The Mechanism · Gated-generation receipt

gen-state-prime-factorization-precisely-and-explain-why-it-matte-5f9b

stable pending witness
2026-05-29 05:38:35 UTC · anthropic/claude-sonnet-4-5

Prompt

State prime factorization precisely and explain why it matters in 2-3 sentences.
80 chars

Generation

**Prime factorization** is expressing any integer greater than 1 as a unique product of prime numbers raised to powers (e.g., 60 = 2² × 3 × 5). It matters because it's fundamental to number theory and reveals the "building blocks" of numbers—the Fundamental Theorem of Arithmetic guarantees this factorization is unique (except for order). This uniqueness underlies critical applications from simplifying fractions and finding GCDs to modern cryptography, where the difficulty of factoring large numbers secures digital communications.
26 tokens in · 124 tokens out · 4677 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
4691.1 ms
Base LLM
4676.7 ms
Verifiers
0.2 ms
Gates
0.0 ms
Cost
$0.0019

Trail

05:38:35 received prompt_chars=80 verifiers=['scripture_anchors', 'theology_doctrine']
05:38:35 red_gate decision=pass reason=no disqualifying input patterns detected
05:38:40 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=26 tokens_out=124 latency_ms=4676.7 cost_usd=0.0019
05:38:40 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
05:38:40 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
05:38:40 floor_gate decision=pass reason=output above protective minimum; no mismatches
05:38:40 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
05:38:40 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
05:38:40 signed hash_algo=sha256 hash=sha256:e673f0a534485ec9efa88b35beb2a4acc5d21eb24f7f560e7bd356bd6eaa3f00

Content hash

sha256:e673f0a534485ec9efa88b35beb2a4acc5d21eb24f7f560e7bd356bd6eaa3f00

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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