The Mechanism · Gated-generation receipt

gen-in-3-4-sentences-explain-the-fundamental-theorem-of-arithmet-c424

stable pending witness
2026-05-29 07:39:29 UTC · anthropic/claude-sonnet-4-5

Prompt

In 3-4 sentences, explain the fundamental theorem of arithmetic. State any key formula.
87 chars

Generation

The Fundamental Theorem of Arithmetic states that every integer greater than 1 can be expressed uniquely as a product of prime numbers, up to the order of the factors. This means any integer n > 1 can be written as: **n = p₁^a₁ × p₂^a₂ × p₃^a₃ × ... × pₖ^aₖ** where p₁, p₂, ..., pₖ are distinct prime numbers and a₁, a₂, ..., aₖ are positive integers. This decomposition is called the prime factorization, and it is unique for each number.
27 tokens in · 155 tokens out · 3548 ms · $0.0024

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
3556.1 ms
Base LLM
3547.7 ms
Verifiers
0.2 ms
Gates
0.1 ms
Cost
$0.0024

Trail

07:39:29 received prompt_chars=87 verifiers=['scripture_anchors', 'theology_doctrine']
07:39:29 red_gate decision=pass reason=no disqualifying input patterns detected
07:39:32 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=27 tokens_out=155 latency_ms=3547.7 cost_usd=0.0024
07:39:32 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
07:39:32 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
07:39:32 floor_gate decision=pass reason=output above protective minimum; no mismatches
07:39:32 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
07:39:32 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
07:39:32 signed hash_algo=sha256 hash=sha256:f2058ae8cb7274e03864c54ea0b3f6f6adcc9632a565e32a08ade3040fc12103

Content hash

sha256:f2058ae8cb7274e03864c54ea0b3f6f6adcc9632a565e32a08ade3040fc12103

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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