Dependency-Boundary Determinism Mismatch

• Dependency-Boundary Determinism Mismatch is the conceptual misalignment where fixed boundaries for assessing outcomes fail to capture the system’s global and graded dependency structure.
• It spans diverse domains like physics, agentic AI, dependency analysis, and fine-grained complexity, illustrating how local boundary tests often overlook holistic constraints.
• Practical insights include re-evaluating initial slices, chain horizons, and theoretical frameworks to ensure that determinism assessments align with true dependency structures.

“Dependency-Boundary Determinism Mismatch” can be treated as an Editor’s term for a recurrent structural problem: the dependencies that actually fix outcomes do not coincide with the boundary relative to which determinism is assessed. In the philosophy of physics, the mismatch appears when entire histories are constrained “all-at-once” rather than generated from an initial slice. In agentic AI, it appears when a task’s required chain length $k$ exceeds what an environment’s per-step determinism $\delta$ can sustain. In dependency analysis, it appears when typing boundaries are meant to control dependence but the relevant semantic structure is obscured by syntax. In fine-grained complexity, it appears when linear-time computation with tiny nondeterministic advice outruns deterministic simulation below conjecture-dependent exponent boundaries. In foundational studies of physics, it appears when existence, uniqueness, or even the identity of a solution depends on the surrounding set-theoretic metatheory (Adlam, 2021, Ding et al., 21 Jun 2026, Choudhury, 2022, Lingas, 2020, Clarke-Doane, 15 Aug 2025).

1. General schema of the mismatch

Taken together, the cited works suggest a common schema. A “boundary” is the interface across which dependence is tracked: an initial slice or region $S$, a chain horizon $k$, a security or stage level $\ell$, a deterministic running-time exponent, or the metatheoretic boundary between physics and set theory. A mismatch arises when the system’s real dependency structure is global, bidirectional, graded, or parameter-sensitive in a way that is not faithfully represented by that boundary.

In “Determinism Beyond Time Evolution,” the relevant contrast is between global constraints on entire Humean mosaics and region-based or initial-value definitions of determinism. In “Grounded Scaling: Why Agentic AI Needs Deterministic Environments,” it is between required long dependency chains and the environment’s agent-consumable correctness level $\delta$. In “Monadic and Comonadic Aspects of Dependency Analysis,” it is between apparent syntactic irregularity and the underlying graded monadic/comonadic semantics that actually determine noninterference. In “Consequences of APSP, triangle detection, and 3SUM hardness for separation between determinism and non-determinism,” it is between a fixed weak nondeterministic resource and deterministic time classes whose separating exponents depend on conjecture and input range parameters. In “Logical Dependence of Physical Determinism on Set-theoretic Metatheory,” it is between the internal physical picture of determinism and the external set-theoretic assumptions that settle regularity, well-posedness, and uniqueness (Adlam, 2021, Ding et al., 21 Jun 2026, Choudhury, 2022, Lingas, 2020, Clarke-Doane, 15 Aug 2025).

2. Constraint-based physical determinism beyond privileged boundaries

“Determinism Beyond Time Evolution” replaces the forward time-evolution picture with a constraint framework on entire histories. Let $\mathcal M$ be the class of all metaphysically possible Humean mosaics, and let a constraint be a subset $C \subseteq \mathcal M$. Each fundamental law $L$ induces a probability measure $\mathbb P_L$ over constraints, and if the world’s laws are $\delta$0 with chosen constraints $\delta$1, then the actual mosaic must satisfy

$\delta$2

Within this framework, holistic determinism means that every fundamental law assigns probability $\delta$3 to a single constraint and zero to all disjoint constraints. Strong holistic determinism adds that the intersection of all law-constraints contains exactly one mosaic; weak holistic determinism allows multiple compatible mosaics and therefore arbitrariness without objective chance; delocalised holistic determinism excludes pairs of compatible mosaics that are identical everywhere except on some small subregion of spacetime.

This reformulation relocates determinism from temporal evolution to modal restriction. “Boundary conditions” become partial specifications of the mosaic on some subset of spacetime, and determination between regions is defined by how far such information narrows the space of allowed mosaics. The paper explicitly criticizes region-based determinism in general relativity because verdicts depend sensitively on which kind of regions $\delta$4 are chosen. It then develops examples in which the dependency structure is not captured by any privileged initial boundary: retrocausal worlds in which initial and final states jointly determine the history; Wheeler–DeWitt models with $\delta$5, where the universal state encodes the whole spacetime at once; Lagrangian and path-integral formulations that weight entire paths rather than evolving states; non-Markovian worlds; and stateless worlds. Closed timelike curves, extendible spacetimes, and spacetimes with holes further show that failures of slice-based determinism can correspond to localized arbitrariness, delocalised arbitrariness, or no objective chance at all. In this setting, the mismatch is precisely the gap between a boundary-specific test of determinism and the actual global dependency structure of the allowed mosaics (Adlam, 2021).

3. Agentic AI: long dependency chains versus environment determinism

“Grounded Scaling: Why Agentic AI Needs Deterministic Environments” defines a Deterministic Agentic Environment $\delta$6 by four properties: (D1) Stability, (D2) Faithful ranking, (D3) Verifiability, and (D4) Bounded latency. Its central quantitative parameter is per-step determinism $\delta$7, defined as the probability that a single environment response is correct, as verified against ground truth. The paper stresses that $\delta$8 measures correctness against an independent ground-truth channel, not self-consistency under replay. A $\delta$9-step chain is a sequence of $S$0 semantically necessary steps, each using the environment, and Proposition 1 gives the Determinism–Efficiency Bound

$S$1

The motivating numbers are explicit: $S$2; $S$3; $S$4; and $S$5.

Within this framework, the dependency boundary is the task horizon $S$6, whereas the environmental determinism boundary is roughly the $S$7 up to which $S$8 remains acceptably high. Retries modify the bound to

$S$9

but if a total retry budget $k$0 is spread over the chain, then $k$1 shrinks as $k$2 grows, so the exponential structure remains. Under correlated session-state success probabilities $k$3, Lemma 1 replaces the independent product with

$k$4

yet the paper states that in all three regimes $k$5 exponentially in $k$6 whenever $k$7.

The paper then extends the mismatch beyond execution to learning and self-improvement. Proposition 2 bounds verifier-induced Goodharting by

$k$8

so increasing horizon without improving verifier quality $k$9 raises proxy optimization pressure. Proposition 3 gives expected drift for environment-side skill evolution,

$\ell$0

with a sufficient condition for monotone improvement

$\ell$1

Operationalization is provided by the Supply Certainty Index,

$\ell$2

and by the five-level Determinism Maturity Model. D2 environments retain personalized rankings and no verifier channel, with $\ell$3 in $\ell$4 for chain tasks and the example $\ell$5. D3 introduces stable rankings and verifier channels, where $\ell$6 is achievable and $\ell$7. The mismatch is therefore not merely rhetorical: it is framed as a falsifiable hypothesis, with explicit retraction conditions such as operator-class deployments achieving $\ell$8 success on $\ell$9 tasks in D2 environments (Ding et al., 21 Jun 2026).

4. Dependency calculi: protection boundaries and semantic determinism

“Monadic and Comonadic Aspects of Dependency Analysis” revisits the Dependency Core Calculus (DCC), where dependency boundaries are represented by levels in a lattice $\delta$0 and by the protection judgment $\delta$1, meaning that terms of type $\delta$2 may depend on level $\delta$3. DCC uses a nonstandard bind rule,

$\delta$4

whose result type is unwrapped to $\delta$5 rather than $\delta$6. This creates an apparent tension: syntactically, the calculus seems to allow elimination of a protection modality, even though the intended semantic result is noninterference for secure information flow and stage separation for binding-time analysis.

The paper resolves that tension by reconstructing DCC in standard categorical terms. A strong graded monad indexed by a preordered monoid yields the DCC modalities, and the protection judgment is shown to mark exactly those types carrying an Eilenberg–Moore algebra for the relevant grade. In the DCC model, if $\delta$7, then there is an isomorphism

$\delta$8

so the nonstandard bind is simply “bind into an EM-algebra.” The key theorem identifies the protected fragment with a full subcategory of the Eilenberg–Moore category $\delta$9. GMCC and GMCC$\mathcal M$0 then make both monadic and comonadic aspects explicit via a single graded modality $\mathcal M$1, subsuming both DCC and Davies’s $\mathcal M$2.

In this literature, the mismatch concerns whether type-level dependency boundaries really line up with deterministic semantic behavior. The paper’s answer is affirmative once the correct graded monadic/comonadic structure is exposed. Its “presence–absence test” constructs observer models that erase data above an observation boundary by mapping it to a terminal object; if an output survives when some input is semantically absent, then it does not depend on that input. Applied to DCC, this yields alternative proofs of noninterference; applied to $\mathcal M$3, it yields a concise proof of staging correctness and time-ordered normalization. A genuine mismatch would arise only if bind ignored the algebra restriction, if the protection judgment marked types that do not carry coherent algebra structure, or if one tried to model essentially comonadic flows with a purely monadic interface (Choudhury, 2022).

5. Fine-grained complexity: simulation boundaries between determinism and nondeterminism

“Consequences of APSP, triangle detection, and 3SUM hardness for separation between determinism and non-determinism” studies the class

$\mathcal M$4

defined as decision problems solvable in time $\mathcal M$5 on a multi-tape Turing machine with access to an $\mathcal M$6-bit nondeterministic oracle, with

$\mathcal M$7

Its central results are negated containments showing that linear-time computation with only $\mathcal M$8 nondeterministic bits cannot be simulated by deterministic algorithms below problem-specific exponent boundaries, assuming fine-grained hardness conjectures.

The boundary exponents depend on both conjecture and input range parameters. Under APSP hardness with polynomial weights, the paper states

$\mathcal M$9

For APSP-type problems with weights in

$C \subseteq \mathcal M$0

the broader boundary becomes

$C \subseteq \mathcal M$1

For 3SUM with the same range parameter,

$C \subseteq \mathcal M$2

For unweighted triangle detection,

$C \subseteq \mathcal M$3

From ETH, the paper derives

$C \subseteq \mathcal M$4

and from SETH via low-dimensional OV,

$C \subseteq \mathcal M$5

Here the mismatch is between a fixed weak nondeterministic resource and the deterministic running-time boundary needed to match it. The paper emphasizes that the separation depends sensitively on problem family, weight range, dimension, and clique size. It also emphasizes that the contribution is mostly conceptual and that all separations are conditional on fine-grained conjectures rather than unconditional lower bounds (Lingas, 2020).

6. Set-theoretic metatheory as a hidden determinism boundary

“Logical Dependence of Physical Determinism on Set-theoretic Metatheory” argues that determinism can depend on which set-theoretic axiom candidates extend $C \subseteq \mathcal M$6, especially $C \subseteq \mathcal M$7 and $C \subseteq \mathcal M$8. The standard physical formalization is Hadamard-style well-posedness: for admissible initial data $C \subseteq \mathcal M$9, existence, uniqueness, and continuous dependence of solutions $L$0 to equations $L$1. Schematically,

$L$2

The paper’s claim is that this verdict is not purely internal to physics, because metatheory determines facts about definable sets of reals that can enter coefficients, parameters, and initial data.

It distinguishes three dimensions of dependence. First, equation coherence: a heat equation on $L$3,

$L$4

with $L$5 for a lightface $L$6 set $L$7, is well-posed under $L$8 because projective sets are measurable under PD, but incoherent in the intended weak formulation under $L$9 if the same $\mathbb P_L$0 is non-measurable. Second, uniqueness: for

$\mathbb P_L$1

the parameter $\mathbb P_L$2 is tied to a $\mathbb P_L$3 sentence that is true in $\mathbb P_L$4 and false in $\mathbb P_L$5. If $\mathbb P_L$6, strict convexity yields a unique minimizer; if $\mathbb P_L$7, the functional has two minimizers $\mathbb P_L$8. Third, identity of the unique solution: both backgrounds can prove existence and uniqueness for the heat equation

$\mathbb P_L$9

while disagreeing on the $\delta$00-coded set $\delta$01 that determines the initial data

$\delta$02

and hence disagreeing on the resulting trajectory.

The same pattern extends to discrete systems. A recurrence $\delta$03 with $\delta$04 becomes coherent or incoherent depending on whether $\delta$05 is defined from an integral over a measurable $\delta$06 set. A discrete lattice functional toggles between unique and double minimizers according to the same $\delta$07-sensitive parameter $\delta$08. A scalar recurrence

$\delta$09

has a unique limit in both metatheories, but the limits differ because $\delta$10. The boundary here is the supposed separation between “physics proper” and foundational set theory; the mismatch is that determinism, well-posedness, and prediction cross that boundary (Clarke-Doane, 15 Aug 2025).

7. Comparative significance and recurrent misconceptions

A first recurrent theme is that determinism is repeatedly separated from mere prediction or operational stability. In the constraint framework, determinism is about how laws constrain entire possible mosaics, not about forward prediction from a present state. In the agentic-AI framework, $\delta$11 is correctness against ground truth, not replay consistency. In dependency analysis, noninterference is not guaranteed by surface syntax alone but by graded monadic/comonadic semantics and Eilenberg–Moore structure. A plausible implication is that many apparent failures of determinism are failures of the chosen boundary representation rather than failures of modal determination itself (Adlam, 2021, Ding et al., 21 Jun 2026, Choudhury, 2022).

A second theme is that boundaries are often indispensable operationally but not fundamental theoretically. Initial slices, Cauchy surfaces, region classes $\delta$12, chain horizons $\delta$13, protection levels $\delta$14, deterministic time exponents, and metatheoretic axioms all function as selection devices for studying dependence. Yet several of the cited works argue that verdicts change when those devices are changed. In physics this motivates holistic, strong, weak, and delocalised determinism; in agentic AI it motivates D3/D4 deterministic agent interfaces rather than human-tolerant D1/D2 surfaces; in dependency analysis it motivates a categorical rather than merely syntactic reading of dependency boundaries (Adlam, 2021, Ding et al., 21 Jun 2026, Choudhury, 2022).

A third theme concerns the plurality of failure modes. Boundary failure can mean objective chance, arbitrariness without chance, localized holes, verifier-induced Goodharting, insufficient environment-side drift, syntactic-semantic misalignment, deterministic inability to simulate weak nondeterminism below a conjectured exponent, or metatheory-relative disagreement about existence, uniqueness, or solution identity. This suggests that “indeterminism” is too coarse a label for several technically distinct phenomena (Adlam, 2021, Ding et al., 21 Jun 2026, Lingas, 2020, Clarke-Doane, 15 Aug 2025).

A final theme is methodological. The complexity-theoretic separations are explicitly conditional, the agentic-AI claims are cast as falsifiable with explicit null results and retraction conditions, and the set-theoretic analysis presents a disjunction between relativizing physical theories to metatheories and allowing empirical input into axiom choice. Across domains, the mismatch is therefore not a single theorem but a structural diagnosis: deterministic assessment becomes unreliable when the dependency structure of the system is richer than the boundary through which it is being measured (Ding et al., 21 Jun 2026, Lingas, 2020, Clarke-Doane, 15 Aug 2025).