The Mechanism · Gated-generation receipt

gen-state-the-infinitude-of-primes-precisely-and-explain-why-it-1a1d

stable pending witness
2026-05-29 04:45:36 UTC · anthropic/claude-sonnet-4-5

Prompt

State the infinitude of primes precisely and explain why it matters in 2-3 sentences.
85 chars

Generation

**The Infinitude of Primes:** There are infinitely many prime numbers; that is, for any finite list of primes, there exists at least one prime not on that list. **Why it matters:** This result, first proven by Euclid around 300 BCE, is fundamental because primes are the multiplicative "building blocks" of all natural numbers (via unique factorization). The theorem guarantees that this structure never terminates, which has profound implications for number theory, cryptography, and our understanding of the distribution of numbers.
28 tokens in · 120 tokens out · 4976 ms · $0.0019

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
4983.8 ms
Base LLM
4976.3 ms
Verifiers
0.1 ms
Gates
0.0 ms
Cost
$0.0019

Trail

04:45:36 received prompt_chars=85 verifiers=['scripture_anchors', 'theology_doctrine']
04:45:36 red_gate decision=pass reason=no disqualifying input patterns detected
04:45:41 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=28 tokens_out=120 latency_ms=4976.3 cost_usd=0.0019
04:45:41 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
04:45:41 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
04:45:41 floor_gate decision=pass reason=output above protective minimum; no mismatches
04:45:41 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
04:45:41 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
04:45:41 signed hash_algo=sha256 hash=sha256:ca40d004a9e1176d2ebcda4aad2173808db025772efae070d8eb03014c8f9b24

Content hash

sha256:ca40d004a9e1176d2ebcda4aad2173808db025772efae070d8eb03014c8f9b24

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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