The Mechanism · Gated-generation receipt

gen-in-3-4-sentences-explain-modular-inverses-state-any-key-form-9ed2

stable pending witness
2026-05-29 03:01:34 UTC · anthropic/claude-sonnet-4-5

Prompt

In 3-4 sentences, explain modular inverses. State any key formula.
66 chars

Generation

A modular inverse of an integer *a* modulo *m* is an integer *x* such that *ax* ≡ 1 (mod *m*). In other words, when *a* and *x* are multiplied together, the result leaves a remainder of 1 when divided by *m*. A modular inverse exists if and only if *a* and *m* are coprime (gcd(*a*, *m*) = 1). The key formula is: **ax ≡ 1 (mod m)**, and the inverse can be found using the Extended Euclidean Algorithm.
26 tokens in · 142 tokens out · 3480 ms · $0.0022

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
3490.0 ms
Base LLM
3480.2 ms
Verifiers
0.1 ms
Gates
0.0 ms
Cost
$0.0022

Trail

03:01:34 received prompt_chars=66 verifiers=['scripture_anchors', 'theology_doctrine']
03:01:34 red_gate decision=pass reason=no disqualifying input patterns detected
03:01:38 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=26 tokens_out=142 latency_ms=3480.2 cost_usd=0.0022
03:01:38 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
03:01:38 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
03:01:38 floor_gate decision=pass reason=output above protective minimum; no mismatches
03:01:38 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
03:01:38 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
03:01:38 signed hash_algo=sha256 hash=sha256:4bb3bcd778b24a092ed47e4eed4c8930cea9cbf1c9c2478c7063debbb6feb93e

Content hash

sha256:4bb3bcd778b24a092ed47e4eed4c8930cea9cbf1c9c2478c7063debbb6feb93e

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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