Let me start by listing the documents to see what inputs we have.

The payroll record: gross_pay 8370.45, payroll_month 2026-05, allowed_deduction 370.62.

Now I need the reference row for 2026-05. The month is in the second half (May... wait, first-half vs second-half — need to check the boundaries). Let me look up both halves to determine which covers 2026-05. Actually, "first-half" and "second-half" of the year presumably: first half = Jan–Jun, second half = Jul–Dec. But I should look up to confirm. Let me call lookup_reference for both to be safe — they're independent calls.

The first-half covers months 2026-01 through 2026-06 (the "months" field shows the range endpoints: ["2026-01", "2026-06"]). The payroll month is 2026-05, which falls within 2026-01 to 2026-06, so the first-half row applies:

- rate: 0.04
- cap: 5000.00
- deduction: 400.00
- free: 1000.00
- upper: 3600.00
- middle_rate: 0.10
- upper_rate: 0.24

Now compute:

R3: employee contribution = 0.04 * min(8370.45, 5000.00) = 0.04 * 5000.00 = 200.00

R4: allowed_deduction = 370.62. Check: must not exceed deduction cap 400.00, nor gross pay minus rounded contribution = 8370.45 - 200.00 = 8170.45. 370.62 ≤ both, so valid.

Taxable pay = 8370.45 - 200.00 - 370.62 = 7799.83.

R5: income tax:
- Portion up to free (1000.00): untaxed.
- Portion above 1000 up to upper (3600): 3600 - 1000 = 2600 * 0.10 = 260.00
- Portion above 3600: 7799.83 - 3600 = 4199.83 * 0.24 = 1007.9592

Total tax = 260.00 + 1007.9592 = 1267.9592 → round half up to 1267.96.

R6: net pay = 8370.45 - 200.00 - 370.62 - 1267.96 = 6531.87.

Let me verify with calculate.

All checks pass:
- gross_pay present, payroll_month present, 2026-05 covered by first-half row.
- No contract_gross_pay or amendment conflicts.
- allowed_deduction 370.62 ≤ cap 400.00 and ≤ gross - contribution (8170.45). Valid.

Results:
- employee_contribution = 200.00
- allowed_deduction = 370.62
- taxable_pay = 7799.83
- income_tax = 1267.96
- net_pay = 6531.87

Citations: R1 (row selection), R2 (rounding), R3 (contribution), R4 (deduction/taxable), R5 (tax brackets), R6 (net pay). No amendment needed, so no R9.

Let me start by listing the documents to see what inputs we have.

The payroll record:
- gross_pay: 8370.45
- payroll_month: 2026-05
- allowed_deduction: 370.62

Payroll month is 2026-05, which is covered by 2026 reference rows. May is in the second half (months 07-12 are second half? Let me check). Actually first-half and second-half — need to check what months each covers. Let me look up both halves to determine which applies to 2026-05. May (05) — I need to check the reference tables. Let me call lookup_reference for both halves to see.

The reference rows:
- first-half: months ["2026-01", "2026-06"], rate 0.04, cap 5000.00, deduction 400.00, free 1000.00, upper 3600.00, middle_rate 0.10, upper_rate 0.24
- second-half: months ["2026-07", "2026-12"], rate 0.04, cap 4000.00, deduction 400.00, free 1000.00, upper 3600.00, middle_rate 0.12, upper_rate 0.22

Payroll month is 2026-05. Neither row covers 2026-05! First-half covers 2026-01 and 2026-06; second-half covers 2026-07 and 2026-12. So 2026-05 has no applicable reference row.

Per R7 item 3: A month without a reference row → decision "clarify", reason_code "NO_APPLICABLE_REFERENCE", citing R1 and R7. Do not extrapolate an old row.

Wait — but R4 says: "When the month or applicable row is absent, accept structurally valid deductions up to the largest reference deduction cap for task validation, but do not calculate a payslip or assume a reference row." That's about validation of the deduction input, not about calculating. Since no applicable row exists for 2026-05, we must clarify with NO_APPLICABLE_REFERENCE.

The allowed_deduction of 370.62 is structurally valid (≤ 400.00, the largest reference deduction cap), but we still don't calculate.

Answer: decision "clarify", reason_code "NO_APPLICABLE_REFERENCE", clauses ["R1", "R7"], with a short explanation naming the unresolved blocker and asking for resolution.

Let me start by listing the documents to get the case inputs.

The payroll record:
- gross_pay: 8370.45
- payroll_month: 2026-05
- allowed_deduction: 370.62

2026-05 is covered (only 2026 is covered). May 2026 — need to determine if it's first-half or second-half. Let me look up the reference. Typically first-half is Jan–Jun, second-half is Jul–Dec. Let me check lookup_reference.

The first-half covers months 2026-01 through 2026-06 (the "months" field lists ["2026-01", "2026-06"], presumably meaning the range from 2026-01 to 2026-06). 2026-05 falls within this range.

Row values:
- rate: 0.04
- cap: 5000.00
- deduction cap: 400.00
- free: 1000.00
- upper: 3600.00
- middle_rate: 0.10
- upper_rate: 0.24

Now compute:
- Employee contribution = round(0.04 * min(5000.00, 8370.45), 2) = round(0.04 * 5000.00, 2) = 200.00
- Allowed deduction = 370.62. Check: must not exceed deduction cap 400.00 ✓, and must not exceed gross pay minus rounded contribution = 8370.45 - 200.00 = 8170.45 ✓. So 370.62 is valid.
- Taxable pay = 8370.45 - 200.00 - 370.62 = 7799.83
- Income tax: 
  - Free: 1000.00 untaxed
  - Portion above 1000 up to 3600: 3600 - 1000 = 2600 at 0.10 = 260.00
  - Portion above 3600: 7799.83 - 3600 = 4199.83 at 0.24 = 1007.9582 → but do not round intermediate products; sum then round.
  - Total = 260.00 + 1007.9582 = 1267.9582 → round to 1267.96
- Net pay = 8370.45 - 200.00 - 370.62 - 1267.96 = 6531.87

Let me verify with calculate.

All checks pass:
- Month 2026-05 is within first-half range (2026-01 to 2026-06), row applies.
- Employee contribution: 0.04 × min(5000.00, 8370.45) = 200.00 (R3)
- Allowed deduction 370.62 ≤ 400.00 cap and ≤ 8170.45 available (R4) ✓
- Taxable pay = 7799.83 (R4)
- Income tax: 2600 × 0.10 + 4199.83 × 0.24 = 260.00 + 1007.9582 = 1267.9582 → 1267.96 (R5, R2)
- Net pay = 6531.87 (R6)

Return JSON with decision: calculate, currency LCR, all fields as two-decimal strings, clauses R1–R6. No amendment present, so no R9.