Sequence & Series Explorer mechanism

Model scope

The listed term formulas are standard, but the instrument specification is incomplete rather than fully verifiable because the partial-sum formulas, recursive-rule contract, indexing constraints, detection algorithm, and honest limitations are not defined.

• The arithmetic-term formula aₙ=a₁+(n−1)d and geometric-term formula aₙ=a₁rⁿ⁻¹ are standard, assuming integer n≥1.
• “Sₙ formulas” is underspecified: the arithmetic formula Sₙ=n(2a₁+(n−1)d)/2 and geometric formula Sₙ=a₁(1−rⁿ)/(1−r) for r≠1 should be stated, along with the r=1 case Sₙ=na₁.
• The specification does not define how recursive rules are represented or evaluated, nor how a closed form is produced for a general rule.
• Arithmetic/geometric detection from finite first terms is underdetermined unless exact matching, tolerance, minimum sample size, and handling of ambiguous sequences are specified.
• The honest_label field is empty; the instrument should disclose that finite-term detection cannot prove a sequence is globally arithmetic or geometric and that floating-point inputs may affect detection.

Units and conventions

• n is a positive integer index; zero or negative indices require an explicit convention.
• a₁, d, and every sequence term use the same units; therefore Sₙ has those same units.
• r is dimensionless, and a₁rⁿ⁻¹ has the same units as a₁.
• For the geometric sum formula with division by 1−r, r≠1; when r=1, use Sₙ=na₁.
• The formulas assume ordinary real or complex arithmetic as appropriate; no modular or integer-overflow semantics are specified.

Inspectable vector

One of 4 source-bound review vectors:

{
  "inputs": {
    "type": "arithmetic",
    "a1": 7,
    "d": 3,
    "n": 5
  },
  "expected_outputs": {
    "term_list": [
      7,
      10,
      13,
      16,
      19
    ],
    "nth_term": 19,
    "partial_sum": 65,
    "closed_form": "aₙ=7+3(n−1)",
    "detected": "arithmetic"
  },
  "working": "Terms: a₁=7; a₂=7+3=10; a₃=10+3=13; a₄=13+3=16; a₅=16+3=19. nth term: a₅=7+(5−1)×3=7+12=19. Sum: S₅=5/2×(2×7+(5−1)×3)=5/2×(14+12)=5/2×26=65."
}

Failure boundary

BREAK_CLAIMED · SELF_FLAGGED

A naive geometric-sum implementation breaks at r=1 because the denominator r-1 becomes zero and produces 0/0. The instrument must switch to the constant-sequence sum branch. Negative signed differences also expose a convention risk if direction is not preserved.

Other records

Sources and evidence

These pointers bind the page to evidence. They do not grant authority or collapse verification into deployment.

• VERIFIED_LEDGER.json
• evidence-tranche4/tranche4-results.json
• BOUNDARY_HAZARD_CATALOG.json