The Mechanism · Gated-generation receipt

gen-what-is-the-significance-of-the-fundamental-theorem-of-arith-9ea7

stable pending witness
2026-05-29 08:04:46 UTC · anthropic/claude-sonnet-4-5

Prompt

What is the significance of the fundamental theorem of arithmetic? Answer concisely in 2-3 sentences.
101 chars

Generation

The Fundamental Theorem of Arithmetic states that every integer greater than 1 can be expressed as a product of prime numbers in exactly one way (excluding order). This theorem is significant because it establishes prime numbers as the unique "building blocks" of all integers, making primes foundational to number theory. It underlies many mathematical concepts including divisibility, greatest common divisors, and cryptographic systems like RSA encryption.
30 tokens in · 91 tokens out · 4141 ms · $0.0015

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
4147.7 ms
Base LLM
4140.5 ms
Verifiers
0.1 ms
Gates
0.0 ms
Cost
$0.0015

Trail

08:04:46 received prompt_chars=101 verifiers=['scripture_anchors', 'theology_doctrine']
08:04:46 red_gate decision=pass reason=no disqualifying input patterns detected
08:04:51 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=30 tokens_out=91 latency_ms=4140.5 cost_usd=0.0015
08:04:51 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
08:04:51 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
08:04:51 floor_gate decision=pass reason=output above protective minimum; no mismatches
08:04:51 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
08:04:51 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
08:04:51 signed hash_algo=sha256 hash=sha256:cfb851a8d6f1b1a06aae86571a6c3d1a8a22655c149eba2e124383437b40eadb

Content hash

sha256:cfb851a8d6f1b1a06aae86571a6c3d1a8a22655c149eba2e124383437b40eadb

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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