The Mechanism · Gated-generation receipt

gen-in-3-4-sentences-explain-congruence-relations-state-any-key-f43a

stable pending witness
2026-05-29 08:00:59 UTC · anthropic/claude-sonnet-4-5

Prompt

In 3-4 sentences, explain congruence relations. State any key formula.
70 chars

Generation

# Congruence Relations A congruence relation states that two integers **a** and **b** are congruent modulo **n** (written **a ≡ b (mod n)**) if they have the same remainder when divided by **n**, or equivalently, if **n** divides their difference. **Key formula:** a ≡ b (mod n) if and only if **n | (a - b)**, meaning there exists an integer k such that **a - b = kn**. Congruence relations are equivalence relations (reflexive, symmetric, and transitive) that partition the integers into equivalence classes, and they preserve addition, subtraction, and multiplication operations.
26 tokens in · 159 tokens out · 4009 ms · $0.0025

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
4016.1 ms
Base LLM
4009.2 ms
Verifiers
0.2 ms
Gates
0.0 ms
Cost
$0.0025

Trail

08:00:59 received prompt_chars=70 verifiers=['scripture_anchors', 'theology_doctrine']
08:00:59 red_gate decision=pass reason=no disqualifying input patterns detected
08:01:03 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=26 tokens_out=159 latency_ms=4009.2 cost_usd=0.0025
08:01:03 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
08:01:03 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
08:01:03 floor_gate decision=pass reason=output above protective minimum; no mismatches
08:01:03 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
08:01:03 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
08:01:03 signed hash_algo=sha256 hash=sha256:2a4888f1b0d782ea633b2a7d9348bcd3deb8dd0360da38531cbaaa1ff4dcc623

Content hash

sha256:2a4888f1b0d782ea633b2a7d9348bcd3deb8dd0360da38531cbaaa1ff4dcc623

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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