Apostol — Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra (Fast Track 0.2) — Audit + Gap Plan

Book: Tom M. Apostol, Calculus, Volume I: One-Variable Calculus, with an Introduction to Linear Algebra (2nd ed., John Wiley & Sons, 1967; SBN 0-471-00005-1). The "second-edition green hardcover" that has been in continuous print since 1967, used as the proof-based first-year calculus text at Caltech, MIT (18.014/18.024 historically), and a long list of honors-track undergraduate programs.

Fast Track entry: 0.2 (the §0 "real prerequisites" slot of the booklist; Apostol is the canonical single-variable proof-based calculus anchor and is paired with Vol. II (entry 0.3) for multivariable + linear algebra; together they form the §0 calculus prerequisite that the Fast Track assumes the reader has internalised before opening Apostol's analysis successors — Rudin, Pugh, etc. — or the §1 differential-geometry / topology stack).

Purpose of this plan: lightweight audit-and-gap pass (P1-lite + P2 + P3-lite of the orchestration protocol). Output is a concrete punch-list of new units to write so that Calculus, Vol. I is covered to the equivalence threshold (≥95% effective coverage of theorems, key examples, exercise pack, notation, sequencing, intuition, applications — see docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4).

This pass is intentionally not a full P1 audit (which would inventory every numbered exercise across Apostol's 16 chapters at the line-number level). It works from the book's chapter-by-chapter organisation and Apostol's distinctive editorial choices, produces the gap punch-list, and stops there.

The audit surface here is the largest of any Fast Track book audited so far. Where Cannas, Hatcher, Donaldson all sit in chapters where Codex already ships ~20 units in the same neighbourhood, Apostol Vol. I sits squarely in single-variable calculus and elementary real analysis — a region where Codex's 02-analysis/ chapter currently ships only topology (02.01.*, 7 units) and functional analysis (02.11.*, 7 units). The entire interior of 02-analysis/ between chapters 01 and 11 — the reals, sequences, series, Riemann integration, differentiation, elementary transcendental functions, Taylor's theorem — is empty. Apostol's coverage will populate roughly seven new sub-chapters of 02-analysis/ as well as one new 00-precalc/ mathematical-induction unit. Expect a large and load-bearing [NEW] punch-list with very few [DEEPEN] items — there is little to deepen because the upstream infrastructure does not yet exist.

§1 What Apostol's book is for

Apostol Vol. I is the canonical English-language proof-based first encounter with calculus. It is the book the Fast Track assumes a reader has either worked through directly or substituted with an equivalent honors-track sequence (Spivak's Calculus, Courant & John Vol. I) — it is not an analysis textbook in the Rudin / Pugh sense, but it is also not a computational textbook in the Stewart / Thomas sense. It occupies the precise pedagogical slot that Codex's existing 02-analysis/01-topology chapter assumes the reader has already crossed: the bridge from "I can compute integrals and derivatives" to "I can read a real-analysis textbook."

The book has 16 chapters across two parts.

Part 1 — Introduction. The famously Apostolian opening: not "limit" or "derivative" but integration of step functions.

Ch. 1 — Introduction. A historical introduction (Eudoxus, Archimedes, Newton, Leibniz). Apostol's preface argues that integration is older, more concrete, and more naturally rigorous than differentiation, and the book is therefore organised integration first.
Ch. 2 — Some basic concepts of the theory of sets. Set-theoretic vocabulary: union, intersection, Cartesian product, functions as graphs.
Ch. 3 — A set of axioms for the real-number system. This chapter is what makes Apostol distinctive at the §0 level. Apostol gives the complete axiomatic specification of $R$ — field axioms (10 axioms), order axioms, and the least-upper-bound axiom (completeness) — and then derives every basic property (Archimedean property, density of $Q$ , integer part, existence of $n$ -th roots of positive reals, the fact that $R$ is uncountable) from these axioms. Apostol does not construct $R$ from $Q$ (no Dedekind cuts, no Cauchy completion); he axiomatises it. Codex must record this pedagogical choice as the canonical proof-based-calculus framing.
Ch. 4 — Mathematical induction, summation notation, and related topics. Mathematical induction (both forms), $Σ$ -notation, the Bernoulli inequality, the binomial theorem, the well-ordering principle, finite vs. countable vs. uncountable sets (with Cantor's diagonal argument).

Part 2 — Differential and integral calculus.

Ch. 1 — The concepts of integral calculus. (Note: chapter numbering restarts at Ch. 1 in Part 2; Apostol calls these "Chapter I.1, I.2, ..." in the table-of-contents convention. We will use Apostol's running chapter numbering — Ch. 1 of Part 2 is Ch. 1.1 in citations below.) The integral of a step function as a finite sum. The integral of a more general function defined as the least upper bound of integrals of step functions below it (lower integral) and the greatest lower bound of integrals of step functions above it (upper integral); a function is integrable when the two agree. This is not the Riemann definition via tagged partitions — it is the Darboux definition via upper and lower step-function integrals. The integral of a monotonic function on a compact interval exists by this definition. The integral of a continuous function exists once continuity-on-a-compact-interval is shown to imply uniform continuity (Ch. 3).
Ch. 2 — Some applications of integration. Area between curves, trigonometric integrals (developed before the trig functions are rigorously defined — Apostol uses a geometric area-of-circular-segment definition of $sin$ and $cos$ here, then revisits the analytical development in Ch. 6), volumes of solids of revolution, work, average value of a function, mean value theorem for integrals.
Ch. 3 — Continuous functions. Definition of limit and continuity (in the $ε$ - $δ$ form), the algebra of continuous functions, the intermediate value theorem, Bolzano's theorem (existence of zero of a continuous function changing sign), boundedness theorem (continuous on $[a, b]$ implies bounded), extreme value theorem (continuous on $[a, b]$ attains its sup and inf), uniform continuity.
Ch. 4 — Differential calculus. Definition of derivative as a limit, algebra of derivatives, chain rule, derivatives of inverse functions, Rolle's theorem, mean value theorem (Lagrange and Cauchy forms), monotonicity test, sufficient conditions for extrema, Taylor's formula with Lagrange remainder (this is Ch. 7 — Apostol defers Taylor until after the elementary transcendental functions in Ch. 6).
Ch. 5 — The relation between integration and differentiation. The two fundamental theorems of calculus stated and proved cleanly. Apostol's framing: FTC1 says differentiation undoes the integral $F (x) = \int_{a}^{x} f$ ; FTC2 says the integral of a derivative recovers the function up to constants. Integration by parts. Integration by substitution. Improper Riemann integrals on $[a, \infty)$ and on intervals where the integrand is unbounded — comparison test and absolute convergence.
Ch. 6 — The logarithm, the exponential, and the inverse trigonometric functions. Apostol's signature analytical construction:
- $lo g x = \int_{1}^{x} \frac{d t}{t}$ for $x > 0$ . From this definition every property — $lo g (x y) = lo g x + lo g y$ , $lo g x \to \infty$ , the bijection $\log : (0, \infty) \to \mathbb{R}$ — is a corollary.
- $exp = lo g^{- 1}$ , with $e = exp (1)$ . The series $e^{x} = \sum x^{n} / n!$ comes later (Ch. 10) as a Taylor expansion.
- $a^{b} = exp (b lo g a)$ for $a > 0$ .
- The trig functions are re-derived analytically as inverses of the corresponding integral $arcsin x = \int_{0}^{x} d t / 1 - t^{2}$ ; the geometric Ch. 2 framing is shown to agree with the analytic one. This integral-first construction of the elementary transcendental functions is one of Apostol's two largest pedagogical signatures (the other being integration-before-differentiation). Codex has zero units here.
Ch. 7 — Polynomial approximations to functions. Taylor's formula with Lagrange remainder, with integral remainder, and with Cauchy remainder. Convergence conditions. Worked Taylor expansions of $exp$ , $lo g (1 + x)$ , $sin$ , $cos$ , $arctan$ , $(1 + x)^{α}$ . L'Hôpital's rule (as application of Taylor).
Ch. 8 — Introduction to differential equations. First-order linear ODEs (integrating-factor method), separable ODEs, applications (radioactive decay, mixing, simple harmonic motion), second-order linear ODEs with constant coefficients (the characteristic-polynomial method), undetermined coefficients for forcing terms, applications to resonance and damping.
Ch. 9 — Complex numbers. Construction $C = R^{2}$ with complex multiplication, polar form, Euler's formula $e^{i θ} = cos θ + i sin θ$ (proved from the Taylor series of $exp$ , $sin$ , $cos$ — closing the loop with Ch. 6 and Ch. 7), de Moivre's theorem, $n$ -th roots of unity, fundamental theorem of algebra (statement; proof deferred to Vol. II). Application: solving second-order constant-coefficient linear ODEs with complex characteristic roots from Ch. 8.
Ch. 10 — Sequences, infinite series, improper integrals. $ε$ -definition of sequence convergence, monotone-convergence theorem, Cauchy sequences, the Bolzano-Weierstrass theorem. Infinite series: comparison, ratio, root, alternating-series, Cauchy condensation tests. Absolute vs. conditional convergence. Riemann's rearrangement theorem (statement). Improper integrals revisited; the integral test for series.
Ch. 11 — Sequences and series of functions. Pointwise vs. uniform convergence (Apostol's $sup$ -norm formulation), the Weierstrass M-test, term-by-term integration and differentiation under uniform convergence, power series and the radius of convergence (with Cauchy-Hadamard), Abel's theorem, Taylor series.
Ch. 12 — Vector algebra. Vectors in $R^{n}$ , dot product, norm, Cauchy-Schwarz inequality, projections, linear independence, bases. (Shorter prelude to Vol. II.)
*Chs. 13-15 — (Brief survey: applications of vector algebra to analytic geometry; calculus of vector-valued functions; linear spaces.) These are bridge chapters into Vol. II and we will not fully audit them here — the proper audit lives in the Apostol Vol. II plan (FT 0.3).
Ch. 16 — Linear transformations and matrices. Same comment; defer to Vol. II audit.

The book ships a substantial exercise pack (~2500 exercises across the volume; ~1800 in the single-variable Chs. 1-11). Many are load-bearing — Apostol's text is one of the few first-encounter books where exercises are routinely cited later in the same book as "Exercise n.m". The exercise layer (FT equivalence Layer 2) is substantial; Codex has zero units in this region today, so the exercise gap is currently 100%.

Distinctive Apostol choices (in roughly the order he develops them):

Axiomatic real numbers. The completeness axiom is axiom 11 (after the 10 ordered-field axioms). Apostol does not construct $R$ from $Q$ ; he axiomatises it and then derives every property. This is the cleanest pedagogical choice for a first-year textbook and is the standard convention assumed by every later Codex unit that quotes "the completeness of $R$ ".
Integration before differentiation. Apostol Part 2 Ch. 1 is the integral; Ch. 4 is the derivative. The motivation is that the integral of a step function is manifestly a finite sum — there is no limiting process, no $ε$ - $δ$ , no derivative to misuse. Apostol uses this to build the integral before the reader has the technical machinery for limits, then circles back to limits in Ch. 3 to handle continuous integrands.
Darboux integral, not Riemann sum. Apostol's "Riemann integral" is actually the Darboux upper/lower integral — agreement of $in f \int s_{+}$ over step functions $s_{+} \geq f$ and $sup \int s_{-}$ over step functions $s_{-} \leq f$ . This is equivalent to the tagged-partition Riemann definition for bounded $f$ on a compact interval, but pedagogically cleaner. Codex must record both formulations and note the equivalence.
Logarithm-as-integral. $lo g x := \int_{1}^{x} d t / t$ is the definition. Every property of the log and the exponential follows by FTC. This is the analytical-purity choice (no a priori construction of " $2^{π}$ " or " $e^{x}$ " — they are defined via integrals and inverses).
Trig analytically. $arcsin x := \int_{0}^{x} d t / 1 - t^{2}$ , then $sin = arcsin^{- 1}$ . The geometric definitions (Ch. 2) are shown to agree.
Taylor with explicit remainder forms. Lagrange, integral, and Cauchy remainders are all proved, not just stated. Applications include L'Hôpital's rule.
Sequences/series after the elementary functions. The monotone-convergence theorem and Bolzano-Weierstrass come in Ch. 10, after the elementary calculus is in place. This sequence reflects Apostol's pedagogical bet that students learn the $ε$ - $δ$ machinery best by seeing it applied to concrete familiar functions before being asked to absorb the abstract sup/inf machinery for sequences.
Historical interludes. Each chapter has historical notes — Eudoxus, Archimedes, Newton, Leibniz, Euler, Cauchy, Riemann, Weierstrass. These are not just colour; Apostol uses the historical sequence to motivate why the modern definitions take the form they do (e.g., Cauchy's $ε$ - $δ$ as the resolution of the "infinitely small quantity" embarrassment).
Exercises as pedagogy. Apostol's exercise sets are graded (computational → proof-based) and many computational exercises set up later proof-based ones. The exercise layer is a substantial fraction of the book's value transfer.

§2 Coverage table (Codex vs Apostol)

Cross-referenced against the current ~217-unit Codex corpus (find content -name "*.md" | sort). ✓ = covered at Apostol-equivalent depth, △ = topic present elsewhere but not at calculus-textbook depth, ✗ = not covered.

The Codex 02-analysis/ chapter currently has units only in 02-analysis/01-topology/ (7 units: topological-space, continuous-map, metric-space, quotient-topology, fibration, cofibration, compact-open topology) and 02-analysis/11-functional-analysis/ (7 units: bounded linear operators, unbounded self-adjoint, Banach, compact operators, normed vector space, inner-product space, Hilbert space). The interior chapters (02 through 10) are empty. Apostol's coverage will populate those interior chapters.

Codex 00-precalc/ currently ships exactly one unit (00.02.05-function, in 00-precalc/02-set-and-function/).

Part 1 — Introduction (Apostol Chs. 1-4)

Apostol topic	Codex unit(s)	Status	Note
Historical introduction (Eudoxus, Archimedes, Newton, Leibniz)	—	△	Codex has no dedicated calculus-history unit; partial coverage in `02.04.*` (Riemann integration, when written) Master / Historical sections will subsume.
Set-theoretic vocabulary, Cartesian product, function as graph	`00.02.05` function	△	The function unit is shipped; missing companion units on union/intersection/Cartesian product (likely deferred to other §0 books). Apostol Ch. 2 is brief; partial coverage acceptable.
Field axioms (10) for $R$	`01.01.01` field	△	The field unit is general (any field); a calculus-specific unit listing the ordered-field axioms exactly as Apostol enumerates them does not exist.
Order axioms	—	✗	Gap. No standalone unit; the ordered-field structure appears in `01.01.01` only by passing mention.
Completeness axiom (least upper bound)	—	✗	Gap (foundational). Apostol's Axiom 11. Load-bearing for every later real-analysis unit in Codex. Currently no unit anywhere in Codex states the LUB axiom as the defining property of $R$ .
Archimedean property, density of $Q$ , integer part	—	✗	Gap. Apostol Ch. 3 §3.10-3.13. Foundational.
Existence of $n$ -th roots, $R$ uncountable	—	✗	Gap.
Mathematical induction (both forms), well-ordering	—	✗	Gap. Apostol Ch. 4. Foundational. Used implicitly throughout Codex.
Summation $Σ$ notation, telescoping sums	—	✗	Generally used silently; not its own unit. Acceptable as part of an induction unit.
Bernoulli inequality	—	✗	Standard — appears in Apostol Ch. 4 exercise; will be a worked example in the induction unit.
Binomial theorem	—	✗	Gap. Apostol Ch. 4. Foundational; used throughout calculus and combinatorics.
Countable vs uncountable, Cantor diagonal	—	✗	Gap. Apostol Ch. 4 §4.10.

Part 2 Ch. 1 — The integral (step functions, Darboux)

Apostol topic	Codex unit(s)	Status	Note
Integral of a step function as a finite sum	—	✗	Gap (foundational). The pedagogical entry point Apostol uses.
Properties of the step-function integral (linearity, additivity over partitions, monotonicity)	—	✗	Gap.
Upper and lower integrals (Darboux) of bounded $f$	—	✗	Gap (foundational).
Riemann/Darboux integrability ( $\overline{\int} f = \underline{\int} f$ )	—	✗	Gap (foundational).
Integrability of monotonic functions on $[a, b]$	—	✗	Gap. Apostol Ch. 1 §1.20.
Linearity of the integral, change-of-interval, additivity	—	✗	Gap.
Mean value theorem for integrals (continuous $f$ )	—	✗	Gap. Apostol Ch. 2 §2.16.

Part 2 Ch. 2 — Applications of integration

Apostol topic	Codex unit(s)	Status	Note
Area between curves	—	✗	Gap (medium). Standard worked-example territory; can fold into a single applications-of-integration unit.
Volume by slicing / disks / shells	—	✗	Gap (medium).
Arc length	—	✗	Gap (medium).
Work, hydrostatic force, centroid	—	✗	Gap (low). Application material; one consolidated unit.
Average value of a function	—	✗	Gap (medium).
Trigonometric integrals (geometric framing of $sin$ , $cos$ )	—	✗	Gap. Apostol's Ch. 2 geometric trig is later replaced by Ch. 6 analytic trig; a single unit can record both.

Part 2 Ch. 3 — Continuous functions

Apostol topic	Codex unit(s)	Status	Note
Limit of a function ( $ε$ - $δ$ )	`02.01.02` continuous-map (general)	△	The continuous-map unit is topological-space general — limits in metric / real-line setting are not its specific subject. Calculus-textbook depth is missing.
Continuity at a point, on an interval	`02.01.02` (general)	△	Same.
Algebra of continuous functions, composition	`02.01.02` (general)	△	Same.
Intermediate value theorem (Bolzano)	—	✗	Gap (high priority). The single most-cited theorem of single-variable calculus. Used silently throughout Codex — every existence-of-a-zero argument relies on it.
Boundedness theorem ( $f$ continuous on $[a, b]$ ⇒ bounded)	—	✗	Gap (high priority).
Extreme value theorem (Weierstrass)	—	✗	Gap (high priority). Used in optimisation, Lagrange multipliers, every "attains its supremum" appeal across Codex.
Uniform continuity on a compact interval (Heine-Cantor)	`02.01.02` (general topological framing — partial)	△	The general topological compactness-implies-uniform-continuity is implicit in the metric-space unit; Apostol's specific calculus-level statement and its application to integrability of continuous functions is not present.

Part 2 Ch. 4 — Differential calculus

Apostol topic	Codex unit(s)	Status	Note
Derivative (definition as a limit)	—	✗	Gap (foundational). No single-variable derivative unit anywhere in Codex.
Algebra of derivatives (sum, product, quotient, chain rule)	—	✗	Gap (foundational).
Chain rule	—	✗	Gap (foundational).
Implicit differentiation	—	✗	Gap.
Inverse function theorem (1D), derivative of inverse	—	✗	Gap. Apostol Ch. 6 §6.7. (Multi-D version belongs in Vol. II.)
Rolle's theorem	—	✗	Gap.
Mean value theorem (Lagrange)	—	✗	Gap (foundational).
Cauchy mean value theorem	—	✗	Gap.
Monotonicity test, sufficient conditions for extrema	—	✗	Gap.
First / second derivative tests	—	✗	Gap.
Concavity, inflection points	—	✗	Gap.

Part 2 Ch. 5 — FTC and techniques of integration

Apostol topic	Codex unit(s)	Status	Note
First fundamental theorem of calculus ( $F (x) = \int_{a}^{x} f$ ⇒ $F^{'} = f$ )	—	✗	Gap (foundational).
Second fundamental theorem of calculus ( $\int_{a}^{b} F^{'} = F (b) - F (a)$ )	—	✗	Gap (foundational).
Integration by parts	—	✗	Gap.
Integration by substitution (change of variables, 1D)	—	✗	Gap.
Partial fractions	—	✗	Gap. Standard technique.
Trigonometric substitution	—	✗	Gap. Standard technique.
Improper integrals (over unbounded intervals; integrand unbounded)	—	✗	Gap. Apostol Ch. 5 §5.10 + Ch. 10.
Comparison test for improper integrals	—	✗	Gap.

Part 2 Ch. 6 — Logarithm, exponential, inverse trig (Apostol's signature analytical construction)

Apostol topic	Codex unit(s)	Status	Note
$lo g x = \int_{1}^{x} d t / t$ as definition	—	✗	Gap (high priority — Apostol signature).
Properties of $lo g$ ( $lo g (x y) = lo g x + lo g y$ , etc.) from FTC	—	✗	Gap.
$exp$ as inverse of $lo g$ ; $e = exp (1)$	—	✗	Gap.
$a^{b} = exp (b lo g a)$ for $a > 0$	—	✗	Gap.
Analytical $sin$ , $cos$ via $arcsin x = \int_{0}^{x} d t / 1 - t^{2}$	—	✗	Gap. Apostol Ch. 6 §6.18-6.20.
Inverse trig functions ( $arcsin$ , $arccos$ , $arctan$ ) and their derivatives	—	✗	Gap.
Hyperbolic functions $sinh$ , $cosh$ , $tanh$	—	✗	Gap (medium). Standard companion topic.

Part 2 Ch. 7 — Polynomial approximations (Taylor)

Apostol topic	Codex unit(s)	Status	Note
Taylor's formula with Lagrange remainder	—	✗	Gap (high priority).
Taylor's formula with integral remainder	—	✗	Gap.
Taylor's formula with Cauchy remainder	—	✗	Gap.
Worked Taylor series ( $e^{x}$ , $sin$ , $cos$ , $lo g (1 + x)$ , $arctan$ , $(1 + x)^{α}$ )	—	✗	Gap. Worked-example density.
L'Hôpital's rule (as Taylor application)	—	✗	Gap.

Part 2 Ch. 8 — Elementary differential equations

Apostol topic	Codex unit(s)	Status	Note
First-order linear ODE, integrating factor	—	✗	Gap.
Separable ODE	—	✗	Gap.
Second-order linear ODE with constant coefficients, characteristic polynomial	—	✗	Gap.
Method of undetermined coefficients (forcing)	—	✗	Gap.
Application to simple harmonic motion, damping, resonance	—	✗	Gap.

Part 2 Ch. 9 — Complex numbers

Apostol topic	Codex unit(s)	Status	Note
Construction of $C$ , real and imaginary parts	—	✗	Gap. No standalone " $C$ as $R^{2}$ " unit; Codex `06-riemann-surfaces/01-*` units assume the reader has this.
Polar form, modulus, argument	—	✗	Gap.
Euler's formula $e^{i θ} = cos θ + i sin θ$ (analytic proof via Taylor)	—	✗	Gap.
de Moivre, $n$ -th roots of unity	—	✗	Gap.
Fundamental theorem of algebra (statement)	—	✗	Gap. Statement-level; analytic proof belongs in `06.01.*` complex analysis (already cited there but the named theorem is missing as its own unit).

Part 2 Ch. 10 — Sequences, series, improper integrals (return)

Apostol topic	Codex unit(s)	Status	Note
$ε$ -definition of sequence convergence	—	✗	Gap (foundational).
Monotone convergence theorem	—	✗	Gap (foundational).
Cauchy sequences, completeness of $R$ as MCT/Cauchy criterion	`02.01.05` metric-space (general — partial)	△	Cauchy sequences appear in the metric-space unit at general level; Apostol's calculus-textbook framing for $R$ specifically is missing.
Bolzano-Weierstrass theorem	—	✗	Gap (foundational). Used silently throughout `02-analysis`.
Limsup, liminf of sequences	—	✗	Gap.
Infinite series, partial sums, convergence	—	✗	Gap (foundational).
Comparison test, ratio test, root test	—	✗	Gap.
Cauchy condensation test	—	✗	Gap.
Alternating series test	—	✗	Gap.
Absolute vs conditional convergence, Riemann rearrangement	—	✗	Gap.
Integral test for series	—	✗	Gap.

Part 2 Ch. 11 — Sequences and series of functions

Apostol topic	Codex unit(s)	Status	Note
Pointwise vs uniform convergence	—	✗	Gap (foundational).
Weierstrass M-test	—	✗	Gap.
Term-by-term integration / differentiation under uniform convergence	—	✗	Gap.
Power series, radius of convergence (Cauchy-Hadamard)	—	✗	Gap.
Abel's theorem on power series	—	✗	Gap.
Taylor series convergence	—	✗	Gap.

Part 2 Chs. 12-16 — Vector algebra / linear algebra prelude

These chapters are the Vol. II handoff. Codex's 01-foundations/01-linear-algebra/ ships 01.01.01 field, 01.01.03 vector-space, and 01.01.15 bilinear-quadratic-form. Vector algebra in Apostol Ch. 12 (dot product, norm, Cauchy-Schwarz, projections, linear independence, bases) overlaps with 01.01.03 vector-space and is partially covered there. The full Apostol-Vol.-II audit (FT 0.3) will cover these chapters in depth; this plan defers them. A single line item:

Apostol topic	Codex unit(s)	Status	Note
Vector algebra (Ch. 12), analytic geometry (Ch. 13), vector calculus prelude (Ch. 14), linear spaces (Ch. 15), linear transformations (Ch. 16)	various `01.01.*`	△	Defer to Apostol Vol. II plan.

Aggregate coverage estimate

Theorem layer: Of Apostol Vol. I's ~120 named theorems / propositions in Chs. 1-11 of Part 2, Codex currently states and proves perhaps 3-5% (IVT/EVT are absent; FTC is absent; MVT is absent; chain rule is absent; all of Ch. 6 and Ch. 7 is absent). After the priority-1 punch-list below this rises to ~70%; after priority-1+2 to ~95%.

Exercise layer: ~0% — Codex has no calculus exercises today. Apostol Vol. I has ~1800 exercises in the single-variable chapters; the exercise-pack pass is a separate (large) follow-up after the priority-1+2 theorem-layer batch closes.

Worked-example layer: ~0% — same.

Notation layer: partial — $\int$ , $\sum$ , $lim$ etc. are used everywhere but Apostol's specific conventions (e.g., upper/lower-integral notation $\overline{\int}$ , $\underline{\int}$ ; the Bachmann-Landau-style $o$ -notation Apostol uses for Taylor; the distinction between $lo g$ and $ln$ ) are not currently catalogued. A notation/apostol-vol1.md crosswalk file should be produced.

Sequencing layer: ~0% reflected in Codex DAG today; the Apostol prerequisite chain (sets → reals → step-function integral → continuity → derivative → FTC → log/exp → Taylor → ODE → series → uniform convergence) is genuinely not represented because the units do not exist. Building the units automatically builds the sequencing.

Intuition layer: Apostol's pedagogical framing (integration-before-differentiation, axiomatic-reals-not-construction, log-as-integral) needs to be preserved in the Beginner / Master sections of the new units.

Application layer: Apostol's applications (work, hydrostatics, simple harmonic motion, radioactive decay, resonance) are physics-flavoured classics; the application unit can fold them in.

§3 Gap punch-list (P3-lite — units to write, priority-ordered)

This is the largest punch-list of any Fast Track audit produced so far. We propose eight new sub-chapters of 02-analysis/ plus one new 00-precalc/ sub-chapter for induction, populated by the units below. Sub-chapter slugs proposed:

00-precalc/03-induction-and-counting/ — induction + binomial.
02-analysis/02-real-numbers/ — axioms, completeness, Archimedean, countability.
02-analysis/03-sequences-series/ — sequence convergence, MCT, Cauchy, Bolzano-Weierstrass, series tests.
02-analysis/04-riemann-integration/ — step functions, Darboux, monotonic / continuous integrability, FTC, techniques, improper.
02-analysis/05-differentiation/ — derivative, chain rule, MVT, Rolle, monotonicity, extrema, inverse function theorem (1D).
02-analysis/06-elementary-functions/ — log-as-integral, exp, analytic trig, hyperbolic.
02-analysis/07-taylor-and-power-series/ — Taylor with three remainders, L'Hôpital, power series, radius of convergence.
02-analysis/08-elementary-odes/ — first-order linear, separable, second-order constant-coefficient, applications.
02-analysis/09-complex-numbers/ — $C$ , polar form, Euler, de Moivre, FTA statement.
02-analysis/10-uniform-convergence/ — pointwise vs uniform, Weierstrass M-test, term-by-term operations, Abel.

The chapter slugs above slot into the empty 02-analysis/ interior between the existing topology (01-) and functional-analysis (11-) chapters; numbering is reserved for them.

Apostol-distinctive priority 1 — the load-bearing real-analysis foundation (~14 units)

These are the units without which Codex cannot rigorously cite "by the IVT", "by the MVT", "by completeness", "by FTC" anywhere in the corpus. They are foundational and currently absent.

00.03.01 Mathematical induction. [NEW] Three-tier; ~1500 words. Statement of induction (weak and strong forms), well-ordering principle, equivalence with each other. Worked examples: $\sum_{k = 1}^{n} k = n (n + 1) /2$ , Bernoulli inequality, factorial-power inequality. Apostol Ch. 4 anchor. Originator citation: Pascal 1654 (treatise on the arithmetic triangle); modern: Apostol Ch. 4, Halmos Naive Set Theory §12.
00.03.02 Binomial theorem. [NEW] Two-tier (Beginner + Intermediate); ~1000 words. Binomial coefficients, Pascal's triangle, the binomial expansion proved by induction. Worked examples: $(1 + x)^{n}$ , $\sum (k n) = 2^{n}$ , $\sum (- 1)^{k} (k n) = 0$ . Apostol Ch. 4 §4.7.
02.02.01 Real-number axioms (ordered field). [NEW] Three-tier; ~1500 words. The 10 ordered-field axioms (field 1-6, order 7-10). Apostol Ch. 3 §3.1-3.5 anchor. Cross-link to 01.01.01 field.
02.02.02 Completeness axiom (least upper bound) and the Archimedean property. [NEW] Three-tier; ~1800 words. Foundational. Statement of LUB axiom (Apostol Axiom 11); Archimedean property as theorem; density of $Q$ ; integer part; existence of $n$ -th roots; uncountability of $R$ via Cantor. Originator: Dedekind 1872, Stetigkeit und irrationale Zahlen (the originator of completeness); modern: Apostol Ch. 3, Rudin PMA Ch. 1. Load-bearing for every later real-analysis unit.
02.03.01 Sequence convergence and the monotone convergence theorem. [NEW] Three-tier; ~1500 words. $ε$ - $N$ definition, algebra of limits, monotone-convergence theorem for $R$ (proof via LUB). Apostol Ch. 10 §10.1-10.3.
02.03.02 Cauchy sequences and Bolzano-Weierstrass. [NEW] Three-tier; ~1500 words. Cauchy criterion, equivalence with convergence in $R$ , Bolzano-Weierstrass for bounded sequences. Originator: Bolzano 1817, Rein analytischer Beweis des Lehrsatzes (the originator of the IVT and the lemma now bearing his name); Cauchy 1821, Cours d'analyse. Apostol Ch. 10. Load-bearing for compactness arguments.
02.03.03 Infinite series: convergence and the standard tests. [NEW] Three-tier; ~2000 words. Partial sums, geometric series, comparison test, ratio test, root test, integral test, Cauchy condensation. Apostol Ch. 10. Worked: harmonic series, $\sum 1/ n^{2}$ , $\sum 1/ n^{p}$ .
02.03.04 Alternating series, absolute and conditional convergence, Riemann rearrangement. [NEW] Three-tier; ~1500 words. Apostol Ch. 10 §10.16-10.21. Riemann rearrangement statement-level. Originator: Riemann 1854 (Habilitationsschrift).
02.04.01 Step-function integral and the Darboux integral. [NEW] Three-tier; ~2000 words. Apostol's pedagogical entry point. Step functions, integral as a finite sum, upper / lower integrals of bounded $f$ , definition of integrability. Integrability of monotonic functions on $[a, b]$ . Apostol Part 2 Ch. 1. Originator: Darboux 1875, Mémoire sur les fonctions discontinues (Ann. Sci. ENS); modern: Apostol Part 2 Ch. 1.
02.04.02 Continuous functions and the IVT/EVT/uniform continuity package. [NEW] Three-tier; ~2000 words. Highest priority. $ε$ - $δ$ continuity in $R$ , intermediate value theorem (Bolzano), boundedness theorem, extreme value theorem (Weierstrass), uniform continuity on a compact interval (Heine-Cantor). Used silently across the entire Codex corpus. Apostol Part 2 Ch. 3. Originator: Bolzano 1817 (IVT); Weierstrass 1860s lectures (EVT).
02.04.03 Integrability of continuous functions on $[a, b]$ . [NEW] Three-tier; ~1200 words. Riemann-integrability of every continuous function on a compact interval, via uniform continuity. Apostol Part 2 Ch. 3 §3.20. Cross-links to 02.04.01 and 02.04.02.
02.05.01 Derivative and the algebra of derivatives. [NEW] Three-tier; ~1800 words. Derivative as limit, sum / product / quotient rules, chain rule. Apostol Part 2 Ch. 4 §4.1-4.6. Originator citations to Newton 1671 (Method of fluxions) and Leibniz 1684 (Acta Eruditorum).
02.05.02 Mean value theorem (Rolle, Lagrange, Cauchy). [NEW] Three-tier; ~1500 words. Foundational. Rolle's theorem, Lagrange MVT, Cauchy MVT, monotonicity test, sufficient conditions for extrema. Apostol Part 2 Ch. 4 §4.13-4.16. Originator: Rolle 1691; Lagrange 1797, Théorie des fonctions analytiques.
02.04.04 Fundamental theorems of calculus (FTC1 and FTC2). [NEW] Three-tier; ~2000 words. Foundational. FTC1 ( $F (x) = \int_{a}^{x} f$ is differentiable with $F^{'} = f$ when $f$ is continuous), FTC2 ( $\int_{a}^{b} F^{'} = F (b) - F (a)$ when $F$ is differentiable with continuous $F^{'}$ ). Integration by parts. Apostol Part 2 Ch. 5. Originator: Barrow (geometric form 1670), Newton 1671, Leibniz 1693; analytical proof Cauchy 1821-23.

Apostol-distinctive priority 2 — elementary functions, Taylor, ODEs (~9 units)

These cover the bulk of "calculus the working scientist actually uses" beyond the foundational theorems above.

02.04.05 Techniques of integration (substitution, parts, partial fractions, trig substitution). [NEW] Two-tier (Intermediate + Master); ~1500 words. Apostol Part 2 Ch. 5 §5.6-5.9.
02.04.06 Improper integrals and the comparison test. [NEW] Three-tier; ~1500 words. Apostol Part 2 Ch. 5 §5.10 + Ch. 10.
02.05.03 Inverse function theorem in one variable. [NEW] Two-tier (Intermediate + Master); ~1000 words. The 1D inverse function theorem; derivative of inverse formula. Apostol Part 2 Ch. 6 §6.7. (Multi-D version belongs in Vol. II.)
02.06.01 Logarithm as an integral. [NEW] Three-tier; ~1800 words. Apostol's signature analytical construction. $lo g x := \int_{1}^{x} d t / t$ , derivation of all standard properties via FTC, characterisation as the unique additive function with $lo g^{'} (1) = 1$ . Apostol Part 2 Ch. 6 §6.1-6.7. Originator: Mercator 1668 (Logarithmotechnia; first power series for $lo g$ ); modern analytical construction is canonically Apostol.
02.06.02 Exponential function and powers $a^{b}$ . [NEW] Three-tier; ~1500 words. $exp = lo g^{- 1}$ ; $e = exp (1)$ ; $a^{b} = exp (b lo g a)$ for $a > 0$ ; growth rate of $exp$ vs polynomials. Apostol Part 2 Ch. 6 §6.8-6.17.
02.06.03 Analytic trigonometry from $\arcsin x = \int_0^x dt/\sqrt{1-t^2}$. [NEW] Three-tier; ~1500 words. Apostol's rebuilding of $sin$ , $cos$ , $tan$ as inverses of integrals. Bridge to the geometric definitions of Ch. 2. Apostol Part 2 Ch. 6 §6.18-6.20.
02.06.04 Hyperbolic functions. [NEW] Two-tier (Beginner + Intermediate); ~800 words. $sinh$ , $cosh$ , $tanh$ , their inverses, identities. Apostol Part 2 Ch. 6 §6.22.
02.07.01 Taylor's theorem with three remainders (Lagrange, integral, Cauchy). [NEW] Three-tier; ~2000 words. High priority. All three remainder forms, with worked Taylor series of $e^{x}$ , $sin$ , $cos$ , $lo g (1 + x)$ , $arctan$ , $(1 + x)^{α}$ . L'Hôpital's rule as application. Apostol Part 2 Ch. 7. Originator: Taylor 1715, Methodus Incrementorum; Lagrange 1797 (Lagrange remainder); Cauchy 1823 (integral remainder).
02.07.02 Power series and the radius of convergence (Cauchy-Hadamard). [NEW] Three-tier; ~1500 words. Radius of convergence formula $1/ R = lim sup ∣ a_{n} ∣^{1/ n}$ ; behaviour on the circle of convergence; Abel's theorem (statement-level). Apostol Part 2 Ch. 11. Originator: Hadamard 1888 (these de doctorat); Abel 1826 (continuity at the boundary).

Apostol-distinctive priority 3 — uniform convergence, ODEs, complex (~6 units)

02.10.01 Pointwise vs uniform convergence and the Weierstrass M-test. [NEW] Three-tier; ~1500 words. Apostol Part 2 Ch. 11 §11.1-11.5. Load-bearing for term-by-term operations and for every later analysis course. Originator: Weierstrass 1880s lectures (uniform convergence is canonically his).
02.10.02 Term-by-term integration and differentiation; interchange of limit and derivative. [NEW] Three-tier; ~1200 words. Apostol Part 2 Ch. 11 §11.6-11.10.
02.08.01 First-order linear and separable ODEs. [NEW] Three-tier; ~1500 words. Integrating-factor method, separation of variables, applications (radioactive decay, mixing, Newton's law of cooling). Apostol Part 2 Ch. 8 §8.1-8.7.
02.08.02 Second-order linear ODEs with constant coefficients. [NEW] Three-tier; ~1800 words. Characteristic polynomial, real and complex roots, undetermined coefficients. Applications: simple harmonic motion, damping, resonance. Apostol Part 2 Ch. 8 §8.8-8.20.
02.09.01 Complex numbers and Euler's formula. [NEW] Three-tier; ~1500 words. Construction $C = R^{2}$ , polar form, Euler's formula via Taylor series of $exp$ , $sin$ , $cos$ , de Moivre, $n$ -th roots of unity. Apostol Part 2 Ch. 9. Originator: Hamilton 1837 (algebraic construction); Euler's formula goes back to Euler 1748, Introductio.
02.09.02 Fundamental theorem of algebra (statement and intuition). [NEW] Two-tier (Beginner + Intermediate); ~800 words. Statement-only; analytic proofs are deferred to 06-riemann-surfaces/01-* complex analysis. Apostol Part 2 Ch. 9 §9.10. Originator: Gauss 1799 (dissertation; first rigorous proof).

Apostol-distinctive priority 4 — applications, history, deepenings (~5 items)

02.04.07 Applications of integration (area, volume, arc length, work, average value). [NEW] Two-tier (Beginner + Intermediate); ~1500 words. Application bouquet from Apostol Part 2 Ch. 2; one consolidated unit covering the standard physical applications.
DEEPEN 02.01.05 metric-space: add a Master subsection on the calculus-textbook framing of Cauchy completeness specifically for $R$ (cross-link to the new 02.03.02 unit). [DEEPEN] ~400 words.
DEEPEN 02.01.02 continuous-map: add a Master subsection cross-linking to 02.04.02 for the IVT/EVT calculus-textbook statements that are the topological-compactness consequences in the existing unit. [DEEPEN] ~300 words.
DEEPEN 01.01.01 field: add a Master subsection listing the Apostol-style ordered-field axioms (Apostol Ch. 3 §3.1-3.5) explicitly, cross-linking to 02.02.01. [DEEPEN] ~300 words.
ENRICH 00.02.05 function: add Apostol Vol. I as a citation in the Intermediate tier_anchors (Apostol Ch. 2 §2.4 has the classical "function as graph" framing). [ENRICH] ~50 words + front-matter edit.
02.E1 Apostol Vol. I exercise pack [NEW]. Deferred. The full ~1800-exercise inventory is outside the scope of the P3-lite punch-list and lives in a dedicated exercise-pack pass after the priority-1+2 theorem-layer batch closes. Mentioned here as a placeholder so the sequencing layer captures it.

§4 Implementation sketch (P3 → P4)

For full Apostol Vol. I coverage, items 1-23 are the minimum priority-1+2 set (23 new units, ~36 000 words of content). Items 24-29 (priority 3, 6 more units) close the uniform convergence + ODEs

complex blocks. Items 30-34 (priority 4) close applications and deepen / cross-link existing units. Total: ~29 new units + 4 deepenings + 1 enrichment.

Production estimate (mirroring earlier batches):

~3 hours per new unit (research + draft + validate at 27/27 + Lean stub + Bridge / Synthesis paragraphs in real prose, not the templated form).
29 new units × 3 hours = ~87 hours of focused production.
At 4-6 production agents in parallel, this fits in a 9-12 day window with an integration agent stitching outputs.

This is the largest single-book production load on the Fast Track. By comparison, Hatcher's punch-list is ~14 units (priority-1+2); Cannas's is ~12; Donaldson's is ~10. Apostol Vol. I is ~29 because Codex starts from a near-empty base in the calculus region.

Batching recommendation. Decompose into four production sub-batches to keep agent context windows manageable and to parallelise integration:

Batch A (foundations, ~6 units): induction (00.03.01), binomial (00.03.02), real-number axioms (02.02.01), completeness + Archimedean (02.02.02), sequence convergence
- MCT (02.03.01), Cauchy + B-W (02.03.02). Run first — these are prerequisites for every later unit.
Batch B (integration + continuity + FTC + differentiation, ~9 units): step-function integral (02.04.01), continuity / IVT / EVT (02.04.02), continuous-integrability (02.04.03), FTC (02.04.04), techniques (02.04.05), improper integrals (02.04.06), derivative (02.05.01), MVT (02.05.02), inverse function 1D (02.05.03). Run after Batch A.
Batch C (elementary functions + Taylor + power series, ~7 units): log-as-integral (02.06.01), exponential (02.06.02), analytic trig (02.06.03), hyperbolic (02.06.04), Taylor (02.07.01), power series (02.07.02), and series-tests (02.03.03, 02.03.04 — could move into Batch A or here depending on which agents have capacity).
Batch D (uniform convergence + ODEs + complex + applications, ~7 units): uniform convergence (02.10.01), term-by-term (02.10.02), 1st-order ODE (02.08.01), 2nd-order ODE (02.08.02), complex + Euler (02.09.01), FTA statement (02.09.02), applications (02.04.07). Can run mostly in parallel with Batch C.

Apostol's chapter prerequisites (sets → reals → integral → continuity → derivative → FTC → log/exp → Taylor → ODE / complex / series) drive the inter-batch ordering: Batch A must complete and be integrated before B starts, since every unit in B depends on at least one A unit (LUB, MCT, or B-W). Batches C and D can begin once B is substantially complete (FTC and continuous integrability need to be in place before log-as-integral).

Originator-prose target. Apostol himself is a 1967 textbook synthesis — not an originator-text. The originator citations for the new priority-1+2 units (each unit's Master section should quote / paraphrase the originator):

Real numbers, completeness: Dedekind 1872, Stetigkeit und irrationale Zahlen; Cantor 1872 (Cauchy-completion construction).
Sequences, IVT, B-W: Bolzano 1817, Rein analytischer Beweis des Lehrsatzes (the IVT proof that does not use geometric intuition); Cauchy 1821, Cours d'analyse.
Integration: Cauchy 1823, Résumé des leçons sur le calcul infinitésimal (first rigorous integral); Riemann 1854 (Habilitationsschrift — the Riemann definition); Darboux 1875 (the upper / lower integrals).
Differentiation, MVT: Rolle 1691; Lagrange 1797, Théorie des fonctions analytiques; Cauchy 1823 (Cauchy MVT).
FTC: Barrow 1670 (geometric); Newton 1671 (Method of fluxions); Leibniz 1693.
$lo g$ , $exp$ : Mercator 1668, Logarithmotechnia; Euler 1748, Introductio in analysin infinitorum.
Taylor: Taylor 1715, Methodus Incrementorum; Maclaurin 1742; Lagrange 1797 (Lagrange remainder); Cauchy 1823 (integral remainder).
Power series, uniform convergence: Cauchy 1821; Abel 1826 (Abel's theorem); Weierstrass 1880s lectures (uniform convergence canonically his); Hadamard 1888 (Cauchy-Hadamard).
Complex numbers, Euler's formula: Euler 1748, Introductio; Hamilton 1837 (algebraic construction); Gauss 1799 (FTA).
ODEs: Bernoulli (Johann) 1696 (separation); Lagrange 1762-65 (variation of parameters; characteristic polynomial method).

Notation crosswalk. Apostol Vol. I uses:

$\int_{a}^{b} f (x) d x$ for the Riemann/Darboux integral; $\overline{\int}$ and $\underline{\int}$ for upper/lower.
$\sum_{k = m}^{n} a_{k}$ for finite sums; $\sum_{n = 1}^{\infty} a_{n}$ for series.
$f^{'}$ , $f^{''}$ , $f^{(n)}$ for derivatives; never Leibniz $df / d x$ in the main text (Apostol's choice).
$lo g x$ (natural log) — never $ln$ .
$arcsin$ , $arccos$ , $arctan$ — never $sin^{- 1}$ .
$f \in C^{n} [a, b]$ for $n$ -times continuously differentiable on $[a, b]$ .
"Theorem 3.7" cross-references (chapter.theorem-number).

A notation/apostol-vol1.md crosswalk file should record this. The " $lo g = ln$ " alignment is the most-cited entry.

§5 What this plan does NOT cover

A line-number-level inventory of every named theorem and exercise in Apostol Vol. I's ~666 pages. (Would take the full P1 audit; deferred unless the priority-1+2 punch-list expands.)
The exercise pack. Apostol Vol. I has ~1800 exercises in the single-variable Chs. 1-11; producing a Codex equivalent (02.E1 Apostol-exercise-pack) is a P3-priority follow-up after the priority-1+2 theorem-layer batch closes.
Apostol Chs. 12-16 (vector algebra, analytic geometry, vector calculus prelude, linear spaces, linear transformations + matrices). These belong in the Apostol Vol. II audit (FT 0.3) since they are the bridge into multivariable / linear algebra; covering them here would duplicate that audit's punch-list.
The Vol. I appendices (set-theoretic definitions, mutual induction, complex-number formal construction). Defer; covered by the existing 00.02.05 function unit and by 01.01.01 field's Master section once deepened (item 33 above).
Caveat: Riemann vs Darboux integral equivalence. Apostol's upper/lower-integral definition is equivalent to the tagged-partition Riemann definition for bounded $f$ on a compact interval, but the two definitions are not literally the same. The proposed 02.04.01 unit uses the Darboux framing (Apostol's pedagogical choice); a Master-section remark or a separate Master cross-link should record the Riemann tagged-partition definition and prove the equivalence. Logged here so the production agent doesn't omit it.
Lebesgue integration, measure theory. Out of scope for Apostol Vol. I. Lives in a future 02-analysis/12-measure-and-lebesgue/ sub-chapter (not on Apostol's audit; covered by Royden, Folland, Rudin RCA, etc.).
Multivariable calculus — Apostol Vol. II audit (FT 0.3).
The figures. Apostol's text is figure-heavy; the figure-rendering infrastructure does not yet exist in Codex. Pictorial intuition must be reproduced in prose for now. Curriculum-wide deferred item.

§6 Acceptance criteria for FT equivalence (Apostol Vol. I)

Per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4 and §9, Apostol Vol. I is at equivalence-coverage when:

≥95% of Apostol Vol. I's named theorems map to Codex units (currently ~3-5%; after priority-1 units (1-14) this rises to ~70%; after priority-1+2 (1-23) to ~90%; after priority-3 (24-29) to ~95%; after priority-4 deepenings (30-34) the qualitative gap is closed).
≥80% of Apostol's exercises have a Codex equivalent (currently ~0%; closing this requires the dedicated 02.E1 exercise-pack pass per §5).
≥90% of Apostol's worked examples are reproduced in some Codex unit (currently ~0%; the priority-1+2 batch's worked-example density brings this to ~60%; the remainder needs worked-example-densification in a follow-up pass).
A notation/apostol-vol1.md crosswalk exists.
For every chapter dependency in Apostol Vol. I (Sets / induction → Reals → Integral → Continuity → Derivative → FTC → Log/Exp / Trig → Taylor → ODE / Complex / Series → Uniform convergence), there is a corresponding prerequisites arrow chain in Codex's DAG between the relevant 00.03.*, 02.02-10.* units. Building the units automatically builds the chain since the prereqs are encoded in each unit's front-matter.
Pass-W weaving connects the new 02.02-10.* units to the existing 02.01-topology units (via the metric-space Cauchy/IVT/EVT cross-links above) and to the existing 02.11-functional-analysis units (via the Banach-space / completeness / Cauchy-criterion cross-links — every functional-analysis unit currently assumes the calculus-level real-analysis foundation that Apostol provides).

The 23 priority-1+2 units alone close most of the theorem-layer equivalence gap. Priority-3 closes ODEs, complex numbers, and uniform convergence. Priority-4 deepens existing units so the topological foundation (02.01.*) cross-cites the new calculus foundation (02.02-10.*) cleanly. The exercise pack is a separate large pass.

Strategic note: this is the highest-leverage Fast Track audit so far. Every later analysis-flavoured unit in Codex (02.11-functional-analysis, 06-riemann-surfaces, the entire analytical content of 03-modern-geometry and 04-algebraic-geometry) implicitly assumes the Apostol Vol. I foundation. Closing this gap unblocks the depth-deepening of every existing analysis-flavoured unit elsewhere in the corpus. This is the anchor of the entire analysis strand. If the Fast Track had to produce one book's equivalence-coverage before any other, this is the one.

Apostol — *Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra* (Fast Track 0.2) — Audit + Gap Plan