Reformalization: Recasting Formal Representations

Updated 6 July 2026

Reformalization is the process of recasting existing formal statements, proofs, and theories into alternative representations while preserving their semantic content.
It spans diverse applications from autoformalization in mathematics to proof assistant transfers, inference invariance, and structural recasting in physics and machine learning.
Iterative and reflective reformalization methods streamline workflows by ensuring semantic fidelity, promoting library alignment, and enabling modular, self-correcting formalizations.

Reformalization is the recasting of an existing statement, proof, inference problem, dynamical law, or learning criterion into a different formal representation while attempting to preserve its underlying content. In recent arXiv usage, the term does not denote a single technique. It appears as a relaxed stage of research-level autoformalization into Lean, as transfer of a machine-checked development from one proof assistant to another, as invariance under equivalent presentations of an inference problem, and as a structural rewriting of theories in physics, computation, and noisy-label learning (Patel et al., 2023, Guilloud et al., 2 Jul 2026, Trésor et al., 19 Jun 2026, Stoyanovsky, 2011, 0809.1790, Gui et al., 2021). This suggests that reformalization is best understood as a family of semantics-preserving or semantics-seeking transformations rather than a domain-specific method.

1. Meanings and scope

The term is used in several distinct senses. In theorem proving, it usually denotes a move from informal or differently formalized mathematics into a target proof language. In inference theory, it denotes equivalent presentations of the same informational content under relabelling, coarsening, or embedding. In physics and machine learning, it can denote a recasting of an existing formalism so that hidden structural assumptions become explicit.

Usage	Source $\rightarrow$ target	Representative papers
Research-math autoformalization	LaTeX + natural language $\rightarrow$ Lean-like syntax	(Patel et al., 2023, Chen et al., 28 Oct 2025)
Proof-assistant transfer	Mizar or HOL Light $\rightarrow$ Lean or Agda	(Guilloud et al., 2 Jul 2026)
Inference reformulation invariance	Equivalent measurable presentations $\rightarrow$ same inferential answer	(Trésor et al., 19 Jun 2026)
Structural scientific recasting	Existing formal theory $\rightarrow$ new algebraic or physical formulation	(Stoyanovsky, 2011, 0809.1790, Gui et al., 2021)

A common misconception is to identify reformalization with direct translation. The proof-oriented literature explicitly separates reformalization from both traditional manual formalization and low-level proof translation. The source may already be formal, the target is expected to be native and library-aligned, and correctness is typically enforced a posteriori by the target kernel rather than by a correct-by-construction compiler (Guilloud et al., 2 Jul 2026). In the autoformalization literature, reformalization can also denote an intentionally incomplete first pass that preserves enough structure for later alignment, rather than immediate integration with a foundational library (Patel et al., 2023).

2. Research-level mathematical autoformalization

In research-level mathematics, reformalization is used to relax the autoformalization problem so that higher-level theorems from arXiv papers can be expressed first in a Lean-like syntax and only later reconciled with an existing library. The central decomposition is: formalize theorems (unlinked), formalize definitions (unlinked), entity linking, type adjustment / type-checking, and optionally proof search. The decisive design choice is unlinked formalization: the generated Lean statement need only be syntactically correct, and may use invented identifiers or ad hoc types such as irrational_number or is_brjuno_number. Library alignment, including mappings such as informal “vector space” to mathlib’s module, is deferred to later steps. The paper introduces arXiv2Formal, a beta benchmark of 50 research-level theorem statements formalized in Lean 3, with average natural-language length 140 tokens and average Lean length 113 tokens (Patel et al., 2023).

The benchmark is statement-level rather than proof-level: LaTeX theorem or lemma statements are paired with Lean declarations ending in := sorry. Construction proceeds by extracting theorem-like statements from arXiv papers, translating them with GPT-3.5 turbo using few-shot in-context learning, and then manually correcting the outputs to obtain gold Lean 3 statements. The authors emphasize that this is more scalable than requiring annotators to fully integrate each theorem into mathlib from the outset, because the initial task is only to produce Lean syntax that logically matches the paper statement.

The empirical baseline is a machine-translation evaluation from LaTeX or natural language to Lean 3. Few-shot prompting improves quality sharply.

Model	BLEU / TER	TrueSkill
GPT-3.5	17.2 / 140.5	10.9
+ICL 1	31.9 / 93.4	25.4
+ICL 5	37.5 / 85.7	28.0
+ICL 10	41.7 / 83.9	33.1

The reported correlation between BLEU/TER improvements and human quality judgments is $R = 0.99$ on a small experiment, and the main observed failure modes are type hallucinations, notation mismatches, and difficulty with multipart theorems. The proposed interpretation is modular: many apparent translation errors are reassigned to later stages such as entity linking and type adjustment rather than being treated as failures of the first-stage reformalization itself (Patel et al., 2023).

3. Reflective and iterative reformalization

A later line of work treats reformalization as an iterative process in which formalization, critique, and repair are jointly optimized rather than executed in a single pass. ReForm defines a reflective loop over a natural-language problem $Q$ and a history $\mathcal H_t = \{(S_1,C_1),\dots,(S_{t-1},C_{t-1})\}$ , where $S_t$ is a candidate formal statement and $C_t$ is a critique of its semantic fidelity. The system uses supervised fine-tuning on 447k multi-iteration trajectories and reinforcement learning with Prospective Bounded Sequence Optimization, in which auxiliary rewards supervise critique faithfulness and terminal rewards require both Lean correctness and semantic consistency. Across four benchmarks, ReForm reports an average improvement of 17.2 percentage points over the strongest baselines; it also introduces ConsistencyCheck, a benchmark of 859 expert-annotated items, and reports that even human-written formalizations in the source corpora are semantically incorrect in 16.4% of miniF2F cases and 38.5% of ProofNet cases (Chen et al., 28 Oct 2025).

A separate reference-free framework casts full-theorem autoformalization as monotonic refinement over four dimensions: Formal Validity, Logical Preservation, Mathematical Consistency, and Formal Quality. It defines a masked composite objective

$\rightarrow$ 0

uses Lean4 for $\rightarrow$ 1, LLM judges for the soft dimensions, and accepts only candidates that strictly improve the estimated objective. The paper gives lower-confidence-bound and convergence results, and reports on miniF2F a progression from 80.33% Formal Validity and 55.69 overall score at $\rightarrow$ 2 to 93.44% Formal Validity and 78.22 overall score at $\rightarrow$ 3; on ProofNet the corresponding numbers are 29.57% and 16.52 at $\rightarrow$ 4, rising to 44.09% and 29.79 at $\rightarrow$ 5 (Zhang et al., 30 Jan 2026).

Trellis pushes the same idea to whole research papers by treating a proof as a Tablet, a DAG of nodes with a LaTeX side and a Lean side, and by enforcing progress through a deterministic kernel, verifier lanes for substantiveness, correspondence, and soundness, fingerprint-based reopening, and a monotone notion of shallow-coarse-closure. In an end-to-end Lean formalization of a recent Ramsey-theory result, the final Tablet contains 156 nodes, 119 theorem-like obligations, and 12,854 lines of Lean; the run takes 140 supervisor cycles, all 119 proof obligations build in the checker with no sorry or extra axioms, and the natural-language proof corpus expands from about 3,700 words in the source paper to about 15,196 words in the Tablet (Pegden, 8 Jun 2026).

Roundtrip verification and repair use a different but related notion of reformalization. A natural-language rule $\rightarrow$ 6 is formalized as $\rightarrow$ 7, back-translated as $\rightarrow$ 8, and reformalized as $\rightarrow$ 9. Formal equivalence $\rightarrow$ 0 is then checked by an SMT solver; disagreement triggers stage diagnosis and targeted repair. On 150 traffic rules, diagnosis-guided repair raises formal equivalence from 44.7% to 85.3% for Claude Opus 4.6 and from 61.3% to 82.7% for GPT-5.2. The paper explicitly treats roundtrip formal consistency as necessary but not sufficient for faithfulness, because the pipeline may stabilize at a semantically different fixed point (Amrollahi et al., 27 Apr 2026).

Taken together, these systems shift reformalization away from one-pass translation and toward self-correcting workflows. This suggests a methodological convergence: semantics preservation is increasingly approximated by iterative agreement between heterogeneous signals such as proof assistants, LLM critics, roundtrip equivalence checks, and structured process constraints.

4. Proof-assistant transfer as reformalization

A distinct usage of the term begins not from natural language but from an existing formal development. Reformalization is defined there as: given a formal development proving a statement $\rightarrow$ 1, produce a new, independently checkable development in a different target proof assistant that establishes the corresponding claim $\rightarrow$ 2. The target is expected to be mathematically faithful, idiomatic in the destination system, library-aligned, and checked from scratch by the target kernel. This is neither standard manual formalization nor low-level logic translation (Guilloud et al., 2 Jul 2026).

The Jordan Curve Theorem is used as the case study, with three reformalizations: Mizar $\rightarrow$ 3 Lean, HOL Light $\rightarrow$ 4 Lean, and HOL Light $\rightarrow$ 5 Agda. The Mizar $\rightarrow$ 6 Lean pipeline extracts file-level and theorem-level dependency graphs from Mizar intermediate artifacts, builds Lean skeletons, aligns source objects with Mathlib definitions, and then fills proofs using the source proof text and dependency metadata. The resulting Lean project contains 413 lemmas and theorems, about 474,000 characters or roughly 11,000 lines, and compiles in 111 seconds with the Mathlib cache; the relevant Mizar source subset is about 775,000 characters or roughly 23,500 lines. Human effort is reported as about 20 hours over about 10 days.

The HOL Light $\rightarrow$ 7 Lean pipeline is more linear. Each section is catalogued in CSV form, source theorems are classified as already in Mathlib, not applicable to Lean, or requiring implementation, and proofs are filled one sorry at a time with continuous compilation. The completed Lean development contains more than 910 theorems and lemmas and 150 definitions, with about 1.4 million characters or roughly 29,000 lines, compared with about 1.7 million characters or roughly 59,000 lines for the original HOL Light development. Of the HOL Light statements, 258 are already in Mathlib, 213 are not applicable to Lean, and 116 new Lean lemmas or definitions have no HOL Light equivalent. Human effort is about 10 hours over about a week, and from section N onward the pipeline is reported to run autonomously. The HOL Light $\rightarrow$ 8 Agda port remains incomplete at the time of writing, with about 27,000 lines covering foundations, Sections A–D, and about half of E; the paper attributes much of the difficulty to type-checker performance, memory pressure, and the mismatch between HOL’s always-inhabited types and Agda’s possibly empty types (Guilloud et al., 2 Jul 2026).

The paper’s main design conclusions are procedural rather than foundational. Heavy bookkeeping, explicit coverage of every source theorem, guardrails against creative proof divergence, systematic library alignment, and a tight edit–compile loop are presented as decisive. A plausible implication is that, for large-scale reformalization, workflow engineering can dominate raw model capability.

5. Reformulation invariance in inference

In statistical inference, reformalization refers neither to theorem proving nor to proof transfer, but to equivalent presentations of the same inferential problem. The foundational principle is reformulation invariance: an inference method should return the same answer whenever the same information is presented in an equivalent form. On finite spaces, measurable maps factor into relabelling, coarsening, and subset embedding; the paper also considers disjoint unions, resolution changes via sub- $\rightarrow$ 9-algebras, and positive reweightings $\rightarrow$ 0 (Trésor et al., 19 Jun 2026).

The inferential problem is written as minimization of a divergence from a reference measure $\rightarrow$ 1 subject to an information set $\rightarrow$ 2: $\rightarrow$ 3 The paper then recasts this numerically familiar picture into order-theoretic form: inference selects least elements under a total preorder on positive measures, and a divergence is only a numerical representation of that preorder. Reformulation invariance is formalized as functoriality from a category of information sets to Čencov’s category of statistical models.

The main representation results form a restriction hierarchy. Invariance under disjoint splitting, coarse-graining, and reference-measure behavior yields $\rightarrow$ 4-divergences. Adding mass–distribution independence forces the admissible family down to $\rightarrow$ 5-divergences and KL. Adding moment–distribution independence forces KL alone. The finite-space representation is then lifted to general measurable spaces by a closure construction. The paper positions this as a replacement for directly postulated consistency axioms of Shore–Johnson and Csiszár, arguing that those axioms follow from reformulation invariance alone and that the measurable-space treatment avoids the finite-alphabet limitation of earlier formulations (Trésor et al., 19 Jun 2026).

This use of reformalization is structurally different from proof-oriented work, but the thematic overlap is strong: equivalent descriptions are required to induce invariant outcomes, and the main object of study is not syntax but semantic stability under changes of representation.

6. Structural recasting in physics and machine learning

In mathematical physics, reformalization can denote a change in the ambient formal structure of a theory. One paper recasts renormalization in quantum field theory as part of deformation quantization of a Poisson algebra $\rightarrow$ 6 of classical Hamiltonians on $\rightarrow$ 7. A quantization $\rightarrow$ 8 of the full algebra, together with a non-canonical embedding

$\rightarrow$ 9

turns the mismatch between naive regular functional differential operators and the full quantum algebra into a renormalization map. For regularized Hamiltonians $\rightarrow$ 0 approximating a singular $\rightarrow$ 1, the map $\rightarrow$ 2 is identified with renormalization; divergences arise because $\rightarrow$ 3 typically diverges as $\rightarrow$ 4 (Stoyanovsky, 2011).

A separate toy-model paper uses the term more polemically. There, a self-interaction Lagrangian is claimed to misformulate the underlying mechanics, producing spurious mass renormalizations; exact renormalization is then interpreted as discarding harmful corrections and recovering the correct equations of permanently coupled constituents. The paper argues that the “right theoretical formulation” is to model those coupled constituents directly, without the initial self-action ansatz. This is presented as a reformulation in which renormalization becomes an artifact of an earlier mathematical error (Kalitvianski, 2011).

In cellular-automaton theory, reformalization means treating CA not as abstract synchronous update rules but as models of closed physical systems. The resulting Closed Cellular Automaton is a 5-tuple $\rightarrow$ 5 with a translation-commutative local interaction $\rightarrow$ 6 and a local update $\rightarrow$ 7, so that one global step factors as $\rightarrow$ 8. This move makes local implementation, memory overhead, and finite-depth locality explicit. The paper proves that any standard CA can be simulated by such a model, but also that some reversible CA, notably shift-right on an infinite 1D lattice, cannot be implemented as reversible CCA or finite-depth local circuits. This use of reformalization therefore exposes physical realizability constraints that are hidden by the standard abstract definition (0809.1790).

In noisy-label deep learning, reformalization refers to a principled rewriting of the small-loss criterion. Under class-conditional noise with a diagonally dominant transition matrix $\rightarrow$ 9, the paper proves that correctly labeled examples with the same observed label $R = 0.99$ 0 have smaller cross-entropy than mislabeled ones, provided the model is sufficiently close to the population minimizer. It then reformalizes the vanilla criterion by replacing single-epoch global ranking with class-wise ranking by mean loss across epochs, together with class-aware selected counts $R = 0.99$ 1. The resulting RSL method, and its semi-supervised extension RSL_WM, improve over prior small-loss baselines on CIFAR-10, CIFAR-100, and WebVision; for example, on CIFAR-10 with uniform noise $R = 0.99$ 2, RSL_WM reaches 93.38% best accuracy, compared with 87.27% for JoCoR and 84.85% for Cross Entropy (Gui et al., 2021).

Across these scientific uses, reformalization does not mean translation between languages. It means rewriting the formal object of analysis so that previously implicit structure—library dependence, equivalence of presentations, local physical implementation, or class-wise noise geometry—becomes explicit and tractable.