A Better Way to Write Prompts for AI

In the paper's words

Prompt engineering for large language models (LLMs) is practiced as a craft of heuristics, yet teams deploying LLM agents need prompts whose behavior can be specified, checked, and reasoned about the way safety-critical software is specified. The gap is formal: no existing framework states prompt quality as a set of machine-checkable constraints on a prompt specification, and no analysis quantifies what such constraints would buy in first-try success, revision cost, or run-to-run consistency. We formalize the KERNEL properties—simplicity, verifiability, reproducibility, narrow scope, explicit constraints, and structured format—as six binary predicates on a prompt specification, prove a composition result that chained narrow, verifiable sub-prompts dominate monolithic prompts under stated independence assumptions, and derive the benchmark design that would test the claim.

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

1. The principal computed results: a monolithic prompt with five independent requirements each met with probability $0.8$ succeeds with probability $0.32768$, while the same workload decomposed into five verified sub-prompts with one bounded repair each succeeds with probability $0.59049$, a $1.80$-fold improvement, at a projected $2.20$-fold reduction in input-token cost per successful completion; a one-at-a-time ablation benchmark over six properties, ten tasks, and three models requires $1{,}800$ runs.

   from Abstract

2. A projected input-token cost per successful completion of $1{,}112$ tokens for the chained design against $2{,}440$ for the monolith, a $2.20$-fold reduction, derived from the same success probabilities and stated prompt-size assumptions (Section 4.4).

   from 1. Introduction

3. A mapping of the framework onto existing verification practice, using contract-extraction cost accounting [11], claim-labeling discipline [10], and tamper-evident execution records [4] as the engineering substrate (Section 7).

   from 1. Introduction

4. Scalable kernel clustering shows that an approximate algorithm can match exact kernel $k$-means in clustering quality while reducing quadratic run-time and memory costs, achieving over $90\%$ speedup in kernel computation on the Imagenet experiments [7].

   from 2. Background and Related Work

5. The analogy disciplines our claims: if full conformance verification of every prompt is expensive, an approximate checker that preserves verdict quality at lower cost is the right engineering target, and the benchmark of Section 4.5 is where the quality equivalence would be measured.

   from 2. Background and Related Work

6. The quantum-decoding analysis labels every quantitative claim as exact computation, complexity arithmetic, or error budget, and reports no new empirical measurements [10]; this paper follows that labeling scheme.

   from 2. Background and Related Work

How it is done

1. We then derive, with shown arithmetic, the quantitative consequences: a state-space coverage count for the six-property compliance lattice, a composition theorem with a worked numerical example, a token-cost comparison, and the exact size of the controlled benchmark that would test the framework's central claim.

   from 1. Introduction

2. The survey of formal specification and verification of separation kernels reports that most spatial-separation verification uses theorem proving because the kernels are small—thousands of lines—making full verification feasible, and that more than $140$ bugs were found in the project of verifying the seL4 kernel [6].

   from 2. Background and Related Work

3. This is an analytical paper: it formalizes, derives, and computes, and it reports no experiment, measurement, or simulation.

   from 3. Methods

4. The number of distinct specification classes is $$ N_{\text{classes}} = 2^{6} = 64. $$ Exactly one class—the all-ones vector—is KERNEL-compliant, so a uniformly drawn specification from the six-bit space is compliant with probability $$ p_{\text{compliant}} = \frac{1}{64} = 0.015625. $$ This count is the prompt-side analogue of the protocol-coverage arithmetic in [12], where a binary protocol over $k$ gated tools covers a $(2/3)^{k}$ fraction of the three-state space; for $k = 5$ gated tools that fraction is $f(5) = 32/243 \approx 0.132$ [12].

   from 4.1 The compliance lattice and its coverage arithmetic

What would falsify it

1. Third, an engineer could decompose a fragile monolithic prompt into narrow, verifiable sub-prompts and predict—before running anything—whether the decomposition helps.

   from 1. Introduction

2. A fully specified benchmark design—$360$ distinct prompt-model configurations and $1{,}800$ runs under a five-repetition protocol—whose regression and ablation analysis would isolate each property's contribution and whose outcomes would confirm or falsify the framework (Sections 4.5 and 7).

   from 1. Introduction

3. Under the model of Section 4.2, the predicted sign and rough magnitude of each coefficient is known in advance for $P_2$ and $P_4$ (they gate the composition mechanism), while $P_1$, $P_3$, $P_5$, $P_6$ are open—the benchmark exists precisely because the framework does not predict their effect sizes.

   from 4.5 Benchmark size arithmetic

Where the claim stops

1. What the results do and do not claim.

   from 6. Discussion

2. R3 and R4 are exact arithmetic on top of assumed inputs ($q = 0.8$, $s = 0.5$, $800$ and $200$ tokens); they are projections, not measurements.

   from 6. Discussion

3. The result rests on the following assumptions, with sensitivity noted for each: Independence of per-requirement success.

   from 6. Discussion

Where to read

1. Abstract208 words · 1 min
2. 1. Introduction629 words · 3 min
3. 2. Background and Related Work1,034 words · 4 min
4. 3. Methods545 words · 2 min
5. 4. Analysis1,487 words · 6 min
6. 5. Results297 words · 1 min
7. 6. Discussion886 words · 4 min
8. 7. Practical Implications and Engineering Pathways965 words · 4 min
9. 8. Conclusion191 words · 1 min
10. Provenance and AI Involvement240 words · 1 min