Finding 001: K2 maximum in the reaching period

Who this is for: Engineers investigating K2 against T4127 (Payroll Deductions Formulas) and PDOC.

When you finish: You can reproduce M-002 and state what is known versus open.

Record: M-002 in CONFORMANCE.md. Vector: on-biweekly-midyear-k2-max PDOC version: 2026-06-11 (retrieved 2026-09-12T03:28:07Z)

T4127 wording

T4127 (Payroll Deductions Formulas) Chapter 3, factor K2 (glossary note): use the maximum base CPP contribution “in that pay period.”

The Chapter 4 Step 2 formula is min(P × C × (0.0495/0.0595), base_max × PM/12). The glossary note is the extra instruction: when year-to-date first reaches the annual CPP maximum, credit base_max in that period, not the still-short annualized figure.

PDOC’s observed behaviour

PDOC’s federal CPP credit base is max(P × C × ratio, D × ratio) capped at base_max × PM/12. It does not force base_max in the reaching period.

On the measured vector, P × C × ratio is well below base_max (this period’s C is only the remaining room, $30.45), and D × ratio is also below base_max. PDOC takes the greater of those two: D × ratio: and stops.

Inversion

Solve T1 from the PDOC period figure, subtract the uncontested EI half, divide by the lowest rate, compare the resulting CPP credit base against each candidate.

1. PDOC federal tax 243.95. Reconstruct annual T1 as 243.95 × 26 = 6342.70 (period rounding; the engine’s unrounded T1 under the matching rule is 6342.78, and round_tax(6342.78 / 26) is still 243.95).
2. Every other input to T3 = (R × A) − K − K1 − K2 − K4 already matched, so the residual is K2.
3. The EI half is uncontested: 0.14 × min(P × EI, 1123.07) with P × 40.75 = 1059.50. Subtract it and divide by 0.14.
4. The resulting CPP credit base is 3494.12, which is D × (0.0495/0.0595) with D = 4200.00 (year-to-date *before* this period).

Candidates from the same vector:

The form that produces that number, and also covers periods after the max is reached (D already at total_max → D × ratio = base_max), is:

min(base_max × PM/12, max(P × C × ratio, D × ratio))

PDOC does not do what the document’s reaching-period sentence says. The difference is 3519.45 − 3494.12 = 25.33 on the CPP credit base, which is $3.54 of annual K2 and thirteen cents of federal tax in that period.

Vector, both outputs, delta

Input (on-biweekly-midyear-k2-max):

{
  "as_of": "2026-01-15",
  "province": "ON",
  "pay_period": 26,
  "gross_pay": "2500.00",
  "cpp_months": 12,
  "federal_claim_code": 1,
  "provincial_claim_code": 1,
  "ytd_cpp": "4200.00",
  "ytd_ei": "1000.00",
  "ytd_pensionable_earnings": "71000.00",
  "ytd_insurable_earnings": "50000.00",
  "pay_periods_elapsed": 20
}

ytd_pensionable_earnings is 71000.00 because PDOC rejects CPP YTD 4200.00 below its pensionable-earnings consistency floor.

PDOC withholds thirteen cents more federal tax, and five cents more provincial, in the reaching period than T4127’s glossary note. After this period both paths credit base_max; the divergence is this one pay.

YearToDate on the same input is a third figure (federal 243.47, K2 649.95). Test 73 asserts all three K2 values differ.

Engine default and the alternative

Default k2_method is pdoc_observed. Customers reconcile against PDOC, so the oracle wins.

To select the document’s reading:

"k2_method": "t4127_literal"

year_to_date remains selectable. annualized is accepted as an alias of pdoc_observed.

Reproduce against PDOC (five minutes)

1. Open https://apps.cra-arc.gc.ca/ebci/rhpd/beta/
2. Province of employment: Ontario. Pay date: 2026-01-15. Pay period: biweekly (26).
3. Salary or wages: 2500.00.
4. Enter Claim codes (TD1 personal amount codes) 1 (federal) and 1 (provincial). Do not type a fixed TD1 dollar amount: a locked $16,452 skips BPAF (Basic Personal Amount formula) phaseout and is a different input.
5. Year-to-date: CPP 4200.00, EI 1000.00, pensionable earnings 71000.00, insurable earnings 50000.00, pay periods elapsed 20.
6. Result screen, PDOC 2026-06-11: Federal tax deduction 243.95, Provincial 132.29, CPP 30.45, EI 40.75, Total 447.44, Net 2052.56.

Independent check without the engine: 4200 × 0.0495 / 0.0595 = 3494.1176… (PDOC’s CPP credit base) vs 3519.45 (T4127’s base_max in this period). Those two bases, plus the uncontested EI half 1059.50, times 0.14, differ by $3.54 of annual K2: thirteen cents once divided by 26 and rounded.

Last reviewed: 2026-09-20
Engine: 0.1.0