John Swygert 

Independent methodological research working paper  |  October 26, 2026

Abstract

This paper proposes a computational methodology in which natural-language reasoning is used to formulate candidate relational constraints, empirical non-occurrence is used to investigate their boundaries, and formal verification is used to prune a search space before expensive numerical computation. The objective is not to replace mathematics or graphics processing units (GPUs) with prose. It is to reduce unnecessary numerical work by discovering and validating exclusion rules that prevent entire families of states or trajectories from entering the computational workload. The proposed pipeline connects the Geometry of Absence method with symbolic constraint solving, reachability analysis, active sampling, and cost-aware evaluation. It distinguishes descriptive compression from certified computational savings, specifies an auditable experiment, and identifies failure modes including false exclusions, selection bias, and language-model hallucination. Its principal claim is a testable efficiency hypothesis: for suitable problems, validated constraint-first workflows can achieve equal or better solution quality with lower total compute than unconstrained baselines.

Keywords: geometry of absence; language-guided reasoning; constraint discovery; symbolic pruning; reachability; search-space reduction; GPU efficiency; verification; active learning

1. Research proposition

A conventional search or simulation may evaluate a large set of candidate configurations before determining that most are infeasible, unreachable, or irrelevant. A constraint-first alternative asks whether the excluded region can be characterized early enough to avoid evaluating it. The key transition is from “compute and reject” to “establish a defensible rule, then do not compute what the rule excludes.”

The proposed role of language is upstream. Natural-language dialogue can help people and computational models express invariants, conditional boundaries, symmetries, causal dependencies, and counterexamples. These descriptions become useful computationally only when translated into explicit predicates and subjected to independent checks.

2. Connection to the Geometry of Absence

The Geometry of Absence begins with observations and examines regions that remain unoccupied despite adequate opportunities for detection. Such regions may suggest boundaries in a candidate state space. The present paper asks a second question: if a boundary survives testing, can it be converted into a safe exclusion rule that reduces subsequent computational effort?

A persistent absence is not automatically a forbidden state. It can reflect undersampling, biased initial conditions, inadequate sensors, or an insufficient observation horizon. Consequently, the computational pipeline must distinguish three statuses: suspected exclusion, statistically supported exclusion under stated conditions, and formally certified exclusion within a defined model.

3. The distinction between language, symbols, and calculation

• Natural language: an efficient medium for proposing and negotiating relational descriptions, including conditions, exceptions, and analogies.
• Symbolic representation: a precise encoding of a candidate rule as a predicate, logical formula, constraint graph, program, or inequality.
• Numerical computation: the evaluation, optimization, integration, or simulation still required after safe pruning.

The thesis does not imply that words inherently require fewer operations than equations. Large language models themselves can be compute-intensive. The possible saving arises when the cost of discovering, checking, and applying constraints is smaller than the work avoided.

4. Formal model of constraint-first reduction

Let Ω be a candidate search space, f(x) the target quantity or objective, and C(x;θ) a constraint predicate parameterized by conditions θ. The admissible candidate set is Ω_C = {x ∈ Ω : C(x;θ) = true}. A solver operating on Ω_C may avoid evaluating Ω \ Ω_C. If C is only provisional, excluded states must remain recoverable for audit and challenge.

Let B be the baseline computational cost. Let D be the cost of constraint discovery, V the cost of validation, P the cost of pruning, and S_C the residual solver cost. Then the constraint-first cost is T_C = D + V + P + S_C. A net saving exists when B − T_C > 0, at equal target accuracy and coverage. A practical efficiency ratio is η = B / T_C; η > 1 indicates lower measured total cost for the constraint-first workflow.

The cost comparison must include model inference, prompt iterations, formalization, counterexample searches, verification, data acquisition, CPU time, GPU time, and human review. GPU-hour reduction alone is not a complete efficiency result if costs are merely transferred elsewhere.

5. From verbal insight to machine-checkable rule

Step 1 — Specify the bounded problem

Define the state variables, allowable actions, initial conditions, time horizon, desired output, and quality threshold. The problem must be small enough to measure and sufficiently structured to permit reproducible baselines.

Step 2 — Collect positive observations and exposure records

Record which states or trajectories occurred and where observations could have occurred. Track sampling intensity, sensor coverage, selection rules, and known blind spots. A blank region without exposure information does not support a pruning decision.

Step 3 — Propose candidate constraints in language

Use a human researcher, an LLM, or both to articulate conditional exclusions: “under conditions θ, no feasible path crosses boundary b,” or “state y cannot follow state x without intermediate event z.” Require each proposal to name its domain and plausible counterexamples.

Step 4 — Formalize the claim

Translate the proposed rule into an executable predicate, satisfiability constraint, invariant, transition guard, or inequality. Preserve a traceable mapping from the original verbal claim to the formal expression. Ambiguous words such as “never,” “near,” and “possible” must be operationalized.

Step 5 — Challenge and validate

Search deliberately for counterexamples using held-out observations, adversarial sampling, boundary-focused simulations, and formal methods where available. A failed rule is revised or rejected rather than silently retained.

Step 6 — Prune conservatively

Only rules meeting the predetermined evidence threshold may exclude candidate work. Less certain rules may rank candidates for later evaluation but should not permanently remove them. Record exactly which regions each rule suppresses.

Step 7 — Measure end-to-end performance

Run the constrained and unconstrained workflows on equivalent tasks. Compare total resource use, solution quality, missed valid states, robustness, and reproducibility. A reduction in raw simulation count is insufficient if it causes unacceptable false negatives.

6. A worked illustrative example: trajectory corridors

Consider a hypothetical agent navigating a discretized environment with 10 million nominal candidate trajectories. Observed successful trajectories concentrate in two corridors, while a central region remains unoccupied across repeated trials. The absence suggests—but does not establish—that the central region is blocked under specified movement and obstacle conditions.

A language-guided hypothesis might state: “With obstacle configuration A and movement limit v, every path from start S to goal G must pass through corridor L or R.” A graph cut or reachability solver can test this statement. If certified, candidate paths violating the cut constraint can be discarded before expensive physical simulation.

For illustration only, suppose an unconstrained workflow evaluates 10 million paths at one cost unit each. A validated rule reduces detailed evaluation to 1 million paths, while discovery and verification cost 500,000 units. Then B = 10,000,000, T_C = 1,500,000, and η ≈ 6.67. These figures are hypothetical, not experimental findings. If the rule wrongly excludes a valid shortcut, the apparent saving is not a valid success.

7. The geometry as a computational object

The relevant geometry need not be a literal three-dimensional surface. It can be a disconnected feasible set, a graph separator, a region of a parameter manifold, a logical incompatibility class, or a reachability boundary in a high-dimensional state space. Its computational value lies in expressing many exclusions compactly.

One inequality may eliminate a continuous region; one invariant may rule out an entire family of trajectories; one conditional independence relation may simplify a probabilistic model. This is the operational meaning of compressing computation into structure: a compact, validated rule substitutes for repeated rejection of individual cases.

8. Candidate methods and their roles

• Constraint programming and SAT/SMT solving: encode and check logical, discrete, and mixed constraints.
• Abstract interpretation and formal verification: establish invariants and conservative reachable-state overapproximations.
• Reachability analysis and control barriers: characterize which states can or cannot be reached under defined dynamics.
• Support estimation and density modeling: identify candidate low-occupancy regions without equating them with impossibility.
• Topological data analysis: detect persistent holes or disconnected regions across sampling scales; use as a hypothesis generator, not proof of exclusion.
• Active learning and counterexample-guided refinement: sample where a proposed boundary is most uncertain or most vulnerable.
• Program synthesis and LLM-assisted formalization: turn relational descriptions into candidate executable constraints, followed by independent verification.

9. The role of multidimensional language-based reasoning

A researcher may describe a four-dimensional trajectory—three spatial coordinates changing through time—without rendering every intermediate frame. An LLM can manipulate the relational description in language and symbolic form without needing to produce a literal internal image. The useful capability is not a claim about subjective visualization; it is the ability to preserve and transform constraints across representations.

The same approach can describe far higher-dimensional spaces. However, natural-language fluency must not be mistaken for geometric fidelity. Every proposed constraint needs a precise interpretation, especially where coordinate choices, temporal ordering, or projection effects can create misleading apparent gaps.

10. Proposed benchmark study

Select three problem families with known or independently checkable feasible sets: grid or graph pathfinding, bounded scheduling/assignment, and low-dimensional dynamical reachability. For each family, construct instances with controlled boundary complexity and varying amounts of misleading observational absence.

Compare four arms under matched accuracy requirements: (A) conventional exhaustive or heuristic baseline; (B) conventional symbolic constraints supplied by an expert; (C) LLM-proposed constraints without automatic pruning, used only to rank candidates; and (D) LLM-proposed, independently validated constraints used for pruning. Hold out test instances and report both successful and failed constraint proposals.

Primary outcomes: total compute cost, wall-clock time, energy where measurable, GPU-hours, valid-solution recall, objective quality, and the fraction of excluded space later found feasible. Secondary outcomes: human review burden, rule complexity, transfer across instances, and sensitivity to sampling bias.

11. Hypotheses and falsification criteria

• H1 — In structured problems with reusable exclusion rules, validated language-guided constraints reduce end-to-end computation at matched solution quality.
• H2 — The benefit increases as the ratio of safely excluded search volume to constraint-discovery overhead increases.
• H3 — Boundary-focused counterexample testing reduces false exclusions relative to relying on observed gaps alone.
• H4 — In weakly structured or rapidly changing domains, discovery and verification overhead may erase the saving.

The main efficiency claim fails for a benchmark if the constrained arm does not improve total measured cost at the prespecified quality threshold. It also fails as a reliable method if pruning introduces false exclusions beyond the accepted error bound, even when runtime improves.

12. Critical failure modes

• False boundary: a gap arises from insufficient sampling, detection failure, or biased experiment design.
• Projection artifact: high-dimensional paths appear separated only because the chosen visualization discards coordinates.
• Unstable rule: a boundary holds in training conditions but disappears when parameters or initial states change.
• LLM overreach: a persuasive verbal claim lacks a valid formal translation or silently changes assumptions.
• Hidden overhead: GPU savings are offset by extensive model inference, verification, or human correction.
• Over-pruning: the fastest result is wrong because rare but valid solutions were excluded.

The appropriate response is not to abandon constraint discovery but to attach provenance, confidence, scope, and revocation procedures to every proposed exclusion.

13. Audit ledger for every proposed boundary

Each rule should have a unique identifier; originating observation or verbal hypothesis; defined variables and conditions; exact formal predicate; exposure and detection assessment; verification method; known counterexamples; validity domain; exclusion volume; measured computational savings; false-exclusion record; and revision history. This ledger makes the method reproducible and allows a later discovery to reopen a previously excluded region.

14. Relationship to the Field of Reality and TSTOEAO

This paper is a methodological companion to The Geometry of Absence within the Reality & Imagination research sequence. It does not require that the Field of Reality be established as a physical entity. The same computational approach can operate on ordinary engineering state spaces, scientific models, or abstract optimization problems.

TSTOEAO may use related reasoning about boundaries and permitted routes, but it remains a distinct research program. A shared method of interpreting reliable absence is not evidence that the programs describe the same underlying ontology.

15. Research roadmap

Phase I: demonstrate correct language-to-constraint translation on benchmark cases with known ground truth. Phase II: add observation-derived candidate boundaries and deliberate counterexample search. Phase III: compare compute efficiency and false-exclusion rates against strong symbolic and numerical baselines. Phase IV: evaluate transfer to larger scientific and engineering domains only after the smaller benchmarks establish reliability.

Publish negative results as well as positive ones. The relevant discovery is not that every absence hides a law; it is whether some well-characterized absences can become reusable, trustworthy computational shortcuts.

16. Conclusion

The proposed research question is precise: can the geometry inferred from reliable non-occurrence be translated, through language-guided constraint discovery and independent verification, into a smaller computational search space without losing valid solutions? If so, some high-dimensional tasks may require fewer numerical evaluations and potentially less GPU computation. The benefit would come not from replacing mathematics with words, but from using words to help discover the mathematical boundaries that make unnecessary calculations avoidable.

The guiding principle is simple: do not calculate every path merely to learn that most paths were never available. First investigate the boundaries. Then calculate what remains.