MATH-CS COMPASS: Curriculum Roadmap & Development Plan (v17)

Author: Yusuke Yokota Last Updated: 8/28/2026 Website: https://math-cs-compass.com

Project Overview

MATH-CS COMPASS is an educational platform bridging pure mathematics and computer science, addressing the gap where CS students struggle with mathematical foundations while math students lack awareness of applications. The primary focus is rigorous mathematical foundations for modern AI/ML, with continuous expansion into adjacent domains (GDL, CDL, cryptography, stochastic analysis).

Total: 254 pages (measured 2026-08-26 from curriculum_index.json). I (linalg) 43 / II (calc) 99 / III (prob) 46 / IV (disc) 46 / V (ml) 20. curriculum.json is authoritative — re-measure it rather than editing this line by hand. (ml-19 / ml-20 published 2026-08-25 / 2026-08-26; disc-41~46 published 2026-08-24; prob-38~46 published 2026-08-18.)

Five tracks (status + full detail in Part 2):

• Phase 2e — stochastic analysis track (III, Øksendal Ch.2–8 body + dispersed app landings) — ✅ mathematical body COMPLETE (Ch.2–8 spine = prob-27~46, 20 pages; Ch.7–8 closed 2026-08-18); all pushes complete (backlink batch pushed 2026-08-19); remaining = trigger-gated app dispersal only
• Rep Theory — GDL continuous leg (I+V) — ✅ complete
• CDL — category theory (IV+V) — ✅ Section IV body complete (Leinster Ch.1–6); ✅ Section V bridge landed (ml-19 + ml-20, cartesian-only); 🔄 third page subject open
• Crypto — through PQC incl. Quantum (IV+I) + Section V landing (ml-18, stage 5) — ✅ complete
• TDL — GDL discrete leg (IV+V) — ✅ landed (optional persistent-homology branch remains)

Part 1 — Application Domains: Pillar vs. Viewpoint

1.1 Two orthogonal axes

• Pillar (vertical thread): a structural thread that multiple Sections converge into; a load-bearing column.
• Viewpoint (re-visit at different heights): a perspective returned to repeatedly at higher mathematical altitude, never “completed.” These are independent. A domain can be Pillar and Viewpoint / Viewpoint only / Pillar only / neither.

1.2 Classification of the three domains

GDL two-leg structure (detailed in Part 3):

• Continuous leg: manifold -> Riemannian -> representation theory -> Equivariant NN (SO(3)/SE(3) continuous symmetry) = Rep Theory track.
• Discrete leg: GNN (ml-13) -> simplicial complexes / Hodge (disc-13~15) -> SNN/TDL (ml-17, complete; higher-order interactions) = TDL track. TDL is a subfield of GDL (confirmed 2026).
• The two legs will eventually rejoin via Hodge theory (continuous Hodge = differential forms / discrete Hodge = simplicial complexes, bridged by DEC).

1.3 Authorial scale

Yusuke holds a US double major in math and CS (non-elite institution), self-described as broad-but-shallow. Prior knowledge varies sharply by topic: Lie theory was familiar (expert-reviewer mode); category theory, stochastic analysis, representation theory, and TDL are learn-while-writing. Track-character is calibrated per topic, not against a global label.

Part 2 — Active-Track Overview (index layer)

A unified view of the five tracks. Per-track detail is authoritative in the individual handouts.

2.1 Track table

2.2 Shared structural pattern

Four of the five tracks share the same shape:

• “The mathematical content is owned by its native Section; identity shifts to Section V at the ML/application point.” Rep (math=I, Equiv NN=V) / CDL (cats=IV, bridge=V) / TDL (Hodge=IV, SNN=V) / Phase 2e (math=III, landing ml-14/15 = existing V; the FP page prob-46 now underwrites that landing from below, 2026-08-18).
• Crypto is the partial exception (updated 2026-07-06, handout §0.9): the quantum half has no Section V landing — Shor/PQC own Section IV (algorithm) and Section I (quantum/lattice geometry), and Section V owns no quantum mathematics (the old quantum “Section V bridge” idea stays dropped, handout §0.8). But the LWE half now does land in Section V: ml-18 (security group, crypto stage 5) reads LWE as noisy regression + ring homomorphism (noise duality), a genuine ML/application landing. Crucially this does not break “mathematics is owner”: ml-18 owns no crypto mathematics — disc-38/39 remain the native owners, ml-18 only ref-links them and adds the ML reading (landing ≠ owner).
• This is a consistent expression of the site principle “application is viewpoint, mathematics is owner” (Part 1).
• Obsolescence-resistance principle (established in Crypto §0.5 / Phase 2e): write the enduring mathematics thickly (continuous-time generation, LWE hardness), not the individual method (diffusion, ML-KEM). When methods are replaced, the foundation survives.

2.3 Start priority (mood-driven; no single order enforced)

No forced order — if one track stalls, others proceed (Part 12 principle 7). With crypto, the Phase 2e body, the CDL Section IV body, and now the CDL Section V bridge (ml-19 + ml-20) all complete, the remaining active work is the third CDL page (subject open), the trigger-gated Phase 2e app dispersal, and the optional TDL persistent-homology branch. Dispatch by interest: CDL still carries the only real deadline pressure (asymmetric prep cost), but the pressure has changed character — the mathematics is pre-positioned and the bridge has landed, so what remains is a choice of subject among four candidates, three of which need machinery the site does not yet own (Part 5.3, stage 4′). The former hard-ordering constraint (Shor after crypto substrate + linalg-41) is fully discharged — nothing blocks anything now.

Part 3 — GDL Pillar (two-leg: continuous Rep + discrete TDL)

GDL stands on two legs (Part 1.2): a continuous leg (Rep Theory) and a discrete leg (TDL).

3.1 Why GDL is a pillar (three independent reasons)

1. Mathematical thickness: Lie groups, Riemannian geometry, fiber bundles, representation theory, and spectral graph theory all converge into GDL.
2. Production maturity (2026): equivariant nets, GNNs, SE(3)-Transformers, EquiformerV3, MACE have moved from research to deployment (AlphaFold, molecular design, equivariant robotics).
3. Independent forward growth: robotics × ML keeps expanding for information-theoretic reasons; architectures lacking equivariance pay a principled sample-efficiency cost.

3.2 Why it is also a viewpoint

A thick pillar looks different at different heights. At ml-13 (GNN) the reader sees only “permutation-equivariant message passing on discrete graphs”; after the manifold series, continuous symmetry; after representation theory, irreducible decomposition; after calc-32, Peter-Weyl bridges to harmonic analysis. The pillar is passed through repeatedly, each time at deeper understanding.

3.3 Continuous leg — Representation Theory track

COMPLETE. Section I (rep theory linalg-31~40) + Section V (Equivariant NN, ml-16). Wiring: Lie groups -> group representations -> irreducible decomposition -> Schur -> Equivariant NN, with Peter–Weyl (linalg-40) reclaiming calc-32 Fourier. GDL-mandatory scope fully satisfied; only the GDL-unnecessary deep-dive (semisimple / Verma / Weyl character formula) remains. Detail: site pages + rep_handout (archival: route, owners, symbol conventions, deep-dive trigger map).

3.4 Discrete leg — TDL (Topological Deep Learning) track

SNN (ml-17) COMPLETE — discrete leg landed. Section IV (Hodge existing in disc-13/15) + Section V (SNN, ml-17). GNN (ml-13) generalized pairwise -> higher-order via the Hodge Laplacian; (\ker L_k \cong H_k) makes the propagation operator report holes. TDL is a subfield of GDL (GNNs = pairwise; TDL = n-body over simplicial/cellular complexes). Reference: TDL book (tdlbook.org, registered) + Edelsbrunner-Harer. Remaining: persistent homology (disc-XX) only — optional TDA branch off disc-15’s other forecast, not GDL-mandatory. Future / mood-driven. Detail: site pages + tdl_track_handout.

3.5 Rejoining of the two legs (Hodge, future deep connection)

Continuous Hodge (differential forms: Lee Ch.14 complete = calc-82/83/84; orientations Ch.15 complete = calc-85~89; integration Ch.16 through Riemannian = calc-90/91/92 complete) and discrete Hodge (simplicial complexes, disc-13/15 existing) share the same structure. DEC (Part 11 deferred) will be the reclamation hub bridging both legs. calc-91 Stokes is the continuous-side origin sitting on the de Rham / DEC critical path (manifold handout v24). Recorded for now; the relevant Part 11 entries are “Spectral Laplacian” and “DEC.”

Both legs’ GDL-mandatory scope is complete (continuous: Rep Theory + Peter–Weyl + ml-16; discrete: SNN ml-17). Standing obligation: each GDL page carries forward-pointers to “the next mathematics.”

Part 4 — Phase 2e: Stochastic Analysis Track (Body COMPLETE 2026-08-18)

A stochastic-analysis track in Section III built from Øksendal Ch.2–8 (the mathematical body: Brownian motion → Itō → SDE → diffusions/generator/Fokker-Planck), with the “with Applications” chapters (Ch.6, 9–12) dispersed as landings/bridges rather than laid out in chapter order. One strand of this — the BM→Itō→SDE→FP path — justifies from below the continuous-time machinery that ml-14 (diffusion) / ml-15 (flow matching) use as given; the track’s range is the analysis itself, not only the generative-model support. The body is complete and published: prob-27~46 (20 pages), spine closed with the Fokker-Planck arc (prob-46), batch pushed 2026-08-18. What remains is the trigger-gated application dispersal of 4.4.

4.1 Scope upgrade (v1 → v2)

v1 scoped Phase 2e narrowly as “justify ml-14/15’s continuous-time objects from below,” 3 pages (BM+Itō / SDE / FP) — only Øksendal Ch.2–5. v2 (after ToC review): one Øksendal volume covers the intended range in full. Of the 12 chapters, Ch.2–8 are the mathematical body; Ch.6, 9–12 are application chapters. Therefore:

• Track body = Øksendal Ch.2–8, owned in Section III (v1’s 3 pages + a diffusions/generator page).
• Application chapters are NOT poured into Section III in chapter order (violates the topicGroup principle: a page is placed by what it is, not what it is used for). They are dispersed as landings/bridges when the site’s callback structure demands them — the same tracks-isomorphic structure as crypto (Peikert not followed in chapter order; SIS/LWE owned, applications landed in V).
• Martingale theory stays “touched only” — Øksendal itself does not systematize it (introduces it minimally as a tool for the Itō integral; Ch.4 “Itô Formula and the Martingale Representation Theorem” is the sole locus, and only to the representation theorem). The site follows suit: Doob’s maximal inequality / optional stopping / convergence theorems are not owned; filtration and adapted processes are owned minimally as Itō-integral scaffolding. A full martingale systematization, if ever done, is Durrett territory and split off as the Part 11 deferred item “Uniform Integrability & Martingale Convergence.” This discipline held to the end: prob-44 avoided general uniform-integrability theory via an elementary L²-bounded transfer, and the representation theorem (prob-33) never needed convergence theory.

4.2 Significance in the Flow Matching era — realized (2026-08-18)

Diffusion is being displaced by FM in practice (FLUX, SD3 are rectified flow), but both are ODE/SDE representations of the same object (Stochastic Interpolants). The generative-support strand is not diffusion-specific but “the continuous-time basis unifying diffusion and FM.” Its significance strengthened, not weakened, in the FM era. Obsolescence-resistance concentrates the value in the self-contained mathematics (Ch.2–8), not the method.

This promise is now a theorem. prob-46 owns T-fp_invariant_family: one probability path satisfies, slice by slice, the forward equation of an entire noise-scale family of dynamics — the FM flow at scale zero, the noisy samplers at every positive scale — with the score correction cancelling the injected diffusion identically. That is exactly the “unifying continuous-time basis” of this section, proved at the level of densities and operators. The two deliberately unproved passages (equation → law: time-inhomogeneous FP attachment, and FP uniqueness) are trust-localized in the page and registered in Part 11. ml-15’s cancellation prose and ml-14’s continuous-time scope note are both underwritten from below; framing of “what is mainstream” stays owned by ml-15.

4.3 Track-body structure (Section III; COMPLETE — Ch.2–8 spine = prob-27~46, closed 2026-08-18)

The planned “Page 1–2” expanded 2 → 20 pages (Page Count Estimation rule; Lie 2→4 precedent). The Ch.7–8 estimate “~2–4 pages” (4A / 4B-FP) itself realized as 9 pages — the same expansion law:

Ch.8.2–8.5 are deliberate non-inclusions: Feynman-Kac is dispersed per 4.4; the martingale-problem / Itō-process-⇒-diffusion cluster and random time change were cut together with their external Stroock-Varadhan / Dynkin dependencies. The “zero new prereqs” prediction held — every page closed on existing foundations, with two declared trust extensions (the product-measure identification (\lambda_m=\lambda_1\otimes\lambda_{m-1}); the equation→law passages of Part 11). The Feller-continuity callback landed as predicted: prob-46’s resolvent remark reads T-start_stability_bound on the horizon (T=t). Anchor registry lives in previews.json; scope-out ledger (S1–S28), design decisions (1–39), and notation conventions are authoritative in phase2e_handout.

4.4 Application-chapter dispersal (body complete — all landings remain trigger-gated; forward-pointers only)

4.5 Reference / start

• primary: Øksendal (SDE proper, on hand — Springer Universitext 6th ed., ISBN 978-3-540-04758-2) + Durrett (measure-theoretic substrate, on hand) + Holderrieth-Erives (FM/diffusion unification, the V-side bridge, free, already used on the ML side).
• No OT book -> FM/rectified flow/OT are forward-pointers only (no href).
• Rigor calibration resolved in practice: the site self-contains proofs where Øksendal leaves gaps (reference-as-ceiling fallacy avoided; e.g. Picard well-definedness induction, L²-limit measurability via filtration augmentation). Karatzas-Shreve not needed so far.
• Status: body COMPLETE and PUBLISHED (2026-08-18). Øksendal 6th ed. on hand (現物照合 protocol held for the whole track: every statement verified against page images at drafting). Ch.2–8 spine = prob-27~46, all live. The backlink-upgrade batch (prob-35/37/38, ml-10/14/15; previews rebuilt for the theorem-block link) is pushed 2026-08-19 — zero pending pushes; only §8 trigger-gated app dispersal remains — detail phase2e_handout.
• track-character: a CS author learning pure math while writing (same class as CDL).
• Detail: phase2e_handout (ledger S1–S28, design decisions 1–39, lessons 1–28).

4.6 Notes (overload / owner facts, see Part 8 ledger)

• D-infinitesimal_generator exists on the Lie/differential-geometry side (owner = calc-64 integral_curves.html) -> SDE generator uses D-sde_generator (owner realized: prob-42); downstream pages receive it by ref-link.
• score_function triple collision (Fisher/data/continuous) -> resolved as planned: prob-46 ref-links ml-14’s D-score_function_data_gradient (Stein’s identity and the invariant family both), defines no new one.
• characteristic is an overloaded word (4 owners incl. ring characteristic D-characteristic) -> resolved harder than planned: Ch.7.5’s characteristic operator is a permanent scope-out (D-characteristic_operator is never created; Øksendal outsources its proofs to Dynkin 1965 and no downstream consumes it).
• prob-18 (markov.html) is a graphical-models (first-order Markov) page, not a stochastic- process Markov-chain page -> held: prob-38~40 built (strong) Markov fresh; prob-18 stays a loose back-link, not a prereq.
• The 2026-07-11 collision check paid off: D-ito_diffusion, D-stopping_time, T-kolmogorov_backward all realized as planned (strong Markov landed as T-bm_strong_markov/T-markov_property_diffusion per the enlargement-vocabulary convention). New in Ch.8: the formal adjoint is deliberately issued for a general pair ((\mathbf{b},a)) (D-formal_adjoint_generator), so the score-corrected frozen coefficients of the invariant family consume it literally; the FA-side T-existence_of_adjoint / D-resolvent_spectrum remain disambiguation ref-links only (Part 8 ledger updated).

Part 5 — CDL: Pre-Pillar Slow-Burn Track

Build category theory in Section IV and name the categorical structure already latent across the site (the culmination of the callback philosophy). Pre-position the mathematics before CDL applications mature.

5.1 Why now (MUST cover but slow)

• Convergence of the mathematics: deepening Section IV (graph -> simplicial -> homology), I (group -> ring -> field -> Lie), II (continuous map -> FA) leads inevitably into the language of categories.
• Asymmetric preparation cost: category-theory preparation is deep and not skippable. If CDL applications mature in 2027-2028, building it at speed is infeasible -> pre-positioning is the only strategy.
• The structural deadline is starting slowly now itself. Unlike other tracks, deferring this one creates a trap later.

5.2 Placement decision (owner Section)

owner = Section IV. Applying the topicGroup principle — a category-theory page’s identity is “a discrete/combinatorial algebraic structure with composition” (a category = objects + morphisms

• composition), which is Section IV’s identity itself. That category theory cites Vect/Hilb as examples is usage, not the object. Corroboration: formal methods (disc-16/17 Lean) = type theory = the twin of category theory (Curry-Howard-Lambek).

“Cross-Section spanning” is not a placement problem but a ref-link direction problem: a category page (IV) citing examples from I/II just means ref-links pointing outward; the owner stays IV (precedents: Peter-Weyl math owned in II while ml-16 lands in V; the same outward-ref-link pattern).

Spanning is the greatest weapon — reclamation hub design: the stage 1 page intentionally bundles ref-links to examples in Sections I/II/IV, letting the reader see at a glance “the groups from Section I, the Banach spaces from Section II — they were all categories.” The Section IV category page becomes the site-wide callback hub. The manifold Q7 (differential (dF_p) = functor; calc-45/46 T-differential_properties/T-global_differential_properties) connects here.

5.3 Track structure (Leinster body complete; Section V bridge landed, cartesian-only)

Progress (2026-08-26): Stages 0–2 and 4 done. The reach verdict resolved as cartesian-only and stage 4 landed as two pages (ml-19 / ml-20; the 2-page split was a verdict of 2026-08-25). Progress (2026-08-24): Stages 0–2 done. disc-18~28 + disc-41~46 published and deployed — Leinster Ch.1–6 is fully consumed and the Section IV work is finished. Ch.6 was estimated at 5–6 pages and landed at 6.

monad / Kan extension handling (unchanged in substance, sharpened 2026-08-24): (a) monad has no dedicated treatment in either Leinster or Fong-Spivak, so it gets no standalone page in IV. The Section IV decision does not automatically carry to Section V: a V page may describe monads without owning them, but defining and using them requires acquiring a rigour source (Mac Lane or Awodey). (b) Kan extension has no chapter in either primary (acquisition flag: Riehl / nLab, not in references.json). New in 2026: applied demand appeared — the 2026 transformer literature uses Kan extensions as a design language, which moves this from “someday” to “a real trigger exists.”

Reach note (measured 2026-08-24). A scan of every anchor the site owns found zero for monoidal category, 2-category, monad, algebra-for-a-monad, lens, optic, coalgebra, Kan extension, topos, subobject classifier, reverse derivative category, string diagram, profunctor, coend, enriched category. What the site does own — categories/functors/naturality, products, limits/colimits, Yoneda, adjunctions, cartesian closure, the Jacobian and chain rule, group actions and equivariant maps — is exactly enough for a cartesian-only account of learning. Everything past that (parametric lenses proper, Para as a 2-category, the ICML position paper’s monad-algebra machinery, the reverse-derivative generality) needs new tooling. The honest first Section V page is therefore cartesian-only, with the rest as declared previews. Detail: cdl_track_handout §11-3.

stage 3 -> 4 is the IV -> V crossing (identity shifts from math to ML application).

5.4 Reference / start

• Leinster Basic Category Theory (rigorous, IV, free arXiv:1612.09375, CC BY-NC-SA, v2 2025/8)
• Fong & Spivak Seven Sketches (applied, free arXiv:1803.05316, CC BY 4.0)
• Both free, no purchase. Leinster himself includes poset/monoid examples in the book (an entry-design option).
• Section V CDL bridge (stage 4) landed 2026-08-25 / 2026-08-26. The rule it was written under is now a standing constraint on every further CDL page: restrict the load-bearing claims to the peer-reviewed layer: Backprop as Functor (LICS 2019) / Categorical Foundations of Gradient-Based Learning (ACT 2021 → ESOP 2022, Springer LNCS) / Category-Theoretical and Topos-Theoretical Frameworks in ML: A Survey (Axioms 2025 14(3):204, the only peer-reviewed map of the field). Frame the ICML 2024 position paper’s “all architectures” claim honestly as an ongoing research direction, not settled theory — the venue is peer-reviewed but the paper is a position paper.
• Classification correction (2026-08-24): Deep Learning with Parametric Lenses (arXiv:2404.00408) was listed here as established layer. It is still unpublished — 2025–2026 citations treat it as a preprint. Its content largely extends the ESOP paper, so cite the ESOP version for anything load-bearing.
• Rigour source for the definitions: Gavranović, Fundamental Components of Deep Learning (Strathclyde thesis 2024, arXiv:2403.13001). The ESOP paper compresses the Para and Lens definitions; the site’s no-external-deferral rule needs the uncompressed version.
• The CDL bridge is where disc-20’s monad foreshadowing pays off, and where disc-19 and disc-46 already carry the forward pointers (“the reference point against which the categorical accounts of learning systems are later positioned” / the monoidal-closed paragraph).
• Detail: cdl_track_handout (§11 = Section V literature and reach).

5.5 Status monitoring (web-verified 2026-08-24)

The field moved between 2024 and 2026, and the move is real but still semantic. In 2024 the picture was one position paper plus two semantics papers. By August 2026 there is (a) a peer-reviewed survey giving the field a map, and (b) theorems — a categorical equivariant universal approximation theorem, with equivariance formulated as naturality in a topological category, unifying group/groupoid, poset/lattice, graph and sheaf networks. 2026 activity splits three ways: generalising equivariance (category-equivariant networks, coalgebraic representations), categorical analysis of transformers (self-attention as a parametric endofunctor with layer-stacking as a free monad; transformers as morphisms in a presheaf topos; Kan extensions as an attention design language), and connections to formal methods (coherent functors, symmetries for theorem proving).

What has not changed: the frontier layers remain conjectural, the position paper’s “all architectures” is still a position, and there is no production deployment (no categorical primitives in PyTorch/JAX). Slow-burn, mathematics-first, stands.

Newly relevant to this site: the transformer-topos work runs directly through the presheaf and Yoneda machinery disc-22~24 and disc-46 now own — the cheapest possible connection point for a further Section V page (the bridge’s two pages did not need it). Status-shift triggers unchanged: a CDL architecture wins a benchmark / a major framework adds categorical primitives / a well-funded second entrant. Paper list with peer-review status and survey date: cdl_track_handout §11-2.

5.6 Notes (Part 8 ledger)

• adjoint triple overload (Lie adjoint representation / FA operator adjoint / CDL adjoint functor) -> CDL uses D-adjoint_functor.
• Category-page entry correction: “lattice in disc-1/2/3” is wrong (actually Graph/Combi/ToC). No poset/lattice page exists; monoid undefined -> entry via Leinster examples or existing assets (Set/Grp/Vect/Ban/Hilb/Top).

Part 6 — Crypto Track (✅ COMPLETE — includes Quantum + Section V landing ml-18)

The former standalone “Quantum” plan is absorbed here — quantum computation shipped as part of the crypto stack (Shor = attack, lattices = defense), not as a separate track.

Complete through PQC. Realized placement below; all policy/lessons/per-page detail live in crypto_track_handout (single source of truth) + curriculum.json/previews.json. Only the placement outcomes that the topicGroup principle produced are kept here, as the canonical worked example of Part 12 principle 10.

6.1 Realized placement

Mainline arc = attack (Shor) -> defense (geometry -> computational problems -> SIS -> LWE -> Ring/Module-LWE + ML-KEM) -> stage 5 Section V landing (ml-18). Stage 5 reads LWE (disc-38) as noisy regression (ref-links ml-2 Ridge/Lasso) + ring homomorphism (ref-links disc-39); it owns no new crypto mathematics — disc-38/39 stay the native owners, ml-18 adds only the ML reading (landing ≠ owner, handout §0.9). New topicGroup security (Section V, holds future FHE/DP/ZKP × ML landings). Primaries: HAC + de Wolf + Regev (courses) + Peikert; all free, all registered in references.json.

6.2 Placement decisions the topicGroup principle forced (the worked example)

• Split by identity, not by track. Lattice geometry is pure math -> Section I (new lattice group); lattice computational problems are complexity -> Section IV computation (joining intractable_problems disc-3/5/6/7/9); the crypto constructions (SIS/LWE) are applications -> Section IV cryptography. Placing everything in Section I would have been wrong on this.
• Shor -> Section IV (algorithm), ref-linking linalg-41 for the quantum substrate. Quantum computation is not ML: no Section V quantum owner exists; Grover/VQE/QEC would also be Section IV (not built — no trigger).
• ECC dropped (isolated, quantum-obsolete, absent from HAC; structural preview via disc-33 suffices).
• No standalone PQC-specs page — ML-KEM/FIPS deployment is thin viewpoint inside disc-38/39 (obsolescence-resistance: write the hardness mathematics thickly, specs thin).
• Stage 5 landing forced a new Section V group (security). ml-18 lands in Section V (its ML element = reading LWE as regression, ref-linking ml-2). But topicGroup must not cross sections: the cryptography group belongs to Section IV, so it cannot be reused on a Section V page. The existing Section V groups (ml-foundations/deep-learning/generative-models) all mismatch by content (ml-18 is neither a base learning method, an NN architecture, nor a generative model). Hence a new Section-V-local group security — sized to hold future landings (FHE/DP/ZKP × ML), not a singleton. Reusing a prereq’s group (ml-2 = ml-foundations) would have been wrong: ml-18 is not a foundation. (landing ≠ owner: disc-38/39 keep the crypto mathematics; handout §0.9.)
• 🔴 Site-wide permanent rule (handout §0.7/§6): never write factoring/DLP as a permanent security anchor, nor lattices as “in-principle quantum-safe” — the known/conjecture register is mandatory on all crypto pages. Enforced; recorded here only as a cross-page invariant.

Part 7 — Reference Acquisition Status

References for the five tracks plus existing ones, by acquisition status. No purchases remain: Øksendal 6th ed. (Phase 2e) and Hall 2nd ed. (Rep Theory) are both on hand. All other references are free.

7.1 Active-track references (status)

7.2 Not yet acquired (trigger-based)

• Amari Information Geometry (when an information-geometry page is planned; entry point calc-81 secured)
• Mac Lane Categories for the Working Mathematician or Awodey Category Theory — monoidal categories and monads. Trigger = a CDL page that defines rather than describes them (Part 5.3 reach note).
• Johnson & Yau 2-Dimensional Categories — 2-categories. Same trigger; needed for Para-as-a-2-category and for string diagrams (stage 3).
• Riehl Category Theory in Context — Kan extensions. Neither primary has a chapter; applied demand now exists (2026 transformer literature).
• All three are textbook-level and would go into references.json if acquired. The CDL papers do not; the Gavranović thesis already is, on book-length grounds (Part 7.1).

Part 8 — Overload Ledger

Collected homonyms (the same symbol used for different concepts across Sections/contexts). Always cross-check before naming a new anchor. The manifold handout §2 overload notes are merged here.

Part 9 — Filename Registry (planned pages)

Completed pages are authoritative in curriculum.json. This table reserves filenames before drafting so cross-page references can be written ahead. IDs assigned at drafting time.

Part 10 — Completed Tracks Log

Completed tracks (on index.html, no planned pages).

Major completed tracks

| Track | Pages | Completed | Notes | |—|—|—|—| | Representation Theory (Hall) — incl. Peter–Weyl + FA block | linalg-31~40 + ml-16 + calc-93~99 | 6/8–6/20/2026 | GDL continuous leg: group/Lie-algebra reps -> irreducible classification -> Schur -> Clebsch-Gordan/Wigner-Eckart -> Peter–Weyl (linalg-40, recovers calc-32 Fourier), landing at Equivariant NN (ml-16). FA block (calc-93~99, functional-analysis) built as Peter–Weyl’s Stone–Weierstrass prerequisite. GDL-mandatory scope complete. Deep-dive deferred (Part 11). Detail: rep_handout. | | Smooth Manifolds (Lee Ch.1–16 through Riemannian integration) | calc-36~81 + calc-42/45/47/52/59 + calc-82~92 | 6/3–6/15/2026 | Manifold spine + differential forms (Ch.14) + orientations (Ch.15) + integration (Ch.16: Haar, Stokes/Green, Riemannian, divergence). Mathematical landing of the GDL continuous leg; Peter–Weyl Haar substrate in calc-90. Corners/Densities deferred (Part 11). Detail: manifold_handout / ch16_integration_handout. | | Crypto Track (through PQC, incl. Quantum + V landing) | disc-29~40 + linalg-41/42/43 + ml-18 | ~7/6/2026 | Full attack->defense arc + ZKP + stage 5 Section V landing (ml-18, LWE as noise-duality, new security group). New topicGroups quantum (linalg-41), lattice (linalg-42/43), security (ml-18, Section V). Placement + deviations in Part 6; full detail in crypto_track_handout. | | Formal Methods | disc-16, disc-17 | 5/14/2026 | Section IV third pillar (disc-4,16,17). Bidirectional bridge with disc-12 (Four Color Theorem). Curry-Howard-Lambek connection point for CDL. |

Completed reference -> page mapping

• Lay Linear Algebra -> linalg-01~10
• Gallian Contemporary Abstract Algebra -> linalg-15~22
• Horn & Johnson Matrix Analysis -> linalg-09~13
• Boyd & Vandenberghe Convex Optimization -> calc-07, calc-08
• O’Searcoid Metric Spaces -> calc-16~22
• Lee Introduction to Smooth Manifolds -> calc-29, calc-36~81, calc-42, calc-82~92 (Ch.14 forms + Ch.15 orientations + Ch.16 integration)
• Conway Functional Analysis -> calc-23~28, calc-30~32, calc-93~99 (FA block: Hahn–Banach, four-pillars/Baire, LCS, separation HB, Krein–Milman, Riesz (C(X)^*=M(X)) [Appendix C], Stone–Weierstrass [V §8])
• Stein & Shakarchi Fourier Analysis -> calc-14, calc-15, calc-32~35
• Stillwell Naive Lie Theory -> linalg-24, linalg-27~30
• Hall Lie Groups, Lie Algebras, and Representations 2nd ed. -> linalg-31~40 (incl. Peter–Weyl, §12.3)
• Durrett Probability -> prob-13, prob-22~26
• Leinster Basic Category Theory -> disc-18~28, disc-41~46 (Ch.1–6 complete; GAFT appendix scoped out)
• Cormen Introduction to Algorithms -> disc-01~11
• Sipser Theory of Computation -> disc-05~09
• Diestel Graph Theory -> disc-01, disc-12
• Merris Combinatorics -> disc-02
• Edelsbrunner & Harer Computational Topology -> disc-14, disc-15
• Avigad et al. Lean -> disc-16, disc-17
• Murphy Book1/Book2 Probabilistic ML -> ml-01~12, prob-16, prob-26 etc.
• Menezes et al. Handbook of Applied Cryptography -> disc-29~34, disc-40 (§10.4 ZKP)
• de Wolf Quantum Computing: Lecture Notes -> linalg-41, disc-35 (Ch.1/4/5/6)
• Regev Lattices in Computer Science (courses) -> linalg-42/43, disc-36
• Peikert A Decade of Lattice Cryptography -> disc-36, disc-37/38/39

Part 11 — Deferred Items (Non-Blocking)

Part 12 — Key Learnings & Development Principles

1. Notation consistency is non-negotiable: calligraphic for spaces (𝒳,𝒴,𝒵,ℋ,𝒳,𝒳**), functionals as φ, operator norm ‖φ‖_𝒳. New Section II pages match calc-23~28.
2. Cross-page linking: verify filenames/anchors against reality before linking. No in-body citations. Forward links use descriptive text (no href) until the target exists.
3. Application-viewpoint philosophy: use Insight Boxes and the Tessera to connect abstract theorems to applications without breaking main-text proofs. Application domains introduced when tools are ready (asymmetric, Parts 1-6).
4. Fisher vs Hessian: the real distinction is reparametrization invariance (Čencov) vs loss-dependence and non-guaranteed positive-definiteness, not “global vs local.”
5. Page count estimation: new conceptual paradigms expand 1.5–4×. Defer ID assignment to drafting. Calibrations: Lie groups 2 -> 4, calc-30 1 -> 2, Phase 2c 2 -> 3, Fourier-PDE 1 -> 3, CDL 6 -> 12 anticipated (actual: 11 pages for disc-18~28 = close to prediction, covering Ch.1–5; more expected at Ch.6+). New strongest data point (Peter–Weyl, 6/2026): a ~2-page target spawned a 7-page prerequisite block (the FA block calc-93~99). A single dependency audit (§12.3 needs Stone–Weierstrass) recursively pulled in SW’s entire Conway chain. The lesson is not just “pages expand” but “a dependency audit can reveal that the prerequisite is the real project” — the audit is what caught it before drafting, exactly as intended (Pre-Writing Dependency Audit pays off).
6. Per-topic prior-knowledge calibration: set track-character per topic (Lie=expert, CDL/Phase2e/Rep/TDL=learn-while-writing).
7. Mood-driven dispatch: no single order enforced. If one track stalls, others proceed. The former hard ordering (layer-2 Shor <- crypto stage 1-2 + layer-1 linalg-41) is now fully discharged — the entire crypto/quantum stack (disc-35~40) shipped + Section V landing ml-18 (stage 5, Part 6). No hard ordering constraint remains in the plan.
8. Tracks-isomorphic structure: “mathematical content owned by its native Section; identity moves to Section V at the ML/application point.” Rep/CDL/TDL/Phase2e take this shape; Crypto is a partial exception (owns IV/I): its quantum half has no Section V landing, but the LWE half landed via ml-18 (security group, stage 5) as noise-duality — owner still native (disc-38/39), landing ≠ owner (Part 2.2, handout §0.9).
9. Obsolescence-resistance principle: write the enduring mathematics thickly, not the individual method. diffusion -> FM, ML-KEM -> next-gen: when methods swap, the foundation survives (Parts 4.2, 6.2).
10. topicGroup is decided by identity: place a page by what it is (its mathematical object), not what it is used for. The crypto/lattice/quantum placement split (Part 6.2), CDL in Section IV, and quantum owner separation are all applications of this principle.
11. Single ownership: each T-/D- anchor has exactly one owner site-wide. grep previews.json + HTML before assigning. Overloads are managed in the Part 8 ledger.
12. Handout-driven continuity: session state is authoritative in this roadmap and the handouts; memory carries protocol/philosophy only. Per-track detail is single-source-of-truth in *_handout_v1.md.

This roadmap is the index layer. Per-track prereq verification, collisions, owner candidates, physical-book inspection items, and resume-time greps are authoritative in the individual handouts for the still-active tracks: phase2e_handout / cdl_track_handout / tdl_track_handout (optional persistent-homology branch only).

Completed-track handouts (archival): crypto_track_handout (crypto through PQC + ZKP + stage 5 Section V landing ml-18 — completion record; v26 added the security-group landing, so the track can extend into Section V security pages (FHE/DP/ZKP × ML) without a new track), rep_handout (Rep Theory incl. Peter–Weyl + FA block; absorbed the spent peter_weyl_handout and fa_block_screening_handout), manifold_handout / ch16_integration_handout (manifold spine through Riemannian integration; only Corners/Densities deferred).

Next active work = the third CDL Section V page (subject open; Part 5.3 stage 4′). The Leinster body closed on 2026-08-24 and the Section V bridge landed on 2026-08-25 / 2026-08-26 as ml-19 + ml-20 (cartesian-only, per the Part 5.3 reach note). Or Phase 2e app dispersal (trigger-gated); the crypto track is closed.