Partial Determinism in Polytopes

• Partial determinism is defined as deterministic laws applying to select observables or scales while allowing residual variability elsewhere.
• The concept demonstrates that systems can exhibit predictable microdynamics combined with emergent randomness at macro levels, impacting fields like computing and physics.
• Applications span performance analysis, quantum foundations, automata theory, and biological systems, highlighting the balance between constraints and unpredictability.

Searching arXiv for the cited literature on partial determinism and closely related uses of the term. Partial determinism denotes a family of positions, models, and formal regimes in which deterministic structure is retained only in a qualified sense: at some scales but not others, for some observables but not the entire state, over finite horizons but not asymptotically, or under constrained interfaces while residual variability remains elsewhere. In the literature, the term is not a single doctrine. In computer performance, it names the gap between deterministic microarchitectural dynamics and limited practical forecastability under high temporal complexity (Garland et al., 2013). In quantum foundations, it denotes deterministic microdynamics with emergent macro-level probabilities and contextual measurement dependence (Vervoort, 2014). In mathematical logic and automata theory, it appears as intermediate strata between full determinism and unrestricted nondeterminism (Bienvenu et al., 2021, Radi et al., 2022). In philosophy of physics, it is naturally represented by laws that fix only some observables, subsystems, or chances across a space of admissible histories (Adlam, 2021), or by determinism relative to a formally specified level of description (Halvorson et al., 7 Mar 2025). Across these uses, partial determinism is best understood as a graded notion of constrained but non-total determination.

1. Conceptual core and recurring schema

A common schema runs through otherwise disparate literatures. Deterministic dynamics or laws are taken to exist at some level, yet prediction, uniqueness, or reproducibility is restricted by complexity, coarse-graining, contextuality, symmetry, or architecture. The resulting system is neither fully deterministic in the strongest operational sense nor wholly indeterministic.

One prominent formulation comes from computer performance analysis. There, computers are treated as deterministic nonlinear dynamical systems, and a scalar performance signal such as cache misses or instructions per cycle is modeled as a smooth measurement of an underlying deterministic state vector. Takens’ embedding theorem then implies that deterministic forecast rules exist in principle, but the paper’s central conjecture is that “in practice, complexity can effectively overwhelm the predictive power of deterministic forecast models” (Garland et al., 2013). This is an explicit statement of partial determinism: determinism in principle, bounded predictability in practice.

A philosophically different but structurally similar formulation appears in Vervoort’s thesis. There, partial determinism is “a layered framework” in which “microdynamics are deterministic,” while probabilities at the macro or experimental level arise from “frequency stabilization under repeatable initiating and probing conditions, coarse-graining, ignorance of $\lambda$, or typical causal averaging” (Vervoort, 2014). The same layered structure recurs in biological and cognitive settings, where constraints remain real but do not select a unique outcome. In Braun’s formulation, the genome provides “a set of constraints on the spectrum of regulatory modes, analogous to boundary conditions in physical dynamical systems,” while actual trajectories remain open and context-dependent (Gilead, 2015).

A more abstract version is given in the constraint-based framework of generalized determinism. There, laws pick out a set $S$ of admissible Humean mosaics, and partial determinism arises whenever some observable $O$ is invariant across all mosaics in $S$ even though $S$ contains more than one mosaic (Adlam, 2021). This suggests a unifying characterization: partial determinism is determinism relative to a restricted target—forecast horizon, observable, subsystem, level of description, or admissible interface—rather than determinism of the entire evolving world.

2. Temporal and computational forms of partial determinism

In empirical and systems contexts, partial determinism often appears as determinism curtailed by temporal complexity, residual implementation variability, or environment reliability.

The computer-performance study provides a canonical case. A three-line microkernel initializing a matrix in column-major order yields chaotic dynamics in L2 cache-miss rates, with “a strange attractor and a positive largest Lyapunov exponent ($\lambda_1 = 8000 \pm 200$ instructions) in a reconstruction space of dimension $\approx 12$” (Garland et al., 2013). Yet deterministic nearest-neighbor forecasting via delay-coordinate embedding performs very differently across workloads. For low-entropy microkernels, normalized RMSE is small; for SPEC applications, prediction error is much larger, and persistent permutation entropy near $0.97$ marks a regime where “the signal’s complexity ‘overshadows’ determinism” (Garland et al., 2013). Determinism persists at the level of dynamics, but practical predictability is only partial.

A directly operational use of the idea appears in code generation with LLMs. Repeated requests to ChatGPT under identical prompts produce variability at semantic, syntactic, and structural levels, and “setting temperature to 0 reduces—but does not eliminate—non-determinism” (Ouyang et al., 2023). On CodeContests, moving from $T=1$ to $T=0$ raises mean OER from $S$0 to $S$1, but “16.36% of tasks still have zero equal outputs at $S$2,” and pass-rate max difference can still reach $S$3 (Ouyang et al., 2023). Here partial determinism means attenuated but nonzero variability under ostensibly deterministic decoding conditions.

A systems-level answer to that problem is presented by LLM-42. Instead of forcing global determinism by disabling dynamic batching or redesigning all kernels, it enforces determinism only where needed through a verify-rollback loop. The fast path remains non-deterministic, while a fixed-shape verifier commits only tokens “guaranteed to be consistent across runs” and repairs KV state when mismatches occur (Gond et al., 25 Jan 2026). This is partial determinism in a literal systems sense: deterministic committed prefixes coexisting with non-deterministic speculative execution.

An analogous reliability-based formulation appears in agentic AI. “Partial determinism is the operating regime in which an agent’s environment is not fully reliable per step, but has a measurable per-step success probability $S$4 strictly between $S$5 and $S$6” (Ding et al., 21 Jun 2026). Under independent per-step verification and no retries, chain success is

$S$7

The paper’s point is not that the environment lacks structure, but that determinism is only partial at the action-interface level, so long chains fail exponentially unless $S$8 is extremely close to $S$9 (Ding et al., 21 Jun 2026).

3. Formalizations in mathematics, logic, and computation theory

Several literatures formalize partial determinism as an intermediate level between complete determinacy and unrestricted nondeterminism.

In algorithmic randomness, partial determinism is embodied by partial computable martingales. A partial martingale is defined only on a prefix-closed domain and may “ignore unfavorable branches,” thereby strengthening deterministic gambling power relative to total computable martingales (Bienvenu et al., 2021). Yet the paper proves that this does not match probabilistic strategies: there exists a sequence that is “partial computably random but not a.e. computably random,” so probabilistic gamblers can succeed with positive measure of oracle tapes where all deterministic partial computable gamblers fail (Bienvenu et al., 2021). Partial determinism here means deterministic strategies with partial domains, strictly stronger than total determinism but strictly weaker than probabilistic computation.

Automata theory offers a closely related hierarchy. The paper “A Hierarchy of Nondeterminism” defines three intermediate levels: determinizable by pruning (DBP), history deterministic (HD), and semantically deterministic (SD) (Radi et al., 2022). DBP means an equivalent deterministic automaton can be obtained by removing transitions; HD means choices can be resolved online from the past; SD means alternative choices lead to equivalent residual languages. These classes sit strictly between deterministic and unrestricted nondeterministic automata, except for certain collapses such as HD–NWW = DBP–NWW (Radi et al., 2022). Partial determinism is thus formalized as controlled nondeterminism with preserved algorithmic utility.

A different but related decomposition appears in multirelation theory. Binary multirelations model two levels of alternation: an outer “angelic” choice of successor sets and an inner “demonic” choice of elements within a chosen set (Furusawa et al., 2023). Determinism can therefore be partial at either level: outer deterministic but inner nondeterministic, or inner deterministic but outer nondeterministic. The paper proves that classes of inner and outer deterministic multirelations form categories under Peleg composition and are isomorphic to $O$0, while more general multirelations are not (Furusawa et al., 2023). The notion of partial determinism is explicit: deterministic behavior may hold at one alternation level but not the other.

A similar structure governs parallel computation. The programming-languages survey defines partial determinism as a property that identifies conditions under which execution yields a deterministic result even though the language or runtime permits nondeterminism elsewhere (Gonnord et al., 2022). Deterministic regions are obtained through FIFO channels with blocking reads, synchronous lock-step semantics, BSP supersteps, ownership or linear types, or actor-local determinism with mailbox-level nondeterminism (Gonnord et al., 2022). This is again determinism relative to a structural boundary.

4. Physics and philosophy: constrained histories, contextuality, and levels of description

In philosophy of physics and foundational physics, partial determinism is typically not a computational or engineering notion but a doctrine about what laws fix, and what they leave open.

The constraint-based framework of “Determinism Beyond Time Evolution” replaces directed time-evolution with global constraints on entire possible histories. Holistic determinism obtains when laws induce trivial probability distributions over constraints. Strong holistic determinism requires that the intersection of these constraints contain exactly one mosaic, while weak holistic determinism allows multiple mosaics without objective chances over them (Adlam, 2021). Partial determinism then arises naturally when only some observables, subsystem restrictions, or event-type chances are invariant across the solution set $O$1 (Adlam, 2021). In this framework, partial determinism is not a defect of the theory; it is a precise modal relation between laws and selected features of admissible histories.

A formally sharper version is given in “Defining Determinism.” There determinism is treated as a property of scientific theories via model-theoretic extension criteria such as Belot’s D1 and D3, rather than via ambiguous possible-worlds “agreement” relations (Halvorson et al., 7 Mar 2025). Partial determinism is introduced by projection: if $O$2 forgets some structure, then one can ask whether agreement at the $O$3-level extends weakly or uniquely. The paper explicitly formulates “partial D1$O$4” and “partial D3$O$5” as determinism relative to coarse-grained variables, qualitative structure, or a subalgebra of observables (Halvorson et al., 7 Mar 2025). This makes partial determinism a rigorously theory-relative property rather than an informal compromise position.

Vervoort’s treatment of Bell’s theorem supplies a different physical interpretation. There, local deterministic hidden-variable models remain viable if measurement independence is relaxed and probabilities are treated as objective frequencies under specified initiating and probing conditions (Vervoort, 2014). The thesis’s “partial determinism” is therefore a layered view: deterministic microphysics together with emergent experimental probabilities. This allows deterministic local models such as spin lattices or background-field models to violate Bell inequalities through measurement dependence while preserving locality in the Clauser–Horne sense (Vervoort, 2014).

Another philosophy-of-physics formulation appears in the deterministic model of free will. That paper rejects the inference from determinism to Big-Bang predestination by introducing “just-in-time” initialization of Planck-scale information and agent-level control over its impact via a deterministic collapse map $O$6 (Palmer, 1 Apr 2025). Partial determinism here means deterministic laws plus time-local initialization of free elements and context-sensitive control parameters $O$7, rather than fully fixed initial conditions at the universe’s origin (Palmer, 1 Apr 2025).

5. General relativity and the geography of deterministic domains

General relativity provides one of the most explicit geometric manifestations of partial determinism: determinism may hold on some spacetime regions or for some classes of models while failing elsewhere.

Isenberg’s account presents the standard Cauchy formulation. For initial data on a spacelike hypersurface $O$8, Einstein evolution is deterministic on the domain of dependence $O$9, and the maximal globally hyperbolic development is unique up to isometry (Isenberg, 2016). Partial determinism appears when this region is proper rather than global. In Taub–NUT spacetime, the Taub region is globally hyperbolic and deterministic, but beyond the Cauchy horizon there are multiple smooth extensions, and “determinism ‘breaks down’ even while remaining intact on $S$0” (Isenberg, 2016). In Gödel spacetime, by contrast, closed timelike curves pass through every point, so the breakdown is global rather than regional (Isenberg, 2016).

A more formal model-theoretic treatment is developed in “Determinism and Asymmetry in General Relativity.” Determinism is defined not absolutely but relative to a collection $S$1 of spacetimes and to increasingly strong notions of agreement: de dicto, de re, and de re$S$2 determinism (Manchak et al., 7 Mar 2025). For the class $S$3 of four-dimensional, inextendible, globally hyperbolic vacuum solutions, every subcollection is de re deterministic by Choquet-Bruhat–Geroch uniqueness, and because GR is rigid, de re and de re$S$4 collapse (Manchak et al., 7 Mar 2025). Stronger asymmetry conditions—“giraffe” for uniqueness of global isometries and “Heraclitus” for uniqueness of local isometric embeddings—yield stronger forms such as de dicto$S$5 and de dicto$S$6 (Manchak et al., 7 Mar 2025). Partial determinism in this setting is explicitly relative to both the model class and the asymmetry conditions imposed.

This geometric picture aligns with the more abstract projection-based accounts. GR may be deterministic on globally hyperbolic subcollections, partially deterministic on others, and non-deterministic in the strongest extension sense when symmetries or horizons permit multiple compatible futures. The upshot is that in relativity, partial determinism is not a vague middle ground but a structured stratification over regions, morphisms, and model classes (Isenberg, 2016, Manchak et al., 7 Mar 2025).

6. Constraint, autonomy, and bounded flexibility in biology and engineered systems

In biological and engineered contexts, partial determinism often names a regime of lawful constraint without unique trajectory selection.

The biological case is stated most sharply in the discussion of Marom and Braun. Organisms and environments “co-determine one another ‘in a nonlinear way,’” and biological systems exhibit “relative autonomy and flexibility in response which could not be predicted. Within the boundaries of some restraints, most of them genetic, this freedom from determinism is well maintained” (Gilead, 2015). Braun’s formulation that the genome provides “a set of constraints on the spectrum of regulatory modes, analogous to boundary conditions in physical dynamical systems” is the paper’s clearest statement of partial determinism in biology (Gilead, 2015). The relevant contrast is not between law and randomness, but between constraints and unique state selection.

This bounded-flexibility picture has a systems analogue in deterministic parallelism. Determinator enforces determinism at synchronization boundaries—through single-threaded “spaces,” pairwise rendezvous, snapshots, and merges—while permitting performance-relevant concurrency between those points (Aviram et al., 2010). The paper explicitly describes deterministic scheduling for legacy pthreads as “a form of partial determinism”: determinism is enforced at synchronization boundaries and in the schedule, guaranteeing repeatable outcomes, “but not full predictability of interleavings within a quantum” (Aviram et al., 2010). The broader survey of parallelism and determinism generalizes this architectural lesson: practical systems often isolate deterministic regions and permit controlled nondeterminism elsewhere, rather than attempting global elimination of all variability (Gonnord et al., 2022).

The same boundary-based logic underlies agentic AI infrastructure. The “Grounded Scaling” paper argues that environment determinism is the binding axis for long-chain agent execution, because even modest nondeterminism causes exponential collapse in chain success (Ding et al., 21 Jun 2026). Its recommended response is not to demand metaphysical determinism from the world, but to measure and raise environment-side determinism through verification channels, stable rankings, typed schemas, and deterministic interfaces (Ding et al., 21 Jun 2026). This suggests an engineering interpretation of partial determinism as interface-grade reliability rather than ontological totality.

7. Misconceptions, distinctions, and scope

A recurrent misconception is that partial determinism is simply weak indeterminism. The cited literature does not support that reduction. In the computer-performance setting, low predictability can arise in fully deterministic systems because sensitive dependence, high dimension, nonstationarity, and finite data overwhelm simple predictors (Garland et al., 2013). In the GR literature, multiple extensions beyond a Cauchy horizon do not mean loss of local determinism inside the domain of dependence (Isenberg, 2016). In automata, HD and SD automata are not “less deterministic” in an informal sense; they satisfy precise structural conditions that preserve important algorithmic behaviors (Radi et al., 2022). In biology, the claim is not absence of law but coexistence of genetic and organizational constraints with context-dependent trajectories (Gilead, 2015).

A second misconception is that partial determinism always means partial knowledge. Some uses are epistemic, but several are not. The constraint-based account defines partial determinism directly as invariance of selected observables across all admissible mosaics, independent of any observer’s ignorance (Adlam, 2021). The model-theoretic account makes it a theory-relative property under projection $S$7 (Halvorson et al., 7 Mar 2025). Vervoort’s account treats probabilities as objective frequencies under specified conditions, not subjective credences (Vervoort, 2014).

A plausible synthesis is that partial determinism functions as a cross-domain family resemblance concept rather than a single definition. What is shared is a rejection of the binary contrast between total determination and unrestricted chance. What varies is the locus of qualification: horizon length in recurrence analysis (Majerová, 2015), prediction success under entropy in computing (Garland et al., 2013), residual batch dependence in LLM inference (Ouyang et al., 2023), verified prefixes under speculative rollback (Gond et al., 25 Jan 2026), local versus global model extension in relativity (Manchak et al., 7 Mar 2025), or deterministic microstructure with emergent probabilities in quantum theory (Vervoort, 2014). This suggests that partial determinism is best treated encyclopedically as a structured family of intermediate determinacy concepts, each anchored to an explicit formal or empirical boundary condition rather than to a single universal doctrine.