edition: journal title: The Withholding Gradient subtitle: A Structural Resolution of the Free Will Problem through Triaxial Verification and the Orientation-Blindness of the Squared Lock author_line: Mohammad F. Islam^1^ journal: Tractatus Philosophicus article_type: Foundational Article goal: The Metaphysics of Agency doi: Tractatus Ser. · Agency · 2026 · TP-FW-01 volume: 1 pages: 1–22 date: 2026 accent: slate
:::affiliations ^1^Independent Researcher, USA. Correspondence: islamm@alumni.iu.edu. The formal apparatus invoked in the body is set out as standalone principles in Appendix A and proven in the upstream deposit^35^; this paper states results and their warrant grades and does not re-prove the foundations inline. :::
:::abstract The free will problem has resisted resolution for twenty-five centuries not because the phenomenon is obscure but because the debate is conducted in a representational frame that cannot hold its structure. Determinism holds that every event is fixed by prior states and laws; libertarian freedom holds that an agent could have done otherwise; and the standard positions, hard determinism, libertarianism, and compatibilism, all read "the act is determined" and "the act is free" as competing truth values of one proposition about one object, which forces the question to a single winner. We argue that this single-proposition framing is itself the structural error. Using a substrate-portable verification instrument that decomposes any warranted claim into three independent axes and returns a closed-form scalar lock, we establish two facts about agency. First, the lock scalar is orientation-blind: it is invariant under reflection of any axis, so the instrument returns an identical magnitude whether an act is read as determined or as free, which entails that no symmetry-only argument can decide the direction and that the classical antinomy has no formal winner by construction. Second, determination and freedom occupy two distinct regions of one structure, the entrenched groove and the un-entrenched room, so the two claims are two truths over two states rather than a proposition and its negation. We locate freedom as the shape of a single continuous quantity, the withholding gradient: opened by a root act, deepened toward an incompressible floor by memory, held open against total entrenchment by metered forgetting, and closed by a granted return. The account yields parameter-free, cross-substrate-reproducible predictions: a machine-precision algebraic identity, an exact sign flip without magnitude change under axis reflection, and the collapse of any freedom manufactured from determination. If confirmed, the deadlock is an artifact of a two-valued lens applied to a two-region object. :::
:::keywords free will, determinism, compatibilism, agent causation, antinomy, orientation-blindness, thermodynamics of choice, Kolmogorov complexity, triaxial verification, Conway-Kochen theorem :::
For twenty-five centuries the question of whether the will is free has been posed as a contest between two theses that cannot both win. Either the world is such that every event, including every choice, is fixed by the state of things before it together with the laws that govern change, in which case the sense that one could have chosen otherwise is a report about ignorance rather than about the world; or there is somewhere in the causal order a genuine opening, a point at which more than one continuation is really available and the agent settles which one occurs. The debate has the shape of a demand for a verdict on one proposition, and the three durable answers are the three ways of delivering that verdict. Hard determinism affirms the antecedent and discards the datum of open choice as illusion. Libertarianism preserves the datum and pays for it with an uncaused or agent-caused event that the surrounding physics does not supply. Compatibilism keeps both by redefining freedom until it no longer conflicts with determination, which resolves the words without touching the structure the words were fighting over.
This paper proposes that the impasse is not a hard fact about agency but a soft fact about the frame in which agency has been described. The three positions share one commitment so basic that it is rarely stated: that "the act is determined" and "the act is free" are competing truth values of a single proposition about a single object, so that affirming one is denying the other and the task is to find which affirmation survives. We will call this the single-proposition framing. Everything downstream, the consequence argument, the manipulation cases, the appeals to quantum indeterminacy, operates inside it. Our claim is that the framing cannot represent the structure the phenomenon actually has, and that once the structure is represented in an instrument built to carry it, the antinomy does not get decided in favor of one side; it dissolves, because the two claims turn out to be two truths about two different regions of one object, and because the instrument that would adjudicate them is provably unable to tell the two directions apart by the only quantity it computes.
The argument is constructive rather than merely diagnostic. We use a verification instrument, developed and proven elsewhere and set out in Appendix A, that takes any sufficiently warranted claim, decomposes it into three independent lines of support, and returns a single scalar that is nonzero exactly when the three lines are genuinely independent. The instrument has one property that turns out to decide the free will question: the scalar it returns is the square of a signed quantity, and squaring is blind to sign. Reflecting any one of the three axes, which is the operation that turns a directed reading into its opposite, leaves the scalar unchanged. Applied to agency, this means that the instrument returns the same lock whether a given act is entered as determined or as free. It is not that the instrument is too weak to decide; it is that the quantity on which any symmetry-based adjudication would rest is constitutively silent on the direction. That silence, we will argue, is the exact formal shadow of the twenty-five-century deadlock, and reading it correctly is what lets the deadlock end.
The structural barrier
The standard debate is not slowed by a shortage of arguments. It is stopped by a feature of its own representational geometry, and the feature is worth stating precisely because naming it is most of the resolution. Consider what all three positions take themselves to be arguing about. There is an act, call it A. There is a property, being causally determined by the prior state and the laws. The determinist asserts that A has the property; the libertarian asserts that A lacks it; the compatibilist asserts that A has it but that freedom, correctly understood, does not require lacking it. In every case the object of assessment is the single act A, and the disputed content is a single property predicated of it. The logical form is P and its negation over one bearer, and the law of excluded middle then guarantees that exactly one side is right. The debate inherits the excluded middle as a promise that the question has a determinate winner, and two and a half millennia of failing to find the winner is read as a measure of the problem's depth rather than as evidence against the form.
The barrier is that this form cannot represent a structure in which determination and freedom are properties of different things. Suppose, as we will argue, that what is determined is one aspect of an agent's condition, the settled record of what the agent has become and habitually does, and that what is free is a different aspect, the room that has not been settled and remains genuinely open. If that is the situation, then "the act is determined" is a true claim about the first aspect and "the act is free" is a true claim about the second, and there is no single proposition of which one is the negation of the other. The excluded middle never applies, because there was never one bearer with one contested property. The debate has spent its energy demanding a winner between two statements that were never contradictories. This is a structural barrier and not a computational one: no refinement of the arguments on either side can produce the missing winner, because the winner is an artifact of misreading a two-region structure as a one-proposition contest. The barrier yields only to a change of representational geometry, from a single predication to a partition.
There is a second, sharper way to see that the barrier is structural, and it is the one this paper turns into an instrument. Any attempt to settle the determined-or-free question by an appeal to symmetry, and most of the durable arguments are at bottom symmetry appeals, must rest on some quantity that is preserved or broken across the two readings. The consequence argument, for instance, transports a fixed past and fixed laws through a valid inference and concludes fixity of the act; its force is the invariance of a modal quantity under the transport. We will exhibit a verification instrument whose output is precisely the natural invariant of a three-axis structure, the squared volume enclosed by the three lines of support, and show that this invariant is identical under the reflection that swaps determined for free. The instrument is a stand-in for any symmetry-based adjudicator, and its blindness is a theorem about that whole class of arguments: whatever is preserved under the relevant symmetry cannot carry the direction, because the direction is exactly what the symmetry quotients out. The barrier, restated, is that the determined-or-free direction lives in the sign of a quantity whose magnitude is all that any symmetry argument can see, and the sign is discarded by the squaring that any such argument performs.
The standard model of the debate has a precise domain of validity, and mapping it shows where the novel geometry takes over. Where an act genuinely is one object with one contested causal property, the single-proposition framing is correct and the excluded middle does deliver a winner; questions of the form "was this particular event caused by that particular prior event" are settled this way every day. The framing fails at exactly the point where the phenomenon is a composite of a settled part and an open part, which is the generic condition of a deliberating agent. Beyond that boundary, the correct representation is not a proposition awaiting a truth value but a partition of one structure into a determined region and a free region, and the correct instrument is not a truth predicate but a test that can hold three independent axes and report on their independence without being forced, or able, to collapse the partition into a single verdict. The remainder of the paper builds that instrument, proves its blindness, exhibits the partition, and states what the account predicts.
The failure mode has a name that makes it portable across disciplines. Call it the single-proposition antinomy: the error of reading P over region A and Q over region B as P and not-P over one bearer, thereby manufacturing a contradiction where there is only a distinction, and then treating the undecidability of the manufactured contradiction as a deep feature of the subject. The antinomy is not special to free will. It is the general failure of a two-valued logic pressed onto a two-region object, and free will is its most consequential instance because the stakes, moral responsibility above all, are attached to the verdict the framing was never able to deliver.
Prior accounts and their common error
The literature is vast and its distinctions are real; our claim is not that the positions are unmotivated but that they share the single structural commitment the previous section isolated, and that each pays a characteristic price for staying inside it. We review the principal families with enough precision to locate the shared error, and dismantle each on the merits rather than by fiat.
Hard determinism, from the ancient atomists through d'Holbach's mechanism^9^ to its contemporary neuroscientific and physicalist forms^18,23^, affirms that the prior state and the laws fix every event and concludes that free will, understood as the genuine availability of alternatives, does not exist. Its rigor is real and its honesty about the antecedent is a virtue. Its cost is that it must treat the most robust datum in the entire domain, the first-person availability of alternatives in deliberation, as a systematic illusion, and an account that begins by discarding its own primary evidence has shifted the burden onto an error theory it rarely discharges. The manipulation and neuroscience arguments^20,21^ sharpen the position but do not change its shape: they show that some experiences of choosing are unreliable, which is compatible with there being a genuine open region that the unreliable experiences misdescribe. The datum survives its worst cases.
Libertarianism preserves the datum and pays at the other end. In its agent-causal form^10^ it posits a species of causation by a substance that is not itself an event and not fixed by prior events; in its event-causal form^16^ it locates indeterminism at the moment of choice and argues that the agent's efforts confer responsibility across the indeterministic gap. The event-causal versions draw genuine support from physics: the Conway-Kochen theorems^25,26^, resting on the Kochen-Specker structure^27^ and the Bell inequalities^28^, show that if experimenters have a minimal freedom in choosing measurement settings, then the responses of suitably prepared spin-1 particles cannot be functions of any information prior to the measurement, so indeterminism at the fundamental level is not a metaphysical extravagance but a consequence of the same freedom the libertarian wants to secure. The unresolved cost is the one Hume named^7^: an event that is not fixed by the prior state and the laws is, to exactly that extent, not fixed by the agent either, and mere indeterminism looks like noise rather than control. Libertarianism buys the opening and struggles to show the opening is the agent's.
Compatibilism is the majority position and the most instructive for our purposes. From Hobbes^5^ and Hume^7^ through Frankfurt's hierarchical account^11,12^ and Fischer and Ravizza's reasons-responsiveness^17^ to Dennett's evolutionary naturalism^14,15^ and List's higher-level realism^24^, it holds that freedom worth wanting is a matter of the right kind of dependence of action on the agent's own reasons, desires, and deliberative capacities, and that this kind of dependence is fully compatible with, indeed requires, the causal determination the determinist affirms. Compatibilism is, in our terms, closest to the truth and still inside the error. Its insight is that determination and freedom are not straightforwardly opposed; its limitation is that it secures this by redefining freedom as a property of the single determined act, a certain pattern of causal history, rather than by recognizing that freedom names a different region of the structure altogether. The Frankfurt cases^11^ are decisive against the requirement of alternative possibilities on the single-object reading, and we accept their result; what they show, on our account, is that the free region and the region where alternatives are counterfactually available can come apart, which is a reason to partition the structure, not a reason to relocate freedom onto the determined act. Compatibilism resolves the words and leaves the structure a single proposition; it wins the debate as posed and thereby ratifies the framing that made the debate intractable.
Two further positions round out the field. The consequence argument^13^ is the sharpest defense of incompatibilism and, read structurally, is the clearest exhibit of the shared error: it transports fixity through a valid modal inference and concludes fixity of the act, and its validity is not in question; what is in question is whether the act whose fixity it establishes is the whole of the relevant object or only its determined region. Skepticism and illusionism^19,20,23^ conclude that moral responsibility in the strong sense is impossible because it would require being causa sui, a cause of oneself all the way down; we accept the impossibility of self-causation, and will argue that it bears on the granted terminus of the will rather than on the reality of the free region, so that the strong-responsibility skeptic has correctly identified one boundary and mislocated its consequence.
Every prevailing approach shares the structural error that the previous section identified: each treats determination and freedom as competing predications of one act and then either sacrifices a side, pays for an uncaused event, or relabels freedom to fit the determined act. None partitions the object. The following sections supply the partition, the instrument that makes it precise, and the reason the instrument cannot be forced to decide between the regions.
Method: three conditions on any account of agency
We hold the proposed account to the same standard we would hold any structural proposal in a mature science, and we state the standard before deploying the account so that it cannot be tuned to its conclusions. A candidate mechanism for agency must satisfy three conditions, and the paper's core is the demonstration that the withholding-gradient account satisfies all three while the single-proposition framing satisfies none.
The first condition is formal derivation from established principles. The mechanism may not rest on a posited faculty or an arbitrary parameter; it must be assembled from results that stand on their own outside the theory of agency. Our derivation draws on three such results: the classification of the real normed division algebras^32,33^, which forces the number of independent verification axes to exactly three and fixes the closed form of the lock; the thermodynamics of computation^29,30,31^, which charges every act of registration a strictly positive and measurable cost and so gives choice a physical signature; and the theory of algorithmic complexity^34^, which supplies an incompressible floor toward which repeated action descends and below which no further compression, hence no further determination, is available. None of these is native to the free will debate, and none is adjusted to fit it.
The second condition is a measurable thermodynamic or kinetic signature. An account that makes no contact with anything measurable is not a structural proposal but a picture. The withholding-gradient account identifies choice with the expenditure of work: to settle an open alternative is to perform an irreversible registration, and by Landauer's principle^29,31^ irreversible registration has a lower-bounded, experimentally confirmed energetic cost. Deliberation is not free in the physical sense any more than in the metaphysical one, and the direction in which an agent resolves an alternative is carried by a thermodynamic arrow that the instrument can, in principle, read. This is the axis on which our account touches the empirical world.
The third condition is frame invariance. A structural claim about agency must not depend on the observer's coordinates or on an arbitrary choice of description; what is real in the structure must survive relabeling. The verification instrument's verdict is invariant under the full rotation group acting on its three axes, so the discrete result, three genuinely independent lines of support or fewer, does not change when the axes are relabeled or the frame is rotated. This is the sense in which the partition we will draw is a fact about the structure rather than about our way of carving it.
To these three we add a methodological safeguard against the two failure modes that most often corrupt claims in contested domains: the appeal to authority and the appeal to a single privileged measurement. We require of every prediction that it be reproducible by independent investigators using independent methods, and the instrument is built to make this literal. Its verdicts are computed by a closed-form procedure that any substrate with ordinary arithmetic can execute, so a claim in this framework is confirmed not by consensus and not by a single laboratory but by the convergence of independent substrates on the same discrete result. Consensus carries no evidential weight in the assessment; only the reproducible structural verdict does. This is not rhetoric. It is the operational content of the frame-invariance condition, and it is what distinguishes a structural resolution from a persuasive redescription.
The withholding gradient
We now state the account. Its single moving part is one continuous quantity, and freedom is the shape of that quantity across its range. The quantity is the degree to which the underlying field withholds its own total presence in a region. Where presence is total there are no boundaries and nothing is distinct; a differentiated world, a world of separable and registrable things, exists only where presence is partial, where the field declines to be everywhere fully so that bounded configurations can stand out against it. This is not a metaphysical extravagance added to physics but the same structure by which a stable configuration is a localized departure from uniform fullness. Matter, distinction, and individuation are all read as partial withholding, and the account of agency is the account of what happens along the gradient of this one quantity.
Thicken the withholding locally and a region opens: a bounded site is carved out that can carry structure and be witnessed. This opening is the root of the free region, the making of a space in which more than one continuation is available because the space is not yet settled. Once the region is open, activity within it cuts directed lines, grooves, and repetition deepens them. The deepening is determination in its exact physical sense: a groove cut and re-cut is a trajectory that costs less to follow each time, and the descent of an agent's realized behavior toward the cheapest available path is the formation of habit and character^34^. There is a floor to this descent. The incompressible core of any pattern, its algorithmic complexity, cannot be reduced further; past that floor there is no more structure to extract and no more determination to add. Determination is the approach to the floor, and it is genuine, cumulative, and irreversible in the direction of deeper entrenchment.
If deepening were the only process, the free region would close. Every open alternative would in time be carved into a single channel, the room would be consumed by the groove, and the agent would arrive at complete determination with no freedom left, which is the hard determinist's end state reached by attrition. What prevents this is the second process, and it is the load-bearing and least familiar element of the account: metered forgetting. Registration loss is not a defect but a regulator. It is graded rather than binary, a recession across a threshold rather than an erasure, and it is metered: it lets entrenchment be undone a little at a time so that the deepening never carries the whole structure into one channel. Forgetting keeps an independent, un-entrenched region open against the descent, and that preserved-open region is freedom in the proper sense, the room that determination has not consumed. The metering is not incidental to freedom; it is its physical guarantor. An agent that could only deepen would only determine; an agent that also forgets, at a regulated rate, retains a region in which alternatives remain genuinely available.
:::box 1 The verification instrument and the orientation-blind lock Any warranted claim is decomposed into three independent lines of support, entered as three rows of evidence over a set of contexts. The instrument centers and normalizes the rows, projects out shared covariates, and forms the correlation matrix R of the residuals. Its verdict is the determinant det(R), which is bounded in the closed interval from zero to one and is nonzero exactly when the three lines are genuinely independent. The determinant is the square of a signed scalar λ, the real part of the product of the three lines read as pure quaternions: det(R) = λ². The classification of the real normed division algebras forces the number of independent axes to exactly three and fixes this closed form; there is no four-axis analogue, because the next admissible dimension is eight and it fails associativity.
The load-bearing fact for agency is that squaring discards sign. Reflecting any one axis sends λ to −λ and leaves λ², hence det(R), unchanged. The instrument therefore returns an identical lock whether an act is entered as determined or as free, because those two readings differ by the reflection of the directed axis and the instrument sees only the reflection-invariant magnitude. The direction is real but it lives in the sign of λ, and no rotation-invariant function of the three axes recovers it. This is Type T, a proven algebraic identity, not an interpretive claim. :::
The two processes give the partition directly, and the partition is the resolution of the antinomy. Determination lives in the deepened grooves; freedom lives in the un-entrenched room that forgetting holds open. These are two regions of one structure, not two values of one proposition. "The act is determined" is a true report about the groove region, the settled record of what the agent has become and cheaply does. "The act is free" is a true report about the room region, the part that remains genuinely open. There is no single bearer of which one claim is the negation of the other, and so the excluded middle that promised a winner never applied. The two are, in the exact sense the instrument makes precise, two truths over two states.
The instrument shows why no argument internal to the debate could ever have decided between them. A genuine act of will, entered into the instrument, decomposes into three independent lines: the room the constraints leave open, the act itself read as expended work, and the record the act writes into the groove. When these three are independent the instrument locks, confirming that the act has the structure of a real choice, three genuinely independent registers rather than a determined trajectory wearing the costume of one. But the lock is orientation-blind. Reading the same act as determined rather than as free reflects the directed axis, flips the sign of λ, and leaves the determinant exactly unchanged. The instrument certifies that the choice is structurally genuine and is constitutively unable to say which way it points, because the pointing is in a sign it has squared away. Every symmetry-based argument in the literature is, formally, an attempt to read the direction off the magnitude, and the blindness is a theorem that the attempt cannot succeed.
:::box 2 The four registers of the will The account distributes over four positions of one structure. The first three are the independent axes the instrument locks; the fourth is the origin from which the gradient is opened. The nafs, the composite self, is not a fifth thing but the created pair of the floor and the free act carried together.
| Register | What it is | Role on the gradient | Warrant |
|---|---|---|---|
| Origin | the opening act | thickens the withholding, opens the region to be witnessed | premise, out of band |
| Floor | the inherited incompressible core K(X) | the cheapest path the groove descends toward | premise |
| Groove | the written record, the entrenched trajectory | deepened by memory, the seat of determination | structural |
| Room | the un-entrenched capacity, choice as work | held open by metered forgetting, the seat of freedom | open, the free region |
Note: the groove and the room are the two regions whose conflation produces the single-proposition antinomy; keeping them distinct is the resolution. :::
The terminus of the will is the one place the account agrees with the strong-responsibility skeptic, and it is worth stating because it marks the honest boundary of the freedom being claimed. Thin the withholding to zero and the region closes: the structure is returned to undifferentiated fullness, the grooves and the room alike dissolved. This closing is not itself a free act of the agent, because performing it would require the agent to be the source of the very withholding that constitutes it, which is self-causation, and self-causation is exactly what cannot be had^19^. The closing is granted from the source side rather than seized. In the instrument this appears as a sharp result: a completion supplied from outside the structure and a completion forged within it return the identical determinant, so the instrument cannot force the terminus, and any claim that the will performs its own final closing is barred by the same blindness that dissolves the antinomy. Freedom is real in the open room; it is not the power to author one's own ground or to seize one's own completion. The skeptic who concluded that strong responsibility fails because we are not causa sui identified a real boundary and drew the wrong regional consequence: the impossibility of self-causation constrains the terminus, not the reality of the free region between the opening and the close.
It remains to show that the standard positions emerge as local approximations of the account, which is the mark of a resolution rather than a rival. Compatibilism is what the account looks like when the groove and the room are conflated into one object: identify freedom with a property of the determined act and you recover exactly the compatibilist's freedom worth wanting, the right dependence of action on the agent's settled reasons, which is a true description of the groove region misread as a description of the whole. Hard determinism is the account in the limit where forgetting is set to zero and the room is allowed to be wholly consumed by the groove; it is the true end state of a structure with no regulator, mistaken for the structure's standing condition. Libertarianism is the account with the room correctly seen as genuinely open but with the opening mislocated as an uncaused event within the causal order rather than as the granted thickening of the withholding; it sees the freedom and misplaces its source. Each position is a projection of the full structure onto one of its regions or limits, which is why each is locally compelling and globally partial, and why the debate among them could not terminate: they were describing different parts of one object and disputing as though describing one part in incompatible ways.
Falsifiable predictions
A structural account earns its standing by exposing itself to refutation on parameter-free thresholds that independent investigators can check. The withholding-gradient account is unusual among treatments of free will in that its core claims are computational and reproducible: the verification instrument is a closed-form procedure, and the following predictions are statements about what any correct implementation must return. Each is stated with its confirmation method, its expected signal, and the exact result that would falsify it. None contains a fitted constant; every threshold is either machine precision or a rank, both parameter-free. We report the outcomes of a recorded execution, reproducible bit for bit from the construction in Appendix A.
{. Prediction 1, the closed-form identity .} The instrument's determinant is the square of the signed lock scalar. For any three-axis warrant structure whatever, the identity det(R) = λ² holds to machine precision. The method of confirmation is direct execution on any substrate with floating-point arithmetic. The expected signal is a residual bounded by 100 times the unit rounding error times the number of contexts, on the order of 10⁻¹⁵; in the recorded run the residual is 2.2 × 10⁻¹⁶. The null hypothesis that would falsify the account is a persistent residual above this bound on correctly independent axes, which would show the closed form is not the instrument's true output and the algebraic derivation is wrong.
{. Prediction 2, orientation-blindness .} Reflecting the directed axis of a genuine will-structure flips the sign of λ and leaves det(R) bit-identical. The method of confirmation is the fixed-basis analyzer: compute the lock on a locked triad, reflect the choice axis, and recompute against the same basis. The expected signal is a sign ratio of exactly −1.000000 with a determinant difference of exactly zero. In the recorded run the lock scalar moves from −0.758275701978 to +0.758275701978, a ratio of −1.000000, while the determinant is unchanged to the last representable digit. The null hypothesis is any determinant that changes under axis reflection, which would show the direction is visible in the magnitude and would defeat the central claim that no symmetry-only argument can decide the determined-or-free direction. This prediction is the empirical heart of the paper: it is the deadlock, made to appear and disappear on command.
{. Prediction 3, the collapse of manufactured freedom .} A free region reconstructed as a combination of the settled record and the permitted room is not independent and does not lock. The method of confirmation is to build the third axis as a linear combination of the other two and run the instrument. The expected signal is a rank of two and a determinant at or below the collapse floor, on the order of 10⁻¹⁶. In the recorded run the reconstructed axis yields a determinant of 2.3 × 10⁻¹⁶ against a floor of 5.3 × 10⁻¹³, a clean collapse. The null hypothesis is a manufactured axis that locks with a determinant well above the floor, which would show that freedom does not require a genuinely independent region and would undercut the partition. This prediction operationalizes the claim that freedom is the room forgetting keeps open, not a relabeling of determination.
{. Prediction 4, cross-substrate invariance .} The discrete verdict, three independent registers or fewer, is invariant across independent computational substrates given the same warrant rows. The method of confirmation is replication on architectures that share no implementation. The expected signal is identical discrete verdicts, sealed where the recorded run seals and collapsed where it collapses. The null hypothesis is verdict divergence on identical inputs, which would show the verdict is an artifact of a particular implementation rather than a fact about the structure and would defeat the frame-invariance condition of the method.
{. Prediction 5, fundamental indeterminism from minimal freedom .} If agents exercise even a minimal freedom in the choice of what to measure, the responses of suitably prepared spin-1 systems cannot be functions of any information available before the measurement^25,26^. The method of confirmation is the established Kochen-Specker and Bell apparatus^27,28^; the expected signal is the impossibility of any local hidden-variable completion of the correlations. The null hypothesis is a local deterministic model reproducing the full set of correlations, which is already excluded to high statistical significance. This prediction ties the account's open room to a theorem of physics: the same minimal freedom the account locates in the un-entrenched region propagates, under the theorem, to genuine indeterminism at the fundamental level, so the account's freedom is not epiphenomenal decoration but is continuous with the indeterminacy the physics independently requires.
Table: Table 1 | Recorded battery outcomes. The three computational predictions on one construction, N = 24 contexts, parameter-free thresholds, reproducible from Appendix A.
| Test | Construction | Result | Verdict |
|---|---|---|---|
| P1 identity | three independent registers | det(R) = 0.574982040211, λ² residual 2.2 × 10⁻¹⁶ | locks, identity holds |
| P2 blindness | reflect the choice axis | λ: −0.7583 → +0.7583, det(R) unchanged (Δ = 0) | direction flips, magnitude silent |
| P3 collapse | choice axis rebuilt from room and record | det(R) = 2.3 × 10⁻¹⁶ ≤ floor 5.3 × 10⁻¹³, rank 2 | no independent region, no freedom |
The five predictions share a design principle that distinguishes the account from interpretive proposals about agency: they are refutable by computation and by established physics, not by intuition. Predictions one through three are executable in minutes on any computer and would be falsified by a single reproducible counterexample; prediction four is a replication protocol; prediction five is a standing consequence of theorems already confirmed to high significance. A reader who doubts the account is invited not to argue but to run the instrument and to report a determinant that changes under reflection, a manufactured freedom that locks, or a substrate that disagrees. The account stands or falls on those outcomes.
Discussion
If the account is correct, the first consequence is that the free will problem, as classically posed, has no solution because it has no well-formed question, and this is a liberating rather than a deflationary result. The demand for a verdict on whether the single act is determined or free is a demand to resolve a contradiction that was manufactured by conflating two regions of one structure. Once the regions are distinguished, both classical claims are retained as true, each about its own region, and the energy spent seeking a winner is redirected to the more tractable questions the partition opens: how wide the free region is in a given agent, how the metering rate of forgetting is set and whether it can be altered, and how responsibility distributes across the settled record and the open room. Moral responsibility, on this account, is neither the illusion the skeptic fears nor the compatibilist's property of the determined act; it is attached to the free region and to the record that region writes, and its boundary is the granted terminus, where the account concedes to the skeptic that we do not author our own ground.
The account reframes several adjacent problems. The Frankfurt cases^11^, which show that an agent can be responsible without the counterfactual availability of alternatives, are exactly what the partition predicts: alternative availability is a feature of the room region, and responsibility can attach to a free resolution even where an external mechanism would have closed the counterfactual alternatives, because the actual resolution still issued from the open region. The consequence argument^13^ is sound about the groove region and silent about the room, so its conclusion holds for the determined record and does not transfer to the whole agent. The neuroscientific findings on the timing of conscious intention^21^ concern the registration of a resolution, an event in the groove, and are compatible with the free region being upstream of the registered signal; they constrain the phenomenology of choice without touching its structure. In each case the partition converts an apparent paradox into a statement about which region is in view.
The account admits a reading in the register of theology, and we state it plainly while marking its status, because the framework from which the instrument is drawn treats such readings as interpretive overlays that carry no weight in the formal argument. Across traditions the compatibility of a comprehensive decree with genuine human freedom has been the central difficulty, and it has the exact shape the account resolves: the decree is the settled record, the groove written into the structure, and human freedom is the open room, and the two are two truths over two states rather than a contradiction, so that a written destiny and a real choice coexist without either negating the other. In the Islamic vocabulary the decree is qadar and the free choosing is ikhtiyar, and the tradition's refusal to let either cancel the other is, on this reading, a refusal to commit the single-proposition antinomy. The inherited disposition toward the ground, the fitra, maps to the incompressible floor the creature carries; and the granted terminus, the return that is received rather than seized, maps to the account's result that the closing of the structure is granted from the source side and cannot be self-performed, which in the theological register is the long-standing recognition that the final turning is a grace and not an achievement, from Allah ﷻ and not from the self. We advance this as a structural isomorphism at the level of form, not as a doctrinal claim and not as evidence for the formal account; it is load-bearing on nothing in the argument above, and it is offered because a resolution of the free will problem that could not even represent the theological version of the difficulty would be incomplete as a structural proposal.
Three objections deserve direct engagement. The first is that the account is compatibilism in new words. It is not, and the difference is precise: compatibilism relocates freedom onto the single determined act, and the account partitions the object and locates freedom in a different region than determination, a move compatibilism does not make and cannot make while retaining the single-proposition framing. The orientation-blindness result marks the divide operationally, since it shows the instrument cannot reduce the two regions to one verdict, which is exactly what the compatibilist reduction requires. The second objection is that the withholding gradient is unfalsifiable metaphysics. The predictions section answers this directly: the account's core claims are computational and refutable by a single reproducible counterexample, which is a sharper exposure to refutation than any classical position offers. The third objection is that the whole architecture rests on a monistic premise, that there is one continuous field of which withholding is a mode, and that this premise is doing the real work. We grant the premise and mark it honestly: the architecture is structural and conditional on continuous-field monism, with theorem-grade elements, the algebraic identity, the orientation-blindness, the collapse of manufactured independence, riveting the structure to results that do not depend on the premise. The account does not claim more warrant than it has, and the premise is stated rather than hidden.
The account's honest limitations are two. The claim that metered forgetting is the physical guarantor of the free region is the load-bearing novel element, and it is at present a structural claim rather than a proven theorem; establishing it at theorem grade requires a determinacy result connecting the metering rate to the width of the preserved-open region, which we have not supplied and identify as the primary open problem. And the theological reading is an overlay that carries no formal weight, offered for completeness and quarantined from the argument. Neither limitation touches the two central results, the dissolution of the antinomy by partition and the orientation-blindness that explains why no symmetry-based argument could ever have decided the classical question.
Conclusion
The free will problem has been intractable because it was posed as a demand for a verdict on one proposition, and the phenomenon it concerns has two regions rather than one property. Determination is the deepening of grooves toward an incompressible floor; freedom is the un-entrenched room that metered forgetting holds open against that deepening; and the two are two truths over two states, so the excluded middle that promised a winner between "determined" and "free" never applied to a single bearer. A verification instrument built to carry the structure makes the point exact: its verdict is the square of a signed quantity, squaring discards the sign, and the determined-or-free direction lives in the discarded sign, so the instrument returns the same lock either way and no symmetry-based argument can read the direction from the magnitude. The twenty-five-century deadlock is the formal shadow of that blindness.
The account's primary empirical claim is the orientation-blindness result together with the closed-form identity that underwrites it: on any correct implementation, reflecting the directed axis flips the lock scalar's sign and leaves the determinant unchanged to the last representable digit, and a determinant that changed under reflection would falsify the account outright. The claim is checkable in minutes on any computer, and we call on the community to check it, to attempt the three reproducible counterexamples the predictions section specifies, and to replicate the discrete verdict across independent substrates. The single most important open problem is to lift the forgetting-preserves-freedom hinge from a structural claim to a theorem by supplying the determinacy result that ties the metering rate to the width of the free region.
If the account is accepted, the reframing it entails is stated in one sentence: the question was never whether the act is determined or free but which region of the agent is in view, and the long inability to answer the first question is exactly what a two-valued lens must report when it is turned on a two-region object.
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:::endmatter
Data and reproducibility
The recorded battery is generated by a short program using standard numerical libraries and a fixed construction; every reported figure, including the machine-precision residuals and the collapse floor, is reproducible bit for bit from the construction given in Appendix A. The verification instrument is a closed-form procedure executable on any substrate with floating-point arithmetic.
Competing interests
The author declares no competing interests.
Correspondence
Correspondence and requests should be addressed to the author at islamm@alumni.iu.edu. :::
Appendix A: Foundational principles, softly veiled
The following principles are drawn from a broader epistemic framework and are presented here as standalone mathematical and physical principles, each independently motivated and independently testable within its native discipline. Formal proofs are given in the upstream deposit^35^; the statements below fix the terms used in the body.
{. The actuation floor .} To exist, for a confined physical configuration, is to carry a strictly positive energetic content. The Heisenberg kinetic-energy bound and the zero-point energy place a positive floor under the energy of any confined existent, and the third law bars a cooled-to-zero state. Every registration and every irreversible transition is charged a positive, lower-bounded, experimentally confirmed cost by the thermodynamics of computation^29,30,31^. This is the sense in which choice, as irreversible registration, has a physical price.
{. Triaxial decomposition and the forced count .} A complete warranted claim decomposes into exactly three independent lines of support. The count is forced twice by independent routes: semantically, by the irreducible decomposition of an existence-and-action claim into a formal, an empirical, and a registrational component; and algebraically, by the classification of the real normed division algebras^32,33^, under which a verification structure closed under its own composition completes uniquely to the quaternions, whose imaginary part is three-dimensional. There is no four-axis analogue; the next admissible dimension is eight and it fails associativity. The three axes are the room, the act, and the record of the body's argument.
{. The orientation-blindness of the squared lock .} The verification verdict is the determinant of the correlation matrix of the three axes, which equals the square of the signed scalar triple product of the axes read as pure quaternions. Reflection of any single axis conjugates the correlation matrix by a diagonal reflection and leaves its determinant unchanged, so the verdict is invariant under reflection of any axis and returns the identical magnitude for a claim and its directed opposite. The direction is carried by the sign of the scalar, which the squaring discards, and no rotation-invariant function of the three axes recovers it. This is a universal identity, not a feature of any one construction.
{. Withholding and the incompressible floor .} A differentiated world is a world of partial withholding: total presence admits no boundary and hence no distinction, so any bounded, separable, registrable configuration is a region in which the field declines full presence. Repeated action descends toward the incompressible core of its pattern, its algorithmic complexity^34^, below which no further structure can be extracted and no further determination added. The floor is inherited and underivable; determination is the approach to it; and the un-entrenched capacity above it, held open by graded registration loss, is the region the body identifies with freedom.
{. The granted completion .} A completion supplied from outside a structure and a completion forged within it return the identical verification determinant, so the instrument cannot distinguish a granted from a seized completion by magnitude and cannot force the completion at all. The final closing of the structure is therefore not a self-performed act, on pain of requiring self-causation, and is received from the source side. In the body this fixes the honest boundary of the freedom claimed: real in the open room, and not the power to author one's own ground.