Choose a debt repayment order that fits your cash flow

SettleADebt R032 PASS-02

Bounded plain-text mathematical addendum to PASS-01, section 6

The supplied six rows and all original numerical conclusions reconcile. No numerical correction is needed; the values remain unchanged. This addendum repairs the displayed formulas, makes the payment sequence explicit, and maps the repair to original C056-C060 and M001-M003.

The only input is the prior-report context supplied in your message. This is not fresh primary-source access or independent review. All original source, claim, coverage, nonbinding-education, visual, edition, legal, contractual, behavioral, internal-tooling and parent limitations remain in force. No component, parent, operational or publication acceptance is asserted.

Every formula and reconciliation below is in a literal ASCII text code block. Tables contain descriptive findings and values, not math-renderer expressions.

1. Supplied-context source/read register

Reference convention: S001 means R032-P2-S001. F001-F004 label formula blocks; V001-V006 label checks of the six original rows. M001-M003 retain their original PASS-01 meanings.

2. Complete assumptions retained

The following restates the supplied assumptions without changing the scenario.

Plain text

ASSUMPTIONS RETAINED FROM ORIGINAL C056

1. There are only two abstract buckets, A and B.
   No person, entity, creditor or account is represented.
   Monetary amounts are measured in the abstract unit u.

2. Initial balances:
   B[A,0] = 20.00 u
   B[B,0] = 40.00 u
   Initial total = 20.00 u + 40.00 u = 60.00 u

3. Constant decimal rates for one model period:
   r[A] = 0.01
   r[B] = 0.03
   These are not annual rates and are not an APR.

4. Available payment budget:
   P[t] = 30.00 u for each modeled period until both buckets are paid.
   This amount is stipulated available only for these two buckets.
   No actual affordability finding is made.

5. Housing, living costs, reserves, legal consequences and all other
   obligations are outside the calculation. Their adequacy is not proved.

6. At the start of each period:
   raw interest = opening balance * the bucket's per-period rate.
   Round EACH bucket's interest separately to the nearest 0.01 u,
   with exact halves rounded upward.
   Add that rounded interest before making payments.
   Carry the resulting rounded closing balances into the next period.

7. Each artificial model minimum is:
   min(5.00 u, the bucket's entire balance after that period's interest).

8. Satisfy BOTH model minimums before selecting an extra-payment target.
   Do not recalculate those minimums during the extra-allocation sequence.

9. Smallest-first:
   Order the positive AFTER-MINIMUM balances from smallest to largest.
   Highest-rate-first:
   Target B while its after-minimum balance remains positive, then A.
   No unresolved positive-balance tie occurs in the supplied paths.

10. Cap every payment at the remaining amount owed in the model.
    Immediately reallocate unused extra to the next target within the
    SAME period. Do not postpone that reallocation to another period.

11. A paid-off bucket accrues no later interest.

12. Retain the 30.00 u total available budget after a bucket closes,
    including the payment capacity freed by that closure.

13. In the final period, pay only what is needed to reach zero.
    Record unused budget without assigning it another use.

14. There are no fees, new borrowing, rate changes, promotions, penalties,
    payment delays, grace periods, balance segments, tax effects or
    external cash flows.

15. These assumptions are exhaustive for this addendum.
    They are mathematical stipulations, not agency instructions or
    any real creditor's terms.

3. Repaired formulas and payment sequence

F001 — Variables, units, indexing and rounding

Plain text

F001: DEFINITIONS

Bucket index:
  j is A or B.

Period index:
  t is 1, 2 or 3 for each supplied path.

All monetary variables below contain numerical amounts measured in u.
Rates are decimal rates for one model period.

B[j,0]       = initial balance of bucket j
B[j,t]       = closing balance of bucket j after period t
O[j,t]       = opening balance of bucket j in period t
I_raw[j,t]   = unrounded interest for bucket j in period t
I[j,t]       = separately rounded interest for bucket j in period t
D[j,t]       = balance after interest, before any payment
m[j,t]       = artificial model minimum for bucket j in period t
R0[j,t]      = balance after the minimum, before extra allocation
P[t]        = total available payment budget for period t
Q0[t]       = extra budget available after BOTH minimums
E[j,t]      = extra ACTUALLY allocated to bucket j in period t
R[j,t]      = remaining balance during extra allocation
Q[t]        = remaining extra budget during extra allocation
payment[j,t] = minimum plus actual extra paid to bucket j
total_paid[t] = total paid to both buckets during period t
unused[t]   = budget not used during period t

Opening-balance indexing:
  O[j,t] = B[j,t-1]

Rounding definition for nonnegative monetary amounts:
  round_001(x) = 0.01 * floor(x / 0.01 + 0.5)

Here:
  x and the returned amount are measured in u.
  floor(y) means the greatest integer not exceeding y.
  The expression uses exact decimal quantities.
  It expresses the original nearest-0.01-u, halves-up rule.

Apply round_001() separately to each bucket's interest.
Do not round the two buckets' combined interest instead.

F002 — Interest, after-interest balance, minimums and extra available

Plain text

F002: PERIOD RECURRENCE BEFORE EXTRA ALLOCATION

For EACH bucket j:

  I_raw[j,t] = O[j,t] * r[j]                  # u
  I[j,t] = round_001(I_raw[j,t])              # u
  D[j,t] = O[j,t] + I[j,t]                    # u

  m[j,t] = min(5.00, D[j,t])                  # u
  R0[j,t] = D[j,t] - m[j,t]                   # u

After computing and satisfying BOTH minimums:

  minimum_total[t] = sum(m[A,t], m[B,t])      # u
  Q0[t] = P[t] - minimum_total[t]             # u

Budget condition for this recurrence:

  minimum_total[t] <= P[t]

That condition holds in all six supplied rows.
No rule for skipping minimums or exceeding the budget is introduced.

Initialize extra allocation:

  E[A,t] = 0.00 u
  E[B,t] = 0.00 u
  R[A,t] = R0[A,t]
  R[B,t] = R0[B,t]
  Q[t] = Q0[t]

F003 — Target timing, cap and same-period reallocation

Plain text

F003: EXTRA-PAYMENT ORDER AND ALLOCATION

Choose the order ONLY AFTER BOTH minimums have been satisfied.

Smallest-first order:
  List the buckets with positive R0[j,t].
  Sort that list by R0[j,t], smallest first.
  No unresolved tie occurs in the supplied paths.

Highest-rate-first order:
  Use B, then A.
  Remove any bucket whose R0[j,t] is already zero.

For each bucket j in that period's order:

  If Q[t] == 0.00 u:
      stop this period's extra-allocation loop.

  step_extra = min(Q[t], R[j,t])             # u
  E[j,t] = E[j,t] + step_extra                # u
  R[j,t] = R[j,t] - step_extra                # u
  Q[t] = Q[t] - step_extra                    # u

  If the target is paid off and Q[t] remains positive:
      proceed immediately to the next target in the SAME period.

The other bucket's balance is unchanged by a payment to bucket j.

If both after-minimum balances are zero:
  the target list is empty;
  E[A,t] = 0.00 u;
  E[B,t] = 0.00 u;
  Q[t] remains unused.

Available extra is not necessarily actual extra paid.
No payment is made merely to exhaust the budget.

F004 — Actual payments, closing balances and unused budget

Plain text

F004: PERIOD-END IDENTITIES

For each bucket j:

  payment[j,t] = m[j,t] + E[j,t]              # u

  B[j,t] = D[j,t] - payment[j,t]              # u
         = R0[j,t] - E[j,t]                  # u
         = R[j,t]                            # u

For the period:

  actual_extra_total[t] = sum(E[A,t], E[B,t]) # u

  total_paid[t] = sum(payment[A,t],
                      payment[B,t])         # u

  unused[t] = P[t] - total_paid[t]            # u
            = Q0[t] - actual_extra_total[t]  # u
            = Q[t]                           # u

Payment caps and budget bounds:

  0.00 u <= payment[j,t] <= D[j,t]
  0.00 u <= total_paid[t] <= P[t]

Carry-forward:

  O[j,t+1] = B[j,t]

Stop after both closing balances are zero.
No additional period or use of unused budget is introduced.

For each complete supplied path:

  total_interest =
      sum(I[A,1], I[B,1],
          I[A,2], I[B,2],
          I[A,3], I[B,3])                    # u

  total_payments =
      sum(total_paid[1],
          total_paid[2],
          total_paid[3])                    # u

  final_combined_balance =
      sum(B[A,0], B[B,0])
      + total_interest
      - total_payments                      # u

  final_unused_budget = unused[3]            # u

4. Verification of the six original rows

All pairs below are ordered (A, B). The checks retain the original opening balances, rates, rounding, minimums, target timing and same-period reallocation.

Original M001 — Smallest-first, V001-V003

Plain text

V001: ORIGINAL M001, PERIOD 1

Opening:
  O = (20.00, 40.00) u

Interest:
  I_raw = (20.00 * 0.01, 40.00 * 0.03) u
        = (0.2000, 1.2000) u
  I = (0.20, 1.20) u

After interest and minimums:
  D = O + I = (20.20, 41.20) u
  m = (min(5.00, 20.20), min(5.00, 41.20)) u
    = (5.00, 5.00) u
  R0 = D - m = (15.20, 36.20) u

Extra available:
  Q0 = 30.00 - sum(5.00, 5.00) = 20.00 u

Order and actual extra:
  A first:
    E[A,1] = min(20.00, 15.20) = 15.20 u
    Q after A = 20.00 - 15.20 = 4.80 u
  B next, in the same period:
    E[B,1] = min(4.80, 36.20) = 4.80 u
    Q after B = 4.80 - 4.80 = 0.00 u
  E = (15.20, 4.80) u

Payments and closing:
  payment = m + E = (20.20, 9.80) u
  B_close = R0 - E = (0.00, 31.40) u
  total_paid = sum(20.20, 9.80) = 30.00 u
  unused = 30.00 - 30.00 = 0.00 u

Result: original period-1 row unchanged.


V002: ORIGINAL M001, PERIOD 2

Opening:
  O = (0.00, 31.40) u

Interest:
  I_raw = (0.00 * 0.01, 31.40 * 0.03) u
        = (0.0000, 0.9420) u
  I = (0.00, 0.94) u

After interest and minimums:
  D = O + I = (0.00, 32.34) u
  m = (min(5.00, 0.00), min(5.00, 32.34)) u
    = (0.00, 5.00) u
  R0 = D - m = (0.00, 27.34) u

Extra available:
  Q0 = 30.00 - sum(0.00, 5.00) = 25.00 u

Order and actual extra:
  Only B remains positive.
  E[B,2] = min(25.00, 27.34) = 25.00 u
  E = (0.00, 25.00) u
  Q after B = 25.00 - 25.00 = 0.00 u

Payments and closing:
  payment = m + E = (0.00, 30.00) u
  B_close = R0 - E = (0.00, 2.34) u
  total_paid = sum(0.00, 30.00) = 30.00 u
  unused = 30.00 - 30.00 = 0.00 u

Result: original period-2 row unchanged.


V003: ORIGINAL M001, PERIOD 3

Opening:
  O = (0.00, 2.34) u

Interest:
  I_raw = (0.00 * 0.01, 2.34 * 0.03) u
        = (0.0000, 0.0702) u
  I = (0.00, 0.07) u

After interest and minimums:
  D = O + I = (0.00, 2.41) u
  m = (min(5.00, 0.00), min(5.00, 2.41)) u
    = (0.00, 2.41) u
  R0 = D - m = (0.00, 0.00) u

Extra available:
  Q0 = 30.00 - sum(0.00, 2.41) = 27.59 u

Actual extra:
  Both buckets are already paid by the capped minimums.
  E = (0.00, 0.00) u
  actual_extra_total = sum(0.00, 0.00) = 0.00 u

Payments and closing:
  payment = m + E = (0.00, 2.41) u
  B_close = R0 - E = (0.00, 0.00) u
  total_paid = sum(0.00, 2.41) = 2.41 u
  unused = 30.00 - 2.41 = 27.59 u
         = 27.59 - 0.00 = 27.59 u

Result: original period-3 row unchanged.

Original M002 — Highest-rate-first, V004-V006

Plain text

V004: ORIGINAL M002, PERIOD 1

Opening:
  O = (20.00, 40.00) u

Interest:
  I_raw = (20.00 * 0.01, 40.00 * 0.03) u
        = (0.2000, 1.2000) u
  I = (0.20, 1.20) u

After interest and minimums:
  D = O + I = (20.20, 41.20) u
  m = (min(5.00, 20.20), min(5.00, 41.20)) u
    = (5.00, 5.00) u
  R0 = D - m = (15.20, 36.20) u

Extra available:
  Q0 = 30.00 - sum(5.00, 5.00) = 20.00 u

Order and actual extra:
  B first:
    E[B,1] = min(20.00, 36.20) = 20.00 u
    Q after B = 20.00 - 20.00 = 0.00 u
  No extra remains for A.
  E = (0.00, 20.00) u

Payments and closing:
  payment = m + E = (5.00, 25.00) u
  B_close = R0 - E = (15.20, 16.20) u
  total_paid = sum(5.00, 25.00) = 30.00 u
  unused = 30.00 - 30.00 = 0.00 u

Result: original period-1 row unchanged.


V005: ORIGINAL M002, PERIOD 2

Opening:
  O = (15.20, 16.20) u

Interest:
  I_raw = (15.20 * 0.01, 16.20 * 0.03) u
        = (0.1520, 0.4860) u
  I = (0.15, 0.49) u

After interest and minimums:
  D = O + I = (15.35, 16.69) u
  m = (min(5.00, 15.35), min(5.00, 16.69)) u
    = (5.00, 5.00) u
  R0 = D - m = (10.35, 11.69) u

Extra available:
  Q0 = 30.00 - sum(5.00, 5.00) = 20.00 u

Order and actual extra:
  B first:
    E[B,2] = min(20.00, 11.69) = 11.69 u
    Q after B = 20.00 - 11.69 = 8.31 u
  A next, in the same period:
    E[A,2] = min(8.31, 10.35) = 8.31 u
    Q after A = 8.31 - 8.31 = 0.00 u
  E = (8.31, 11.69) u

Payments and closing:
  payment = m + E = (13.31, 16.69) u
  B_close = R0 - E = (2.04, 0.00) u
  total_paid = sum(13.31, 16.69) = 30.00 u
  unused = 30.00 - 30.00 = 0.00 u

Result: original period-2 row unchanged.


V006: ORIGINAL M002, PERIOD 3

Opening:
  O = (2.04, 0.00) u

Interest:
  I_raw = (2.04 * 0.01, 0.00 * 0.03) u
        = (0.0204, 0.0000) u
  I = (0.02, 0.00) u

After interest and minimums:
  D = O + I = (2.06, 0.00) u
  m = (min(5.00, 2.06), min(5.00, 0.00)) u
    = (2.06, 0.00) u
  R0 = D - m = (0.00, 0.00) u

Extra available:
  Q0 = 30.00 - sum(2.06, 0.00) = 27.94 u

Actual extra:
  Both buckets are already paid by the capped minimums.
  E = (0.00, 0.00) u
  actual_extra_total = sum(0.00, 0.00) = 0.00 u

Payments and closing:
  payment = m + E = (2.06, 0.00) u
  B_close = R0 - E = (0.00, 0.00) u
  total_paid = sum(2.06, 0.00) = 2.06 u
  unused = 30.00 - 2.06 = 27.94 u
         = 27.94 - 0.00 = 27.94 u

Result: original period-3 row unchanged.

Six-row verification table

All monetary values are in u; pairs are (A, B). Extra available, per-bucket actual extra and unused budget are displayed separately to prevent a final-period interpretation that spends the entire budget.

5. Complete reconciliations in ASCII

Original M001 / C057 — Smallest-first totals

Plain text

SMALLEST-FIRST INTEREST

Total interest =
    sum(0.20, 1.20, 0.00, 0.94, 0.00, 0.07) u

Omitting only the zero terms:
    0.20 u + 1.20 u + 0.94 u + 0.07 u = 2.41 u


SMALLEST-FIRST PAYMENTS

Total payments =
    sum(30.00, 30.00, 2.41) u

    30.00 u + 30.00 u + 2.41 u = 62.41 u


SMALLEST-FIRST BALANCE RECONCILIATION

Initial total + total interest - total payments = final total balance

    60.00 u + 2.41 u - 62.41 u = 0.00 u


SMALLEST-FIRST FINAL UNUSED BUDGET

Final available budget - final actual payment = final unused budget

    30.00 u - 2.41 u = 27.59 u

Final extra available - final actual extra allocated = final unused budget

    27.59 u - 0.00 u = 27.59 u

Original M002 / C058 — Highest-rate-first totals

Plain text

HIGHEST-RATE-FIRST INTEREST

Total interest =
    sum(0.20, 1.20, 0.15, 0.49, 0.02, 0.00) u

Omitting only the zero term:
    0.20 u + 1.20 u + 0.15 u + 0.49 u + 0.02 u = 2.06 u


HIGHEST-RATE-FIRST PAYMENTS

Total payments =
    sum(30.00, 30.00, 2.06) u

    30.00 u + 30.00 u + 2.06 u = 62.06 u


HIGHEST-RATE-FIRST BALANCE RECONCILIATION

Initial total + total interest - total payments = final total balance

    60.00 u + 2.06 u - 62.06 u = 0.00 u


HIGHEST-RATE-FIRST FINAL UNUSED BUDGET

Final available budget - final actual payment = final unused budget

    30.00 u - 2.06 u = 27.94 u

Final extra available - final actual extra allocated = final unused budget

    27.94 u - 0.00 u = 27.94 u

Original M003 / C059 — Conditional differences and closure timing

Plain text

INTEREST DIFFERENCE

Smallest-first interest - highest-rate-first interest:

    2.41 u - 2.06 u = 0.35 u


PAYMENT DIFFERENCE

Smallest-first payments - highest-rate-first payments:

    62.41 u - 62.06 u = 0.35 u


FIRST CLOSURE

Smallest-first:
    A first reaches 0.00 u at the end of period 1.

Highest-rate-first:
    B first reaches 0.00 u at the end of period 2.

Difference in first-closure period:
    2 - 1 = 1 model period.


FINAL CLOSURE

Smallest-first:
    Both closing balances are 0.00 u at the end of period 3.

Highest-rate-first:
    Both closing balances are 0.00 u at the end of period 3.

The original numerical comparison is therefore unchanged: the highest-rate-first path costs 0.35 u less within this stipulated model, while the smallest-first path closes its first bucket one model period earlier. No conclusion about motivation, adherence, actual affordability or real-world savings follows.

6. Atomic R032-P2 claim register

All input claims trace to S001. Formula and numerical findings are checked against the visible blocks above, not against newly retrieved agency material.

7. Correction crosswalk to the original report

No row, input or dependent total requires numerical correction. “Repair” in this table means replacement display and clarification of notation, not alteration of the immutable original.

8. Compact coverage table

9. Remaining gaps and researcher self-check

Explicit remaining gaps

Researcher self-check — not independent review

END R032 PASS-02