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Short answer: a tightly defined text-only physics task may cost cents to a few dollars in model inference, but that is rarely the full price. Web searches, file retrieval, code containers, symbolic checks, simulations, cloud GPUs, data preparation, retries and human verification can exceed the API bill. A defensible estimate names the model and version, input and output tokens, iterations, tools, compute and labor, then reports both direct cash spend and an all-in cost.
What “per problem” actually means
A physics “problem” can mean a single numerical answer, a literature-backed derivation, a plotted analysis of experimental data or a simulation that must survive independent checking. Those workflows have very different cost profiles. Before pricing anything, define the boundary of the unit you are measuring:
- Inference: prompt tokens, retrieved context, output tokens, reasoning usage where billed, and every retry or correction pass.
- Tools: web search, file search, code execution, plotting, symbolic algebra and external APIs.
- Compute: local CPU/GPU time, rented cloud instances, simulation runs, self-hosting, fine-tuning and batch jobs.
- Human work: prompt design, experiment setup, debugging, citation checking, physical interpretation and sign-off.
- Shared overhead: software subscriptions, storage, data licences, laboratory access and the amortised cost of hardware.
If you quote only the model invoice, call it metered inference spend, not the cost of the research problem. A project estimate should state what is excluded.
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A metered-inference example
Use the same calculation for any provider:
(input tokens ÷ 1,000,000 × input price) + (output tokens ÷ 1,000,000 × output price)
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Suppose one problem consumes 100,000 input tokens and 20,000 output tokens. Using the listed OpenAI rates in the pricing snapshot, the arithmetic is:
| Model and listed rate | Input charge | Output charge | Total before tools |
|---|---|---|---|
| GPT-6 Sol: $2 per million input tokens; $10 per million output tokens | 0.1 × $2 = $0.20 | 0.02 × $10 = $0.20 | $0.40 |
| GPT-6 Astra: $10 per million input tokens; $50 per million output tokens | 0.1 × $10 = $1.00 | 0.02 × $50 = $1.00 | $2.00 |
These are transparent examples of API spend, not an average physics-research bill. If the same workload is run three times, the inference totals become $1.20 for Sol or $6.00 for Astra before tools. A long paper, a large dataset or repeated failed attempts can push token volume several times higher.
Record the date, model version, pricing tier, token counts and whether cached or batch rates apply. Prices and model names change, so an undated number is not a reproducible estimate.
Tool charges can overtake tokens
Web search and retrieval
OpenAI lists web search at $10 per 1,000 calls, plus the search-content tokens supplied to the model. Ten calls therefore add $0.10 in tool fees before counting the retrieved text. A literature question may need many searches, repeated queries and full-paper retrieval, so count calls separately from prompt tokens.
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Hosted code containers
OpenAI-hosted containers are listed from $0.03 to $1.92 per 20-minute session, depending on memory. That is an additive session charge; model tokens and any other tools still apply. A notebook that runs through several sessions, retries a failed install or processes a large dataset can make container spend more significant than the initial prompt.
Provider-specific extras
Anthropic documents model-dependent token prices, cache-write and cache-read multipliers, a 50% batch discount for input and output tokens, regional-processing premiums and additional server-side tool charges. Do not transfer one Claude model’s price to another, and do not assume that a batch discount applies to an interactive run. Read the current rate card for the exact model, region and context length.
Why physics problems need a wider budget
Iterations and verification
A first response may omit a boundary condition, misuse units or choose a mathematically valid but physically irrelevant branch. A reliable workflow asks for an independent derivation, dimensional checks, limiting cases and a comparison with known results. Every pass adds tokens and often another tool call. The relevant quantity is not the cost of the first answer but the cost of reaching an answer that a qualified reviewer accepts.
Symbolic, numerical and visual work
Deriving an equation, simplifying it with a computer-algebra system, plotting residuals and running a simulation are separate activities. They may require file uploads, code execution and CPU or GPU time. If a simulation runs on your workstation, its electricity and hardware are still part of an all-in estimate even though no vendor invoice appears.
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Multimodal accuracy is uneven
Polverini and Gregorcic (2025) evaluated 15 multimodal models from Anthropic, Google and OpenAI on 102 image-based physics items. Reported performance ranged from 81.5% to 21%, and expensive models did not consistently outperform cheaper ones. Token price is therefore a poor proxy for cost per accepted result: a cheaper model that needs extensive correction can consume more reviewer time than a costlier model that is usually right.
Direct cash cost versus all-in cost
Publish two numbers when the decision matters:
- Direct cash spend: API tokens, tool calls, cloud compute, storage and paid data access attributable to the problem.
- All-in estimate: direct spend plus the value of researcher and reviewer time, software, infrastructure and a fair share of capital costs.
The LCOAI paper notes that token bills, GPU-hour pricing and conventional total-cost-of-ownership measures each omit parts of the lifecycle. Its proposed approach normalises capital expenditure (CAPEX) and operating expenditure (OPEX) by the volume of valid inferences. In practice, divide the monthly cost of a dedicated machine or service by the number of accepted, checked results it produces, rather than by raw requests. This prevents a heavily used but error-prone system from looking artificially cheap.
A simple project worksheet is:
all-in cost = inference + tools + compute + (researcher hours × loaded hourly rate) + (reviewer hours × loaded hourly rate) + allocated overhead
Keep the labour rates and allocation period visible. If you cannot establish a value, label it “not stated” instead of silently treating it as zero.
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How to build an estimate step by step
- Define acceptance. Specify whether the result needs a numerical tolerance, a reproducible script, cited primary literature, an uncertainty estimate or expert sign-off.
- Set the workload. Estimate input tokens, output tokens, attached papers, data size and the number of expected retries. Include context repeated in every call.
- Choose a dated rate card. Record provider, model version, region, context tier and whether caching or batch processing is available.
- Calculate baseline inference. Apply the token formula to one run, then multiply by the planned number of iterations. Add a separate contingency for failed calls rather than hiding it in rounded token counts.
- Inventory tools. Count search calls, retrieval operations, container sessions, plotting jobs, symbolic-algebra calls and external API requests. Include per-call fees and the tokens returned by those tools.
- Price compute. Measure local runtime or use the cloud provider’s CPU/GPU-hour and storage rates for the required region and instance. Include setup, queue and data-transfer time when they are material.
- Value human work. Log prompt engineering, data cleaning, debugging, verification and interpretation separately. A five-minute automated run can still require an hour of expert checking.
- Run sensitivity cases. Produce low, expected and high cases by changing iterations, context size, tool use and reviewer time. State which assumption drives the range.
Comparing API, batch and self-hosted approaches
| Option | What you pay for | Advantages | Risks or unknowns |
|---|---|---|---|
| Interactive vendor API | Per-token inference plus tool and container charges | Low setup effort; costs track usage | Retries, long contexts and tool calls can multiply the bill; current rates vary by model |
| Batch processing | Token charges under the provider’s batch rules | Anthropic documents a 50% input/output batch discount | Higher latency and eligibility limits; exact model and regional terms must be checked |
| Self-hosted model | Hardware or cloud GPU CAPEX/OPEX, power, storage, maintenance and engineering time | Control over data residency and predictable throughput at high volume | Break-even depends on utilisation and validated-result rate; no universal price is established here |
| Hybrid workflow | API calls for difficult steps plus local tools or simulation | Can keep sensitive data local while using a stronger model selectively | Pipeline integration, transfer and debugging add labour |
Compare these options on verified accuracy, tokens per accepted result, number of iterations, context capacity, latency, batch eligibility, privacy requirements and recovery time—not on headline token price alone.
Is AI research cheaper than hiring a physicist?
There is no general yes-or-no answer. API spend can be tiny compared with an expert’s time, but an unsupervised answer is not equivalent to professional research. The fair comparison is the cost of a checked deliverable: AI usage plus the human hours needed to formulate, test, correct and interpret it versus the hours a physicist would spend doing the same work with conventional software.
For a break-even calculation, estimate:
AI cost per accepted result = (all AI and infrastructure spend + review labour) ÷ accepted results
Compare that with the loaded labour cost and turnaround time for a physicist, including their software, data and laboratory access. AI is more likely to win on repetitive literature triage, parameter sweeps and boilerplate code when validation is straightforward. A novel derivation, an experiment with safety implications or a result that informs a publication may remain labour-dominated because interpretation and accountability cannot be outsourced to a token counter.
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Why current prices should be treated as moving targets
Stanford’s AI Index 2025 reports that the cost of querying a model with GPT-3.5-level MMLU performance (64.8) fell from $20.00 per million tokens in November 2022 to $0.07 per million tokens by October 2024. The report also says inference prices fell between ninefold and 900-fold per year depending on the task. Those are historical capability-matched figures, not a promise about any particular physics model or today’s invoice. Date every budget and re-check prices before committing to a long project.
Common budgeting mistakes
- Counting only the first answer: include retries, independent checks and failed tool calls.
- Mixing token and tool units: search calls, container minutes and output tokens are different meters.
- Ignoring retrieved context: full papers and repeated system instructions can dominate input tokens.
- Assuming the most expensive model is best: the 2025 visual-physics study found no consistent price-performance ordering.
- Calling an unverified result “solved”: report cost per accepted result and the reviewer effort required.
- Using an old rate card: retain the model version, date, region and billing mode with every estimate.
- Treating local compute as free: allocate hardware, power, storage and maintenance when they support the workflow.
A practical reporting template
For each problem, keep a small ledger with these fields:
- Problem identifier and acceptance test
- Provider, model version, date and region
- Input, output and cached-token counts
- Number of attempts and total latency
- Search calls, retrieved files and container sessions
- CPU/GPU runtime, storage and data-transfer usage
- Researcher and reviewer minutes
- Direct cash total, all-in total and accepted-result status
This record makes a “cost per problem” figure auditable and lets you see whether optimisation should target prompts, model choice, retrieval, code, simulation or human review.
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