The Mechanism · Gated-generation receipt

gen-explain-quadratic-residues-as-you-would-to-a-capable-student-04e9

stable pending witness
2026-05-29 08:02:06 UTC · anthropic/claude-sonnet-4-5

Prompt

Explain quadratic residues as you would to a capable student, in 3-4 sentences.
79 chars

Generation

A quadratic residue modulo n is a number that can be expressed as a perfect square in modular arithmetic. For example, modulo 7, the number 2 is a quadratic residue because 3² ≡ 9 ≡ 2 (mod 7), but 3 is not a quadratic residue since no integer squared gives you 3 modulo 7. When working modulo a prime p, exactly half of the non-zero numbers are quadratic residues, and these residues have important applications in cryptography and number theory. You can think of them as the "perfect squares" in the world of modular arithmetic.
29 tokens in · 149 tokens out · 3871 ms · $0.0023

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
3877.6 ms
Base LLM
3870.9 ms
Verifiers
0.2 ms
Gates
0.0 ms
Cost
$0.0023

Trail

08:02:06 received prompt_chars=79 verifiers=['scripture_anchors', 'theology_doctrine']
08:02:06 red_gate decision=pass reason=no disqualifying input patterns detected
08:02:10 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=29 tokens_out=149 latency_ms=3870.9 cost_usd=0.0023
08:02:10 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
08:02:10 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
08:02:10 floor_gate decision=pass reason=output above protective minimum; no mismatches
08:02:10 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
08:02:10 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
08:02:10 signed hash_algo=sha256 hash=sha256:62e82f60e6123c26d5ab011c79224c95b3bdf96e5663b087660f3b8099e9efe3

Content hash

sha256:62e82f60e6123c26d5ab011c79224c95b3bdf96e5663b087660f3b8099e9efe3

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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