The Mechanism · Gated-generation receipt

gen-in-3-4-sentences-explain-euler-s-totient-function-state-any-01df

stable pending witness
2026-05-29 07:54:00 UTC · anthropic/claude-sonnet-4-5

Prompt

In 3-4 sentences, explain Euler's totient function. State any key formula.
74 chars

Generation

Euler's totient function, denoted φ(n), counts the number of positive integers up to n that are relatively prime to n (i.e., share no common factors with n except 1). For a prime number p, φ(p) = p - 1 since all numbers less than p are relatively prime to it. The key formula for computing φ(n) when n has prime factorization n = p₁^k₁ · p₂^k₂ · ... · pₘ^kₘ is: **φ(n) = n · (1 - 1/p₁) · (1 - 1/p₂) · ... · (1 - 1/pₘ)** This function is fundamental in number theory and cryptography, particularly in the RSA encryption algorithm.
28 tokens in · 195 tokens out · 3462 ms · $0.003

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
3468.7 ms
Base LLM
3461.8 ms
Verifiers
0.2 ms
Gates
0.0 ms
Cost
$0.003

Trail

07:54:00 received prompt_chars=74 verifiers=['scripture_anchors', 'theology_doctrine']
07:54:00 red_gate decision=pass reason=no disqualifying input patterns detected
07:54:04 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=28 tokens_out=195 latency_ms=3461.8 cost_usd=0.003
07:54:04 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
07:54:04 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
07:54:04 floor_gate decision=pass reason=output above protective minimum; no mismatches
07:54:04 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
07:54:04 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
07:54:04 signed hash_algo=sha256 hash=sha256:65b633f76a2147fb6dbc05d94188621f470e8998a9324fb1e66dc2f948a707bd

Content hash

sha256:65b633f76a2147fb6dbc05d94188621f470e8998a9324fb1e66dc2f948a707bd

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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