Author: Yusuke Yokota Last Updated: 6/3/2026 Website: https://math-cs-compass.com
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 practical applications. The primary focus is providing rigorous mathematical foundations for modern AI/ML, with continuous expansion into adjacent domains (geometric deep learning, categorical deep learning, cryptography, stochastic analysis).
Total: 169 pages as of 6/3/2026. (linalg 30 / calc 81 / prob 26 / disc 17 / ml 15; per curriculum.json, which is authoritative for the count.)
An earlier framing treated GDL, CDL, and Quantum uniformly as “viewpoints, not terminal goals.” The three domains are not symmetric in 2026, and this roadmap reflects that asymmetry.
A clean separation of two concepts that were previously conflated:
These are independent. A domain can be:
| Domain | Pillar? | Viewpoint? | Production maturity (2026) | Site treatment |
|---|---|---|---|---|
| GDL | ✅ Yes | ✅ Yes | High (AlphaFold, MACE, equivariant robotics) | Pillar ∧ Viewpoint — full track |
| CDL | ⚠️ Pre-pillar (will likely become pillar within ~2–3 yr) | ✅ Yes | R&D stage (Coend $31M funding; no deployed product) | Slow-burn parallel track — start now, do not wait for production maturity |
| Quantum | ⚠️ Latent | ✅ Yes | Limited (error-correction codes; molecular simulation niches) | Viewpoint via Insight Boxes — no dedicated track planned |
The asymmetry reflects empirical reality, not stylistic preference.
Background: Yusuke holds a US double major in mathematics and computer science from a non-elite institution; self-described as broad-but-shallow undergraduate coverage.
Operational implication: prior knowledge varies sharply by topic. Lie theory was familiar (linalg-27~30 ran in expert-reviewer mode); category theory and continuous-time stochastic analysis are not (CDL and Phase 2e are learn-while-writing). Track-character assignments must be calibrated per topic, not against a global expertise label.
Three independent reasons (any one would suffice; together they are decisive):
Mathematical thickness: Lie groups, Riemannian geometry, fiber bundles, representation theory, and graph spectral theory all converge into GDL. No other application domain on the site has this many distinct mathematical threads naturally pointing at it.
Production maturity (2026): equivariant networks, GNNs, SE(3)-Transformers, neural operators on manifolds have moved from research stage into deployed applications (AlphaFold, MACE/Allegro for molecular design, SE(3)-equivariant robotics policies). This is empirical fact, independent of any “Physical AI” marketing framing.
Independent forward growth: robotics × ML will continue expanding for reasons grounded in information theory, not market trend. Manipulation tasks have geometric structure that is physical, not coordinate-system-dependent — architectures lacking equivariance pay a sample-efficiency cost for principled reasons. This argument does not require predicting any specific company or product trajectory.
The “pillar” framing does not contradict the original viewpoint principle. The two coexist because a thick pillar looks different at different heights:
Each return visit happens at a higher mathematical altitude. The pillar is not climbed-and-finished; it is passed through repeatedly, at progressively deeper levels of understanding.
Every GDL-related page must include forward-pointers to “the next mathematics” the reader could pursue from there:
This obligation is not stated explicitly in each page. Readers will infer the open-endedness from the forward-pointer structure across pages without need for a meta-declaration.
ml-13 (GNN) → manifold series (calc-XX, ~3 pages) → representation theory (linalg-XX, ~3–4 pages) → ml-XX (Equivariant NN) → ml-XX (GDL Overview)
The earlier-planned calc-32 (Fourier in Hilbert Spaces) has been completed and remains a prerequisite capstone for the Peter-Weyl bridge later in the sequence. With ml-13 and calc-32 already complete, the active ordering from here is: manifold series → representation theory → ml-XX (Equivariant NN). As of 6/3/2026 the manifold series is complete through Lee Ch.1–13 (calc-36~81 published), reaching Riemannian metrics, the distance/metric-space structure, and the musical isomorphisms that ground the natural gradient; see Part 7. Next active track is representation theory.
Two independent arguments converge:
(a) Convergence of the mathematics: Continuing to deepen Section IV (graph theory → simplicial complexes → homology), Section I (groups → rings → fields → Lie groups), and Section II (continuous maps → functional analysis) inevitably leads into the language of categories. The site already implicitly contains category-theoretic structure (functors between Set, Grp, Ring, Vect_F, Top, Ban, Hilb are all latent); not naming these is a deliberate scope discipline that becomes harder to maintain as the curriculum deepens.
(b) Asymmetry of preparation cost: The mathematical preparation for CDL is deep and not skippable. There is no shortcut to the mathematics. If CDL applications begin maturing in 2027–2028 (a plausible scenario given the Coend trajectory), retroactively building category theory at speed would be infeasible. Pre-positioning the mathematics is the only viable strategy.
These arguments together establish CDL as MUST cover, but slow. It cannot be added to Section V proactively (no production applications yet to reference), but the mathematical foundations in Section IV must be built starting now.
Important context: Yusuke does not currently know category theory. Unlike the Lie group series (where Yusuke had prior knowledge and operated in fact-check mode), the CDL track requires Yusuke to learn alongside the writing process.
Pace strategy adopted: (γ) Trial parallel mode — start with Stage 1’s first page in parallel-learning mode (Yusuke reads Leinster Ch.1, Claude proposes site adaptation, Yusuke reviews as learner-reviewer rather than expert-reviewer). After this first page, reassess whether to continue parallel or shift mode.
Rationale: Yusuke’s own learning rhythm with category theory is unknown; cannot be predicted from Lie theory experience. The first page acts as calibration probe.
Both primary references are freely available as official PDFs (verified):
| Reference | Status | Access |
|---|---|---|
| Leinster, Basic Category Theory (CUP 2014) | Already in references.json, url field populated |
https://arxiv.org/abs/1612.09375 |
| Fong & Spivak, Seven Sketches in Compositionality (CUP 2019) | In references.json, but url field missing — needs update |
https://arxiv.org/abs/1803.05316 (also: https://dspivak.net/7Sketches.pdf, http://brendanfong.com/fong_spivak_an_invitation.pdf) |
Action item: Add Fong & Spivak url field to references.json (low priority; cosmetic).
Roles:
Both books are necessary; they are not redundant. Standard pairing in the ACT community.
Six stages, ~6–9 total pages, estimated 6 months to 1+ year of slow-burn parallel work:
| Stage | Est. pages | Content | Site location |
|---|---|---|---|
| Stage 0 | 0 | Yusuke reads Leinster Ch.1–2 (most of investment cost lives here) | — |
| Stage 1 | 2–3 | Categories, functors, natural transformations / Yoneda / adjunction (intro level) | disc-XX (Section IV) |
| Stage 2 | 1–2 | Limits & colimits / monads / Kan extensions (as needed) | disc-XX |
| Stage 3 | 1–2 | Applied flavor: quivers as functors / database functors / string diagrams | disc-XX |
| Stage 4 | 1 | CDL bridge: 2-categories of Para, lenses, monads on Para — primer for the Gavranović paper | ml-XX |
| Stage 5 (future) | 1 | CDL Overview: revisit the entire site from a categorical viewpoint | ml-XX |
Total: 6–9 pages, larger than the Lie group series (4 pages). Page-count expansion is expected — the Lie series went 2 → 4; the CDL series may go 6 → 12. Page IDs (XX) deferred until drafting.
A non-obvious strategic advantage: when Stage 1 begins, the site already contains a rich library of category-theoretic examples. The Stage 1 pages can use these as already-familiar concrete instances, dramatically lowering the explanatory burden:
This is an originality opportunity: rather than “introducing categories abstractly then giving examples,” the site can introduce categories as the language that names what readers have already been working with for 100+ pages. This framing is rare in standard textbooks and aligns with the site’s principle of using delayed payoff and retroactive recognition as a primary source of pedagogical originality.
Yusuke’s stated commitment: periodic monitoring of CDL project trajectory (Coend, ACT conferences, ICML/NeurIPS CDL papers). When production deployment signals appear, the CDL track shifts from slow-burn to active, and Section V CDL pages move from [deferred] to [next] status.
Trigger candidates for status shift:
Until then: parallel slow-burn, mathematics-only.
“Quantum” as a single application domain is misleading: what is called “Quantum” is actually three sub-domains with very different mathematical requirements and very different production-maturity levels. Treating them uniformly would either over-invest in immature areas or under-invest in already-deployed ones.
| Sub-domain | Mathematical core | 2026 production status |
|---|---|---|
| A — Quantum theory (physics) | Hilbert space + spectral theory + Fourier; unitary evolution; observables as self-adjoint operators; measurement | 100-year-established classical theory; no novelty |
| B — Quantum computation | Sub-domain A + tensor product structure (qubits) + quantum circuits + algorithms (Shor, Grover, VQE) + error correction | NISQ era; first documented practical quantum advantage cases (e.g., IonQ × Ansys medical-device simulation, 2025); Harvard/QuEra: fault-tolerant systems plausibly arrive late this decade. No production deployment yet. |
| C — Post-quantum cryptography (PQC) | Lattice theory + LWE / Module-LWE problem + coding theory + hash functions + elliptic curves | Already production-deployed: NIST finalized FIPS 203 (ML-KEM), 204 (ML-DSA), 205 (SLH-DSA) in August 2024; Google has enabled ML-KEM in Chrome; Microsoft deployed PQC in Azure and Windows; AWS likewise. End users already use it daily. CNSA 2.0 mandates quantum-safe systems for new US national-security deployments by January 2027. |
The asymmetry is decisive: A is mathematically settled but applied; B is research-active but not deployed; C is mathematically novel for the site AND already deployed at planet scale.
Sub-domain A + B: combined into a single Section V viewpoint page (“Quantum Information Science Overview” or similar — ml-XX).
Sub-domain C: belongs in Section I, as an extension of the existing crypto entry point at linalg-26 (Finite Fields).
references.json (currently tagged niche — this tag must be lifted if the crypto track is activated).ml-XX (Quantum Overview) separate from the A+B combined page. One page covers the viewpoint.The Section V quantum page (A+B) must explain Shor’s algorithm, which requires the reader to know what Shor breaks. This forces a dependency:
Before the Section V quantum page can be written, the Crypto Track must have covered classical encryption + public-key cryptography + at minimum a description of what RSA / ECDH protect.
This is a hard prerequisite. Without it, Shor’s algorithm reduces to “this breaks something” with no concrete referent, and the page fails its viewpoint role.
The reverse is not required: the Crypto Track does not need to reach PQC before the quantum page is written. Coverage stages 1–3 (and stage 4’s Shor description) of the Crypto Track suffice.
The Crypto Track sits at low priority in the roadmap — below GDL main-track, below CDL slow-burn — and is mood-driven in the precise sense: it is neither scheduled nor trigger-based, but driven by Yusuke’s interest level on a given day. It functions as a switch destination when CDL slow-burn fatigue accumulates, or when GDL main-track work needs a contrast.
Self-assessment (recorded explicitly, not a passing remark): cryptography is, by any reasonable measure, the most under-covered area of the entire site relative to its importance. It is unambiguously mathematics; it is unambiguously CS; it is unambiguously a math ↔ CS bridge — exactly the kind of content the site name MATH-CS COMPASS points at. That it is currently untouched reflects an honest accident of curriculum sequencing, not a principled exclusion. The roadmap acknowledges this asymmetry explicitly so future decisions are anchored against it: “low priority” does not mean “low importance” — it means “deferred for reasons of bandwidth, not principle.”
Three reasons converge:
Long mathematical runway: PQC requires understanding what classical cryptography is, why it works, what RSA / ECDH are, and only then does the lattice-based path become meaningful. Without classical cryptography first, there is nowhere to start — the path is unexpectedly long. Like CDL, this track has no shortcut: each layer requires the one below it.
No structural blocker: unlike CDL (which the site’s deepening curriculum will inevitably collide with), the Crypto Track can be deferred indefinitely without breaking other tracks. The site’s other content does not flow toward cryptography in the way it flows toward category theory.
Hard dependency only with Section V quantum page: see §4.4. The Crypto Track must reach a minimum coverage point before the Section V quantum page can be written, but that quantum page itself is also low-priority, so the constraint propagates naturally without forcing premature work.
The full track, if completed end-to-end, has seven natural stages. Page counts are not estimated — Yusuke may stop the track at any stage, and the right cutoff depends on developments not visible in 2026/5/9:
| Stage | Topic | Mathematical content | Section placement |
|---|---|---|---|
| 1 | Classical cryptography foundations | Symmetric encryption, encryption / decryption duality, Kerckhoffs’s principle, computational-complexity-based security definitions | Section I (extension of linalg-26 or new dedicated page) |
| 2 | Public-key cryptography & number-theoretic foundations | Diffie-Hellman, RSA, discrete logarithm problem, integer factorization problem, modular arithmetic in cryptographic context | Section I |
| 3 | Elliptic curve cryptography (ECC) | Elliptic curve groups, ECDH, ECDSA, modern usage (TLS, Bitcoin) | Section I |
| 4 | Quantum threat & Shor’s algorithm | Shor’s algorithm sketch, what it breaks (RSA, ECDH), Grover’s bound on symmetric ciphers — interfaces with Section V quantum page | Section I (with cross-link to Section V) |
| 5 | Lattice theory foundations | Lattices in ℝⁿ, SVP / CVP problems, LLL algorithm, lattice-based hardness assumptions | Section I |
| 6 | LWE / Module-LWE | Learning With Errors problem, Module-LWE, security reductions, structured-lattice variants | Section I |
| 7 | NIST PQC standards overview | ML-KEM (FIPS 203), ML-DSA (FIPS 204), SLH-DSA (FIPS 205); deployment context | Section I (capstone) |
Realistic stop-points (any of these is a legitimate end-state for the track):
Already in references.json:
Not yet in references.json, likely needed when track reaches stage 5+:
Promotion: trigger-based → active slow-burn track. Joins CDL as the second concurrent slow-burn track running parallel to the GDL main track.
Yusuke’s familiarity level (β): name-recognition and concept-level grasp of Brownian motion, Itō integral, SDE — but no prior structured reading of Durrett’s later chapters or Karatzas-Shreve. Yusuke’s self-assessment: trivia-level familiarity, not domain expertise. Yusuke is a CS author, not a probabilist by training.
Mode: Trial Parallel (same protocol as CDL):
Starting point rationale: BM is the natural foundation because
Combining BM with the Itō integral on a single nominal page reflects the structural logic: the pathological path properties of BM (nowhere differentiability, non-zero quadratic variation) are exactly what motivate the L² construction of the Itō integral.
Track scope: nominally three pages, but the page count is not fixed — the track follows the precedent of recent multi-page tracks (Lie groups 2→4, Lp spaces 1→2, Fourier-PDE 1→3) where conceptual paradigm shifts expand the initial estimate by a factor of 1.5–3×. The expectation here is that one or more of the three nominal pages will split as the material is developed. The structure below states scope and sequence, not a page-count commitment.
| Nominal page | Filename | Scope |
|---|---|---|
| 1 | brownian_motion_ito.html |
Axiomatic BM (Gaussian process + independent increments + continuity); existence (Kolmogorov extension + Čentsov, statement-level with proof pushed to Durrett); path properties (quadratic variation, nowhere differentiability); Itō integral construction via L² isometry; Itō formula (1D). May split into BM page + Itō page if the L² isometry construction expands. |
| 2 | sde_diffusion.html |
SDE definition; existence-uniqueness (Lipschitz drift/diffusion); canonical examples (geometric BM, Ornstein–Uhlenbeck, Langevin); infinitesimal generator; Dynkin’s formula; Girsanov theorem (statement + intuition). |
| 3 | fokker_planck_diffusion_model.html |
Fokker-Planck equation as the generator’s adjoint; heat equation as the special case (calc-33 back-link); score function and Stein’s identity (with optional ml-11 NGD back-link); reverse-time SDE (Anderson 1982); diffusion-model bridge to ml-14. |
First page priority: brownian_motion_ito.html. Even if it splits during writing, BM + Itō are conceptually one unit (path pathology motivates the L² construction), so they are written together and split only if section count forces it.
Application priority anchor: diffusion models are currently in production (Stable Diffusion, DALL-E, Sora, Veo), and the site already covers them at the discrete-DDPM/DDIM level in ml-14 (Intro to Diffusion Models). Phase 2e provides the continuous-time foundations — Brownian motion, Itō calculus, Fokker-Planck — that ml-14 deliberately defers; together the two complete what Murphy MLBook2 Ch.25 compresses.
Track-character note (recorded explicitly): Phase 2e is structurally a CS-author-learns-pure-math project, not a “CS author writes pure math from prior knowledge” project. This places it in the same category as the CDL slow-burn track:
| Track | Out-of-specialty domain for Yusuke | Authoring mode |
|---|---|---|
| CDL slow-burn | Category theory | Learning = writing (trial parallel) |
| Phase 2e slow-burn | Stochastic analysis | Learning = writing (trial parallel) |
Both contrast with the Lie group series (linalg-27~30), where Yusuke had prior knowledge and operated in expert-reviewer / fact-check mode. The two slow-burn tracks share a different rhythm: slower per-page progress, more dialog at the conceptual layer, and page-count expansion expected (CDL: 6→9 anticipated; Phase 2e: page splits anticipated as the L² construction, Fokker-Planck adjoint argument, and reverse-time SDE bridge each surface their own complexities).
Soft prereq from the Fourier-PDE pages (calc-33/34/35, completed): the Fokker-Planck page builds directly on calc-33 (heat equation) — Fokker-Planck is a parabolic PDE, the heat equation generalized with a drift term. The completed heat equation page already covers the spectral and kernel machinery the Fokker-Planck page will cite.
Bandwidth note: with Phase 2e promoted, the active track inventory is:
This is a 4-track configuration. Sustainability check is implicitly delegated to Yusuke’s mood-driven prioritization across sessions; if any one track stalls, the others continue. The roadmap does not enforce a single sequence.
Status: complete through Lee Ch.1–13 (calc-36~81 published, 6/3/2026). Series
scope, notation overload notes, future-track open items, and the Lee 2nd-ed. full
table of contents are in manifold_series_design_handout_v19.md.
The series spans topological/smooth manifolds, tangent vectors, immersions and
embeddings, submanifolds, Sard/Whitney, Lie groups, vector fields and flows, vector
bundles, the cotangent bundle, tensors, and Riemannian metrics, closing at the
tangent–cotangent (musical) isomorphism and pseudo-Riemannian metrics. Related
extension pages (calc-42 Homotopy and the Fundamental Group; supporting topology
calc-47) sit outside the spine. Per-page IDs, splits, prereqs, and topicGroups are
authoritative in curriculum.json; this roadmap no longer enumerates them.
Downstream now unblocked: Ch.13 supplies the Riemannian foundation for the GDL pillar (round metric on (S^n), (S^2) not flat as the curvature motivation), the retroactive strengthening of ml-12 NGD (Fisher metric as a Riemannian metric; musical isomorphism and (\operatorname{grad} f = (df)^\sharp)), and the entry point to information geometry. Curvature, geodesics, the exponential map, and the Levi-Civita connection are out of scope here (Lee Introduction to Riemannian Manifolds territory), to be acquired when GDL-pillar curvature work begins.
Group representations, subrepresentations, irreducibility, Maschke, Schur, character theory (finite groups), Lie group representations, Peter-Weyl. Finite group representations fill ≥2 pages; Lie group representations add 1–2 more. Connection: Peter-Weyl bridges back to calc-32 (Fourier as harmonic analysis on groups). Prereqs: linalg-27~30, linalg-22.
equivariant_nn.html)Generalize from permutation invariance (GNN) to continuous group equivariance (SO(3), SE(3)). Key insight: “The architecture encodes the symmetry.” Prereqs: linalg-27~30, linalg-XX (rep theory), ml-13.
At this point, readers have seen permutation equivariance (GNN), rotation/translation equivariance (Equivariant NN), and Riemannian structure (NGD). The unifying language is GDL — which becomes not a lesson to teach but a pattern the reader has already experienced.
Forward-link target reservations for planned pages. Completed pages are tracked in curriculum.json (authoritative); this registry exists to lock in filenames before drafting so cross-page references can be written ahead of time.
| Track | Est. Pages | Planned Filenames | Trigger / status |
|---|---|---|---|
| Representation Theory (linalg-XX) | ~3–4 | TBD | After manifold series (now complete); next active track |
| Equivariant NN (ml-XX) | 1 | equivariant_nn.html |
After rep theory |
| Section V Quantum Page (ml-XX) | 1 | TBD | After Crypto Track stages 1–4 |
| Stochastic Calculus (prob-XX) | 3+ | brownian_motion_ito.html, sde_diffusion.html, fokker_planck_diffusion_model.html (splits anticipated) |
Active slow-burn (Part 6) |
| CDL Track (disc-XX, ml-XX) | ~6–9 | TBD | Active slow-burn (Part 3) |
| Crypto Track (linalg-XX) | varies | TBD | Mood-driven (Part 5) |
| Regular Conditional Distributions (prob-XX) | ~1 | regular_conditional_distributions.html |
Phase 2e prerequisite (SDE / path-space measures) |
| Advanced VI topics (prob-XX) | ~1–2 | TBD | Triggered individually by ML-application pressure |
| DEC (disc-XX) | ~1–2 | TBD | Backlog |
| GDL Overview (ml-XX) | 1 | TBD | Backlog |
This map tracks primary usage for development planning. The site-wide reference index (data/references.json) is authoritative for bibliography format.
| Book | Pages (existing → planned) | Role |
|---|---|---|
| Conway — A Course in Functional Analysis | calc-23~28, calc-30~32 | Primary for all of Section II advanced analysis |
| Durrett — Probability: Theory and Examples | prob-13, prob-22~26 → Phase 2e | Primary for measure-theoretic probability and stochastic calculus |
| Stein & Shakarchi — Fourier Analysis | calc-14, calc-15, calc-32, calc-33, calc-34, calc-35 | Primary for Fourier and classical PDE applications |
| Stillwell — Naive Lie Theory | linalg-24, linalg-27~30 | Primary for Lie group series; supplement with Lee Ch.7+ for rigorous treatment |
| Leinster — Basic Category Theory | → CDL track Stages 1–2 | Primary for rigorous-pure side of CDL track; free PDF on arXiv |
| Fong & Spivak — Seven Sketches in Compositionality | → CDL track Stages 3 (and motivation throughout) | Primary for applied / intuitive side of CDL track; free PDF on arXiv |
| Menezes et al. — Handbook of Applied Cryptography | linalg-26 → Crypto Track (Part 5) | Primary for Crypto Track; previously tagged niche, lifted as of 5/9/2026 |
| Bronstein et al. — Geometric Deep Learning | Insight Boxes site-wide, ml-13 → ml-XX (Equivariant NN), ml-XX (GDL overview) | “Destination viewpoint” text; referenced not followed |
| Murphy Book 1 — Probabilistic ML: Introduction | ml-01~08, ml-13 | General ML reference |
| Murphy Book 2 — Probabilistic ML: Advanced | ml-09~12, ml-14, prob-16, prob-26 → Phase 2e (continuous-time diffusion) | Information geometry at applied level; sufficient until Amari is needed |
manifold_series_design_handout_v19.md| Track | Pages | Completed | Notes |
|---|---|---|---|
| Formal Methods | disc-16 (formal_methods.html), disc-17 (lean_in_practice.html) |
5/14/2026 | Section IV third pillar (disc-4, 16, 17) established. Bidirectional bridge with disc-12 (Four Color Theorem) via T-four_color_theorem anchor. Full continuity in formal_methods_track_handout_v3.md. |
| Smooth Manifolds (Lee Ch.1–13) | calc-36~81 (spine) + calc-42, calc-47 (related/supporting) | 6/3/2026 | Phase 3 complete. Topological/smooth manifolds → tangent vectors → immersions/embeddings → submanifolds → Sard/Whitney → Lie groups → vector fields/flows → vector bundles → cotangent bundle → tensors → Riemannian metrics (Ch.13: calc-78~81). Landing point for the GDL pillar; retroactively strengthens ml-12 NGD. New topicGroup riemannian-geometry for calc-78~81. Continuity in manifold_series_design_handout_v19.md. Curvature/geodesics deferred to a future LeeRM-based series. |
Topics explicitly deferred — not forgotten, but not on the critical path.
| Item | Trigger to Revisit |
|---|---|
| Schwartz Space & Distributions | If a PDE or generalized function page is planned beyond calc-33/34/35 |
| Pontryagin Duality (Fourier on groups) | After linalg-27~30 + calc-32 + linalg-XX (rep theory), if harmonic analysis track emerges |
| Spectral Theory of the Laplacian (continuous) | After calc-XX (Riemannian Metrics), as bridge to geometric spectral theory |
| Regular Conditional Distributions | Phase 2e companion — required for SDE / Itô filtration. Not blocking prob-26 (VI works under density assumption). |
| Fiber Bundles & Gauge Theory | If GDL viewpoint page demands gauge equivariance machinery |
| String Diagrams | After CDL Stage 4 (CDL bridge page); part of Stage 4 if applicable |
| Advanced VI topics (normalizing flows, IWAE, EP, wake-sleep, hierarchical / structured / implicit posteriors) | Triggered individually by ML-application pressure |
| Uniform Integrability & Martingale Convergence | Triggered by RL theory / stochastic approximation / bandit algorithms. Resolves prob-23 UI forward-reference. |
| Variational Representations & f-Divergences | Triggered by contrastive learning / MI estimation / generalization theory (foundation for MINE, f-GAN, InfoNCE, PAC-Bayes) |
| Characteristic Functions & CLT (Rigorous) | Triggered by advanced asymptotic statistics; prereq: calc-32 |
Notation Consistency is Non-Negotiable: Calligraphic letters for spaces ((\mathcal{X}, \mathcal{Y}, \mathcal{Z}, \mathcal{H}, \mathcal{X}^, \mathcal{X}^{**})); functionals as (\varphi); operator norm as (|\varphi|_{\mathcal{X}^}). All new Section II pages match the notation in calc-23~28.
Cross-Page Linking & References: Links verified against actual filenames and anchor IDs before inclusion. No in-body citations — references handled in the site-wide reference index only. Forward links use descriptive text only (no <a href>) until target page exists.
Application Viewpoint Philosophy: Use Insight Boxes and the map’s Tessera to connect abstract theorems to applications without breaking formal proofs in main text. Application domains introduced when tools are ready — not forced prematurely. (Asymmetric across domains: see Parts 1–5.)
Fisher Information vs. Hessian Distinction: The real distinction is reparametrization invariance (Čencov’s theorem) vs. loss-dependence and non-guaranteed positive definiteness — not “global vs. local.”
Page Count Estimation: Topics requiring new conceptual paradigms consistently expand 1.5–4× initial estimates. Plan for expansion and defer ID assignment until drafting begins. Calibrations: Lie group series (planned 2 → actual 4), calc-30 split (1 → 2), Phase 2c (planned 2 → actual 3), Fourier-PDE pages (Claude initially proposed 1 → ratified as 3, delivered as calc-33/34/35). The CDL track is expected to follow this pattern (proposed ~6–9, may grow to 9–12+).
Per-topic prior-knowledge calibration (added 2026/5/9): Yusuke’s prior knowledge varies sharply by topic. Lie theory was familiar (linalg-27~30 ran in expert-reviewer mode); category theory and continuous-time stochastic analysis are not (CDL and Phase 2e are learn-while-writing). Track-character assignments must be calibrated per topic, not against a global expertise label.
Mood-driven dispatch over forced sequencing (added 2026/5/9): the active 4-track configuration (GDL main + CDL + Phase 2e + Crypto) does not enforce a single sequence. Yusuke prioritizes per session; if any one track stalls, the others continue.