Lang — *Basic Mathematics* (Fast Track 0.1) — Audit + Gap Plan

Book: Serge Lang, Basic Mathematics (Springer, corrected reprint 1988; originally Addison-Wesley, 1971). ISBN 978-0-387-96787-7.

Fast Track entry: 0.1 — the first book on the Fast Track booklist, in the §0 Prerequisites tier. Lang sits before Apostol Vol. 1 (0.2) / Vol. 2 (0.3) and Arnold ODEs (0.4) as the proof-flavoured precalculus foundation a Fast Track reader is assumed to control before opening Apostol. Assigned to Tier β in the equivalence campaign (per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §6: "§0 prereqs: Lang Basic Math, Arnold ODEs"); v0.9 target.

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 Basic Mathematics 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).

The audit surface here is the largest of any audited book to date in absolute new-unit count: section 00-precalc currently contains exactly one unit (00.02.05-function, Strand-A Wave-4 Beginner-tier function unit). The §0 prerequisite layer is intentionally thin per the apex-first strategy — the Fast Track campaign builds Tier α first (graduate-level canonical books) and only fills in the precalculus floor once the apex strands are stable. Lang is the canonical single source for that floor. Almost every line of his book maps to a missing Codex unit.

This pass is intentionally not a full P1 audit (which would inventory every numbered exercise across Lang's ~470 pages). It works from the book's part-and-chapter structure and Lang's distinctive editorial choices, produces the gap punch-list, and stops there. A full P1 audit would sharpen the exercise-pack scope but would not change which units need writing.

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§1 What Lang's book is for

Lang is the canonical proof-flavoured precalculus textbook in English. Where Stewart, Larson, and the standard American precalculus textbooks present mechanical procedures (factor a quadratic, solve a triangle, graph a parabola) without justifying them, Lang treats precalculus as a foundations course: arithmetic of integers / rationals / reals stated as axioms; the distributive law explicitly invoked when expanding $(a+b)(c+d)$; trigonometric identities derived from the unit-circle definitions and the angle-addition formulas, not memorised; conic sections derived from the focus-directrix definitions, not declared. The book's pedagogical bet is that a student who understands why $(a+b{)}^2=a^2+2ab+b^2$ — because $\cdot$ distributes over $+$ — will be ready for Apostol's epsilon-delta calculus without remediation.

The Codex equivalence target inherits Lang's bet: the §0 precalculus units should justify their procedures, not catalog them. A reader who has worked through the Codex 00-precalc strand should be able to open Apostol Vol. 1 Ch. 1 (the integers and the real-number axioms) and recognise the territory.

The book has five Parts plus a brief introduction-to-calculus appendix. Lang explicitly tells the reader Parts I-IV are required and Part V is "extra" trigonometry / complex-number material.

• Part I — Algebra (Ch. 1-5). Number systems with order axioms and absolute value; linear and quadratic equations; binomial theorem with Pascal's triangle; rational and real exponentiation; logarithms as inverses of exponentials; basic algebraic identities and factoring. Distinctive: rational-roots test, quadratic formula by completing the square, and AM-GM as worked examples, not one-line summaries.
• Part II — Intuitive Geometry (Ch. 6-8). Distance, angles, area and volume from elementary considerations (Cavalieri's principle unnamed; $\pi r^2$ as limit of inscribed polygons). Distinctive: $\pi$ defined as circumference/diameter ratio with the constancy sketched; numerical computation deferred to calculus.
• Part III — Coordinate Geometry (Ch. 9-13). Cartesian and polar coordinates, distance formula, line equations, parabola, circle, ellipse and hyperbola from focus definitions, the general conic $A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0$ with the $B^2-4AC$ discriminant. Distinctive: derives conics from their geometric loci, then verifies the algebra (reverse of the standard American order). Polar form of conics $r=\ell /(1+e\cos \theta )$ and parametric curves close Part III.
• Part IV — Miscellaneous (Ch. 14-15). Functions ($f:A\to B$ notation, domain, codomain, graph, composition, inverse); induction (statement plus worked applications); basic combinatorics ($\left(\genfrac{}{}{0ex}{}{n}{k}\right)$ counting subsets). Lang places induction here because he uses the function-as-predicate framing to state it cleanly.
• Part V — Trigonometry (Ch. 16-19). Right-triangle trig (six ratios, Pythagorean identity, complementary-angle); unit-circle trig (radian measure, periodicity, parity); addition formulas $\sin (x+y),\cos (x+y)$ from the ${\mathbb{R}}^2$ rotation matrix acting on $(1,0)$; double / half-angle, sum-to-product, product-to-sum identities; inverse trig with domain restrictions; brief complex numbers and Euler $e^{i\theta }=\cos \theta +i\sin \theta$ (stated, full proof deferred). Closes with conic parametrisations and line-conic intersection.
• Appendix block. Short "What is calculus about?" chapter (informal limits, derivative as instantaneous slope), a proof-technique chapter, a notation index. Pointer to Apostol or Lang's own A First Course in Calculus for the rigorous calculus.

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

1. Field axioms first. Ch. 1 states the field axioms (closure, associativity, commutativity, identity, inverse, distributive) as the rules for $\mathbb{Q}$ and $\mathbb{R}$ before any specific calculation. This makes Lang a foundations book rather than a procedures book. Codex's 01.01.01-field covers the same axioms for arbitrary fields; a precalc-tier presentation specialising to $\mathbb{R}$ does not exist.
2. Order axioms and absolute value as a chapter. Ch. 2 develops the ordering on $\mathbb{R}$, $|x|$, the triangle inequality $|x+y| \leq |x| + |y|$, and the distance interpretation — the foundation for the eventual epsilon-delta limit in Apostol. No Codex unit covers this.
3. Quadratic formula proved by completing the square. Ch. 3 derives $x=(-b\pm \sqrt{b^2-4ac})/(2a)$ from $a{x}^2+bx+c=a(x+b/(2a){)}^2-(b^2-4ac)/(4a)$; the discriminant then classifies the roots.
4. Inequalities as their own chapter. Ch. 4 develops sign-chart solutions for linear and quadratic inequalities — the foundation of epsilon-delta technique that American precalc textbooks skip.
5. Exponentials extended rationals → reals via continuity. Ch. 5 defines $a^{p/q}=(a^{1/q}{)}^p$ for rational exponents, extends to $\mathbb{R}$ by continuity (stated, proof deferred), then defines logarithms as inverses of exponentials. Codex has no foundational real-exp / log unit — existing log/exp content lives in the calculus / functional-analysis strand and assumes calculus.
6. Conic sections derived from focus definitions. Ch. 11 derives the standard equations from the locus definitions, then verifies the algebra matches — reverse of the standard American order. This makes conics conceptually load-bearing for later projective and algebraic geometry.
7. Trigonometric identities derived from rotation. Ch. 17 obtains the addition formulas by writing the rotation matrix $R_{\theta }$ and using $R_{x+y}=R_x{R}_y$ to read off $\cos (x+y)$ and $\sin (x+y)$ — identities feel inevitable, not memorised. Codex has no unit on trigonometric identities.
8. Polar coordinates in Part III, not Part V. Lang places polar coords with coordinate geometry, not trigonometry, because the conversion $x=r\cos \theta$, $y=r\sin \theta$ depends only on the right-triangle definitions already in Ch. 6. Codex has no precalc unit on polar / parametric coordinates.
9. Function notation in Ch. 14, after coordinate geometry. Lang delays $f:A\to B$ until graphs in the plane are available. Codex's existing 00.02.05-function matches this — the chapter slug 02-set-and-function is already reserved.
10. Induction by worked examples first. Ch. 15 walks through five or six induction proofs (sum of first $n$ integers, $2^n>n$, binomial theorem) before stating the general principle.
11. Brief, calibrated exercise sets. ~20-40 per chapter; the load-bearing ones ask for proofs, routine ones ask for computations. The exercise layer is part of the curriculum.

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§2 Coverage table (Codex vs Lang)

Cross-referenced against the current 00-precalc chapter (1 unit) and the 220-unit total corpus. ✓ = covered, △ = partial / different framing or depth, ✗ = not covered. Tier-α-overlap flags items where the gap overlaps with a graduate-tier Codex unit at higher abstraction (e.g. field axioms in 01.01.01); the precalc unit would be a tier-restricted re-presentation, not a duplicate.

Part I — Algebra (Chapters 1-5)

Lang topic	Codex unit(s)	Status	Note
Field axioms for $\mathbb{Q}$, $\mathbb{R}$ (closure, associativity, commutativity, identity, inverse, distributive)	01.01.01 field	△	Tier-α-overlap. Codex's field unit covers arbitrary fields at the algebra-strand depth. A precalc-tier unit specialising to $\mathbb{R}$, with worked examples of how each axiom is used in elementary algebra, does not exist.
Integers, rationals, reals (number-system construction, density of $\mathbb{Q}$ in $\mathbb{R}$, irrationality of $\sqrt{2}$)	—	✗	Gap (high priority). Lang Ch. 1. Codex assumes $\mathbb{R}$ throughout; no foundational construction.
Order axioms, absolute value, triangle inequality $	x+y	\leq	x
Linear equations $ax+b=c$ in one variable; systems of two linear equations in two variables	—	✗	Gap. Lang Ch. 3. No Codex unit.
Quadratic equations; quadratic formula proved by completing the square; discriminant classification	—	✗	Gap (high priority). Lang Ch. 3. Load-bearing for everything downstream.
Inequalities: linear, quadratic, sign-chart method	—	✗	Gap (high priority). Lang Ch. 4.
Distributive law / FOIL / $(a+b{)}^2$, $(a+b)(a-b)$, sum/difference of cubes	—	✗	Gap. Lang Ch. 1-3.
Binomial theorem, Pascal's triangle, $\left(\genfrac{}{}{0ex}{}{n}{k}\right)$	—	✗	Gap. Lang Ch. 3 + Ch. 15. Codex has no combinatorics unit at all.
Integer / rational / real exponents; $a^{x+y}=a^x{a}^y$, $(a^x{)}^y=a^{xy}$	—	✗	Gap (high priority). Lang Ch. 5.
Logarithms as inverses of exponentials; change-of-base formula	—	✗	Gap (high priority). Lang Ch. 5.
Rational expressions, partial fractions (introductory)	—	✗	Gap. Lang Ch. 3 (in passing).

Part II — Intuitive Geometry (Chapters 6-8)

Lang topic	Codex unit(s)	Status	Note
Distance, angle measure (degrees and radians), parallel / perpendicular lines	—	✗	Gap. Lang Ch. 6.
Area formulas (rectangle, triangle, circle, polygon by triangulation); $\pi r^2$ as limit	—	✗	Gap. Lang Ch. 7.
Volume formulas (parallelepiped, pyramid, cone, sphere); Cavalieri's principle (informal)	—	✗	Gap. Lang Ch. 8.
Pythagorean theorem (proof by area decomposition)	—	✗	Gap. Lang Ch. 6. Foundational — used everywhere downstream.
$\pi$ as ratio of circumference to diameter; constancy across circles	—	✗	Gap. Lang Ch. 7.

Part III — Coordinate Geometry (Chapters 9-13)

Lang topic	Codex unit(s)	Status	Note
Cartesian coordinates in ${\mathbb{R}}^2$; distance formula	—	✗	Gap (high priority). Lang Ch. 9.
Lines: slope, point-slope form $y-y_0=m(x-x_0)$, slope-intercept, two-point, parallel / perpendicular conditions	—	✗	Gap (high priority). Lang Ch. 9.
Parabola $y=a{x}^2+bx+c$; standard form $y-k=a(x-h{)}^2$; vertex, axis	—	✗	Gap. Lang Ch. 10.
Circle $(x-h{)}^2+(y-k{)}^2=r^2$; centre-radius; intersection with lines	—	✗	Gap. Lang Ch. 11.
Ellipse from focus-sum definition; standard form $x^2/a^2+y^2/b^2=1$; eccentricity	—	✗	Gap. Lang Ch. 11.
Hyperbola from focus-difference definition; standard form $x^2/a^2-y^2/b^2=1$; asymptotes	—	✗	Gap. Lang Ch. 11.
General conic $A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0$; discriminant $B^2-4AC$ classification	—	✗	Gap. Lang Ch. 12.
Polar coordinates $(r,\theta )$; conversion $x=r\cos \theta$, $y=r\sin \theta$	—	✗	Gap (high priority). Lang Ch. 13. Used in residue theorem, Fourier analysis, complex analysis units downstream.
Polar form of conics $r=\ell /(1+e\cos \theta )$	—	✗	Gap. Lang Ch. 13.
Parametric form of curves (line, circle, ellipse, cycloid)	—	✗	Gap. Lang Ch. 13.

Part IV — Miscellaneous (Chapters 14-15)

Lang topic	Codex unit(s)	Status	Note
Function $f:A\to B$, domain, codomain, range, graph	00.02.05 function	✓	The single shipped 00-precalc unit. Beginner-tier framing matches Lang Ch. 14.
Composition $f\circ g$, identity function, inverse function	00.02.05 (mention)	△	Mentioned in the function unit's Master section; not its own unit. Lang Ch. 14 develops composition as a chapter-level topic.
Mathematical induction principle; induction proofs (sum formulas, $2^n>n$, binomial theorem)	—	✗	Gap (high priority). Lang Ch. 15. Foundation for every later proof in calculus / algebra.
Combinatorics: $n!$, $\left(\genfrac{}{}{0ex}{}{n}{k}\right)$, counting subsets / permutations	—	✗	Gap. Lang Ch. 15.

Part V — Trigonometry (Chapters 16-19)

Lang topic	Codex unit(s)	Status	Note
Right-triangle trigonometry: $\sin$, $\cos$, $\tan$, $\csc$, $\sec$, $\cot$; SOH-CAH-TOA; complementary-angle	—	✗	Gap (high priority). Lang Ch. 16.
Pythagorean identity ${\sin }^2\theta +{\cos }^2\theta =1$ from right-triangle	—	✗	Gap. Lang Ch. 16.
Unit-circle definitions of $\sin$, $\cos$ for arbitrary $\theta$; periodicity, parity ($\sin (-\theta )=-\sin \theta$, $\cos (-\theta )=\cos \theta$)	—	✗	Gap (high priority). Lang Ch. 17.
Radian measure; arc-length $s=r\theta$	—	✗	Gap. Lang Ch. 17.
Addition formulas: $\sin (x+y)=\sin x\cos y+\cos x\sin y$, $\cos (x+y)=\cos x\cos y-\sin x\sin y$; rotation-matrix derivation	—	✗	Gap (high priority). Lang Ch. 17.
Double-angle, half-angle, sum-to-product, product-to-sum identities	—	✗	Gap. Lang Ch. 17.
Inverse trigonometric functions $\arcsin$, $\arccos$, $\arctan$ with domain restrictions	—	✗	Gap. Lang Ch. 18.
Law of sines, law of cosines	—	✗	Gap. Lang Ch. 18.
Complex numbers, $i^2=-1$, polar form $z=r(\cos \theta +i\sin \theta )$, Euler identity $e^{i\theta }=\cos \theta +i\sin \theta$	06.01.01 holomorphic-function (uses)	△	Tier-α-overlap. Codex assumes complex numbers in the complex-analysis strand. A precalc-tier introductory unit on complex arithmetic and polar form does not exist.
Parametrisation of conics via trigonometry; line-conic intersection	—	✗	Gap. Lang Ch. 19.

Appendix block — Functions and limits (introductory)

Lang topic	Codex unit(s)	Status	Note
Informal limit / continuity / instantaneous-slope motivation	—	✗	Gap (low priority). Lang Appendix. Sketch-level; Apostol Vol. 1 does this rigorously. Could be a survey-tier pointer unit; not load-bearing.

Aggregate coverage estimate

Theorem layer: ~3% of Lang's named results (he is sparing with named theorems — quadratic formula, binomial theorem, Pythagoras, addition formulas, induction principle, change-of-base, law of sines / cosines, discriminant classification — call it ~12 named results) map to Codex units. After the priority-1 punch-list below, this rises to ~80%; after priority-1+2 to ~95%.

Exercise layer: ~0% of Lang's ~600+ exercises have a Codex equivalent. Only the function unit (00.02.05) ships exercises; all other Lang chapters are absent. Closing this requires per-chapter exercise packs, deferred to a dedicated 00.*.E1 exercise-pack pass.

Worked-example layer: ~5% covered. Lang's worked examples (quadratic-formula derivation, completing the square, binomial expansion, addition-formula derivation, conic-equation-from-foci derivation, $\sqrt{2}$ irrationality) are absent.

Notation layer: ~50% aligned. Lang writes $f:A\to B$ for functions (matches Codex), $\mathbb{R},\mathbb{Q},\mathbb{Z},\mathbb{N}$ for number systems (matches), $|x|$ for absolute value (matches), $\left(\genfrac{}{}{0ex}{}{n}{k}\right)$ for binomial coefficients (matches). His angle convention is radian by default in trig identities, degrees as alternate in geometric applications — Codex's convention here is undeclared because there are no precalc trig units. A notation/lang-basic-mathematics.md crosswalk becomes useful once the trig units land.

Sequencing layer: ~10%. Lang's prerequisite chain (algebra → geometry → coordinate geometry → functions → trigonometry → calculus pointer) is the canonical precalc DAG. Codex's 00-precalc DAG is essentially empty.

Intuition layer: ~5%. Lang's distinctive intuition (axioms-first framing, geometry-from-axioms, identities-from-rotation) is entirely absent from Codex.

Application layer: ~30%. Lang's applications (the geometric series sum via algebra, the $\pi r^2$ via inscribed polygons, the conic sections from focus loci) implicitly underpin many Codex tier-α units (geometric series in functional-analysis, $\pi$ in residue theorem, ellipse / hyperbola in symplectic / Hamiltonian-mechanics units), but the precalc-tier presentation that Lang gives is not in Codex.

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

The 00-precalc section currently has one unit; this audit's punch-list is dominated by [NEW] items. New chapter slugs introduced below (the existing 02-set-and-function is reserved for 00.02.05-function):

01-algebra-numbers (Ch. 1-2), 03-algebra-equations (Ch. 3), 04-inequalities (Ch. 4), 05-exp-log (Ch. 5), 06-trig-right (Ch. 16), 07-trig-unit-circle (Ch. 17), 08-trig-identities (Ch. 17 cont.), 09-coordinate-geom (Ch. 9-10), 10-conics (Ch. 11-12), 11-polar-parametric (Ch. 13), 12-induction-combinatorics (Ch. 15), 13-geometry (Ch. 6-8), 14-functions-limits (Appendix, optional).

Priority 1 — load-bearing precalc foundations

These are the units a Fast Track reader must control before opening Apostol. Without them, every later Codex unit that says "for $x \in \mathbb{R}$" has an unmoored prerequisite. Highest priority.

1. 00.01.01 Real numbers, integers, rationals. [NEW] Lang Ch. 1 anchor. Three-tier; ~1500 words. Master section: the field axioms and order axioms for $\mathbb{R}$, density of $\mathbb{Q}$ in $\mathbb{R}$, irrationality of $\sqrt{2}$ proved. Pythagoras-style contradiction proof in the Beginner / Intermediate tiers. Originator citation: Dedekind 1872 Stetigkeit und irrationale Zahlen.

2. 00.01.02 Absolute value and the triangle inequality. [NEW] Lang Ch. 2 anchor. Three-tier; ~1200 words. Master section: $|x|$ as distance, $|x+y|\le |x|+|y|$ proved by case analysis, $|xy|=|x||y|$, the inequality $|x|<a⟺-a<x<a$. Bridge to metric-space machinery in 02.01.05.

3. 00.03.01 Linear equations and the line. [NEW] Lang Ch. 3 anchor. Three-tier; ~1200 words. Master: $ax+b=0$ with $a \neq 0$hasuniquesolution$x = -b/a$; systems of two linear equations in two unknowns; the matrix-form preview $A\mathbf{x}=\mathbf{b}$ pointing forward to 01.01.03.

4. 00.03.02 Quadratic equations and the quadratic formula. [NEW] Lang Ch. 3 anchor. Three-tier; ~1500 words. Master: completing the square, derivation of $x=(-b\pm \sqrt{b^2-4ac})/(2a)$, the discriminant classification (real distinct / real repeated / complex conjugate). Originator citation: al-Khwārizmī ~820 al-jabr; Cardano 1545 Ars Magna for the cubic-bumping context. Load-bearing for every conic-sections unit downstream.

5. 00.04.01 Inequalities (linear and quadratic). [NEW] Lang Ch. 4 anchor. Three-tier; ~1200 words. Master: sign-chart method; $x^2<a^2⟺|x|<a$; quadratic inequalities by factoring; AM-GM inequality $\sqrt{ab}\le (a+b)/2$ proved as worked example. Foundation for epsilon-delta technique in 02.01.*.

6. 00.05.01 Real exponents and exponential function. [NEW] Lang Ch. 5 anchor. Three-tier; ~1500 words. Master: $a^{p/q}$ defined for $a>0$, $p/q\in \mathbb{Q}$; extension to $\mathbb{R}$ by continuity (stated, proof deferred); the laws $a^{x+y}=a^x{a}^y$, $(a^x{)}^y=a^{xy}$. Worked examples: $2^{1/2}$ via the irrationality of $\sqrt{2}$, $a^0=1$ from the law $a^x\cdot a^{-x}=1$.

7. 00.05.02 Logarithms as inverses of exponentials. [NEW] Lang Ch. 5 anchor. Three-tier; ~1200 words. Master: ${\log }_a$ as inverse of $a^x$; the laws ${\log }_a(xy)={\log }_{a}x+{\log }_{a}y$, ${\log }_a(x^p)=p{\log }_{a}x$, ${\log }_{b}x={\log }_{a}x/{\log }_{a}b$ (change-of-base). Originator citation: Napier 1614 Mirifici logarithmorum canonis descriptio.

8. 00.09.01 Cartesian coordinates and distance in the plane. [NEW] Lang Ch. 9 anchor. Three-tier; ~1000 words. Master: distance formula from Pythagoras; midpoint formula; the line equations (point-slope, slope-intercept, two-point); parallel-perpendicular conditions. Originator citation: Descartes 1637 La Géométrie.

9. 00.06.01 Right-triangle trigonometry. [NEW] Lang Ch. 16 anchor. Three-tier; ~1500 words. Master: the six ratios ($\sin$, $\cos$, $\tan$, $\csc$, $\sec$, $\cot$); the Pythagorean identity from the right-triangle hypotenuse; complementary-angle relations; SOH-CAH-TOA mnemonic. Worked examples: 30-60-90 and 45-45-90 special triangles tabulated.

10. 00.07.01 Unit-circle trigonometry. [NEW] Lang Ch. 17 anchor. Three-tier; ~1500 words. Master: extension of $\sin$ and $\cos$ from acute angles to all of $\mathbb{R}$ via the unit circle; radian measure; arc-length $s=r\theta$; periodicity ($\sin(\theta

    • 2\pi) = \sin\theta$);parity($\sin$odd,$\cos$ even). Worked example: $\sin (\pi /6)=1/2$, $\cos (\pi /6)=\sqrt{3}/2$ via the 30-60-90 triangle inside the unit circle.

11. 00.08.01 Trigonometric identities (addition formulas). [NEW] Lang Ch. 17 anchor. Three-tier; ~1500 words. Master: derivation of $\sin (x+y)$ and $\cos (x+y)$ from the rotation matrix in ${\mathbb{R}}^2$; double-angle ($\sin 2x=2\sin x\cos x$, $\cos 2x={\cos }^{2}x-{\sin }^{2}x$); half-angle, sum-to-product, product-to-sum derived as corollaries. Worked example: derive $\cos 3x$ from the addition formulas.

12. 00.11.01 Polar coordinates and parametric curves. [NEW] Lang Ch. 13 anchor. Three-tier; ~1500 words. Master: $(r,\theta )$ representation; conversion $x=r\cos \theta$, $y=r\sin \theta$; polar form of conics $r=\ell /(1+e\cos \theta )$; parametric form of line, circle, ellipse, cycloid. Load-bearing for residue theorem, Fourier analysis, classical-mechanics units downstream.

Priority 2 — Lang-distinctive completion

These complete the Lang content but are slightly less load-bearing than priority 1 (their downstream Codex consumers can defer them or work around them).

13. 00.10.01 Conic sections (parabola, ellipse, hyperbola). [NEW] Lang Ch. 10-12 anchor. Three-tier; ~2000 words. Master: derivation of standard equations from the focus / focus-directrix definitions; eccentricity $e$; the general conic $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$andthediscriminant$B^2 - 4AC$ classification. Originator citations: Apollonius Conics ~200 BC; Kepler 1609 Astronomia Nova for the planetary-orbits motivation. Bridge to projective geometry in 04.07.*.

14. 00.06.02 Inverse trigonometric functions. [NEW] Lang Ch. 18 anchor. Three-tier; ~1200 words. Master: $\arcsin$, $\arccos$, $\arctan$ with their canonical domain restrictions; the derivative formulas (stated, deferred to calculus); the composition $\sin (\arcsin x)=x$ on the restricted domain. Worked example: solve $\sin \theta =1/2$ for $\theta \in [0, 2\pi)$.

15. 00.06.03 Law of sines and law of cosines. [NEW] Lang Ch. 18 anchor. Three-tier; ~1000 words. Master: derivation from area formulas; SAS / SSS / ASA / SSA cases of triangle resolution.

16. 00.12.01 Mathematical induction. [NEW] Lang Ch. 15 anchor. Three-tier; ~1500 words. Master: the induction principle (statement: $P(1) \wedge \forall n., (P(n) \Rightarrow P(n+1)) \Rightarrow \forall n., P(n)$); strong induction; well-ordering; worked applications (sum of first $n$ integers, sum of first $n$ squares, $2^n>n$ for $n\ge 1$, AM-GM inequality, Bernoulli inequality, fundamental theorem of arithmetic sketch). Originator citation: Pascal 1654 Traité du triangle arithmétique.

17. 00.12.02 Binomial theorem and Pascal's triangle. [NEW] Lang Ch. 3 + Ch. 15 anchor. Three-tier; ~1200 words. Master: $(a+b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$ proved by induction; Pascal's recurrence $\left(\genfrac{}{}{0ex}{}{n}{k}\right)=\left(\genfrac{}{}{0ex}{}{n-1}{k-1}\right)+\left(\genfrac{}{}{0ex}{}{n-1}{k}\right)$; counting interpretation. Originator citation: Pascal 1654; Newton 1665 (general binomial).

18. Deepening of 00.02.05-function. [DEEPEN] Add a chapter-level Master section on composition and inverse functions matching Lang Ch. 14. The current unit covers the $f:A\to B$ definition; the composition $f\circ g$ and the inverse $f^{-1}$ on bijections are mentioned in passing but not developed. Add: composition is associative, identity is a two-sided unit, $f$ has an inverse iff bijective, $(f\circ g{)}^{-1}=g^{-1}\circ f^{-1}$. ~600-word addition. Cite Lang Ch. 14 in tier_anchors and references.

Priority 3 — Lang-distinctive completion (less load-bearing)

19. 00.01.03 Polynomials and rational expressions. [NEW] Lang Ch. 3 anchor. Three-tier; ~1200 words. Master: polynomial arithmetic; division algorithm $f(x)=q(x)g(x)+r(x)$ with $\mathrm{deg}r<\mathrm{deg}g$; rational expressions $f(x)/g(x)$; partial fractions (introductory: distinct linear factors only). Bridge to 01.01.* for the algebra-strand polynomial-ring depth.

20. 00.13.01 Plane geometry (distance, area, $\pi$). [NEW] Lang Ch. 6-7 anchor. Three-tier; ~1500 words. Master: Pythagorean theorem proved by area decomposition (the standard inside-and-outside-the-square proof); area formulas for triangle, parallelogram, circle (as limit of inscribed polygons); definition of $\pi$ as circumference/diameter ratio. Originator citations: Euclid Elements Books I, IV, XII; Archimedes Measurement of a Circle.

21. 00.13.02 Solid geometry (volume). [NEW] Lang Ch. 8 anchor. Three-tier; ~1200 words. Master: parallelepiped, prism, pyramid, cone, sphere volumes via Cavalieri's principle (informal); surface-area formulas. Originator citations: Euclid Elements Book XII; Archimedes On the Sphere and Cylinder.

22. 00.05.03 Complex numbers (introductory). [NEW] Lang Ch. 19 anchor. Three-tier; ~1500 words. Master: $\mathbb{C}$ as ordered pairs $(a,b)$ with $i^2=-1$; modulus, argument, polar form $z=r(\cos \theta +i\sin \theta )$; multiplication geometrically as rotation-and-scaling; Euler identity $e^{i\theta} = \cos\theta

    • i\sin\theta$ stated; de Moivre's formula. Originator citations: Bombelli 1572; Wessel 1797; Argand 1806; Hamilton 1837 for the $(a,b)$-formalism. Load-bearing for 06.01.* complex-analysis strand.

23. 00.11.02 Conic-section parametrisations and intersections. [NEW] Lang Ch. 19 anchor. Three-tier; ~1000 words. Master: parametric form of conics via trigonometry (ellipse: $x = a\cos t$,$y = b\sin t$;hyperbola:$x = a\sec t$,$y = b\tan t$); line-conic intersection by substitution; tangent line at a point.

Priority 4 — Bibliography-only / survey items

24. [ENRICH] 01.01.01-field. Add Lang Ch. 1 to tier_anchors.beginner and references as the precalc-tier presentation of the field axioms. Cross-link from the Beginner section: "for the same axioms applied to $\mathbb{R}$ specifically, see 00.01.01."

25. [ENRICH] 02.01.05-metric-space. Add Lang Ch. 2 to tier_anchors.beginner and references as the precalc-tier introduction to absolute value as distance on $\mathbb{R}$. The triangle inequality on the line is the single-source ancestor of the metric-space triangle inequality.

26. [ENRICH] 06.01.01-holomorphic-function. Add Lang Ch. 19 to tier_anchors.beginner and references as the precalc-tier introduction to complex arithmetic and polar form. Cross-link to new 00.05.03 if produced.

27. [ENRICH] 00.02.05-function. Add Lang Ch. 14 to tier_anchors.beginner (currently lists "Strogatz-style input-output machine" only). The function unit's existing prose should cite Lang as the canonical precalc anchor — currently it cites Halmos and Bourbaki at the Master tier but no precalc anchor at the Beginner tier.

28. [NEW] 00.14.01 Functions and limits (informal preview). Optional pointer unit; Lang Appendix anchor. Master-only; ~700 words. Informal limit, derivative as instantaneous slope, area as integral — pointer to Apostol Vol. 1. Defer unless the §0 strand expands or a "what is calculus" survey unit is requested.

29. Notation crosswalk (notation/lang-basic-mathematics.md). Worth producing once priority-1 units land. Records: $\mathbb{R}, \mathbb{Q}, \mathbb{Z}, \mathbb{N}$usage;$|x|$forabsolutevalue;$\binom{n}{k}$ for binomials; radian as default angle unit in Part V; degree as alternate in Part II. Defer until trig units (priority-1 items 9-11) are integrated.

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§4 Implementation sketch (P3 → P4)

Minimum Lang-equivalence batch = priority 1 only (items 1-12): 12 new units across 9 chapter slugs. Realistic production estimate (mirroring earlier batches):

• ~2.5 hours per typical precalc unit (research + draft + validate at 27/27 + Lean stub + Bridge / Synthesis prose). Precalc units are shorter than tier-α units (~1200-1500 words vs 1800-2500), but the Beginner-tier prose density matters more.
• Priority 1 totals: 12 × 2.5 h = ~30 hours.
• Priority 1+2 totals: 12 + 6 new units × 2.5 h + 1 deepening × 1.5 h = ~46.5 hours.
• Priority 1+2+3 totals: 12 + 6 + 5 × 2.5 h + 1.5 h = ~59 hours.

At 4-6 production agents in parallel, priority-1 fits in a 2-3 day window with one integration agent stitching outputs.

Batch structure.

• Batch A (algebra core, items 1-7, ~17.5 h): opens 01-algebra-numbers, 03-algebra-equations, 04-inequalities, 05-exp-log. Item 7 depends on 6; item 4 depends on 1; otherwise parallel. 5-6 agents, one per chapter slug.
• Batch B (trigonometry, items 9-11, ~7.5 h): opens 06-trig-right, 07-trig-unit-circle, 08-trig-identities. Items 10, 11 depend on 9; serial within batch.
• Batch C (coordinate geometry, items 8 + 12, ~5 h): opens 09-coordinate-geom, 11-polar-parametric. Item 12 depends on Batch B.
• Batch D (priority 2, items 13-18, ~13 h): conics, inverse trig, law of sines/cosines, induction, binomial, function-deepening. Runs after A-C land.
• Batch E (priority 3, items 19-23, ~12 h): polynomials, plane and solid geometry, complex numbers, conic parametrisations. Optional.

Originator-prose targets (priority-1 unit Master sections cite originator + Lang). Most ancient-arc topics in Lang have multi-author provenance; the canonical first-hits are:

• Real numbers (1): Dedekind 1872, Cantor 1872. Quadratic formula (4): al-Khwārizmī ~820; Cardano 1545 (cubic context). Logarithms (7): Napier 1614; Euler 1748 Introductio for the inverse-exp formulation.
• Cartesian coordinates (8): Descartes 1637 La Géométrie; Fermat 1636.
• Right-triangle trig (9): Hipparchus ~150 BC; Ptolemy Almagest. Unit-circle trig (10): Euler 1748 (modern real-variable form).
• Addition formulas (11): Ptolemy (chord form); Vieta 1591 (modern algebraic form). Polar coordinates (12): Newton 1671 manuscript; Bernoulli 1691; Euler 1748.

The full citation list is recorded per-unit in tier_anchors and references at production time.

Notation crosswalk. Lang's notation is the canonical American precalc notation: $f:A\to B$ for functions; $f(x)$ for values; $\mathbb{R},\mathbb{Q},\mathbb{Z},\mathbb{N}$ for number systems ($\mathbb{N}$ as positive integers, not including 0 — Lang's convention, matching the Bourbaki "natural numbers from 1" tradition, not the modern set-theoretic "natural numbers from 0" tradition); $|x|$ absolute value; $[a,b]$ closed interval; $(a,b)$ open interval (Lang uses parentheses, not the European $]a,b[$); $\left(\genfrac{}{}{0ex}{}{n}{k}\right)$ binomial coefficient; $\sin ,\cos ,\tan ,\cot ,\sec ,\csc$ trig functions; radian as default in Part V. The $\mathbb{N}$-from-1 convention is the only notation point that needs flagging in the crosswalk file — Codex-elsewhere uses $\mathbb{N}=\{0,1,2,\dots \}$.

DAG edges to add. New prerequisites for the priority-1+2 batch:

• 00.01.01 (real numbers) ← {} (root of 00-precalc)
• 00.01.02 (absolute value) ← 00.01.01
• 00.03.01 (linear equations) ← 00.01.01
• 00.03.02 (quadratic formula) ← {00.01.01, 00.03.01}
• 00.04.01 (inequalities) ← {00.01.02, 00.03.02}
• 00.05.01 (real exponents) ← 00.01.01
• 00.05.02 (logarithms) ← 00.05.01
• 00.06.01 (right-triangle trig) ← {00.01.01, 00.13.01}
• 00.07.01 (unit-circle trig) ← 00.06.01
• 00.08.01 (trig identities) ← 00.07.01
• 00.09.01 (Cartesian coordinates) ← {00.01.01, 00.13.01}
• 00.11.01 (polar / parametric) ← {00.07.01, 00.09.01}
• 00.10.01 (conic sections) ← {00.09.01, 00.03.02}
• 00.12.01 (induction) ← 00.01.01
• 00.12.02 (binomial theorem) ← 00.12.01
• 00.02.05 (function — existing) — add explicit link from 00.01.01 as prerequisite

Successor edges from §0 to §1+:

• 00.01.01 → 01.01.01-field (Codex's algebra-strand field unit)
• 00.04.01 → 02.01.05-metric-space (the triangle-inequality ancestor)
• 00.05.03 (complex numbers, priority-3) → 06.01.01-holomorphic-function
• 00.07.01 → tier-α units that assume unit-circle trig (e.g. 06.01.03-residue-theorem)
• 00.11.01 → 06.01.03-residue-theorem (polar form for contour integrals)

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§5 What this plan does NOT cover

• Line-number-level inventory of every named result and exercise across Lang's ~470 pages. Defer unless priority-1+2 expands.
• Lang's ~600+ exercises. The exercise layer is ~0% covered; closing it requires per-chapter exercise packs (00.01.E1, 00.03.E1, etc.). Dedicated precalc-exercise-pack pass is a P3 follow-up.
• Apostol Vol. 1 overlap. Apostol Ch. 1 re-covers Lang's Part I (real numbers, induction) at calculus depth. The Lang priority-1 units serve as the precalc-tier prerequisite layer; Apostol's per-book plan should cite Lang's units in prerequisites rather than re-derive the real-number construction.
• Stewart / Larson-style applied word problems (loans, exponential growth in epidemiology, navigation with law of cosines). Codex follows Lang's proof-foundations framing; applied-problem density is deprioritised.
• Special-angle memorisation tables ($\sin 30°=1/2$, etc.) are derived in items 9-10 rather than catalogued. A separate cheat-sheet artifact is out of scope here.
• Calculus. Anything past Lang's Appendix preview lives in 02.* (Apostol territory).
• Number theory at the precalc tier (divisibility, GCD, primes, modular arithmetic). Lang touches these only in passing. Defer unless an arithmetic-foundations sub-strand is added.
• Set theory at the precalc tier (union, intersection, Venn diagrams). Lang assumes minimal set theory. Defer unless a §0 set-foundations sub-strand is added.

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§6 Acceptance criteria for FT equivalence (Lang)

Per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4 and §9, Basic Mathematics is at equivalence-coverage when:

• ≥95% of Lang's named results map to Codex units (currently ~3%; after priority-1 this rises to ~80%; after priority-1+2 to ~95%; after priority-3 deepenings to ~98%).
• ≥80% of Lang's exercises have a Codex equivalent (currently ~0%; closing this requires the dedicated precalc-exercise-pack pass per §5).
• ≥90% of Lang's worked examples are reproduced in some Codex unit (currently ~5%; the priority-1+2 batch's required worked-example inclusions bring this to ~80%; full coverage requires worked-example densification across the new units).
• A notation/lang-basic-mathematics.md crosswalk exists (priority-4 item 29; defer until priority-1 trig units land).
• For every chapter dependency in Lang (Part I → Part II → Part III → Part IV → Part V → Appendix), there is a corresponding prerequisites arrow chain in Codex's DAG. The 00.01.01 → 00.03.02 → 00.10.01 (real numbers → quadratic formula → conic sections) and 00.01.01 → 00.06.01 → 00.07.01 → 00.08.01 → 00.11.01 (real numbers → right-triangle trig → unit-circle trig → identities → polar coords) chains are the load-bearing ones.
• Pass-W weaving connects the new 00-precalc units to the existing 00.02.05-function and forward to 01.01.01-field, 02.01.05-metric-space, 06.01.01-holomorphic-function, and 06.01.03-residue-theorem via lateral connections — the precalc strand becomes the prerequisite floor for the algebra, analysis, and complex-analysis strands.

The 12 priority-1 units alone close most of the theorem-layer equivalence gap and produce the load-bearing precalc DAG. Priority-2 items round out Lang's content and close the function-unit deepening. Priority-3 items handle the geometry and complex-number bridges. Priority-4 items are bibliography-only enrichments plus the optional notation crosswalk and Apostol-bridge survey.

Honest scope. The largest absolute new-unit count of any audited book to date — 00-precalc is intentionally near-empty per the apex-first strategy, and Lang is the canonical single source for filling it. Most of the work is new content at the precalc tier, not depth rewrites. The priority-1 batch (12 new units, ~30 hours) is the minimum viable §0 strand and is the prerequisite for auditing Apostol Vol. 1 honestly — without the Lang precalc floor, Apostol's "for $x\in \mathbb{R}$" goes unanswered in Codex.

Composite Lang + Apostol-Vol-1 recommendation. Defer Apostol Vol. 1's audit until the Lang priority-1 batch lands; Apostol's per-book plan then cites Lang's items 1, 2, 5, 16, 9-11 as prerequisites instead of re-deriving them.

Largest single Lang-distinctive gap: the trigonometry block (items 9-11). Codex has zero precalc trigonometry; addition formulas, unit-circle definitions, and identities are silently assumed in every downstream unit that uses Fourier analysis, residue theorem, or classical mechanics.