The Mechanism · Gated-generation receipt

gen-in-3-4-sentences-explain-the-infinitude-of-primes-state-any-b63e

stable pending witness
2026-05-29 08:03:45 UTC · anthropic/claude-sonnet-4-5

Prompt

In 3-4 sentences, explain the infinitude of primes. State any key formula.
74 chars

Generation

The infinitude of primes means there are infinitely many prime numbers, with no largest prime. This was proven by Euclid around 300 BCE using proof by contradiction: assume finitely many primes p₁, p₂, ..., pₙ exist, then consider the number N = (p₁ × p₂ × ... × pₙ) + 1. This number N is either prime itself or divisible by a prime not in our original list, contradicting the assumption of finitely many primes. Therefore, the sequence of primes 2, 3, 5, 7, 11, ... continues forever.
28 tokens in · 147 tokens out · 3358 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
3364.9 ms
Base LLM
3358.0 ms
Verifiers
0.2 ms
Gates
0.0 ms
Cost
$0.0023

Trail

08:03:45 received prompt_chars=74 verifiers=['scripture_anchors', 'theology_doctrine']
08:03:45 red_gate decision=pass reason=no disqualifying input patterns detected
08:03:48 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=28 tokens_out=147 latency_ms=3358.0 cost_usd=0.0023
08:03:48 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
08:03:48 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
08:03:48 floor_gate decision=pass reason=output above protective minimum; no mismatches
08:03:48 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
08:03:48 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
08:03:48 signed hash_algo=sha256 hash=sha256:58dbc2658ab378325a0d8b0ff0656bc85da8a920a13e6fe0bc9a7bea4eb19ab2

Content hash

sha256:58dbc2658ab378325a0d8b0ff0656bc85da8a920a13e6fe0bc9a7bea4eb19ab2

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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