The Mechanism · Gated-generation receipt

gen-state-the-fundamental-theorem-of-arithmetic-precisely-and-ex-0111

stable pending witness
2026-05-29 08:48:46 UTC · anthropic/claude-sonnet-4-5

Prompt

State the fundamental theorem of arithmetic precisely and explain why it matters in 2-3 sentences.
98 chars

Generation

## Fundamental Theorem of Arithmetic **Statement:** Every integer greater than 1 can be expressed as a product of prime numbers in exactly one way, up to the order of the factors. **Why it matters:** This theorem establishes prime numbers as the unique "building blocks" of all integers, making them fundamental to number theory. It guarantees that prime factorization is well-defined and unique, which underpins critical applications in cryptography, computer science, and advanced mathematics where we need to decompose numbers reliably.
27 tokens in · 115 tokens out · 3113 ms · $0.0018

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
3120.2 ms
Base LLM
3113.5 ms
Verifiers
0.1 ms
Gates
0.0 ms
Cost
$0.0018

Trail

08:48:46 received prompt_chars=98 verifiers=['scripture_anchors', 'theology_doctrine']
08:48:46 red_gate decision=pass reason=no disqualifying input patterns detected
08:48:50 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=27 tokens_out=115 latency_ms=3113.5 cost_usd=0.0018
08:48:50 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
08:48:50 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
08:48:50 floor_gate decision=pass reason=output above protective minimum; no mismatches
08:48:50 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
08:48:50 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
08:48:50 signed hash_algo=sha256 hash=sha256:91786491be5e9ac177428bb7df7d15d0eb0c24bd683a84b09ccd915b16419cb7

Content hash

sha256:91786491be5e9ac177428bb7df7d15d0eb0c24bd683a84b09ccd915b16419cb7

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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