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Substantive Agency and Computational Non-Anticipability: An Axiomatic Route to a Conditional Separation of P and N P

Jérome Clech

arXiv:2609.28040v1cs.LO

TL;DR

The paper asks how to distinguish free action from deterministic execution or chance, represent that distinction formally, and identify the consequences for search complexity. It separates substantive agency from additional deployment and certification bridges, then derives a conditional complexity result when those bridges hold.

  • Problem

    Free-will debates often conflate the conceptual distinction between free action and alternatives such as deterministic execution with its mathematical representation and computational consequences.

    Source: §1 · p. 1

  • Method

    The paper axiomatizes substantive agency separately from computational assumptions, then connects actualisation to search complexity through modal deployment and certified-actualisation bridges.

    Source: §12 · pp. 25–26

  • Results

    With polynomially bounded, polynomially verifiable certificates and extensional unique projection, the associated relation lies in TFNPsearch \ FPsearch, and a self-contained search argument conditionally yields P != NP.

    Source: §12 · p. 26

  • Takeaways & Limitations

    The paper offers a conditional transfer architecture whose premises and points of refusal are explicitly identified, rather than a generic lower-bound technique or a reduction of humans to a machine model.

    Source: §12 · p. 26

    The classification is exhaustive only within its declared domain, and certified actualisation is an independent property not derived from generic recording technology.

    Source: §9 · p. 18, §10.2 · p. 21

Abstract

from arXiv · show

This paper characterises substantive agency and identifies the additional bridges under which it has a standard complexity-theoretic consequence: conditionally, P __ = N P . The constitutive core separates coherent plurality, causal openness, anticipatory non-pointing, act-level singularisation, and endogenous sourcehood. Its temporal claim yields a two-sided modal result: exact passive pre-act selection is incompatible with jointly retaining singularising priority and invariance under causally inert informational extension. If every standard polynomial procedure is deployable in that passive form, the actualisation map $\alpha$ has no polynomial-time selector. On an effective presentation, $\alpha$ is therefore either computable outside FP or noncomputable. A separate certifiedactualisation protocol supplies polynomially bounded, polynomially verifiable traces with extensional unique projection. This is the constructive interface that turns historical actualisation into a total standard search relation; combined with the modal result, it places that relation in TFNP search \ FP search , from which a self-contained search argument yields P __ = N P . The result is a domain-relative conditional transfer theorem: it neither assumes polynomial non-anticipability in the definition of agency nor proves that human decision-making satisfies the bridge and certification premises.

1 Introduction

The introduction separates conceptual, representational, and computational questions about free action, then presents a stratified bridge from substantive agency to a conditional complexity result. Its central distinction is between coherent possibilities and the act that first makes one continuation historical.

1 Introduction

  • The paper distinguishes what makes an act free, how that distinction can be represented mathematically, and what follows if an act is verifiable afterward but not efficiently selectable beforehand.These questions are connected but not interchangeable.Source: §1 · p. 1
  • The agency core separates causal openness, anticipatory non-pointing, act-level singularisation, and endogenous sourcehood, with countermodels distinguishing their roles.The account also requires coherent plurality; without it, there is nothing to singularise.Source: §1 · pp. 2–3
  • In an actually access-complete frame, singularising priority entails actual anticipatory non-pointing before the act.The introduction treats this as a pre-pointing explication, not as a computational premise.Source: §1 · pp. 2–3
  • Singularising priority, passive-extension invariance, and deployable exact pre-act selection are jointly incompatible; effective deployment then excludes FP selectors.The exclusion follows only with the additional claim that every standard polynomial procedure can be implemented in passive form.Source: §1 · pp. 2–3
  • Short retrospective certificates with extensional unique projection provide a separate interface from actualisation to a total search relation without a polynomial selector.The introduction also identifies a realisation gap: computability alone does not provide polynomially verifiable certificates.Source: §1 · p. 3
  • The conditional separation theorem keeps constitutive agency distinct from computational bridges and treats the result as conditional rather than an empirical claim about human decisions.The paper states that rejecting its core commitments requires abandoning at least one of them, while passive deployment and certified actualisation remain independent additions.Source: §1 · pp. 2–3

2 Five levels that must not be conflated

The paper separates five levels of its argument and treats their logical relations as distinct. Polynomial non-anticipability, traceability, and effective computability enter through separate bridge principles, not as constitutive assumptions.

2 Five levels that must not be conflated

  • Polynomial non-anticipability, polynomial traceability, and effective computability are not built into the constitutive definition or conflated with one another.They enter through distinct bridge principles.Source: §2 · p. 3
  • The argument is organized into five logically separated levels.Table 1 summarizes the levels and their separation.Source: §2 · p. 3, Table 1 · p. 4

3 Related work and differential contribution

The paper’s contribution is a formally stratified, auditable bridge among substantive agency, modal constraints, certification, and standard search complexity. It distinguishes this conditional interface from prior work on prediction, sourcehood, agency, sheaves, and total search.

3 Related work and differential contribution

  • Computational irreducibility work [17] [27] establishes prediction limits but does not identify agency with act-level pointing or provide the certified unique-projection interface.The cited work concerns deterministic-process prediction and limits on inference by embedded devices, respectively.Source: §3 · p. 4
  • Krumm and Mueller’s computational-sourcehood account [13] differs from this paper’s criterion, which requires separately justified modal closure and certification for complexity transfer.The paper says its substantive core alone supplies no complexity lower bound.Source: §3 · p. 4
  • Azadi derives autonomous-behaviour conditions from computational hardness, whereas this paper starts from an independently characterized agency model and adds computational interfaces.The paper explicitly distinguishes computational difficulty from its A3-C and A5 conditions.Source: §3 · p. 4
  • No single row of the contribution comparison yields the main result; the contribution is the proved bridge across rows.The paper presents the bridge as an auditable chain of premises and theorems connecting substantive singularisation to a computability bifurcation and, conditionally, standard search complexity.Source: Table 2 · p. 5, §3 · p. 6
  • Durham’s free-choice model [7] does not derive anticipatory non-pointing from singularising priority or connect post-act certification to standard search complexity.The paper presents its axioms as specifying commitments needed for the computational bridge, not as replacing the philosophical literature.Source: §3 · p. 5
  • List’s functional account of agency and control does not require indeterminism or unpredictability; this paper targets substantive sourcehood and singularising priority.The paper frames the difference as one of explanatory target, not as a dispute over the usefulness of functional agency.Source: §3 · p. 5
  • The sheaf-theoretic use differs from contextuality’s obstruction to a global section [1]: the issue here is selecting among multiple global sections.The sheaf language organizes local compatibility and symmetry; it does not establish a complexity lower bound.Source: §3 · p. 5
  • The paper’s total-search claim is not that FNP or total search is new, but that unique projection links certified actualisation to the map excluded from FP.Under the separately justified agency and certification premises, any solver of the associated relation computes the actualisation map.Source: §3 · p. 5

4 Substantive agency: desiderata and formal core

The formal core distinguishes causal determination from exact pre-act pointing and treats the act as the first point at which an admissible continuation becomes historical. Under actual access-completeness, substantive singularising priority entails anticipatory non-pointing; further computational consequences require separate bridges.

4 Substantive agency: desiderata and formal core

  • Singularising priority concerns where the exact distinction first enters the relevant frame, not merely whether the continuation eventually occurs.An act may physically realise a continuation already pointed in advance without supplying its differentiating point.Source: §4 · p. 6
  • In an actually access-complete frame, substantive singularising priority entails that the realised continuation was not exactly pointed before the act.A coupled predictor counts within the frame; wholly isolated, operationally unavailable foreknowledge does not.Source: §4 · pp. 6–7
  • Causal non-determination is model-relative: no continuation is forced across all admissible evolutions under the declared pre-act causal interpretation.The interpretation and its system boundary must be fixed independently of the realised value; adequacy requires completeness relative to the causal theory and resistance to opportunistic omission.Source: §4 · p. 7
  • The formal frame distinguishes the causally forced set D(q) from Pact(q), the continuations exactly and operationally pointed before the act.Causal forcing is not defined by running time, while actual pointing concerns informational presence.Source: §4 · p. 7
  • The core requires coherent plurality, causal pre-act non-determination, anticipatory non-pointing, act-level actualisation, and endogenous sourcehood.A3-P and A4 jointly formalize substantive singularisation; A5 attributes the transition to the represented agency rather than an exogenous intervention or randomiser.Source: §4 · p. 8
  • The choice-frame cannot establish endogenous authorship by itself; applications need an independently defensible agency locus and causal attribution.That locus may be individual, distributed, collective, or transindividual, and changing it requires reassessing the causal class and sourcehood attribution.Source: §4 · p. 8

4.1 Five meanings of non-predetermination

  • Causal non-determination, anticipatory non-pointing, structural non-pointing, and computability-level non-anticipability are distinct notions without automatic implications.The paper separates these predicates and says effective-pointing closure supplies a distinct modal bridge from exact efficient computation to deployability.Source: §4.1 · pp. 8–9

4.2 Openness and sourcehood are independent

  • Pre-act openness and endogenous sourcehood do not imply one another; their conjunction, rather than either alone, excludes deterministic unfolding and pure chance.The paper also distinguishes both from anticipatory non-pointing, which follows from singularising priority under access-completeness.Source: §4.2 · p. 9

5 Countermodels and the role of each requirement

The countermodels test what each requirement rules out, showing that plurality, coherence, openness, pointing, sourcehood, and actualisation play distinct roles. These are diagnostic role tests, not an exhaustive proof of independence or a definition covering every use of “free will”.

5 Countermodels and the role of each requirement

  • Without plurality, a deterministic automaton has only one coherent continuation to reveal or execute, so there is no alternative to singularise.The paper treats this as deterministic agency, not substantive choice in its account.Source: §5 · p. 10
  • A list of mutually incompatible or physically impossible continuations does not establish genuine alternatives; coherence constrains plurality.Formal plurality also does not ensure openness when complete pre-act data already distinguishes one continuation.Source: §5 · p. 10
  • A causally independent coin can supply openness without authorship, while an agent-owned deterministic policy can supply sourcehood without causal openness.The latter is the paper’s principal compatibilist countermodel when the policy is fixed by the complete prior state.Source: §5 · p. 10
  • Super-polynomial evaluation does not turn an internal deterministic policy into causal openness; it remains a determination map in D.Practical or polynomial-time prediction may fail even while the condition violates A3-C.Source: §5 · p. 10
  • An informationally coupled infallible observer makes the frame already pointed, but wholly isolated foreknowledge does not violate anticipatory non-pointing.The distinction depends on membership in the access-complete decision frame.Source: §5 · p. 10
  • Without actualisation, non-singleton possibility remains deliberative possibility rather than completed history; the examples form a dependency graph, not an exhaustive independence proof.The passage cautions that the role tests do not exhaust every use of “free will”.Source: §5 · pp. 9–10

6 Compatibilism, individuation, and the sourcehood boundary

The section distinguishes functional attribution and practical responsibility from the stronger claim that an act’s realised choice is not wholly fixed by complete pre-act conditions. It argues that views retaining real alternatives, ultimate sourcehood, and the act’s priority in selecting an alternative must meet the paper’s substantive-agency criteria, while the computational bridges remain separate.

6.1 What compatibilism preserves

  • Compatibilist accounts preserve reasons-responsive control and attribution, but ultimate sourcehood additionally requires that complete intrinsic pre-act conditions do not already select the realised continuation.The paper does not demand self-creation or independence from every causal condition; its criterion is relative to the model’s declared pre-act causal description.Source: §6.1 · p. 11

6.2 The problem of “the agent’s own reasons”

  • An agent’s reasons being internal or functionally owned does not establish ultimate sourcehood when prior conditions or externally configured dispositions determine the deliberative mechanism.The paper treats manipulation and consequence arguments as motivating an additional inference, not as conclusive refutations of compatibilism.Source: §6.2 · pp. 11–12

6.3 The boundary of the agent

  • The ordinary individual boundary is not assumed to mark where a determined causal chain changes its ontological status; substantive agency can be framed at individual, distributed, collective, or transindividual scales.Each proposed locus requires its own agency frame and justification; changing the boundary alone cannot create causal non-determination.Source: §6.3 · p. 12

6.4 Practical responsibility and ultimate responsibility

  • Practical responsibility practices can be important without deductively establishing ultimate sourcehood, and denying ultimate sourcehood does not logically require abandoning every practical responsibility regime.The paper distinguishes social loci for praise, blame, and answerability from the stronger metaphysical claim about the act’s source.Source: §6.4 · pp. 12–13

6.5 Progressive closure and the exact transmission boundary

  • Among the positions considered, retaining real alternatives, ultimate sourcehood, and the act’s singularising priority requires the substantive core; available exact prescience supplies no intermediate position within the access-complete frame.The paper describes this closure as progressive, not exhaustive of every conceivable philosophy.Source: §6.5 · p. 13
  • Causal non-reducibility and singularising priority do not by themselves imply α ∉ FP, noncomputability, or certified actualisation; effective-pointing closure and a certified search interface are additional bridges.A determined reasons-responsive act fails the substantive-agency criterion when complete intrinsic pre-act conditions and mechanisms already license a selector for its realised continuation.Source: §6.5 · p. 13

7 Choice-frame and sheaf-theoretic local-to-global representation

The choice-frame representation separates a coherent set of pre-act possibilities from the post-act selection of the continuation that becomes actual. Its sheaf-theoretic layer represents local compatibility and global continuations, while symmetry can rule out an intrinsic symmetry-preserving selector.

7.1 Abstract representation

  • A substantive-agency frame corresponds to coherent, non-singleton, causally undetermined and anticipatorily unpointed pre-act objects, followed by endogenous post-act pointing.The representation lemma is elementary, not a classification theorem or complexity lower bound.Source: §7.1 · p. 14

7.2 Local-to-global realisation

  • Compatible local sections glue into coherent global continuations, whereas actualisation is a distinct operation that selects which continuation became historical.The distinction separates the local-to-global construction of possibilities from singularisation; sheaf theory alone does not establish the complexity conclusion.Source: §7.2 · p. 14

7.3 Symmetry and intrinsic non-pointing

  • When the symmetry condition holds, no equivariant map from the trivial one-point set to the global continuations exists, excluding an intrinsic symmetry-preserving selector but not every representation-dependent algorithm.Externally labelled or representation-dependent selection remains possible.Source: §7.3 · p. 15

8 Prediction, selection, and the bridge principle

The section separates exact pre-act prediction from its physical deployment and states a modal incompatibility among prediction, singularising priority, and invariance under causally inert information. With effective-pointing closure, the incompatibility excludes polynomial-time selectors, conditionally on the deployment bridge.

8 Prediction, selection, and the bridge principle

  • A standard exact pre-act algorithm induces a uniform selector, but that algorithm-to-selector step alone establishes no causal conclusion.The selector returns the algorithm’s exact output on the fixed encoding.Source: §8 · p. 15
  • Exact pre-act prediction cannot coexist with both substantive singularising priority and passive-extension invariance when the selector can be added as a causally inert informational extension.The selector’s exact output points to the realised continuation before the act, while invariance requires the act to retain its first-pointing status.Source: §8 · p. 16
  • The passive-extension dilemma requires relinquishing either singularising priority or invariance of that priority under causally inert informational extension.Effective-pointing closure selects the invariant branch, turning the modal result into an exclusion of efficient exact selectors.Source: §8 · pp. 16–17
  • Effective-pointing closure combines passive-extension invariance with effective deployment, which requires every standard polynomial-time procedure to admit a preserving passive implementation.The deployment premise is counterfactual and abstracts from physical resource limits; it is stronger than polynomial computability alone.Source: §8 · pp. 16–17
  • Under effective-pointing closure on a complete uniform encoding, α has no FP selector; the polynomial threshold enters through deployment, not through the definition of causation or substantive agency.The argument is conditional on connecting an abstract polynomial procedure to an admissibly passive pre-act implementation.Source: §8 · p. 17
  • Polynomial-time intertranslations preserving admissible sets and uniquely projected certified actualisation preserve whether α belongs to FP across the two encodings.Representation robustness does not discard representation-dependent polynomial algorithms from the standard class.Source: §8 · pp. 15, 17

9 Certified realisation as a standard search relation

The section constructs a standard search relation from certified actualisation: valid occurrences have short, verifiable traces whose accepted continuations project uniquely to the historical outcome. This interface is an independent assumption, not a consequence of agency or computability, and the section distinguishes the search classification from the provenance needed to interpret witnesses as historical records.

9 Certified realisation as a standard search relation

  • The certified-actualisation interface uses finite encodings, a polynomially decidable unbounded valid domain, polynomially balanced witnesses, and fixed polynomial verification.For each valid occurrence, accepted certificates must exist and all accepted certificates must project to the same continuation; the input includes an outcome-independent occurrence identifier.Source: §9 · p. 18
  • Condition (O5) independently requires accepted certificates for every valid occurrence and extensional unique projection to its realised continuation.Generic authentication or a physical post-act trace does not establish this property; applications must bind the accepted language to one committed post-act state.Source: §9 · p. 18
  • The search witness is the pair (s, τ), and the extensional relation uses the fixed verifier; provenance conditions (P1)–(P2) are not needed for its complexity classification.Those additional conditions govern whether accepted strings faithfully represent records caused by completed acts rather than a stipulated answer graph.Source: §9 · pp. 18–19
  • Totalisation accepts ⊥ exactly on invalid inputs, while valid inputs have accepted witnesses; total search therefore depends on the protocol’s validity and certification conditions.The standard search classes distinguish total relations from relations that have witnesses only on some inputs.Source: §9 · pp. 17–19
  • Conditions (O1)–(O5) make α computable by exhaustive enumeration of bounded certificates, but this procedure may take exponential time.Existence of an accepted certificate and unique projection ensure that the returned continuation is α(q).Source: §9 · pp. 17, 19
  • Computability alone does not guarantee a polynomially balanced, polynomial-time decidable total relation with unique projection; the paper gives a computable non-polynomial function for which no such relation exists.Thus certified actualisation is substantive interface work, not a consequence of computability alone.Source: §9 · p. 19

10 Exhaustive bifurcation and companion interfaces

The paper classifies effectively presented substantive agency into computable-but-not-polynomial and noncomputable branches, then connects certified actualisation to conditional complexity and ontological results. These classifications are exhaustive only within their stated premises.

10.2 The level at which exhaustiveness holds

  • Under effective presentation and effective-pointing closure, substantive agency falls into exactly one branch: computable outside FP or noncomputable.The classification is exhaustive only within its declared domain; the theorem does not make certified actualisation automatic.Source: §10.2 · pp. 20–21

10.3 Computational interface

  • Conditions (O1)–(O5) provide polynomially balanced, polynomial-time verifiable witnesses whose continuation projection is total and uniquely equals α(q).Several traces may certify the same projected value; the unique projection, rather than a unique trace, is what the search argument uses.Source: §10.3 · pp. 21–22
  • With effective-pointing closure and certified actualisation (O1)–(O6), the associated total search relation lies in TFNPsearch \ FPsearch, yielding P̸ = NP conditionally.The proof uses a padded prefix-search argument to show that P = NP would give every FNP search relation a polynomial-time selector.Source: §10.3 · p. 22

10.4 Ontological interface

  • For the noncomputable branch, effective equivalence to sequential causal frames transfers uniform noncomputability; structural non-singularisation additionally supports the ontological interface.The stronger model and near-maximal incompressibility require additional assumptions, including AIK; LAnc alone does not establish near-maximal incompressibility.Source: §10.4 · p. 23

10.5 Domain-relative classification

  • The effective-domain classification separates failure of substantive agency or effective-pointing closure from the two computability branches.It is a logical partition, not an independent lower-bound theorem; the P̸ = NP consequence requires branch (2) together with (O1)–(O6).Source: §10.5 · p. 23

11 Scope, correspondence, and points of refusal

The conclusions concern exact uniform selection over unbounded encoded families, not prediction in general. Human applications require a separate correspondence principle, and the computational route depends on contestable bridge and certification premises.

11.1 Exact versus statistical anticipation

  • The target is exact, passive, uniform selection across an unbounded family, not statistical prediction, typical cases, practical heuristics, or distribution prediction.The argument remains compatible with bounded rationality, incomplete pre-act models, and practical unpredictability from chaos or computational expense.Source: §11.1 · p. 24

11.2 Human correspondence and empirical limits

  • Formal results apply to encoded families, not directly to humans; finite behavioural data cannot prove an asymptotic lower bound over an infinite family.Empirical evidence may support representation adequacy, defeat particular predictors, or constrain proposed mechanisms.Source: §11.2 · p. 24

11.3 Limits and exact points of refusal

  • The FP exclusion requires effective deployment, while the complexity consequence additionally requires certified actualisation; either bridge remains contestable.The paper does not establish that humans satisfy these premises, and neither ordinary recording nor authentication supplies the unique-projection certification condition.Source: §11.3 · p. 25

12 Conclusion

The paper combines a modal incompatibility result with a separate certified-search interface. Together, and only under stated premises, they yield a conditional P̸ = NP consequence while leaving human realisation unestablished.

Conclusion

  • Substantive singularising priority, passive-extension invariance, and passive deployment of an exact pre-act selector are jointly incompatible.Under effective deployment, choosing invariance excludes every FP selector of the actualisation map.Source: §12 · pp. 25–26
  • Polynomially bounded, verifiable certificates with extensional unique projection turn historical actualisation into a total standard search relation in TFNPsearch \ FPsearch.A self-contained search argument then conditionally yields P̸ = NP.Source: §12 · p. 26
  • The substantive core does not assume complexity conditions; the lower bound enters only through modal and certification bridges.On an effective presentation, the actualisation map is either computable outside FP or noncomputable.Source: §12 · p. 26
  • The paper does not establish human satisfaction of effective-pointing closure or certified actualisation, and LAeff alone does not entail verifiable traces.The result is a conditional architecture that identifies the premises needed for each route and where they may be refused.Source: §12 · p. 26
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