Free Data Formalism Overview

• Free Data Formalism is a conceptual framework where one component of a system is freely specified, while the remaining structure is defined by constraints, dynamics, or algebraic transformations.
• It is applied across fields such as general relativity, formal mathematics, model-free computational mechanics, and structural operational semantics, each adapting the idea to its unique requirements.
• The approach offers a controlled balance between flexibility and rigor, allowing essential parts to be data-driven while preserving necessary structural consistency via imposed mathematical or physical conditions.

Searching arXiv for papers related to “free data formalism” and adjacent usages of “free data,” “free formalism,” and “free-fermionic formalism.” The available literature suggests that free data formalism is not a single standardized doctrine but a family of technically distinct constructions in which some component of a system is treated as freely specifiable, directly data-driven, or released from exhaustive proof obligations, while the remaining structure is imposed by constraints, dynamics, or algebraic transformations. In current arXiv usage, this expression and closely related ones appear in at least four research settings: Sachs’ characteristic initial value problem in general relativity, a communication-first “free approach” to formal mathematics, model-free data-driven computational mechanics, and structural operational semantics with data (Paoli et al., 2017, Farmer, 21 Mar 2026, Ciftci et al., 2023, Gebler et al., 2013).

1. Free data in the characteristic initial value problem

In Sachs’ characteristic initial value formulation, free data denotes the part of the gravitational initial data on a null hypersurface that is not fixed by the constraint equations and can be specified arbitrarily on an initial null slice, up to corner data. In the metric formulation, one uses Bondi-type coordinates

$x^\mu=(u,r,x^A),\qquad g^{00}=0,\qquad g^{0A}=0,$

with null hypersurfaces $u=\mathrm{const}$ and null generator

$l^\mu=-g^{\mu\nu}\partial_\nu u.$

If $\gamma_{AB}$ is the induced 2-metric on the transverse surfaces $S$, Sachs’ freely specifiable bulk data are the conformal 2-metric

$\check\gamma_{AB}:=\gamma^{-1/2}\gamma_{AB},$

equivalently the shear of the null geodesic congruence, together with corner data on the initial 2-surface $S_0$ (Paoli et al., 2017).

The geometric link between the induced metric and the shear is

$$\frac12 \mathcal L_l \gamma_{AB} =\frac{1}{2N}\partial_r \gamma_{AB} =\sigma_{AB}+\frac12 \gamma_{AB}\theta.$$

Hence the traceless part of $\mathcal L_l\gamma_{AB}$ is the shear $\sigma_{AB}$. Up to zero modes and corner terms, the formalism carries two physical degrees of freedom. This identifies “free data” not with unconstrained metric data in general, but specifically with the conformal geometry or, equivalently, the shear of the outgoing null congruence.

A common misconception is that null free data are simply arbitrary metric coefficients on a null slice. The formulation is narrower: the free sector is the conformal 2-metric or shear, while the remaining fields are constrained, and the initial specification is incomplete without the corner data.

2. Real connection variables and the ISO(2) reinterpretation

A first-order reformulation recasts Sachs’ free data in real connection variables on a null foliation. Starting from the Einstein–Cartan action,

$u=\mathrm{const}$0

the null hypersurface normal is encoded by an internal direction $u=\mathrm{const}$1, with $u=\mathrm{const}$2 on a null hypersurface. After decomposing the canonical variables, the dynamical connection data are the components of $u=\mathrm{const}$3 orthogonal to $u=\mathrm{const}$4. These are the translation generators of the little group ISO(2) stabilizing $u=\mathrm{const}$5, with stabilizer split into a helicity rotation $u=\mathrm{const}$6, two null-rotation or translation generators $u=\mathrm{const}$7, and a radial boost $u=\mathrm{const}$8. The free data are the translational or null-rotation components, namely the helicity $u=\mathrm{const}$9 projection

$l^\mu=-g^{\mu\nu}\partial_\nu u.$0

entering through $l^\mu=-g^{\mu\nu}\partial_\nu u.$1 (Paoli et al., 2017).

On shell, these connection components become the shear of the null congruence. With

$l^\mu=-g^{\mu\nu}\partial_\nu u.$2

and decomposition

$l^\mu=-g^{\mu\nu}\partial_\nu u.$3

the Newman–Penrose relation is

$l^\mu=-g^{\mu\nu}\partial_\nu u.$4

Two second-class light-cone simplicity constraints $l^\mu=-g^{\mu\nu}\partial_\nu u.$5 enforce the equivalence between the connection quantity and the metric shear. The paper further shows canonical equivalence by matching symplectic potentials: the first-order symplectic structure reproduces the metric symplectic structure, with the traceless part of the momentum exactly equal to the shear of $l^\mu=-g^{\mu\nu}\partial_\nu u.$6.

A distinctive feature of the first-order null Hamiltonian theory is that Sachs’ propagating equations for the shear become tertiary constraints,

$l^\mu=-g^{\mu\nu}\partial_\nu u.$7

Their role is to preserve, under retarded-time evolution, the equality between connection shear and metric shear established on the initial hypersurface. The conversion of propagating equations into constraints is enabled by the algebraic Bianchi identity

$l^\mu=-g^{\mu\nu}\partial_\nu u.$8

The same framework also analyzes torsion by writing

$l^\mu=-g^{\mu\nu}\partial_\nu u.$9

which modifies shear, expansion, twist, non-geodesicity, and non-affinity. For minimally coupled fermions, torsion induces an axial-current contribution to twist; for non-minimal coupling it can also affect expansion and non-geodesicity. This establishes that the connection-based free data formalism is not merely a change of variables, but a canonical reinterpretation of radiative degrees of freedom with a precise constraint structure.

3. The “free approach” to formal mathematics

In formal mathematics, the phrase free approach denotes an alternative to the standard proof-assistant workflow. It is called “free” because it is free of the obligation to formally prove and mechanically check all details of a mathematical development using a proof assistant. The underlying claim is not that mathematics should cease to be formal, but that proofs need not always be machine-certified. The paper defines four requirements for this approach: a fully formal logic that supports standard mathematical practice; proofs that may be traditional, formal, or mixed; support for the little theories method; and several levels of software support, ranging from minimal LaTeX support to proof assistants or interactive mathematics laboratories (Farmer, 21 Mar 2026).

The framework is explicit that this is not informal mathematics. Mathematics remains fully formal at the level of logic and representation, but not necessarily fully certified by machine-checked proofs. The motivation is that the standard proof-assistant model prioritizes certification, whereas many practitioners prioritize communication. Traditional proofs are therefore retained because they are generally shorter, easier to read, easier to write, and better suited for explaining ideas. The tradeoff is also explicit: the free approach cannot achieve the same level of assurance as the standard approach.

Its implementation is based on Alonzo, described as a practice-oriented classical higher-order version of predicate logic, a version of Church’s simple type theory, and closely related to Q0 and LUTINS. Alonzo is intended to remain close to ordinary mathematical practice: it supports higher-order quantification, definite description, functions, sets, tuples, lists, structures, and partial functions with undefined expressions handled by the “traditional approach to undefinedness.” It also distinguishes a formal syntax for machines from a compact notation for humans. At the semantic level, it has one semantics for mathematics based on standard models and one for logic based on Henkin-style general models.

The framework’s organizational principle is the little theories method. A theory is a pair $\gamma_{AB}$0, where $\gamma_{AB}$1 is a language of the logic and $\gamma_{AB}$2 is a set of sentences of $\gamma_{AB}$3, called axioms. A theory morphism is a meaning-preserving translation between theories, and mathematical knowledge is organized as a theory graph whose nodes are theories and directed edges are theory morphisms. This supports the transport of theorems from an abstract theory to more concrete ones.

The implementation uses Alonzo modules to construct theories, define theory morphisms, develop theories by definitions and theorems, and transport knowledge between theories. The present software support is intentionally minimal, consisting of LaTeX macros for types and expressions and LaTeX environments for modules. The paper closes with a call to develop logics, software, libraries of formal mathematical knowledge, and practitioner training for this communication-oriented style of formalization. A central point of clarification is that the free approach is not presented as a replacement for proof assistants, but as an additional path for settings where communication matters more than full certification.

4. Model-free and free-data computational mechanics

In computational mechanics, a closely related usage appears in model-free data-driven computational mechanics. Here the constitutive law is not parameterized by a phenomenological model such as

$\gamma_{AB}$4

but replaced by a finite set of observed material states in strain-stress space,

$\gamma_{AB}$5

The mechanics problem is reformulated as the selection of an admissible state that is closest to the data set:

$\gamma_{AB}$6

The admissible set $\gamma_{AB}$7 enforces compatibility, equilibrium, and boundary conditions, while the distance is induced by the norm

$\gamma_{AB}$8

In this sense the formulation is model-free because no constitutive equation is prescribed; the solution is the point in the admissible set that best matches the observed data (Ciftci et al., 2023).

The cited work embeds this paradigm in a physics-informed GAN. The generator is a PINN with trainable parameters $\gamma_{AB}$9,

$S$0

and the strain is computed kinematically,

$S$1

Thus

$S$2

The generator is constrained by strong-form residuals for equilibrium and boundary conditions,

$S$3

with physics-informed loss

$S$4

The discriminator operates on strain-stress pairs encoded in Voigt notation. Its distinctive mechanism is nearest data point selection:

$S$5

Rather than comparing a generated state with an arbitrary target, it compares against the closest state in the observed material database. The adversarial objective is modified accordingly; in Wasserstein form,

$S$6

and with gradient penalty,

$S$7

This use of “free-data” does not mean physics-free computation. Compatibility, equilibrium, and boundary conditions remain explicit, and the paper even hard-enforces boundary conditions in its quarter-plate example by modifying the generator outputs. The formal novelty is instead that the data set replaces the constitutive model while the field equations remain active constraints.

5. Processes with data and algebraic currying

In the theory of structural operational semantics with data, a “free data formalism” is described as one in which data is treated as an add-on that can be pushed into labels, reasoned about algebraically, and then abstracted away again. The formal setting uses a multisorted signature $S$8 with two distinguished sorts, $S$9 for processes and $\check\gamma_{AB}:=\gamma^{-1/2}\gamma_{AB},$0 for data, split into process operators $\check\gamma_{AB}:=\gamma^{-1/2}\gamma_{AB},$1 and data operators $\check\gamma_{AB}:=\gamma^{-1/2}\gamma_{AB},$2. Operational transitions are defined on pairs of process and data terms,

$\check\gamma_{AB}:=\gamma^{-1/2}\gamma_{AB},$3

so a transition rule can describe both process evolution and store evolution (Gebler et al., 2013).

The central device is the currying transformation. A rule

$\check\gamma_{AB}:=\gamma^{-1/2}\gamma_{AB},$4

is transformed into

$\check\gamma_{AB}:=\gamma^{-1/2}\gamma_{AB},$5

The resulting curried system has process terms as states and triples $\check\gamma_{AB}:=\gamma^{-1/2}\gamma_{AB},$6 as labels. This permits the direct reuse of ordinary SOS meta-theorems, including rule formats guaranteeing commutativity, associativity, idempotence, zero elements, unit elements, and distributivity.

The behavioral notion for the original system is stateless bisimilarity. A relation $\check\gamma_{AB}:=\gamma^{-1/2}\gamma_{AB},$7 is a stateless bisimulation if it is symmetric and, whenever $\check\gamma_{AB}:=\gamma^{-1/2}\gamma_{AB},$8,

$\check\gamma_{AB}:=\gamma^{-1/2}\gamma_{AB},$9

The crucial theorem is that stateless bisimilarity for the original data-aware system coincides with strong bisimilarity for the curried system with closed labels. This bridge is what makes the algebraic meta-theory reusable.

For axiomatization, the framework extends BCCSP with auxiliary store operators $S_0$0 and $S_0$1. Intuitively, $S_0$2 requires that the current store equals $S_0$3, while $S_0$4 updates the store to $S_0$5 after a process step. The algebraic laws include

$S_0$6

$S_0$7

$S_0$8

$S_0$9

and

$$\frac12 \mathcal L_l \gamma_{AB} =\frac{1}{2N}\partial_r \gamma_{AB} =\sigma_{AB}+\frac12 \gamma_{AB}\theta.$$0

Under the restriction that $$\frac12 \mathcal L_l \gamma_{AB} =\frac{1}{2N}\partial_r \gamma_{AB} =\sigma_{AB}+\frac12 \gamma_{AB}\theta.$$1 is a finite set of constants, the resulting axiomatization is sound and ground-complete.

This setting illustrates a different sense of “free”: data is not hardwired into the state space of the meta-theory. Instead, it can be moved between states and labels, exposing store handling as equational structure and allowing existing process-algebraic machinery to be applied without redesigning the whole framework.

6. Comparative interpretation and recurrent limitations

Across these literatures, the factual commonality is not a shared formal calculus but a shared structural move. In general relativity, the freely specifiable object is the conformal 2-metric or shear on a null hypersurface. In formal mathematics, what is “free” is the absence of an obligation to formally prove and mechanically check every proof detail. In data-driven mechanics, the constitutive law is omitted and replaced by direct selection from observed strain-stress data. In SOS with data, store information is freed from a rigid state-based presentation by currying it into labels (Paoli et al., 2017, Farmer, 21 Mar 2026, Ciftci et al., 2023, Gebler et al., 2013).

These usages also share precise limits. In the Sachs setting, free data are specified only up to corner data, and their propagation is controlled by equations that in the first-order null Hamiltonian treatment become tertiary constraints. In the free approach to formal mathematics, the gain in accessibility is purchased by lower assurance than full proof-assistant certification. In model-free mechanics, there is no constitutive law, but compatibility, equilibrium, and boundary conditions remain compulsory. In the SOS-with-data framework, the ground-completeness result relies on a finite set of data constants, and extension to arbitrary data terms is stated to be open.

A plausible implication is that, in these contexts, free does not signify absence of structure. It denotes a controlled release from one specific source of rigidity: constraint equations for part of the initial data, proof certification for every detail, phenomenological constitutive modeling, or a fixed state-level treatment of data. The retained structure is then supplied by geometry, symplectic and constraint analysis, theory graphs and logic, physics-informed optimization, or algebraic metatheory. That pattern explains why the same adjective can recur across otherwise unrelated fields without implying a single cross-disciplinary formalism.