The Mechanism · Gated-generation receipt

gen-in-3-4-sentences-explain-counting-with-repetition-state-any-caf2

stable pending witness
2026-05-29 08:48:36 UTC · anthropic/claude-sonnet-4-5

Prompt

In 3-4 sentences, explain counting with repetition. State any key formula.
74 chars

Generation

# Counting with Repetition Counting with repetition involves selecting items from a set where each item can be chosen multiple times. The most common scenario is finding the number of ways to distribute **n** indistinguishable objects into **k** distinguishable bins (or choosing **n** items from **k** types with repetition allowed). The key formula is: **C(n+k-1, k-1) = (n+k-1)! / (n!(k-1)!)** This is also known as "stars and bars" and equals the number of **k-combinations with repetition** from a set of **n** elements.
26 tokens in · 148 tokens out · 5587 ms · $0.0023

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
5594.4 ms
Base LLM
5586.6 ms
Verifiers
0.1 ms
Gates
0.0 ms
Cost
$0.0023

Trail

08:48:36 received prompt_chars=74 verifiers=['scripture_anchors', 'theology_doctrine']
08:48:36 red_gate decision=pass reason=no disqualifying input patterns detected
08:48:42 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=26 tokens_out=148 latency_ms=5586.6 cost_usd=0.0023
08:48:42 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
08:48:42 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
08:48:42 floor_gate decision=pass reason=output above protective minimum; no mismatches
08:48:42 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
08:48:42 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
08:48:42 signed hash_algo=sha256 hash=sha256:f0557ada384408757de36c018512045a2237afb758fc980a2aa907e2362c9f12

Content hash

sha256:f0557ada384408757de36c018512045a2237afb758fc980a2aa907e2362c9f12

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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