The Mechanism · Gated-generation receipt

gen-in-3-4-sentences-explain-prime-factorization-state-any-key-f-62e1

stable pending witness
2026-05-29 08:08:39 UTC · anthropic/claude-sonnet-4-5

Prompt

In 3-4 sentences, explain prime factorization. State any key formula.
69 chars

Generation

Prime factorization is the process of breaking down a composite number into a product of prime numbers. Every composite number can be expressed uniquely as a product of prime factors, which is formalized by the **Fundamental Theorem of Arithmetic**. **Key formula**: For any composite number n, we can write: **n = p₁^a₁ × p₂^a₂ × p₃^a₃ × ... × pₖ^aₖ**, where p₁, p₂, ..., pₖ are distinct prime numbers and a₁, a₂, ..., aₖ are positive integers representing their exponents. For example, 60 = 2² × 3 × 5.
26 tokens in · 172 tokens out · 3300 ms · $0.0027

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
3307.8 ms
Base LLM
3299.7 ms
Verifiers
0.2 ms
Gates
0.0 ms
Cost
$0.0027

Trail

08:08:39 received prompt_chars=69 verifiers=['scripture_anchors', 'theology_doctrine']
08:08:39 red_gate decision=pass reason=no disqualifying input patterns detected
08:08:42 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=26 tokens_out=172 latency_ms=3299.7 cost_usd=0.0027
08:08:42 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
08:08:42 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
08:08:42 floor_gate decision=pass reason=output above protective minimum; no mismatches
08:08:42 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
08:08:42 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
08:08:42 signed hash_algo=sha256 hash=sha256:1e022cd8e2d7271909994f8ad56ce5de7305a3d1da46b0622c345cd5e8c5006c

Content hash

sha256:1e022cd8e2d7271909994f8ad56ce5de7305a3d1da46b0622c345cd5e8c5006c

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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