The Mechanism · Gated-generation receipt

gen-in-3-4-sentences-explain-a-bijective-proof-state-any-key-for-be13

stable pending witness
2026-05-29 04:29:57 UTC · anthropic/claude-sonnet-4-5

Prompt

In 3-4 sentences, explain a bijective proof. State any key formula.
67 chars

Generation

A bijective proof demonstrates that two sets have the same size by constructing a bijection (one-to-one correspondence) between them. Instead of computing cardinalities algebraically, you show the equality |A| = |B| by explicitly pairing each element of set A with exactly one element of set B and vice versa. This technique is particularly powerful for proving combinatorial identities—for example, showing that C(n,k) = C(n,n-k) by constructing a bijection between k-element subsets and their complements. The key principle is: **|A| = |B| if and only if there exists a bijection f: A → B**.
26 tokens in · 150 tokens out · 3515 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
3522.2 ms
Base LLM
3515.0 ms
Verifiers
0.2 ms
Gates
0.0 ms
Cost
$0.0023

Trail

04:29:57 received prompt_chars=67 verifiers=['scripture_anchors', 'theology_doctrine']
04:29:57 red_gate decision=pass reason=no disqualifying input patterns detected
04:30:01 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=26 tokens_out=150 latency_ms=3515.0 cost_usd=0.0023
04:30:01 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
04:30:01 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
04:30:01 floor_gate decision=pass reason=output above protective minimum; no mismatches
04:30:01 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
04:30:01 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
04:30:01 signed hash_algo=sha256 hash=sha256:d1dd79da2bf351d9d27e1bb6597413ee29f2541282f2f3077c63f8fd89b77910

Content hash

sha256:d1dd79da2bf351d9d27e1bb6597413ee29f2541282f2f3077c63f8fd89b77910

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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