The Mechanism · Gated-generation receipt

gen-define-modular-inverses-and-give-one-short-worked-example-0117

stable pending witness
2026-05-29 06:32:36 UTC · anthropic/claude-sonnet-4-5

Prompt

Define modular inverses and give one short worked example.
58 chars

Generation

# Modular Inverses ## Definition The **modular inverse** of an integer *a* modulo *m* is an integer *x* such that: **ax ≡ 1 (mod m)** We say *x* is the multiplicative inverse of *a* modulo *m*, often written as *a*⁻¹ (mod m). **Important:** The modular inverse exists if and only if gcd(*a*, *m*) = 1 (i.e., *a* and *m* are coprime). ## Worked Example **Find the modular inverse of 3 modulo 7.** We need to find *x* such that 3*x* ≡ 1 (mod 7). Testing values: - 3 × 1 = 3 ≡ 3 (mod 7) ✗ - 3 × 2 = 6 ≡ 6 (mod 7) ✗ - 3 × 3 = 9 ≡ 2 (mod 7) ✗ - 3 × 4 = 12 ≡ 5 (mod 7) ✗ - 3 × 5 = 15 ≡ 1 (mod 7) ✓ **Answer:** The modular inverse of 3 modulo 7 is **5**. We can verify: 3 × 5 = 15 = 2 × 7 + 1 ≡ 1 (mod 7) ✓
19 tokens in · 388 tokens out · 6431 ms · $0.0059

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
6438.0 ms
Base LLM
6431.3 ms
Verifiers
0.2 ms
Gates
0.1 ms
Cost
$0.0059

Trail

06:32:36 received prompt_chars=58 verifiers=['scripture_anchors', 'theology_doctrine']
06:32:36 red_gate decision=pass reason=no disqualifying input patterns detected
06:32:42 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=19 tokens_out=388 latency_ms=6431.3 cost_usd=0.0059
06:32:42 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
06:32:42 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
06:32:42 floor_gate decision=pass reason=output above protective minimum; no mismatches
06:32:42 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
06:32:42 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
06:32:42 signed hash_algo=sha256 hash=sha256:dc83337d6f87a153cf6b1cbc2b0ad09d1c1bc78dfd6fadef73e18201ef21ed65

Content hash

sha256:dc83337d6f87a153cf6b1cbc2b0ad09d1c1bc78dfd6fadef73e18201ef21ed65

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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