The Mechanism · Gated-generation receipt

gen-explain-counting-with-repetition-as-you-would-to-a-capable-s-c2fc

stable pending witness
2026-05-29 06:54:23 UTC · anthropic/claude-sonnet-4-5

Prompt

Explain counting with repetition as you would to a capable student, in 3-4 sentences.
85 chars

Generation

Counting with repetition deals with selecting items from a set where you're allowed to pick the same item multiple times. For example, if you're choosing 3 scoops of ice cream from 5 flavors, you could pick chocolate three times, or chocolate twice and vanilla once—repetitions are allowed. The formula for this is C(n+r-1, r) where n is the number of types of items available and r is the number of selections you're making. This is also called "combinations with repetition" or the "stars and bars" method, since it's equivalent to distributing r identical items into n distinct bins.
29 tokens in · 139 tokens out · 3954 ms · $0.0022

The four gates

RED
pass
no disqualifying input patterns detected
FLOOR
pass
output above protective minimum; no mismatches
WAY
pass
no declared way_path (Way check NA); no coercion language
EXECUTION
deferred
no witnesses available (Witness Roll not yet constituted)

Verifier results

scripture_anchors NOT_APPLICABLE
0 Scripture citation(s) extracted
theology_doctrine NOT_APPLICABLE
no doctrine keyword patterns detected

Metrics

Total latency
3961.2 ms
Base LLM
3954.5 ms
Verifiers
0.2 ms
Gates
0.0 ms
Cost
$0.0022

Trail

06:54:23 received prompt_chars=85 verifiers=['scripture_anchors', 'theology_doctrine']
06:54:23 red_gate decision=pass reason=no disqualifying input patterns detected
06:54:27 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=29 tokens_out=139 latency_ms=3954.5 cost_usd=0.0022
06:54:27 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
06:54:27 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
06:54:27 floor_gate decision=pass reason=output above protective minimum; no mismatches
06:54:27 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
06:54:27 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
06:54:27 signed hash_algo=sha256 hash=sha256:adeaad33e94c3d8525a832deb143b3477afb5b7802156a80bbba9b5bbe9188b2

Content hash

sha256:adeaad33e94c3d8525a832deb143b3477afb5b7802156a80bbba9b5bbe9188b2

SHA256 over the canonical JSON (excluding this field). Tamper detection. Ed25519 signing planned for v2 once the operator’s signing key is provisioned on disk.

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