Where your loan payment goes: interest, fees, and principal

SettleADebt R004 PASS-02 — Appendix J legal-method investigation

Result: partial recovery; the current legal APR hold remains.

This pass recovered an official Federal Reserve rendering with readable mathematical text for Appendix J’s symbols, general equation, iteration procedure, regular-payment form, and odd-final-payment form. Those equations permit a precise algebraic mapping to the reviewed E05 cash flows.

The recovered rendering, however, references §226.22, does not establish its own edition or effective date, and contains apparent truncation in two iteration-result figures. The current Part 1026 equation assets still returned unreadable placeholders. The targeted recodification sources establish the relationship between the Board’s Regulation Z and the Bureau’s Regulation Z, but do not finish the comparison of the recovered equations against the current operative panels. The reviewed numerical results therefore remain numerical finance-math results; they are not relabeled as legal APRs. federalreserve.gov+3

Actual source-check date: October 27, 2026, America/Los_Angeles. Observed session clock readings: 13:46:55–13:53:02 PDT, corresponding to 20:46:55–20:53:02 UTC.

Foundation preserved: The coordinator’s supplied review at 2026-10-27T20:31:51.105871Z is treated as the stated same-coordinator separate review—not independent review. P01’s six models, cash totals, numerical roots, raw history, and model-provenance limits remain unchanged. E01–E04 and E06 were not rerun, and no new amortization schedules were generated.

No assertion is made that the researcher verified unseen UI model family or power. No files, subagents, other chats, APIs, account or entity searches, contacts, filings, or security workarounds were used.

Reference convention: S001 below means R004-P2-S001; C001 means R004-P2-C001. All claim-register entries belong to A004.

1. What was recovered—and what remains missing

The new recovery is more than a title or search snippet: the Federal Reserve page’s body supplied literal mathematical markup. That is a textual equation read, not an inspection of image or PDF pixels. The separate current-image and PDF-screenshot attempts supplied only placeholders. federalreserve.gov+3

2. Recovered symbols and equations

2.1 Transcription status and notation

The following transcriptions come from S003, the Federal Reserve rendering referencing Part 226. They are not represented as a completed read of the current Part 1026 equation images.

Multiplication is written as *, powers as ^, and subscripts with underscores. For readability, a_due(x,i) represents the source’s annuity-due symbol: the present-value sum beginning with an immediate unit payment. This is an explicit plain-text normalization, not a new official symbol.

Symbols in (b)(7)

The source’s annualization is:

Plain text

I = 100 * w * i

Notation collision: P01 called its numerical monthly rate q. Appendix J’s q_k instead counts whole periods before an advance. In the mapping below:

Plain text

i = P01's numerical monthly rate q
q_1 = 0 whole periods

Confusing those two uses of q would corrupt the equation. federalreserve.gov

2.2 General equation, (b)(8)

The recovered expression is:

Plain text

sum over k=1..m:
    A_k / [(1 + e_k*i) * (1+i)^q_k]

=

sum over j=1..n:
    P_j / [(1 + f_j*i) * (1+i)^t_j]

Two operator details matter:

• Whole periods use the exponential factor (1+i)^q_k or (1+i)^t_j.
• Fractional periods use the linear factor 1 + e_ki or 1 + f_ji.

The latter is not the same expression as (1+i)^e_k or (1+i)^f_j. For E05’s stipulated zero fractions, the fractional factors equal one. That eliminates this particular difference in E05; it does not establish a general irregular-date IRR method. The current readable instruction in (b)(6) also specifies proportional treatment for a fractional unit period. federalreserve.gov+1

2.3 Regular single-advance form, (c)(1)

The recovered form is:

Plain text

A = [P * a_due(n,i)] / [(1 + f*i) * (1+i)^t]

Here, the relevant example uses t and f for the whole and fractional unit periods before the first payment.

For an ordinary first payment one full unit period after the advance:

Plain text

t = 1
f = 0

A = P * a_due(n,i) / (1+i)

Because the annuity-due sum starts at exponent zero, division by (1+i) moves its payments to periods 1 through n.

2.4 Odd-final-payment form, (c)(3)

The recovered form is:

Plain text

A = [P * a_due(n-1,i) + P_n/(1+i)^(n-1)]
    / [(1 + f*i) * (1+i)^t]

The first bracketed component represents n-1 ordinary payments. The second represents the distinct final payment. This is the relevant candidate form for E05’s eleven payments of $888.49 followed by $888.47. federalreserve.gov+1

Relevant official examples actually read

These are the examples as reported by the recovered Federal Reserve rendering, not newly calculated current-law examples.

The 1978 dates are example inputs, not the rendering’s publication, adoption, or currentness dates. federalreserve.gov+1

3. Iteration instructions: recovered procedure and transcription defect

3.1 Readable update equation

The source’s (b)(9) example concerns a $1,000 advance repaid by 36 monthly payments of $33.61:

Plain text

A(i) = 33.61 * a_due(36,i) / (1+i)

Its update procedure uses annual percentage-number estimates:

Plain text

i_1 = I_1 / (100*w)
A_prime = A(i_1)

I_2 = I_1 + 0.1
i_2 = I_2 / (100*w)
A_double_prime = A(i_2)

I_new = I_1
        + 0.1 * (A_target - A_prime)
                / (A_double_prime - A_prime)

Then set I_1 = I_new and repeat.

The increment 0.1 is 0.1 annual percentage point in this notation. It is not a 0.1 monthly decimal rate. The numerator and denominator signs must remain as written; the formula does not call for absolute values.

3.2 What the retrieved example actually supplied

The text also contained stray formatting characters around some numbers. Without an inspectable equation image, I cannot determine whether those defects originate in the page or its extraction. I have not represented inferred repairs as an official transcription. federalreserve.gov+2

This pass therefore establishes the readable update expression and its units, but does not claim a clean, complete transcription of every printed intermediate figure.

P01’s bisection results are preserved as reviewed numerical results. No rerun of those roots was needed to demonstrate the algebraic mapping below.

4. Current timing and applicability predicates

The following current prose was freshly read. These instructions govern the proposed mapping; they are not a substitute for the missing current equation comparison.

These are APR timing conventions. They do not establish a note’s servicing day-count convention or receipt/posting rule. Consumer Financial Protection Bureau+2

Actuarial method versus United States Rule

This investigation follows Appendix J’s actuarial-method expressions. Current §1026.22 and its commentary also recognize the United States Rule and distinguish the treatment of unpaid accrued interest. Those are computation descriptions; neither establishes permission to capitalize interest under a particular contract.

There is no need here to reopen E06 or investigate negative-amortization law. No separate United States Rule implementation is claimed. Consumer Financial Protection Bureau+1

5. E05 mapping and retained results

5.1 Conditional symbol mapping

This is a candidate mapping to the recovered expression, subject to current-version verification and the stated legal and timing predicates.

5.2 Stepwise equation reduction

Substitute the E05 payment structure into the recovered odd-final-payment form:

Plain text

A = [888.49 * a_due(11,i) + 888.47/(1+i)^11] / (1+i)

Expand the annuity-due sum:

Plain text

a_due(11,i)
= 1 + 1/(1+i) + ... + 1/(1+i)^10

Therefore:

Plain text

A = 888.49/(1+i)
    + 888.49/(1+i)^2
    + ...
    + 888.49/(1+i)^11
    + 888.47/(1+i)^12

Equivalently:

Plain text

sum over j=1..12 [P_j/(1+i)^j] - A = 0

That is algebraically the reviewed P01 numerical equation after substituting:

Plain text

i = q_P01
A = V

Operator checks

• a_due(11,i) contains eleven terms, not twelve.
• The outside division by (1+i) moves the ordinary payments to months 1–11.
• The final payment has total exponent 11+1=12.
• Substituting the all-equal-payment form without adjustment would replace the actual final $888.47 with $888.49.
• The extra two cents would enter as 0.02/(1+i)^12, not as an undiscounted origination charge.
• No fee is added twice.

These are new algebraic checks of the recovered form against the supplied inputs. They do not establish that the recovered rendering is a completely verified current legal method.

5.3 Classification inputs remain conditional

The following figures are reused from the reviewed foundation, not newly sourced borrower data or newly generated schedules.

The included branch requires the charge actually to qualify as a prepaid finance charge. The excluded branch requires an actual applicable exclusion, such as the stipulated qualifying all-applicants application-fee condition. There is no classification election.

The reviewed cash identities remain:

Plain text

Included branch:
9700.00 + 961.86 = 10661.86
300.00 + 10661.86 = 10961.86

Excluded branch:
10000.00 + 661.86 = 10661.86
300.00 + 10661.86 = 10961.86

A withheld fee is an alternative structure; it does not create another separate $300 payment. Financing the fee on top of the advance, producing a $10,300 note, is outside this pass and would require a different schedule.

5.4 Reviewed numerical roots—not relabeled

The two annualizations remain distinct:

Plain text

nominal annualized percentage = 1200 * i

mathematical effective annualized percentage
= 100 * [(1+i)^12 - 1]

For the excluded-fee branch, the economic cash-flow calculation may still use net $9,700, while the conditional disclosure-input comparison uses $10,000. That distinction survives the formula recovery.

The algebraic match is evidence of consistency between the reviewed mathematics and the recovered expression. It is not evidence that the current-version and legal-classification requirements have all been satisfied.

6. Precision and accuracy allowances

The current §22 clauses and relevant commentary were freshly read for these propositions. The §18 dollar-tolerance premise is explicitly reused from the reviewed P01 foundation. Consumer Financial Protection Bureau+2

7. Currentness and the Part 226–Part 1026 connection

The targeted history supplied useful evidence, but did not close the remaining comparison.

1. The 2011 republication: The FederalRegister.gov rendering states that the Bureau’s new Part 1026 substantially duplicated the Board’s Regulation Z with specified nonsubstantive, technical, formatting, and stylistic changes. It identifies an effective date of December 30, 2011. Its Appendix J prose is readable, but the equation-page content is absent from the retrieved text. The large official PDF could not be fetched because of a reported content-length limit.
2. The 2016 finalization: The final rule, effective April 28, 2016, adopted the December 2011 interim final rules subject to intervening amendments. Its operative issuance paragraph specifically includes the Regulation Z republication. This establishes a legal-history connection, not a certification that every equation in the undated Federal Reserve rendering is identical to every operative October 2026 panel. govinfo.gov+1
3. Current displays: The current CFPB Appendix J page identifies itself as the current regulation. The eCFR response obtained in this pass displayed Title 12 as updated through October 23, 2026, but exposed Appendix J only through (b)(6). This fresh access does not contradict or erase the coordinator’s separately reported unsuccessful §22 access. They were different observed retrievals. Consumer Financial Protection Bureau+1
4. Annual CFR: The recovered PDF is explicitly the January 1, 2025 edition. Its production footer includes July 31, 2025; that is not a new substantive effective date. The candidate 2026 annual-CFR URL did not return readable content, so its publication status and contents were not established.

The remaining version predicate is precise: verify the recovered symbols, operators, iteration instructions, and relevant examples against an actually readable current Part 1026 rendering—or against an adequately established official edition and its relevant subsequent amendment history. Neither a part-wide amendment date nor a successful numerical root supplies that verification.

8. Source table — R004-P2-S001 onward

All fresh checks below occurred 2026-10-27 PDT. “Text read” does not mean visual inspection. Sources grouped within a row are alternative renderings or related access routes, not invented independent corroboration.

Printed-page cross-reference limits

For the 2025 PDF, the extracted page headers establish the printed-page/index pairing. The graphic identifiers align with the CFPB asset stems:

Plain text

ER22DE11.062 -> printed 438 / PDF index 2
ER22DE11.063 -> printed 439 / PDF index 3
ER22DE11.064 -> printed 440 / PDF index 4
ER22DE11.066 -> printed 442 / PDF index 6

This is a structural cross-reference from extracted identifiers, not a claim that the formula pixels on those pages were seen. govinfo.gov

9. Atomic claim table — A004

10. Granular coverage table

11. Bounded read/failure log

All actions in this table occurred 2026-10-27 PDT. No response was treated as readable merely because it contained an image label.

No CAPTCHA or access-control challenge was bypassed. No source was obtained through an API or security workaround.

12. Self-check, precise next action, and preserved parent limits

Self-check

Precise next public-source action

The next bounded action is to obtain a genuinely readable current Part 1026 equation rendering covering the six identified panels: both symbol panels, general equation, iteration, regular-payment example, and odd-final-payment example. That requires either accessible official mathematical text or a supported public image/PDF response that actually exposes the pixels.

Once available, compare those contents against the transcriptions above, resolve the truncated iteration figures, and establish the relevant edition/currentness connection. Then apply the confirmed method to the existing E05 inputs under explicit term-start and monthly-timing assumptions.

Repeating these unchanged placeholder responses would not advance that task. No account, provider, case, record request, or contact is needed.

Remaining parent dimensions

END R004 PASS-02.