The Mechanism · Gated-generation receipt

gen-define-a-recurrence-relation-and-give-one-short-worked-examp-ce45

stable pending witness
2026-05-29 07:04:40 UTC · anthropic/claude-sonnet-4-5

Prompt

Define a recurrence relation and give one short worked example.
63 chars

Generation

# Recurrence Relation ## Definition A **recurrence relation** is an equation that defines a sequence where each term is expressed as a function of previous terms, along with initial conditions. General form: **aₙ = f(aₙ₋₁, aₙ₋₂, ..., aₙ₋ₖ)** for n ≥ k ## Worked Example **Find the first 5 terms of the sequence defined by:** - aₙ = 2aₙ₋₁ + 3 (recurrence relation) - a₁ = 1 (initial condition) **Solution:** - a₁ = 1 (given) - a₂ = 2a₁ + 3 = 2(1) + 3 = **5** - a₃ = 2a₂ + 3 = 2(5) + 3 = **13** - a₄ = 2a₃ + 3 = 2(13) + 3 = **29** - a₅ = 2a₄ + 3 = 2(29) + 3 = **61** The sequence is: **1, 5, 13, 29, 61, ...**
19 tokens in · 318 tokens out · 5005 ms · $0.0048

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
5012.0 ms
Base LLM
5005.1 ms
Verifiers
0.3 ms
Gates
0.0 ms
Cost
$0.0048

Trail

07:04:40 received prompt_chars=63 verifiers=['scripture_anchors', 'theology_doctrine']
07:04:40 red_gate decision=pass reason=no disqualifying input patterns detected
07:04:45 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=19 tokens_out=318 latency_ms=5005.1 cost_usd=0.0048
07:04:45 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
07:04:45 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
07:04:45 floor_gate decision=pass reason=output above protective minimum; no mismatches
07:04:45 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
07:04:45 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
07:04:45 signed hash_algo=sha256 hash=sha256:f6cc0fe2fd0044454bfef8db8ec0836b5a9c52ef5ccdf1fce8cf938fa31dcea2

Content hash

sha256:f6cc0fe2fd0044454bfef8db8ec0836b5a9c52ef5ccdf1fce8cf938fa31dcea2

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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