi} P^{(1)}_{ij}(x) + \cdots \right] \otimes f_j(x, M^2),$$07, if $$\frac{\mathrm{d} f_i(x, M^2)}{\mathrm{d}\ln M^2} = \sum_j \left[ P^{(0)}_{ij}(x) + \frac{\alpha_s(M^2)}{2\pi} P^{(1)}_{ij}(x) + \cdots \right] \otimes f_j(x, M^2),$$08 is the smallest positive integer with $$\frac{\mathrm{d} f_i(x, M^2)}{\mathrm{d}\ln M^2} = \sum_j \left[ P^{(0)}_{ij}(x) + \frac{\alpha_s(M^2)}{2\pi} P^{(1)}_{ij}(x) + \cdots \right] \otimes f_j(x, M^2),$$09, then removing $$\frac{\mathrm{d} f_i(x, M^2)}{\mathrm{d}\ln M^2} = \sum_j \left[ P^{(0)}_{ij}(x) + \frac{\alpha_s(M^2)}{2\pi} P^{(1)}_{ij}(x) + \cdots \right] \otimes f_j(x, M^2),$$10 lattice points from $$\frac{\mathrm{d} f_i(x, M^2)}{\mathrm{d}\ln M^2} = \sum_j \left[ P^{(0)}_{ij}(x) + \frac{\alpha_s(M^2)}{2\pi} P^{(1)}_{ij}(x) + \cdots \right] \otimes f_j(x, M^2),$$11 yields a maximal zero sequence. The broader significance is that Fock-space zero sets are highly unstable: adding a finite number of points can convert a zero set into a uniqueness set, in sharp contrast to Hardy- and Bergman-space behavior (Zhu, 2011).

This mathematical usage is conceptually distant from the ML and QCD ones, but it preserves the core semantic motif of “zero” as a structural extremum: vanishing kernels, vanishing gradients, or maximal vanishing sets.

8. Overall significance

Across disciplines, FFzero consistently denotes an attempt to remove a difficult intermediate object from the formalism. In QCD that object is the NLO splitting function; in LLM optimization it is the backward pass and its activation/gradient storage; in physical learning it is the full reverse-mode computational graph; in federated optimization it is the communicated gradient vector; in few-/zero-shot segmentation it is the dependence on visual pre-training; and in Fock-space analysis it is the maximal zero set beyond which the zero-sequence property collapses (Kolar, 2012, Dang et al., 10 Jun 2025, Guo et al., 25 Mar 2026, Neto et al., 2024, Mhanna et al., 2024, Shu et al., 2023, Wang et al., 12 Nov 2025, Zhu, 2011).

The common methodological lesson is more nuanced than the shared label might suggest. In several cases the zero ideal is attractive but unattainable in full generality. Approximately ZERO factorization schemes cannot simultaneously satisfy full-range applicability and literal smallness of $$\frac{\mathrm{d} f_i(x, M^2)}{\mathrm{d}\ln M^2} = \sum_j \left[ P^{(0)}_{ij}(x) + \frac{\alpha_s(M^2)}{2\pi} P^{(1)}_{ij}(x) + \cdots \right] \otimes f_j(x, M^2),$$12 (Kolar, 2012). Zeroth-order optimization can approach Adam-scale speed, but only with careful batching, variance control, and systems-level implementation (Dang et al., 10 Jun 2025). Forward-only physical learning can be stable, but only when objectives are localized layer-wise rather than treated globally (Guo et al., 25 Mar 2026). Federated zero-order methods can be communication-efficient, but they must explicitly manage estimator variance, heterogeneity, quantization, or Byzantine failures (Neto et al., 2024, Mhanna et al., 2024, Shu et al., 2023).

FFzero is therefore best understood not as a single theory but as a recurring research pattern: the design of systems that operate effectively after excising an expensive, unstable, or physically implausible higher-order mechanism.