The Mechanism · Gated-generation receipt

gen-define-the-fundamental-theorem-of-arithmetic-and-give-one-sh-457c

stable pending witness
2026-05-29 02:57:18 UTC · anthropic/claude-sonnet-4-5

Prompt

Define the fundamental theorem of arithmetic and give one short worked example.
79 chars

Generation

# Fundamental Theorem of Arithmetic **Definition:** Every integer greater than 1 can be expressed uniquely as a product of prime numbers, up to the order of the factors. In other words, each integer n > 1 either is prime itself or can be written as a product of primes in exactly one way (ignoring the order). ## Worked Example Find the prime factorization of 84. **Solution:** ``` 84 = 2 × 42 = 2 × 2 × 21 = 2 × 2 × 3 × 7 = 2² × 3 × 7 ``` Therefore, **84 = 2² × 3 × 7** This is the *unique* prime factorization of 84. No other combination of primes will multiply to give 84.
20 tokens in · 201 tokens out · 3768 ms · $0.0031

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
3776.9 ms
Base LLM
3768.2 ms
Verifiers
0.1 ms
Gates
0.0 ms
Cost
$0.0031

Trail

02:57:18 received prompt_chars=79 verifiers=['scripture_anchors', 'theology_doctrine']
02:57:18 red_gate decision=pass reason=no disqualifying input patterns detected
02:57:21 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=20 tokens_out=201 latency_ms=3768.2 cost_usd=0.0031
02:57:21 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
02:57:21 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
02:57:21 floor_gate decision=pass reason=output above protective minimum; no mismatches
02:57:21 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
02:57:21 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
02:57:21 signed hash_algo=sha256 hash=sha256:43d4fa244103029ee01928fcfdba911b4f18035a6b950989616b3b0c98cf970c

Content hash

sha256:43d4fa244103029ee01928fcfdba911b4f18035a6b950989616b3b0c98cf970c

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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