The Mechanism · Gated-generation receipt

gen-in-3-4-sentences-explain-quadratic-residues-state-any-key-fo-c67a

stable pending witness
2026-05-29 03:52:07 UTC · anthropic/claude-sonnet-4-5

Prompt

In 3-4 sentences, explain quadratic residues. State any key formula.
68 chars

Generation

A quadratic residue modulo n is an integer a such that the congruence x² ≡ a (mod n) has a solution. In other words, a is a quadratic residue if there exists some integer x whose square leaves remainder a when divided by n. For an odd prime p and integer a not divisible by p, the **Legendre symbol** determines whether a is a quadratic residue: **(a/p) = a^((p-1)/2) (mod p)** where (a/p) = 1 if a is a quadratic residue mod p, and (a/p) = -1 if it is not.
26 tokens in · 153 tokens out · 3860 ms · $0.0024

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
3867.5 ms
Base LLM
3860.4 ms
Verifiers
0.2 ms
Gates
0.0 ms
Cost
$0.0024

Trail

03:52:07 received prompt_chars=68 verifiers=['scripture_anchors', 'theology_doctrine']
03:52:07 red_gate decision=pass reason=no disqualifying input patterns detected
03:52:11 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=26 tokens_out=153 latency_ms=3860.4 cost_usd=0.0024
03:52:11 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
03:52:11 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
03:52:11 floor_gate decision=pass reason=output above protective minimum; no mismatches
03:52:11 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
03:52:11 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
03:52:11 signed hash_algo=sha256 hash=sha256:b4133e5d826ab4ac9c8eac3387a94114a21a03d9639792ed8f1161e759067515

Content hash

sha256:b4133e5d826ab4ac9c8eac3387a94114a21a03d9639792ed8f1161e759067515

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.

Bring another teaching · Run a discernment