The Mechanism · Gated-generation receipt

gen-define-modular-arithmetic-and-give-one-short-worked-example-d5a4

stable pending witness
2026-05-29 08:45:28 UTC · anthropic/claude-sonnet-4-5

Prompt

Define modular arithmetic and give one short worked example.
60 chars

Generation

# Modular Arithmetic **Modular arithmetic** is a system of arithmetic where numbers "wrap around" after reaching a certain value called the **modulus**. We say two integers *a* and *b* are congruent modulo *n* if they have the same remainder when divided by *n*. **Notation:** *a* ≡ *b* (mod *n*) This means *n* divides (*a* - *b*), or equivalently, *a* and *b* leave the same remainder when divided by *n*. ## Worked Example **Find 17 + 23 (mod 12)** Solution: - First, add normally: 17 + 23 = 40 - Now divide by the modulus 12: 40 ÷ 12 = 3 remainder **4** - Therefore: 17 + 23 ≡ **4** (mod 12) *Think of this as clock arithmetic: starting at 5 o'clock and moving forward 23 hours lands you at 4 o'clock.*
18 tokens in · 254 tokens out · 4668 ms · $0.0039

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
4674.5 ms
Base LLM
4667.6 ms
Verifiers
0.3 ms
Gates
0.0 ms
Cost
$0.0039

Trail

08:45:28 received prompt_chars=60 verifiers=['scripture_anchors', 'theology_doctrine']
08:45:28 red_gate decision=pass reason=no disqualifying input patterns detected
08:45:32 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=18 tokens_out=254 latency_ms=4667.6 cost_usd=0.0039
08:45:32 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
08:45:32 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
08:45:32 floor_gate decision=pass reason=output above protective minimum; no mismatches
08:45:32 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
08:45:32 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
08:45:32 signed hash_algo=sha256 hash=sha256:92b4043e4473150d2353470be0692ca6ed41e3010a3eaf41d2594fa9cc59ab4e

Content hash

sha256:92b4043e4473150d2353470be0692ca6ed41e3010a3eaf41d2594fa9cc59ab4e

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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