Free Will Equation: A Multidisciplinary View

• The Free Will Equation is a conceptual framework that unites probabilistic independence, impossibility theorems, and dynamical formulations across quantum mechanics and philosophical models.
• It encapsulates measurement independence in Bell-type analyses, where violating this condition can reproduce quantum correlations and challenge local determinism.
• Applications range from quantum foundations and AGI decision policies to Newtonian force laws, highlighting both structured agency and recursive, uncomputability-based models.

Searching arXiv for relevant papers on "Free Will Equation", measurement independence, and Free Will Theorem to ground the article in current arXiv literature. “Free Will Equation” is not a standard technical term with a single accepted definition across physics, philosophy, and mathematical modeling. In the arXiv literature, the expression designates several distinct formal roles: a probabilistic independence condition in Bell-type analyses, a theorem-level impossibility structure in the Conway–Kochen tradition, an extended Newtonian force law for interacting agents, a quantum-inspired decision policy for AGI, and a family of meta-theoretic or uncomputability-based formalisms. Across these usages, the recurring theme is the attempt to formalize the relation between choice, determination, randomness, and higher-order structure. In foundational quantum work, the relevant condition is typically measurement independence, written as $P(a,b|\lambda)=P(a,b)$ or equivalently $P(\lambda|a,b)=P(\lambda)$, while other papers explicitly propose dynamical or recursive equations under the same label (Lorenzo, 2011, Giuffrida et al., 2022, Kabali, 4 Jul 2025).

1. Quantum-foundational meaning: measurement independence as the canonical “free will equation”

In Bell-type discussions, “free will” usually means that experimenters’ measurement settings are statistically independent of the hidden variables that influence outcomes. In probability form, the condition is typically written

$P(a,b|\lambda)=P(a,b)$

or equivalently

$P(\lambda|a,b)=P(\lambda).$

A stronger factorized form also appears: $P(a,b,\lambda)=P(a,b)\,P(\lambda).$ These expressions are the clearest candidates for a foundational “free will equation” in quantum mechanics because they encode measurement independence directly (Lorenzo, 2011, Bisognano, 2012).

In Di Lorenzo’s “Free will and quantum mechanics” (Lorenzo, 2011), the term “free will” is not introduced as a named equation, but the paper constructs a concrete classical model in which this independence is explicitly violated. Adam’s and Eve’s settings are determined by internal pendulums, while Aunt Angler’s source behavior depends on the same pendulum states. In the notation extracted from the model, if $a\in\{c,v\}$, $b\in\{w,s\}$, and $\lambda$ denotes the relevant hidden variables, then the model realizes

$P(a,b,\lambda)\neq P(a,b)\,P(\lambda),$

equivalently

$P(a,b|\lambda)\neq P(a,b).$

This is the operational negation of the free-will condition (Lorenzo, 2011).

The same independence condition appears in broader discussions of experimenters’ free choice. Bisognano’s “Experimenters’ Free Will and Quantum Certainty” (Bisognano, 2012) takes

$P(\lambda|a,b)=P(\lambda)$0

as the standard formal expression of experimenters’ free will, and argues that rejecting it opens a route to deterministic, globally self-consistent accounts of single outcomes. Palmer’s “A Deterministic Model of Free Will” (Palmer, 1 Apr 2025) denotes the same assumption as Measurement Independence,

$P(\lambda|a,b)=P(\lambda)$1

and then studies structured violations of it while maintaining that such violations do not invalidate experimenter free choice in his framework (Palmer, 1 Apr 2025).

This suggests a useful distinction. In quantum foundations, the “free will equation” is usually not an equation of motion but an independence constraint. Its violation is mathematically sufficient to reopen deterministic or local-hidden-variable constructions that Bell-type arguments ordinarily exclude (Lorenzo, 2011, Bisognano, 2012).

2. Bell correlations, CH/CHSH structure, and what follows when the equation is violated

The most explicit quantitative consequences of violating measurement independence are given in Di Lorenzo’s toy model (Lorenzo, 2011). The model defines the Clauser–Horne–type parameter

$P(\lambda|a,b)=P(\lambda)$2

with correlator

$P(\lambda|a,b)=P(\lambda)$3

The output assignments are $P(\lambda|a,b)=P(\lambda)$4 for $P(\lambda|a,b)=P(\lambda)$5 and $P(\lambda|a,b)=P(\lambda)$6 for $P(\lambda|a,b)=P(\lambda)$7 (Lorenzo, 2011).

Under the usual Bell assumptions, one has the local hidden-variable bound $P(\lambda|a,b)=P(\lambda)$8. Quantum mechanics allows up to $P(\lambda|a,b)=P(\lambda)$9. Di Lorenzo’s construction shows that a classical local model with violated free will can reproduce

$P(a,b|\lambda)=P(a,b)$0

and, with modified deck compositions, can even reach

$P(a,b|\lambda)=P(a,b)$1

the algebraic maximum (Lorenzo, 2011).

The physical significance is narrow but precise. The model is local in the sense that there is no superluminal causal influence during a run, yet the Bell bound fails because the source variables are correlated with future settings through common hidden structure. A plausible implication is that Bell inequalities rule out local hidden-variable models only together with measurement independence; they do not, by themselves, exclude sufficiently fine-tuned measurement dependence (Lorenzo, 2011).

This same point is echoed in later discussions. Bisognano argues that no-go theorems such as Bell and Kochen–Specker become obstructive only if experimenters’ free will is retained as a primitive assumption (Bisognano, 2012). Palmer likewise treats violation of Measurement Independence as compatible with a broader notion of free choice (Palmer, 1 Apr 2025). The random-variable reformulation in “Systems of random variables and the Free Will Theorem” (Dzhafarov et al., 2020) goes further: it argues that the “experimenters’ free will” assumption is not needed for the proof once contextuality of the relevant compound system is taken as a premise.

3. The Free Will Theorem family: theorem-level “equations” and impossibility conditions

In the Conway–Kochen line of work, the closest analogue of a “free will equation” is not a scalar law but an incompatibility schema. The Strong Free Will Theorem is summarized as

$P(a,b|\lambda)=P(a,b)$2

with “free” meaning “not a function of what has happened at earlier times (in any inertial frame)” for experimenters, and analogously for particles (0807.3286).

The SPIN axiom states that measurements of the squared components of spin in three orthogonal directions for a spin-1 particle yield one of

$P(a,b|\lambda)=P(a,b)$3

The TWIN axiom enforces equality of corresponding squared-spin outcomes for twinned spin-1 particles when matching directions are measured. MIN asserts that in a spacelike-separated arrangement, each experimenter can freely choose settings, and the distant particle’s response is independent of that choice (0807.3286).

Kochen’s later strengthening “On the Free Will Theorem” (Kochen, 2022) formulates the core contradiction in terms of deterministic response functions $P(a,b|\lambda)=P(a,b)$4, with the central value-matching condition

$P(a,b|\lambda)=P(a,b)$5

Combined with SPIN, this requires a function $P(a,b|\lambda)=P(a,b)$6 on a Kochen–Specker set of 33 directions such that for each of 40 orthogonal triples,

$P(a,b|\lambda)=P(a,b)$7

but the Kochen–Specker theorem asserts that no such function exists (Kochen, 2022).

The result can therefore be compactly expressed as an impossibility statement: $P(a,b|\lambda)=P(a,b)$8 or, in Kochen’s stronger formulation,

$P(a,b|\lambda)=P(a,b)$9

This is theorem-level rather than dynamical. It functions like an equation by specifying a set of jointly unsatisfiable constraints (Kochen, 2022, 0807.3286).

Gisin’s “The Free Will Theorem, Stochastic Quantum Dynamics and True Becoming in Relativistic Quantum Physics” (Gisin, 2010) reframes the same issue in terms of frame-dependent response functions. In one ordering,

$P(\lambda|a,b)=P(\lambda).$0

while in the reversed ordering,

$P(\lambda|a,b)=P(\lambda).$1

Demanding covariance of actual outcomes leads to

$P(\lambda|a,b)=P(\lambda).$2

$P(\lambda|a,b)=P(\lambda).$3

and these conditions cannot reproduce Bell-violating quantum correlations if free settings are retained (Gisin, 2010).

A different extension is given by Suarez’s “The General Free Will Theorem” (Suarez, 2010), which uses Bell-type functions

$P(\lambda|a,b)=P(\lambda).$4

for local models, and time-ordered nonlocal variants such as

$P(\lambda|a,b)=P(\lambda).$5

then argues via the before-before experiment that time-ordered nonlocal determinism also fails. The paper condenses its conclusion into

$P(\lambda|a,b)=P(\lambda).$6

This is again an implication schema rather than a differential or probabilistic law (Suarez, 2010).

4. Dynamical proposals explicitly labeled as free-will equations

Outside quantum foundations, several papers use “free will equation” for genuine state-update or force equations rather than independence conditions.

In “Extending Newton’s Laws of Motion with Free Will” (Giuffrida et al., 2022), a free will system consists of agents $P(\lambda|a,b)=P(\lambda).$7 with masses, positions, and velocities. Free will is represented by chosen inter-agent force functions $P(\lambda|a,b)=P(\lambda).$8. The central equation is

$P(\lambda|a,b)=P(\lambda).$9

together with Newton’s second law

$P(a,b,\lambda)=P(a,b)\,P(\lambda).$0

The same form is given in both discrete and continuous time (Giuffrida et al., 2022).

The authors interpret the minus sign on $P(a,b,\lambda)=P(a,b)\,P(\lambda).$1 as a built-in Golden Rule: if agent $P(a,b,\lambda)=P(a,b)\,P(\lambda).$2 exerts $P(a,b,\lambda)=P(a,b)\,P(\lambda).$3 on $P(a,b,\lambda)=P(a,b)\,P(\lambda).$4, then $P(a,b,\lambda)=P(a,b)\,P(\lambda).$5 experiences $P(a,b,\lambda)=P(a,b)\,P(\lambda).$6 itself. In this literature, the “free will equation” is therefore the decomposition of net force into freely chosen inter-agent forces subject to a structural ethical constraint (Giuffrida et al., 2022).

The paper also introduces bounded free will: $P(a,b,\lambda)=P(a,b)\,P(\lambda).$7 and conjectures that bounded free will may imply a speed bound

$P(a,b,\lambda)=P(a,b)\,P(\lambda).$8

These are not presented as established theorems, but as definitions and conjecture internal to the model (Giuffrida et al., 2022).

A more recent use appears in “The Free Will Equation: Quantum Field Analogies for AGI” (Kabali, 4 Jul 2025), where the term refers to a quantum-inspired stochastic policy over actions. The agent’s “cognitive state” is written as

$P(a,b,\lambda)=P(a,b)\,P(\lambda).$9

and action selection is defined by

$a\in\{c,v\}$0

Here $a\in\{c,v\}$1 is extrinsic value, $a\in\{c,v\}$2 is intrinsic motivation, $a\in\{c,v\}$3 weights intrinsic versus extrinsic terms, and $a\in\{c,v\}$4 is a temperature controlling stochasticity (Kabali, 4 Jul 2025).

The intrinsic term is instantiated as

$a\in\{c,v\}$5

and the temperature adapts to surprise: $a\in\{c,v\}$6 In this setting, “free will” is explicitly non-metaphysical and denotes adaptive spontaneity or controlled stochasticity in decision making (Kabali, 4 Jul 2025).

These two examples show a strong bifurcation in the term’s usage. In the Newtonian extension, free will is encoded as agent-chosen force functions within a dynamical law. In the AGI proposal, it is encoded as an intrinsically motivated softmax policy with adaptive entropy control (Giuffrida et al., 2022, Kabali, 4 Jul 2025).

5. Deterministic, computational, and meta-theoretic formulations

A separate body of work treats free will as a structural feature of recursion, uncomputability, or meta-theoretic hierarchy rather than as stochastic independence or a physical force law.

Palmer’s “A Deterministic Model of Free Will” (Palmer, 1 Apr 2025) introduces the state-update

$a\in\{c,v\}$7

embedded in a generalized 2-adic shift-map framework. Here $a\in\{c,v\}$8 is a just-in-time initialized microstate, $a\in\{c,v\}$9 is a control parameter representing macro-level evaluation, and $b\in\{w,s\}$0 is a deterministic “collapse” map designed so that under ignorance of the microstate,

$b\in\{w,s\}$1

This is presented as the central “free will equation” of the model (Palmer, 1 Apr 2025).

The paper’s conceptual move is that determinism need not imply predestination because crucial sub-Planck information is initialized “just in time” rather than at the Big Bang. A plausible implication is that the model seeks to preserve “could have done otherwise” by allowing alternative just-in-time microstates consistent with the same macroscopic history (Palmer, 1 Apr 2025).

In “Uncomputability and free will” (Nayakar et al., 2012), free will is modeled by a stochastic evolution

$b\in\{w,s\}$2

where $b\in\{w,s\}$3 and $b\in\{w,s\}$4 are computable matrices representing constraints and guidance, while $b\in\{w,s\}$5 is in general uncomputable (Nayakar et al., 2012). This is complemented by a diagonal argument. If an algorithm $b\in\{w,s\}$6 predicts the choice of an agent $b\in\{w,s\}$7,

$b\in\{w,s\}$8

one constructs

$b\in\{w,s\}$9

and obtains contradiction when $\lambda$0. The paper interprets this as showing that free-will dynamics must be uncomputable within the base theory (Nayakar et al., 2012).

The meta-theoretic line is developed much further in “Gödel, Tarski, Turing and the conundrum of free will” (Nayakar et al., 2014) and “The concept of free will as an infinite metatheoretic recursion” (Hashim et al., 2015). The starting point is the idea that a physical law $\lambda$1 determines ordinary dynamics,

$\lambda$2

while free will requires deviations from such laws caused by higher-order parameters. In the “FUN model,” a binary choice is written as

$\lambda$3

with $\lambda$4 representing Nature, $\lambda$5 Understanding, and $\lambda$6 a freedom parameter (Nayakar et al., 2014).

The more elaborate recursion paper defines will functions $\lambda$7 and preference distributions $\lambda$8 across causal levels $\lambda$9. The basic recursive clause is

$P(a,b,\lambda)\neq P(a,b)\,P(\lambda),$0

leading to the nested expression

$P(a,b,\lambda)\neq P(a,b)\,P(\lambda),$1

This is explicitly treated as the paper’s effective “free will equation,” with physical choice $P(a,b,\lambda)\neq P(a,b)\,P(\lambda),$2 generated by an infinite composition of higher-order will functions (Hashim et al., 2015).

That same paper introduces a quantitative measure

$P(a,b,\lambda)\neq P(a,b)\,P(\lambda),$3

the correlation between prime intent and actual action, and summarizes its rationalist-compatibilist position as

$P(a,b,\lambda)\neq P(a,b)\,P(\lambda),$4

This usage is highly nonstandard within physics, but it is mathematically explicit in the source (Hashim et al., 2015).

6. Controversies, limits, and competing interpretations

One persistent source of ambiguity is that “free will equation” designates incompatible things in different subliteratures. In Bell and Free Will Theorem contexts, the key mathematical object is an independence assumption or an impossibility theorem, not a dynamical law (Lorenzo, 2011, 0807.3286). In AGI or Newtonian extensions, by contrast, the label attaches to an explicit state-update or force equation (Giuffrida et al., 2022, Kabali, 4 Jul 2025). In meta-theoretic approaches, it refers to recursive schemas or uncomputability structures rather than empirically grounded physical dynamics (Nayakar et al., 2014, Hashim et al., 2015, Nayakar et al., 2012).

A second controversy concerns whether “free will” in foundational physics is best understood probabilistically or functionally. Di Lorenzo, Bisognano, and Palmer all work with measurement independence or its violation, using formulas such as $P(a,b,\lambda)\neq P(a,b)\,P(\lambda),$5 and $P(a,b,\lambda)\neq P(a,b)\,P(\lambda),$6 (Lorenzo, 2011, Bisognano, 2012, Palmer, 1 Apr 2025). Kochen, Conway, and Gisin instead formulate the issue as nonexistence of frame-compatible or Kochen–Specker-compatible response functions (Kochen, 2022, 0807.3286, Gisin, 2010). The random-variable reformulation by Dzhafarov and Kujala argues that the “free will of experimenters” assumption is redundant once contextuality is properly stated, and that the more fundamental point concerns the impossibility of representing a contextual compound system as a mixture of non-signaling deterministic systems (Dzhafarov et al., 2020).

A third dispute is over whether violating measurement independence necessarily destroys experimenter freedom. Bisognano suggests denial of free choice may be the only workable route to deterministic single-outcome explanations (Bisognano, 2012). Palmer argues instead that violating Measurement Independence does not invalidate the free-will conclusion of his deterministic model (Palmer, 1 Apr 2025). This suggests that “free will” is not being used uniformly: in one literature it means statistical independence of settings from hidden variables, while in another it refers to structured agency or control even when such independence fails.

Finally, many proposals are explicitly limited in scope. Di Lorenzo’s pendulum-and-deck construction is a contrived toy model for a CHSH scenario (Lorenzo, 2011). The Newtonian extension is a conceptual framework rather than an empirically validated physical theory (Giuffrida et al., 2022). The AGI paper demonstrates its mechanism only in a non-stationary multi-armed bandit environment (Kabali, 4 Jul 2025). The recursive and uncomputability-based models are primarily conceptual and do not supply direct experimental tests (Nayakar et al., 2014, Hashim et al., 2015, Nayakar et al., 2012).

7. Synthesis

The phrase “Free Will Equation” therefore has no single canonical referent. In quantum foundations, the most defensible core usage is the measurement-independence condition

$P(a,b,\lambda)\neq P(a,b)\,P(\lambda),$7

or equivalently

$P(a,b,\lambda)\neq P(a,b)\,P(\lambda),$8

whose violation allows classical local systems to reproduce or exceed Bell-type correlations (Lorenzo, 2011). In the Free Will Theorem tradition, the corresponding formal object is an impossibility structure: under SPIN, TWIN, and MIN, no deterministic response functions of past variables can exist (0807.3286, Kochen, 2022). In relativistic refinements, covariance constraints on response functions play the same role (Gisin, 2010).

Beyond foundations, the term is repurposed for explicit dynamical constructions. The Newtonian extension writes

$P(a,b,\lambda)\neq P(a,b)\,P(\lambda),$9

embedding free will in agent-chosen forces constrained by the Golden Rule (Giuffrida et al., 2022). The AGI proposal writes

$P(a,b|\lambda)\neq P(a,b).$0

treating free will as adaptive stochastic policy modulation (Kabali, 4 Jul 2025). Deterministic and meta-theoretic models instead locate free will in update maps such as

$P(a,b|\lambda)\neq P(a,b).$1

or in infinite recursive constructions like

$P(a,b|\lambda)\neq P(a,b).$2

or in uncomputable mixtures of guidance and constraint

$P(a,b|\lambda)\neq P(a,b).$3

(Palmer, 1 Apr 2025, Hashim et al., 2015, Nayakar et al., 2012).

Taken together, these works do not converge on a unified theory. What they do establish is a taxonomy. A “free will equation” may mean an independence postulate, a no-go theorem, a state-update law, a force decomposition, a stochastic control policy, or an infinite recursion. The term is therefore best understood as a family resemblance concept rather than a settled technical object.