The Mechanism · Gated-generation receipt

gen-explain-the-fundamental-theorem-of-arithmetic-as-you-would-t-1d64

stable pending witness
2026-05-29 04:19:21 UTC · anthropic/claude-sonnet-4-5

Prompt

Explain the fundamental theorem of arithmetic as you would to a capable student, in 3-4 sentences.
98 chars

Generation

The Fundamental Theorem of Arithmetic states that every integer greater than 1 can be expressed as a product of prime numbers in exactly one way (ignoring the order of factors). For example, 12 = 2 × 2 × 3, and there's no other combination of primes that multiply to give 12. This uniqueness is what makes prime numbers the "building blocks" of all whole numbers—just like how every molecule can be broken down into a unique combination of atoms. The theorem guarantees both that such a prime factorization exists for every number and that it's the only one possible.
30 tokens in · 134 tokens out · 5016 ms · $0.0021

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
5024.1 ms
Base LLM
5015.7 ms
Verifiers
0.2 ms
Gates
0.0 ms
Cost
$0.0021

Trail

04:19:21 received prompt_chars=98 verifiers=['scripture_anchors', 'theology_doctrine']
04:19:21 red_gate decision=pass reason=no disqualifying input patterns detected
04:19:26 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=30 tokens_out=134 latency_ms=5015.7 cost_usd=0.0021
04:19:26 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
04:19:26 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
04:19:26 floor_gate decision=pass reason=output above protective minimum; no mismatches
04:19:26 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
04:19:26 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
04:19:26 signed hash_algo=sha256 hash=sha256:a679319bbb5a3daf92a63dad96e5a47630bbd8584310cffb5b0f52d54a90f607

Content hash

sha256:a679319bbb5a3daf92a63dad96e5a47630bbd8584310cffb5b0f52d54a90f607

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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