The Mechanism · Gated-generation receipt

gen-state-mathematical-induction-precisely-and-explain-why-it-ma-edcf

stable pending witness
2026-05-29 05:03:36 UTC · anthropic/claude-sonnet-4-5

Prompt

State mathematical induction precisely and explain why it matters in 2-3 sentences.
83 chars

Generation

## Mathematical Induction **Principle:** For a statement P(n) about natural numbers, if (1) P(1) is true (base case), and (2) P(k) true implies P(k+1) true for any k ≥ 1 (inductive step), then P(n) is true for all natural numbers n ≥ 1. **Why it matters:** Induction provides a rigorous method to prove infinitely many statements with just two finite steps, making it essential for validating formulas, algorithms, and properties across all natural numbers. It transforms an impossible task—checking infinitely many cases individually—into a manageable logical framework that underpins much of discrete mathematics and computer science.
25 tokens in · 164 tokens out · 5285 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
5292.6 ms
Base LLM
5285.2 ms
Verifiers
0.2 ms
Gates
0.0 ms
Cost
$0.0025

Trail

05:03:36 received prompt_chars=83 verifiers=['scripture_anchors', 'theology_doctrine']
05:03:36 red_gate decision=pass reason=no disqualifying input patterns detected
05:03:42 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=25 tokens_out=164 latency_ms=5285.2 cost_usd=0.0025
05:03:42 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
05:03:42 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
05:03:42 floor_gate decision=pass reason=output above protective minimum; no mismatches
05:03:42 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
05:03:42 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
05:03:42 signed hash_algo=sha256 hash=sha256:962b5e07c7ec3d0563b756bdb7d568877cf78d9fc064a91614231e7f2259c775

Content hash

sha256:962b5e07c7ec3d0563b756bdb7d568877cf78d9fc064a91614231e7f2259c775

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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