Appendix D: Exercises, Hints, and Selected Solutions

The book’s 128 pre-existing exercise banks contain 939 primary problems. Closed problems have worked solutions; laboratories and research problems may instead end with acceptance criteria, milestones, or guidance. Reprinting that material here would create a second, quickly drifting book. This appendix instead supplies a chapter-indexed atlas, a set of curated gateways, and five new synthesis problems with progressive help.

The count treats each numbered or titled top-level task as one primary problem, even when a task has separately solved subparts. It therefore counts Formal WKB Problem 2 once rather than counting its two solution panels as two problems.

The source page remains authoritative for every local exercise. The semantic identifiers introduced here are authoritative for the synthesis set and may be used in syllabi, issue reports, or citations.

Use the least revealing aid first

Freeze the equation, normalized bases, paths or cycles, and claim status before calculating. Make a serious first attempt, reveal Hint 1, try again, and reveal Hint 2 only if the obstruction remains. Read a selected solution last, then write down which missing idea—not which algebraic line—blocked the first attempt.

Problem labels are part of the deliverable

The four types refine the exercise levels introduced in How to Use This Book.

Type	Principal task	A complete submission contains
C · Concept or diagnostic	Decide what an object or claim means	Assumptions, invariant content, and a countercheck
D · Derivation	Establish a formula or limiting relation	Every convention-sensitive step and at least one check
L · Laboratory	Compute a declared datum	A reproduction record, refinement table, and independent audit
R · Research	Formulate a controlled extension	Known input, unknown target, falsification test, and stop rule

Difficulty levels are workload guides, not prerequisites:

• Level 1: one local idea, usually 10–20 minutes;
• Level 2: a working derivation, usually 30–60 minutes;
• Level 3: a cross-page synthesis, usually 1–3 hours;
• Level 4: a multi-session frontier investigation.

For a laboratory, also declare the intended evidence level: computed, internally converged, cross-verified, or certified. These terms have the precise meanings fixed in Appendix C.

Stable identifiers do not encode mutable metadata

An identifier such as ALODE.SPECTRUM.DIVISOR-TRANSPORT names mathematical content. It does not contain a difficulty, exercise number, solution status, or source-page slug. Those attributes may change; the identifier will not. Identifiers are never recycled.

The canonical URL targets this appendix. A secondary source link may move if a chapter heading is improved, but the semantic anchor remains fixed. For example:

• exercise: ALODE.SPECTRUM.DIVISOR-TRANSPORT;
• selected solution: solution to ALODE.SPECTRUM.DIVISOR-TRANSPORT.

The 939 page-local problems retain their visible local numbering. Those numbers are useful locators, but they are not promoted here to permanent semantic IDs because doing so without adding aliases to every frozen source page would create false stability.

The chapter-indexed exercise atlas

This table inventories the source banks before the five synthesis problems on this page. “Entry” opens the overview bank; the gateway lists below route to more specialized banks.

Scope	Banks	Problems	Entry
Front matter	2	8	Conventions
Chapter 1 · Complex linear ODEs	8	39	Chapter 1
Chapter 2 · Global connection data	8	56	Chapter 2
Chapter 3 · Heun laboratories	8	56	Chapter 3
Chapter 4 · Wronskians and recurrences	8	59	Chapter 4
Chapter 5 · Isomonodromy	8	61	Chapter 5
Chapter 6 · CFT and ODEs	8	57	Chapter 6
Chapter 7 · Block connection formulae	8	53	Chapter 7
Chapter 8 · Formal WKB	8	52	Chapter 8
Chapter 9 · Exact WKB	9	74	Chapter 9
Chapter 10 · Seiberg–Witten/NS	8	80	Chapter 10
Chapter 11 · AGT dictionary	8	63	Chapter 11
Chapter 12 · ODE/IM	8	63	Chapter 12
Chapter 13 · TBA	8	64	Chapter 13
Chapter 14 · Quantum spectra	8	62	Chapter 14
Chapter 15 · Black holes and holography	9	72	Chapter 15
Appendices A–C	3	15	Appendix A
Epilogue · Beyond the book	1	5	Epilogue
Total	128	939

Part I · Analytic ODE foundations

• Chapter 1. Start with the overview bank, derive the Liouville and oper transformations, audit the worked case files, and stretch with confluence.
• Chapter 2. Start with monodromy representations, derive connection and Wronskian identities, compute the hypergeometric benchmark, integrate the chapter in the global-data problem set, and stretch with Riemann–Hilbert problems.
• Chapter 3. Start with the local-model bank, derive the general-Heun normal form, audit parameter crosswalks, and stretch with polynomial and QES sectors.
• Chapter 4. Start with Wronskian ratios, derive minimal-solution recurrences, reproduce the verified-numerics audits, and stretch with Green functions and QNM recurrences.

Part II · Monodromy, CFT, and connection formulae

• Chapter 5. Start with monodromy moduli, derive the Schlesinger equations, audit series and Fredholm realizations, and stretch with inverse monodromy.
• Chapter 6. Start with Virasoro blocks, derive the BPZ equation, audit the four-point normalization, and stretch with irregular states.
• Chapter 7. Start by separating the two block regimes, derive the general-Heun connection matrix, reproduce the recurrence comparison, and stretch by rebuilding the full coefficient.

Part III · Formal and exact WKB

• Chapter 8. Start with the Riccati equation, derive the even–odd recursion, compute the Airy, Weber, and Mathieu examples, integrate cycles and covariance in the chapter problem set, and stretch with regularized periods.
• Chapter 9. Start with Borel–Laplace calibration, derive the local sectorial formulae, run the Borel–Padé spectral tests, and stretch with Voros wall crossing.

Part IV · Gauge theory and the AGT dictionary

• Chapter 10. Start with the gauge-theory background, derive special geometry, audit the WKB/SW-period identification, and stretch with nonperturbative completion.
• Chapter 11. Start with the operational AGT content, derive the regular-puncture dictionary, audit the normalization-complete example, and stretch with irregular decoupling limits.

Part V · ODE/IM and nonlinear integral equations

• Chapter 12. Start with spectral determinants, derive the TQ and Y-system relations, audit exact quantization, and stretch with scope and failure modes.
• Chapter 13. Start with the TBA derivation, derive the kernel and contour data, run the iteration laboratory, and stretch by deciding when two integral equations coincide.

Part VI · Spectral and physical capstones

• Chapter 14. Start with the oscillator calibrations, derive Floquet and Hill data, reproduce the three controlled capstones, and stretch by separating QES from generic exact WKB.
• Chapter 15. Start with separated wave equations, derive Wronskian and recurrence QNM conditions, reproduce the named-equation capstones, and stretch with the open-problem selector.

The reference calibrations are Appendix A’s convention problems, Appendix B’s analytic problems, and Appendix C’s computational problems.

Five cross-chapter synthesis problems

Identifier	Type	Level	Typical time	Support
ALODE.CLAIM.CHAIN-AUDIT	C	3	90 min	Hints · selected solution
ALODE.LIMIT.COMMUTING-SQUARE	D	3	120 min	Hints · solution outline
ALODE.SPECTRUM.DIVISOR-TRANSPORT	D	3	90 min	Hints · selected solution
ALODE.LAB.INDEPENDENCE-DOSSIER	L	3	2–4 h	Hints · acceptance criteria
ALODE.RESEARCH.EXCEPTIONAL-STRATUM	R	4	Multi-session	Hints · guidance

Each problem is followed by two collapsible hints and then by separately collapsed solution support. The HTML heading identifier—not its visible wording—is the canonical target.

ALODE.CLAIM.CHAIN-AUDIT · Audit an equality chain

Canonical exercise link · Hints · Selected solution

Type: C · Level: 3 · Prerequisites: Chapters 2, 7, 10, 12, and 13

A manuscript compresses its argument into the slogan

boundary Wronskian = conformal block = exponential of an NS period = TBA determinant.

Choose one named scalar boundary problem for which at least two links can be made precise. Then:

1. replace the equality chain by a directed diagram of mathematical objects and dictionary maps;
2. classify every arrow as universal, theorem under stated hypotheses, conjectural, numerical, or unsupported;
3. write the minimum normalization passport needed at each arrow;
4. decide whether values, zero divisors, logarithmic derivatives, or only asymptotic expansions can legitimately be compared; and
5. give one inexpensive falsification test for every non-universal arrow.

A complete answer must distinguish a conformal block from a full connection coefficient, a formal period from a resummed period, and a functional relation from the analyticity data needed to derive a TBA equation.

Hint 1 — replace equals signs by typed arrows

Begin with the differential operator and its two normalized boundary lines. The Wronskian is then a boundary section. Ask what extra data turn a CFT block, NS quantity, or TBA solution into that same type of object.

Hint 2 — compare the weakest invariant first

Multiplication by a nowhere-zero holomorphic function preserves a zero divisor but not a numerical value or residue. Try to establish a divisor identity before attempting a normalization-complete value identity.

Selected solution — a modified-Mathieu audit

There is no meaningful equality chain until every term has the same input parameter, codomain, and normalization. For a named test case, take the modified-Mathieu boundary problem

$$\left[ -\hbar_{\mathrm m}^2\frac{\dd^2}{\dd x^2} +2\Lambda^2\cosh x \right]\psi = E\psi,$$

with $\hbar_{\mathrm m},\Lambda>0$, and normalize $\psi_-$ and $\psi_+$ by their subdominant asymptotics as $x\to-\infty$ and $x\to+\infty$. The direct ODE section is

$$D_{\mathrm W} (E;\hbar_{\mathrm m},\Lambda) = \Wr[\psi_-,\psi_+].$$

This produces the defensible starting diagram

$$(L_E,\ell_-,\ell_+) \longrightarrow D_{\mathrm W}(E) \longrightarrow \operatorname{div}D_{\mathrm W}.$$

The arrows in the manuscript’s slogan now receive explicit labels.

• Boundary problem to Wronskian — theorem under the stated endpoint hypotheses. Existence and uniqueness of the normalized subdominant solutions make $D_{\mathrm W}$ well defined. Rescaling either endpoint frame changes its value by a holomorphic unit but preserves its zero divisor.

• Irregular conformal block to this $D_{\mathrm W}$ — unsupported by the results established in this book. The book develops formal and normalization-conditional confluent block kernels, but it does not prove a unit-complete block formula for this real-line modified-Mathieu boundary determinant. A block alone would still lack fusion or braiding factors, three-point normalization, exponent lifts, and a continuation path.

• NS quantity to quantum period — checked only as a finite-order formal relation here. The holomorphic NS parameter is not the positive mechanical parameter. The passport must include

  $$\epsilon_1 = \hbar_{\mathrm{NS}} = -\ii\hbar_{\mathrm m},$$

  together with the energy and coupling map, oriented cycle, polarization, additive constants, Borel ray, and chamber continued along that phase rotation. Chapter 10 checks the WKB–NS relation coefficientwise to the retained order after this continuation; it does not establish equality of the fully resummed analytic functions or their exact divisors.

• ODE/IM functional relation to TBA and then to $D_{\mathrm W}$ — unsupported in the present chain. This book proves neither a model-specific $TQ/Y$ system for this boundary problem nor the analyticity, zero-divisor, asymptotic, logarithm-sheet, contour, source-term, and normalization data needed to turn it into this determinant. Relevant model-specific literature cannot be imported by an equals sign; it needs a separate passport and status audit.

Thus the strongest common target is often

$$\operatorname{div}D_{\mathrm W} = \operatorname{div}D_{\mathrm alt},$$

not equality of values. Equality of Green-function residues requires the numerator normalization as well.

The cheap falsification tests are representation-specific: compare Wronskian zeros with direct eigenvalues; test the first block coefficient and the fusion limit; recover the classical period and weak-coupling limit of the NS expression; and check both the functional-relation residual and the off-grid TBA residual while monitoring zeros in the analyticity strip. Agreement only at one final root does not validate the intervening arrows.

ALODE.LIMIT.COMMUTING-SQUARE · Does confluence commute with the calculation?

Canonical exercise link · Hints · Solution outline

Type: D · Level: 3 · Prerequisites: Chapters 1, 2, 3, 5, and 7

For constant matrices $B_1,C_1$ and a nonzero semisimple matrix $B_2$ with distinct eigenvalues, consider the coefficient matrix of $Y'=A_\epsilon(z)Y$,

$$A_\epsilon(z) = \frac{A_0(\epsilon)}{z} + \frac{A_1(\epsilon)}{z-\epsilon},$$

with

$$\begin{aligned} A_1(\epsilon) &= \frac{B_2}{\epsilon}+B_1+O(\epsilon), \\ A_0(\epsilon) &= -\frac{B_2}{\epsilon}+C_1+O(\epsilon). \end{aligned}$$

1. Prove uniform convergence on compact subsets of a punctured neighborhood of zero to a system with a double pole, and identify both limiting polar coefficients.
2. Explain why coalescing two fixed residues does not by itself create this irregular limit.
3. Formulate a commuting-square test: compute connection data before taking $\epsilon\to0$, or first take the operator and basis limit and then compute the irregular connection/Stokes data.
4. State the frame renormalizations, $z$-sectors, loop order, scalar factors, and a fixed $\epsilon$-ray or simply connected parameter sector with continued branches that must be frozen for the two routes to be comparable. Exclude resonant parameter values, or replace the failing eigen-Frobenius frames by resonant frames that extend on the resulting domain.
5. Explain why convergence of the product of the two regular monodromy matrices is not, by itself, convergence of the full wild monodromy data.

Hint 1 — expand away from the collision

On a compact set with $|z|>|\epsilon|$, use

$$\frac1{z-\epsilon} = \frac1z + \frac{\epsilon}{z^2} +O(\epsilon^2).$$

The divergent simple-pole residues cancel in their sum but leave a finite first polar moment.

Hint 2 — raw bases rarely have a finite limit

Introduce explicit right frame factors before comparing connection matrices. At the irregular point, declare a $z$-sector and lateral normalization. Also approach $\epsilon=0$ through a fixed parameter ray or sector so branch labels and dominance order do not jump across a parameter Stokes wall. Detect resonant values accumulating toward the limit; either remove them or use a Levelt/logarithmic frame that extends through them.

Solution outline

On a compact set $K$ with $0\notin K$, the geometric expansion is uniform for sufficiently small $\epsilon$. Hence

$$\begin{aligned} A_\epsilon(z) ={}& \frac{A_0+A_1}{z} + \frac{\epsilon A_1}{z^2} +O(\epsilon) \\ ={}& \frac{B_1+C_1}{z} + \frac{B_2}{z^2} +O(\epsilon). \end{aligned}$$

The limit therefore has double-pole coefficient $B_2$ and simple-pole coefficient $B_1+C_1$. If both original residues remain bounded, then $\epsilon A_1\to0$ and no double pole survives. Collision of locations is not enough; a scaled polar moment must survive.

Let $C_\epsilon$ denote a regular connection matrix between declared local frames. Those frames generally contain divergent powers or exponentials. Choose right normalization factors $H_-(\epsilon)$ and $H_+(\epsilon)$ and test the finite object

$$\widehat C_\epsilon = H_-^{-1}(\epsilon) C_\epsilon H_+(\epsilon).$$

Take $\epsilon\to0$ on a fixed ray, or inside a simply connected parameter sector on which every logarithm, eigenvalue label, and frame is continued from one base value. Rotating the parameter path can cross a wall, exchange dominance labels, or change lateral branches, so the parameter approach is part of the datum.

A ray or sector alone is not enough if resonant values accumulate at zero. Either delete those values and state the connected approach domain, or resolve them with a resonant Levelt/logarithmic basis. In both cases the normalized frames and the factors $H_\pm(\epsilon)$ must extend on the declared domain.

The other route constructs sectorial bases for the limiting irregular system. The square commutes only if $\widehat C_\epsilon$ converges to the matrix assembled from those bases in the same $z$-sectors, lateral convention, path order, scalar gauge, and continued parameter branches.

The product of the two regular monodromies may approach the total local return, but the irregular return factorizes into formal monodromy and ordered Stokes factors. Its product does not determine the factors separately. A valid audit therefore compares the total return and enough sectorial connection data to recover the claimed wild-monodromy tuple.

ALODE.SPECTRUM.DIVISOR-TRANSPORT · Follow one spectral divisor

Canonical exercise link · Hints · Selected solution

Type: D · Level: 3 · Prerequisites: Chapters 2, 4, 9, and 12

On a simply connected parameter chart $U$, suppose three holomorphic boundary functions satisfy

$$D_{\mathrm R}=uD_{\mathrm W}, \qquad D_{\mathrm F}=vD_{\mathrm W}, \qquad u,v\in\mathcal O^\times(U).$$

Interpret the subscripts as a recurrence residual, a boundary Wronskian, and a normalized Fredholm or canonical determinant. Let $N$ be holomorphic on $U$, and let a Green-function entry be

$$G(\lambda) = \frac{N(\lambda)}{D_{\mathrm W}(\lambda)}.$$

1. Prove that the three boundary functions define the same zero divisor on $U$.
2. At a simple zero $\lambda_*$, derive the residue of $G$ and its transformation under $D_{\mathrm W}\mapsto wD_{\mathrm W}$ for $w\in\mathcal O^\times(U)$.
3. State what must happen to the numerator for the Green function—not merely its pole set—to remain unchanged.
4. Give counterexamples showing what fails when a multiplier has a zero or a pole in $U$.
5. Explain how a pole of a scalar recurrence ratio can be removed by changing projective chart without creating a physical spectral pole.

Hint 1 — use orders of vanishing

For $u\in\mathcal O^\times(U)$, $\operatorname{ord}_{\lambda_*}(uD)= \operatorname{ord}_{\lambda_*}D$. If $u$ is not a unit, add its order.

Hint 2 — separate a quotient from its denominator

At a simple zero, expand numerator and denominator to first order. Then repeat after scaling the denominator alone and after scaling numerator and denominator together. A recurrence ratio is only one affine chart on a projective line.

Selected solution

For every $\lambda_*\in U$,

$$\begin{aligned} \operatorname{ord}_{\lambda_*}D_{\mathrm R} &= \operatorname{ord}_{\lambda_*}u + \operatorname{ord}_{\lambda_*}D_{\mathrm W}, \\ \operatorname{ord}_{\lambda_*}D_{\mathrm F} &= \operatorname{ord}_{\lambda_*}v + \operatorname{ord}_{\lambda_*}D_{\mathrm W}. \end{aligned}$$

Because $u$ and $v$ are units, their orders vanish. All three functions have the same zeros with the same multiplicities.

If $D_{\mathrm W}(\lambda_*)=0$ and $D_{\mathrm W}'(\lambda_*)\neq0$, then

$$\operatorname{Res}_{\lambda=\lambda_*}G = \frac{N(\lambda_*)}{D_{\mathrm W}'(\lambda_*)}.$$

Replacing $D_{\mathrm W}$ by $wD_{\mathrm W}$ with $w$ a unit changes the denominator derivative at the zero to $w(\lambda_*)D_{\mathrm W}'(\lambda_*)$. If $N$ is held fixed, the residue is divided by $w(\lambda_*)$. If the same frame change sends $N\mapsto wN$, then the quotient and its residue are unchanged. Pole locations alone do not fix response amplitudes.

If $N(\lambda_*)=0$, the displayed residue is zero and the putative simple pole cancels; higher Taylor coefficients determine the regular value.

If $w(\lambda)=\lambda-\lambda_*$, multiplication creates one extra zero. If $w=(\lambda-\lambda_*)^{-1}$, it removes one zero or makes the boundary function meromorphic. Neither multiplier is a unit on a neighborhood of $\lambda_*$, so neither preserves the divisor.

Finally, a recurrence ratio $R_n=u_n/u_{n-1}$ has a pole when the chosen denominator coordinate vanishes. The projective pair $[u_n:u_{n-1}]$ may remain regular. Switching to the reciprocal chart, or evaluating a homogeneous endpoint determinant, removes that coordinate pole. Only a zero of the chart-independent boundary section is spectral.

ALODE.LAB.INDEPENDENCE-DOSSIER · Build two computations that can disagree

Canonical exercise link · Hints · Acceptance criteria

Type: L · Level: 3 · Prerequisites: Chapter 4 and Appendix C

Choose one benchmark:

• a hypergeometric connection-matrix entry;
• a Weber eigenvalue or connection coefficient; or
• a modified-Mathieu connection or spectral datum.

Compute the same invariant datum with two representations having materially different failure modes—for example, exact special-function evaluation and pathwise ODE transport, or a minimal recurrence and an endpoint Wronskian. Add one exact invariant such as an Abel determinant, symmetry relation, known spectrum, or monodromy trace.

Submit the full Appendix C reproduction record, independent refinement ladders, a conditioning estimate, raw discrepancies, and an evidence label. The target is cross-verification, not merely agreement at one precision.

Hint 1 — independence means different failure modes

Two implementations of the same recurrence mainly test transcription and arithmetic. Pair a recurrence with ODE transport, a local-series connection with a closed special-function formula, or Borel–Padé with a direct spectral determinant.

Hint 2 — predeclare the acceptance gate

Choose target digits before running the comparison. State which cutoff, order, contour, precision, and match-point controls are material at that tolerance, and refine every material control. A posteriori digit selection is not an independent test.

Acceptance criteria

A submission passes only if all of the following are present.

• The two methods compute the same normalized object, not merely values with similar names.
• Every cutoff, discretization, contour, match-point, and precision control material at the target tolerance is refined independently. If a genuine closed-form evaluator has only precision as an active numerical control, explain why no truncation or discretization control is hidden.
• The exact invariant is evaluated from raw outputs rather than imposed by a normalization step.
• Agreement is comfortably inside the separate internal error estimates and survives changes in every material auxiliary choice. The two representations must still have genuinely different failure modes.
• Inputs are parsed from exact strings, the command and dependency versions are archived, and unreported digits are retained in the raw data.
• The final label is no stronger than the evidence. In particular, agreement of two floating computations is cross-verification, not certification.

If the methods disagree, the dossier is still successful when it localizes the failure to a convention, truncation, conditioning, or theorem-scope gate. Concealing the discrepancy fails the exercise.

ALODE.RESEARCH.EXCEPTIONAL-STRATUM · Design a falsifiable exceptional-limit project

Canonical exercise link · Hints · Guidance and stop rule

Type: R · Level: 4 · Prerequisites: Chapters 1–4 and one advanced branch

Choose one exceptional stratum: resonant exponents, a confluence limit, an exceptional point, an extremal horizon, or a TBA divisor crossing. Prepare a research dossier that:

1. states the generic theorem or dictionary and the hypothesis that fails;
2. identifies the correct local replacement basis or coordinate;
3. formulates one global connection or spectral quantity that should extend;
4. proposes two representations with genuinely different failure modes;
5. lists known exact, asymptotic, or numerical milestones; and
6. gives a predeclared stop rule that would count against the proposed extension.

Do not ask whether the generic formula “still works.” State which normalized section, divisor, Stokes product, or observable is proposed to extend and in what topology or error norm.

Hint 1 — locate the first failed hypothesis

Separate local basis failure from global spectral failure. Resonance may destroy a generic Frobenius basis while leaving a logarithmic basis regular; an extremal limit may change the singularity type and therefore the boundary problem itself.

Hint 2 — make failure informative

A useful stop rule is quantitative: loss of convergence under a declared renormalization, incompatible limiting divisors in two representations, or a certified invariant that contradicts the proposal. “The computation became difficult” is not a stop rule.

Guidance and stop rule

There is deliberately no model solution. A research-grade dossier should reach four milestones:

1. a local normal form or replacement basis that remains meaningful on the exceptional stratum;
2. a globally normalized candidate section, divisor, or Stokes datum;
3. two independent limiting constructions with declared topology and error norm; and
4. one exact invariant or certified enclosure capable of disproving the proposal.

Predeclare an operational stop such as: two normalized estimates remain separated by more than their combined empirical uncertainties under two successive refinements. Reaching that threshold is useful negative evidence and triggers a scope or method audit; by itself it is not a mathematical disproof, because both calculations may still be preasymptotic.

A certified negative result requires more: for example, nonoverlapping validated enclosures for the limiting quantities, a change of divisor multiplicity under a purported holomorphic-unit frame change, or an exact monodromy, Wronskian, or root-count invariant that contradicts the proposed continuation. State in advance which kind of stop the project uses.

A computational submission ends with a claim label

Before submitting any laboratory, verify that the equation, boundary data, branches, paths or cycles, finite model, arithmetic, refinement table, conditioning estimate, independent representation, and raw outputs are all archived. The complete field list is in Appendix C’s reproduction record.

End with one sentence of the form:

The reported datum is [computed / internally converged / cross-verified / certified] for the declared operator and normalization because …

That sentence forces the evidence to match the claim and is part of the answer, not an editorial afterthought.