Appendix J — Reader Exercises and Answer Notes

Exercises to test transfer, not just recall. Answer notes follow each set.

#J.1 — Mathematical exercises

1. Using the base cases $F_0=0,F_1=1$, compute $F_{11}$ through $F_{15}$ by hand, keeping a derivation column.
2. Verify Cassini's identity for $n=7$: compute $F_6{F}_8-F_7^2$ and confirm it equals $(-1{)}^7$.
3. Use Binet's formula to estimate $F_{20}$ via ${\phi }^{20}/\sqrt{5}$, and confirm the nearest integer is the exact value.
4. Compute the ratio $F_{13}/F_{12}$ and its signed error from $\phi \approx 1.6180339887$.
5. Explain why naive recursive computation of $F_{40}$ is impractical, and state the running-time class of iteration and of matrix exponentiation.

Answer notes. (1) $F_{11}=89,F_{12}=144,F_{13}=233,F_{14}=377,F_{15}=610$, each the sum of its two predecessors. (2) $F_6{F}_8-F_7^2=8\cdot 21-{13}^2=168-169=-1=(-1{)}^7$. ✓ (3) ${\phi }^{20}/\sqrt{5}\approx 6765.00$; nearest integer $6765=F_{20}$. ✓ (4) $233/144\approx 1.6180556$; error $\approx +0.0000216$. (5) Naive recursion is exponential ($\sim {\phi }^n$) because it recomputes overlapping subproblems; iteration is linear ($\sim n$); matrix exponentiation is logarithmic ($\sim logn$).

#J.2 — Method exercises

1. For a system you work on, complete the RFPA Framing Canvas (Appendix C). Which of the four moves is weakest?
2. Take a recent output your team accepted. Rewrite its acceptance as an AVPT Execution Record (Appendix D). Was validation independent of how it was produced?
3. Identify one derived artifact your team edits in place. Describe how you would convert it to a regenerate-not-patch pipeline.
4. Design a termination policy (success / resource / failure) for a repeated process you run.
5. Classify three "X follows a pattern" claims you have heard using the Claim Matrix (Appendix F).

Answer notes. These are open-ended; a good answer (6) names a specific weakest move and why; (7) honestly identifies whether any check could have failed if the output were wrong; (8) names the canonical source and the read-only derived artifacts and a deterministic build; (9) states all three termination types with concrete thresholds; (10) assigns each claim a category and a refutation test, downgrading any claim with no possible refutation.

#J.3 — Integration exercise

1. Run the Reader's Implementation Workbook (Chapter 28 / Appendix uses) on one small real problem over a two-week horizon. Complete at least one full frame → cycle → gate → terminate loop, preserve the evidence, and write a one-paragraph review comparing your recovery capability before and after.

Answer note. Success is not a perfect outcome; it is a completed loop with preserved evidence and an explicit gate decision. If your Recover move failed at the start (no preserved prior state existed), that discovery is itself a valuable result — it is the most common finding, and fixing it is often the whole win.