Portion of Lost Tokens (PLT) Metric

• The PLT metric is a domain-specific measure quantifying the portion of wasted, lost, or inefficient tokens in fields like algebraic geometry, decentralized finance, and language processing.
• It applies distinct methodologies—using adjoint ideals, deterministic financial models, and token counting—to assess singularity purity, liquidity loss, or token inefficiency.
• Its practical applications include optimizing liquidity strategies, enhancing model tokenization, and supporting cost auditing in large language model APIs.

The “Portion of Lost Tokens” (PLT) metric is a technical construct whose meaning and methodology depend on the domain in which it arises. The term “PLT” appears as an acronym in research spanning several specialties: in birational algebraic geometry, PLT refers to “purely log terminal” singularities and associated measurement procedures; in decentralized finance (DeFi), it denotes the proportion of tokens lost by a liquidity provider due to systematic trading inefficiencies; and in natural LLM tokenization and auditing, it serves as a measure of inefficient or unjustifiable token usage. Across these domains, the essential function of the PLT metric is to quantify the portion of tokens that is either irretrievably lost (in the algebraic or financial sense) or wasted (in the computational sense) relative to an ideal or optimal scenario. This article reviews the PLT metric in depth, organizing the discussion by its formal definitions, domain-specific methodologies, mathematical formulations, practical applications, comparative analysis, and ramifications for future research.

1. Formal Definitions and Domain Interpretations

The PLT metric is formally defined in context-dependent ways, reflecting the technical nuances of each field:

• Birational Geometry: Here, PLT is not an independent metric but rather a property measured via the triviality of adjoint ideals associated to singularities. A pair $(R,D+A)$ is “purely BCM–regular” if $\operatorname{adj}_{B_c}(R, D+A) = R$. This is interpreted as the pair being “purely log terminal” (PLT), and the adjoint ideal functions as a measurement tool for how singular or “pure” a pair is (Ma et al., 2019).
• Decentralized Finance: In optimal liquidity provision, the “portion of lost tokens (PLT)” quantifies the percentage of assets lost by a liquidity provider (LP) owing to predictable, systematic depreciation in value resulting from the pool’s convex trading function and market dynamics. The PLT metric is closely linked to, and often coincides with, the “predictable loss (PL)” term (Cartea et al., 2023).
• LLM Tokenization: The PLT metric is used to express the fraction of tokens rendered inefficient—those that do not effectively contribute to valid or correct output due to suboptimal tokenization or grammar constraint mismatches. Mathematically, this takes the form $$\mathrm{PLT} = \frac{N_\mathrm{lost}}{N_\mathrm{total}}$$, where $N_\mathrm{lost}$ is the count of tokens wasted, and $N_\mathrm{total}$ is the total number required to represent the output (Hamilton et al., 20 Feb 2025). In the context of LLM APIs, PLT likewise expresses the number of tokens charged (including hidden or intermediate tokens) that cannot be justified by the observable output (Wang et al., 29 Jul 2025).

2. Methodological Frameworks and Measurement Procedures

Measurement of the PLT metric varies by technical context:

• Geometry via Adjoint Ideals: In mixed characteristic settings, the adjoint ideal $\operatorname{adj}_{B_c}(R, D+A)$, constructed with perfectoid big Cohen–Macaulay algebras, is used to determine whether a singularity is PLT. The procedure is invariant under choices of divisor presentations, and the equivalence $\operatorname{adj}_{B_c}(R, D+A) = R$ indicates “complete purity” (Ma et al., 2019). In K-stability theory, δ-invariants are approximated using lc places of plt complements, where the “plt metric” becomes the ratio $A_{X,\Delta}(E) / S_{X,\Delta}(E)$ as $E$ ranges over lc places associated to plt complements (Zhou, 2020).
• Token Loss in DeFi: LP position loss is modeled as a deterministic process $$\mathrm{PL}_t = -\frac{\sigma^2}{2} \int_0^t \frac{\tilde{x}_s}{\delta_s} ds$$, where $\sigma$ is volatility, $\tilde{x}_s$ is position value, and $\delta_s$ is spread width; higher volatility and narrower ranges increase PLT proportionally (Cartea et al., 2023). This framework is used to inform liquidity provision strategies by controlling the width and skew of the liquidity range to minimize PLT.
• Tokenization and LLM Auditing: PLT is measured by counting tokens that result from grammar-constrained decoding and comparing the count to an ideal representation. In predictive auditing of LLM APIs, frameworks such as PALACE predict hidden reasoning token counts using specialized domain routers and Group Relative Policy Optimization (GRPO) adaptation modules, with the goal of estimating unjustifiable, hidden token usage and quantifying PLT as the excess charged relative to observable output (Hamilton et al., 20 Feb 2025, Wang et al., 29 Jul 2025).

3. Mathematical Formulations and Quantitative Properties

Quantitative aspects of the PLT metric are directly traceable to domain-specific mathematical models:

• Geometry: PLT conditions relate to the triviality of adjoint ideals, equivalently the condition $adj_{B_c}(R, D+A) = R$, which functions as a threshold and diagnostic criterion for purity or singularity type. In K-stability estimation, the limit

$$\delta(X, \Delta) = \lim_{i\to\infty} \frac{A_{X,\Delta}(E_i)}{S_{X,\Delta}(E_i)}$$

as $E_i$ runs over relevant lc places of plt complements, serves as a plt-type approximation metric (Zhou, 2020).

• Decentralized Finance: Key quantitative expressions for PLT are

$$\mathrm{PL}_t = - \frac{\sigma^2}{2} \int_0^t \frac{\tilde{x}_s}{\delta_s} ds,$$

and the optimal liquidity spread is

$$\delta^*_t = \frac{2\gamma + \mu_t^2 \sigma^2}{4(\pi_t - \eta_t) + \epsilon},$$

where $\pi$ is the fee rate, $\gamma$ the concentration cost, $\mu$ the drift term, and $\eta$, $\epsilon$ threshold parameters. These formulas support self-financing strategies for LPs, balancing fee revenues against PLT penalties (Cartea et al., 2023).

• Tokenization: In structured output models, PLT can be written as

$$\mathrm{PLT} = \frac{N_\mathrm{lost}}{N_\mathrm{total}},$$

where $N_\mathrm{lost}$ quantifies subword inefficiencies or tokenization mismatches. In PALACE, token usage is estimated using reward objectives such as

$$R(\hat{y}, y) = \max(0, 1 - \frac{|\hat{y} - y|}{|y|}),$$

with overall model training minimizing average prediction error in token counts (Wang et al., 29 Jul 2025).

4. Comparative Analysis Across Domains

A salient feature of the PLT metric across fields is its use as a threshold or control parameter:

• Geometry: The condition $adj_{B_c}(R, D+A) = R$ for a pair to be PLT serves both as a precise geometric criterion and as a bridge between mixed and equal characteristic theories. PLT-centric measurement unifies multiplier ideals, test ideals, and adjoint ideal approaches (Ma et al., 2019).
• DeFi: PLT quantifies exposure to impermanent loss; optimal strategies minimize PLT by dynamically adjusting liquidity spread and responding to market volatility, as empirically validated in Uniswap v3 data (Cartea et al., 2023).
• LLM Tokenization and Auditing: PLT reflects inefficiency due to poor alignment between grammar-constrained output formats and pre-trained model subword representations. Using real-number formats with leading whitespace tokens is shown to reduce PLT and improve output accuracy, especially for small models; empirical results suggest PLT differences up to 10% across format choices (Hamilton et al., 20 Feb 2025). In LLM auditing, PALACE achieves higher accuracy in estimating hidden, potentially overbilled tokens, supporting cost auditing and inflation detection (Wang et al., 29 Jul 2025).

A cross-domain summary table:

Domain	PLT Metric Definition	Primary Function
Birational Geometry	Triviality of adjoint ideal	Diagnose PLT singularities
DeFi	Ratio of predictable/systematic loss	Benchmark LP trading loss
LLM Tokenization	Portion of inefficient/wasted tokens	Improve structured outputs
LLM Auditing	Predicted hidden reasoning token excess	Audit and detect overbilling

5. Practical Applications and Diagnostic Roles

Applications of the PLT metric are domain-specific but share diagnostic and optimization roles:

• Geometry: The PLT criterion via adjoint ideals is used to certify normality of divisors, to establish inversion of adjunction, to test openness of singularity properties in families, and to derive uniform Briançon–Skoda type results (Ma et al., 2019). For K-stability, PLT-based approximation of δ-invariants yields tractable computational pathways to identifying destabilizing divisors (Zhou, 2020).
• DeFi: PLT is instrumental in designing liquidity provision algorithms that balance competitive fee income against systematic losses; LPs calibrate the liquidity range width to minimize PLT while maintaining profitability (Cartea et al., 2023). Model-driven PLT monitoring informs real-time trading adjustments.
• Tokenization and LLM Auditing: Researchers optimize output formats (e.g., favoring real-numeric representations with leading whitespace tokens) to reduce PLT and enhance model performance in grammar-constrained decoding environments, particularly for low-capacity models (Hamilton et al., 20 Feb 2025). In cost auditing, PALACE provides predictive estimates of concealed token usage, supporting transparency and accountability in LLM API billing records (Wang et al., 29 Jul 2025).

6. Recent Advancements and Future Research Directions

PLT-centric approaches are evolving, reflecting both theoretical advances and practical exigencies:

• Geometry: Advances include uniform analogs of equal characteristic theorems in mixed characteristic, new functoriality results for perfectoid big Cohen–Macaulay algebras, and integration with test ideals and valuation theory. The development of bounded families of plt complements strengthens the tractability of K-stability computations (Ma et al., 2019, Zhou, 2020).
• DeFi: Empirical evaluation of optimal PLT-minimizing strategies and dynamic algorithmic liquidity management continues, with growing datasets and greater sophistication in market modeling (Cartea et al., 2023).
• LLM Auditing: Emerging frameworks such as PALACE leverage domain routing and reward-guided adaptation to achieve high-fidelity prediction of reasoning token counts. Future directions include the incorporation of richer auxiliary datasets, integration of additional metadata, and live deployment in commercial API environments (Wang et al., 29 Jul 2025).
• Tokenization: Ongoing research seeks finer-grained control of grammar-constrained decoding, the design of tokenization schemes that further reduce PLT, and deeper analysis of model-specific sensitivities to token format.

7. Limitations and Misconceptions

Several misconceptions and practical limitations are noted:

• In birational geometry, PLT is not a metric in the classical sense but a diagnostic tool via the adjoint ideal. There is no direct analogy to the “portion of lost tokens” found in computational contexts.
• In DeFi, PLT as a metric is only as accurate as the underlying model’s fit to market microstructure and trading function convexity.
• In language modeling, PLT does not measure semantic correctness but tokenization efficiency; optimal PLT does not guarantee optimal task performance if grammar constraints are misaligned with model pre-training.
• In auditing, PALACE’s performance depends on the quality of available data and the granularity of domain differentiation; it infers from prompt–answer pairs and may be limited in adversarial or heavily obfuscated scenarios.

Conclusion

The Portion of Lost Tokens (PLT) metric is a core measurement construct with technical implementations that vary by application domain. Whether diagnosing singularities via adjoint ideals in algebraic geometry, benchmarking loss in automated market-making, or auditing hidden reasoning tokens in LLM APIs, the PLT metric functions as a threshold for purity, efficiency, or accountability. Its value arises from precise mathematical formulation, diagnostic utility, and capacity to inform optimization strategies and transparent accounting. Recent advances signal further integration of PLT-centric methodologies in research programs across algebraic geometry, financial engineering, and LLM deployment, with significant scope for refinement and broader application.