The Mechanism · Gated-generation receipt

gen-in-3-4-sentences-explain-counting-lattice-paths-state-any-ke-ed49

stable pending witness
2026-05-29 05:31:23 UTC · anthropic/claude-sonnet-4-5

Prompt

In 3-4 sentences, explain counting lattice paths. State any key formula.
72 chars

Generation

# Counting Lattice Paths Counting lattice paths involves finding the number of ways to move from one point to another on a grid using only specified directions (typically right and up). For a path from $(0,0)$ to $(m,n)$ using $m$ steps right and $n$ steps up, the total number of paths is given by the **binomial coefficient**: $$\binom{m+n}{m} = \binom{m+n}{n} = \frac{(m+n)!}{m! \cdot n!}$$ This formula counts the ways to arrange $m$ right steps and $n$ up steps in a sequence of $m+n$ total steps.
26 tokens in · 163 tokens out · 2845 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
2852.2 ms
Base LLM
2844.8 ms
Verifiers
0.3 ms
Gates
0.0 ms
Cost
$0.0025

Trail

05:31:23 received prompt_chars=72 verifiers=['scripture_anchors', 'theology_doctrine']
05:31:23 red_gate decision=pass reason=no disqualifying input patterns detected
05:31:25 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=26 tokens_out=163 latency_ms=2844.8 cost_usd=0.0025
05:31:25 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
05:31:25 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
05:31:25 floor_gate decision=pass reason=output above protective minimum; no mismatches
05:31:25 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
05:31:25 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
05:31:25 signed hash_algo=sha256 hash=sha256:42e5e61306869d5daa0e06e963fcd07913f70d73fae6214d17e973d6634a4f6d

Content hash

sha256:42e5e61306869d5daa0e06e963fcd07913f70d73fae6214d17e973d6634a4f6d

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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