There is no universal per-problem price. A tightly bounded text-only workflow may consume only cents to a few dollars in model inference, while web search, code execution, simulations, retries, data preparation and human checking can become the larger bill. A defensible estimate names the model and version, input and output tokens, tool calls, compute time, labor and shared infrastructure, then reports both direct cash spend and the all-in cost of producing a result you are willing to accept.
Define what counts as one physics problem
“One problem” can mean a single textbook calculation, an answer supported by several papers, a parameter sweep, or a result that must survive independent reproduction. Those are different workloads. Set the boundary before estimating: identify the question, the evidence the model may use, the required output, the number of attempted solutions, and the acceptance test.
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Choose an explicit stopping point
- Draft answer: the model produces an explanation or derivation, with no independent check.
- Checked answer: equations, units, numerical values and citations are reviewed by a person or a trusted calculation.
- Reproducible result: code, input data, environment details and random seeds are retained so another researcher can rerun the work.
- Research finding: the result also includes literature review, uncertainty analysis, sensitivity tests and interpretation suitable for a report or paper.
A quote for a draft answer is not comparable with a budget for a reproducible finding. State which boundary your number uses.
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For an API model, calculate inference as:
inference cost = (input tokens ÷ 1,000,000 × input price) + (output tokens ÷ 1,000,000 × output price)
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Count every request, including retries, follow-up prompts, long documents sent again after a failed attempt, and any reasoning or long-context usage that the provider bills. Do not estimate from the final answer’s word count: a short response may have been preceded by several large prompts.
Illustrative rates and a 100,000/20,000-token workload
The following arithmetic uses the listed rates for two OpenAI models. It is an example of metered API spend, not an average observed physics project. Provider prices and model names can change, so verify the live schedule before committing funds.
| Model and listed rate | Input charge for 100,000 tokens | Output charge for 20,000 tokens | Inference subtotal |
|---|---|---|---|
| GPT-6 Sol: $2 per million input tokens; $10 per million output tokens | $0.20 | $0.20 | $0.40 |
| GPT-6 Astra: $10 per million input tokens; $50 per million output tokens | $1.00 | $1.00 | $2.00 |
For the same rates, doubling both token volumes doubles the subtotal. A workload of 250,000 input tokens and 50,000 output tokens would therefore be $1.00 on GPT-6 Sol or $5.00 on GPT-6 Astra before tools. These are straightforward scale calculations, not guarantees about the number of tokens a particular question will use.
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Why token totals are easy to underestimate
- Sending a full paper or dataset description in every turn repeats input tokens.
- Asking for alternative derivations creates additional output and often additional checking turns.
- Tool results pasted back into the conversation become new input tokens unless the workflow uses a billed retrieval or cache mechanism.
- Failed code, timeout recovery and corrected prompts are real billable requests.
- Batch discounts, cached input, regional processing and long-context multipliers can change the effective price. Anthropic documents cache-write and cache-read multipliers, a 50% batch discount for input and output tokens, regional-processing premiums and separate server-side tool charges; the applicable Claude rate depends on model and context length.
Tool, retrieval and compute charges
Inference is only the first line item when the model must find sources, inspect files or run calculations.
Web search and hosted containers
OpenAI lists web search at $10 per 1,000 calls, plus the search-content tokens returned to the model. At the call-only rate, one search call is $0.01 before those content tokens; a dozen calls would be $0.12 before content. Search snippets that trigger another model turn can therefore raise both tool and inference charges.
Hosted containers are listed from $0.03 to $1.92 per 20-minute session, depending on memory tier. A symbolic-algebra check may fit the smallest tier, while a memory-heavy simulation may require a higher one. Record the tier, number of sessions and whether sessions were fully used; a 20-minute allowance is not the same as 20 minutes of useful computation.
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Other tool categories to include
| Activity | What to measure | Typical cost consequence |
|---|---|---|
| Literature retrieval | Number of searches, retrieved pages or papers, and tokens returned | Search-call fees plus extra input tokens; subscription or document-licence fees may be separate |
| File or data processing | File size, parsing passes, storage and transfer | Retrieval, storage and egress charges in addition to model tokens |
| Code execution and plotting | Session count, memory tier, wall time and failed runs | Container or notebook charges and additional model turns for debugging |
| Symbolic or numerical services | External API calls, solver time and precision settings | Per-call fees or CPU/GPU time; high precision can increase runtime |
| Simulation | CPU/GPU hours, parallel workers, checkpoints and reruns | Often the dominant direct cash cost for parameter sweeps or large models |
Cloud GPU pricing is not specified by a single universal rate. Use the actual instance, region, reservation or spot status and runtime in your estimate. If a simulation runs locally, assign an internal rate for electricity, maintenance and machine depreciation rather than treating it as free.
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Turn direct charges into an all-in cost
Token invoices and cloud bills omit the work required to obtain a trustworthy answer. Add the labor and shared overhead that your organization would otherwise pay.
Labor categories
- Problem formulation: translating the physical question into measurable assumptions and acceptance criteria.
- Prompt and workflow design: selecting models, context windows, tools and retry rules.
- Data preparation: cleaning units, converting file formats, labeling images and documenting provenance.
- Verification: checking derivations, dimensions, limiting cases, numerical stability and citations.
- Debugging: diagnosing failed code, malformed outputs, retrieval errors and model hallucinations.
- Interpretation: deciding whether a result is physically meaningful and communicating uncertainty.
Multiply hours in each category by the loaded hourly rate for the people doing the work. If a student, staff scientist and reviewer have different rates, keep them separate. Include review time even when the model’s first answer looks plausible.
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Shared infrastructure and capital
Software licences, storage, data licences, lab access, security controls and platform administration are shared overhead. For self-hosted models, allocate hardware purchase or lease, power, cooling, maintenance and model-serving staff across the valid inferences the system produces. The LCOAI framework argues that token prices, GPU-hour billing and conventional total-cost-of-ownership measures each miss parts of the lifecycle; it recommends normalizing capital expenditure and operating expenditure by valid inference volume. In practice, report two numbers:
- Direct cash spend: API, tool, storage and compute invoices attributable to the problem.
- All-in accepted-result cost: direct spend plus allocated labor, infrastructure and the cost of failed or rejected attempts.
Accuracy changes the economics
A low token price is not automatically a low cost per accepted result. A 2025 study evaluated 15 multimodal models from Anthropic, Google and OpenAI on 102 image-based physics items. Reported performance ranged from 81.5% to 21%, and the authors found that more expensive models did not always outperform cheaper ones. The task mix was image-based physics, so its percentages should not be generalized to every research domain, but it demonstrates why verification belongs in the budget.
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Use this operational measure:
cost per accepted result = (all attempts + tools + compute + human review) ÷ number of results that pass your acceptance test
If a workflow costs $2 in API calls but only one of five attempts survives checking, its API-only cost per accepted result is $10 before labor. A cheaper model may require more correction; a pricier model may reduce iterations. Measure both token spend and the number of attempts to a checked answer on a representative pilot set.
Compare two workflows on the same axes
When choosing between models, an API and a local deployment, compare the complete workflow rather than headline token prices.
| Axis | Questions to answer |
|---|---|
| Verified accuracy | How often does the option pass the relevant physics checks, including diagrams, units and edge cases? |
| Iterations | How many prompts, retries and correction cycles reach an accepted result? |
| Tool and code charges | Are search, retrieval, containers, symbolic tools or external APIs billed separately? |
| Context handling | Can the system ingest the needed papers or datasets without repeated chunking and resubmission? |
| Latency and batching | Can jobs run in batch, and does a batch discount apply? |
| Privacy and residency | Do policy or funder rules require a specific region or prohibit sending data to a vendor? |
| Human review | How much expert time is needed to validate, debug and document the output? |
| Deployment economics | What capital, maintenance and utilization assumptions apply to local or self-hosted hardware? |
A repeatable seven-step estimate
- Write the acceptance test. Specify required equations, numerical tolerance, citation standard, uncertainty analysis and reproducibility artifacts.
- Record the model and version. Include provider, model name, context mode, region and pricing date.
- Forecast tokens by phase. Separate planning, retrieval, solving, tool-result interpretation, revision and final reporting. Estimate input and output tokens for each phase.
- Price inference. Apply the token formula to every phase, including retries and rejected drafts.
- List tool and compute events. Count search calls, container sessions, solver calls, CPU/GPU hours, storage and data transfer.
- Add human time. Log formulation, preparation, verification, debugging and interpretation at loaded hourly rates.
- Divide by accepted results and review variance. Report direct spend, all-in cost, assumptions, range and the number of accepted outputs. Re-estimate after a pilot of similar problems.
When is AI cheaper than hiring a physicist?
There is no universal break-even point. A comparison with a physicist must use the same acceptance test and include the physicist’s briefing, literature search, coding, checking, meetings and revision time. Compare that loaded labor cost with the AI workflow’s API, compute and review cost. If an expert must inspect every line and rerun every simulation, AI may mainly shift work rather than remove it. If the workflow handles many similar, well-specified problems and the verification step is short, automation can reduce marginal cost.
Do not use a model’s fluent explanation as a substitute for domain accountability. For safety-critical experiments, grant claims or publishable results, retain a qualified human sign-off regardless of the invoice total.
Why historical prices should not anchor a new budget
Stanford’s AI Index 2025 reports that the cost of querying a model scoring the equivalent of GPT-3.5’s 64.8 on MMLU fell from $20.00 per million tokens in November 2022 to $0.07 per million tokens by October 2024. The report also describes annual inference-price declines ranging from 9-fold to 900-fold depending on the task. Those figures describe historical capability-matched prices, not a promise about a particular physics model today. Date every price table, save the provider’s rate card, and rerun the estimate when model versions, context policies or tool fees change.
Common budgeting mistakes and fixes
- Mistake: budgeting only the final response. Fix: export request logs and count every attempt.
- Mistake: treating search as free. Fix: count calls and the search-content tokens returned.
- Mistake: hiding verification in overhead. Fix: record reviewer minutes and failed-result rates explicitly.
- Mistake: comparing models at different quality levels. Fix: run the same physics test set and acceptance criteria.
- Mistake: assuming local hardware has zero cost. Fix: allocate capital, power, maintenance and utilization across valid inferences.
- Mistake: quoting a promotional or batch rate as normal. Fix: label one-time allowances, discounts, regional premiums and eligibility conditions beside the number.
- Mistake: ignoring privacy constraints until deployment. Fix: decide permitted data residency and redaction before selecting a provider or architecture.
A practical decision rule
Use token pricing to establish the lower bound, then add tools, compute, labor and shared infrastructure until the number represents an accepted, reproducible result. For every estimate, publish the model and version, token volumes, iteration count, tool calls, compute assumptions, human hours, pricing date and acceptance rate. That transparent ledger is more useful than a single headline price—and it remains interpretable when models and tariffs change.
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Product prices and availability are accurate as of the date/time indicated and are subject to change. Any price and availability information displayed on Amazon at the time of purchase will apply.
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