Harvest — project-file sweep (2026-09-12)

Harvest — project-file sweep

Source: everything currently in the project (algebra-geometry-dictionary.md §1–60, CASE-01…CASE-07, the three standalone notes, README.md + R*.py, the five procedure docs, the four course docs) Date: 2026-09-12 Kind of run: harvest (finished material, no blind step — per the SOP's own second option) Checked against: claude_map-index.md §2 registry, §3 graph, §4 false friends

Nothing below is written to any file. Per the SOP, the triage call is yours; what follows is the bookkeeping plus my proposed mechanism sentences, sorted so the mechanical items are separated from the judgment ones. Each item is numbered so you can accept/reject by number.

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0. The headline

The map is missing a node for which a complete course already exists, and it is the node that four other nodes' edge labels are already quietly leaning on. That's item B1, and it's not a judgment call — it's a filing error.

After that, the material splits cleanly: three genuinely new node candidates with theorem-backed mechanisms, about five new edges to existing nodes, five false-friend entries (one of which upgrades §4 from a list into a test), and an answer to the probe-registry question that's already sitting in the project as README.md.

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1. Section B — bookkeeping (no triage needed, these are just unrecorded)

B1. p-adic Numbers has a complete course and no node

padic-numbers-course.md is a full course doc by §1.2's definition — motivation, history, foundations, eight modules with problem sets, answer key, resources, and its own §6 local connection graph — plus a 218-line flashcard deck. It appears nowhere in §2 or §3.

Meanwhile the map already references it from three directions without naming it:

• the HARM → CFT edge label reads "Tate's local zeta integral IS the p-adic Mellin transform" — an object that lives on $\mathbb{Q}_p^\times$;
• the HARM note in §3 refers to "the $p$-adic side of the map" as though it were a node;
• st/padic-reduction-and-chaos is cited in the §4 Lyapunov entry.

Proposed row:

Edges I'd draw, each with a mechanism sentence that survives §1.1:

• PA → CFT. Local class field theory is the theory of abelian extensions of $\mathbb{Q}_p$; the invariant map $\mathrm{Br}(F)[n] \cong \mathbb{Z}/n$ is stated about the local field, and the global theory is assembled from the local ones by the idele class group. (Course §2, Module 7; dictionary §37.)
• PA ↔ HARM. Not a new mechanism — the existing HARM → CFT label. Tate's local zeta integral is the Mellin transform on $\mathbb{Q}_p^\times$; the local Euler factor is its value. The edge currently jumps HARM straight to CFT over a node that isn't drawn.
• PA → EC. Three joins, and the course says so itself: good/bad reduction, Chabauty's method (which bounds rational points using $p$-adic analysis directly), and $\mathbb{Z}_p$ as the coefficient ring of the Galois representations in EC Module 7.
• PA → LANG, via LCFT, same route the map already uses for CFT → LANG.

I would not draw PA → RMT on the strength of the function-field analogy alone; the Katz–Sarnak material is over $\mathbb{F}_q$, not over $\mathbb{Q}_p$, and treating "local" as the shared object would be exactly the §1.1 failure mode. See E5 for the edge I think is real there.

B2. The cases, probes and procedures are invisible to §2

map-index.md describes itself as "single source of truth for the whole project," but seven Differential cases, five compute recipes, and five procedure docs live in the project without appearing in it. Minimal fix: a §2b table listing them with the node each attaches to. My attribution:

Note for the memory of this project: the video-intake procedure is no longer unrecovered. claude_procedure-video-intake.md is here, alongside an article-intake doc that wasn't previously recorded as existing at all.

B3. Two recipes named in README.md are not in the project

R2_katz_sarnak.sage and R6_krasner_degree_growth.sage appear in the README table but no file exists. R6 is the one that settles CASE-07, and R2 is the only probe attached to the map's most-developed node. Either they were never written or they didn't make the upload.

B4. The §4 Lyapunov entry has a fifth instance

The existing entry tracks four uses. Dictionary §38 adds a fifth: the degree-growth rate of algebraic approximants to a transcendental element of $\mathbb{C}_p$, described there as "a Lyapunov-like exponent for [completion, algebraic closure], zero at ∞," and carried into CASE-07 as the case's whole question. Same word, and this time not even the same kind of object (a growth rate in precision, not an orbit separation rate). Worth appending to the entry rather than letting it be re-checked.

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2. Section N — new node candidates

Ordered by how much I'd defend them. Each gives the mechanism sentence §1.1 demands.

N1. RAMIF — Ramification, depth, and the two wildnesses

Sources: CASE-04, note-accidental-symmetries-as-ramification.md, dictionary §5a, §7, §17.

Mechanism to LANG: the local Langlands correspondence for tori preserves Moy–Prasad depth exactly when the torus is tame (Yu); for a wildly ramified torus the depth shifts by an amount governed by the Hasse–Herbrand function of the splitting extension (Mishra–Pattanayak), and the shift is computed from $v_E(\sigma u - u) \ge v_E(u) + b$ — the Galois action being trivial to depth $b$.

Mechanism to PA: the ramification filtration $G_i = \{\sigma : v_E(\sigma x - x) \ge i+1\}$ is defined by the valuation of a local field, i.e. by the object PA is about. This is a genuine prerequisite arrow, not a resemblance.

Mechanism to GAL: $G_i$ is a filtration of a Galois group; tame/wild is a statement about its image.

Why it's a node and not three statements: it carries a stated open invariant question (is the depth shift a defect of the definition or a real invariant of the outer action?), an explicitly identified two-sources structure (form-wildness vs parameter-wildness, with Swan = irregularity of $p$-curvature as the characteristic-$p$ common yardstick), and it's the hinge that N5 and the accidental-symmetry note both hang off.

The physics edge is the one I'd make you decide. The note identifies the EFT filtration by breaking dimension, $d(S) = \min\{d : \mathcal{L}_d \text{ not } S\text{-invariant}\}$, with the ramification filtration, and reads the depth-shift theorem as the statement that a symmetry of depth $b$ is violated at order $(v/\Lambda)^b$ — neutrino mass at depth 1, proton decay at depth 2. The note's own status line is honest about what this is: "a definition on the physics side and a theorem-backed structure on the arithmetic side… no new theorem." So the mechanism sentence exists, but it's a translation, not a shared object. My read: it passes §1.1 in letter and is exactly the kind of thing §1.1 was written to catch in spirit. I'd draw it dashed, or hold it as a statement inside RAMIF rather than as a graph edge, and let the Herbrand-function open item (upper vs lower numbering ↔ UV vs IR operator dimension) decide it — if that lands, it's a real edge; if it doesn't, the identification stays a mnemonic.

N2. LOCGLOB — Local-to-global obstructions, and their two mechanisms

Sources: note-chromatic-vs-anonymous-impossibility.md, dictionary §28–31, §53–54, CASE-05.

Mechanism to PA: the local–global principle is Module 7 of the $p$-adic course; Grunwald–Wang is its canonical failure, and the note computes it — $\Sha^1(G,\mu_8) = H^1(G,\mu_8) = \mathbb{Z}/2$ for $K = \mathbb{Q}(\sqrt7)$, so 16 is an 8th power in every completion and not in $K$.

Mechanism to CFT: the classical Grunwald–Wang special-case condition ($K(\zeta_{2^t})/K$ noncyclic with all decomposition groups cyclic) is exactly the note's §2 lemma — a $G$-set with elementwise fixed points and no global fixed point exists iff $G$ is noncyclic — and Tate's $H^3(G,L^\times) \cong \mathbb{Z}/(n/m)$ is the same noncyclicity seen one rung up. Same theorem, two statements.

That lemma is what makes this a node rather than an analogy: it is a specific, proved, elementary group-theoretic statement that both the arithmetic and the distributed-computing side instantiate, not a generic appeal to orbits and stabilizers.

The edge I'd reject. LOCGLOB → SYMM looks tempting — "a distinguished element can be selected from a symmetric family iff the residual symmetry has a fixed point" would give SYMM its first outward edge, which it wants. I don't think it survives §1.1. Landau-theory vacuum selection and Franklin's involution's stabilizers use orbit-stabilizer, which every group action uses; that's the Lyapunov failure mode with better vocabulary. What would earn the edge is the finer distinction — fixed-point-free vs elementwise-fixed-but-globally-free — actually discriminating two physical selection problems. It doesn't yet. Log as checked-and-rejected until it does.

CASE-05 sits here, and is the strongest test of whether this node is load-bearing: it asks directly whether the distributed reading of the problem of time yields anything the standard 3+1 analysis doesn't, and names a candidate (boosts must break completely, rotations may survive). That's a probe, not a node claim.

N3. FRACTAL — Fractal geometry and topological matter

Sources: dictionary §21–27, CASE-01, R1/R5/R7.

This is not FRAC. FRAC is fractional calculus; this is Hausdorff/spectral dimension, Sierpinski carpets and gaskets, Chern bands on fractal lattices, and whether topological order survives on a space with no plane in it. Keeping them as one node would bury the single most interesting thing in the cluster (E1 below), which is that there's a real, computed edge between them.

Mechanism to FRAC: the Caffarelli–Silvestre extension realizes $(-\Delta)^s$ as scattering on $\mathbb{H}^{n+1}$ (Chang–González), the first scattering pole sits at $s = \delta$ = the Hausdorff dimension of the limit set, and Laskin's fractional Schrödinger operator $(-\Delta)^{\alpha/2}$ generates Lévy paths of dimension $\alpha$. Fractional order equals fractal dimension, by theorem, in three independent settings. That's the edge; the shared word is not.

Mechanism to RMT: dictionary §21's Lapidus row — lattice self-similar strings have complex dimensions periodic in $2\pi i/\log r$ (the $\mathbb{F}_q$ zeta shape), non-lattice ones are quasiperiodic (the number-field shape), and Lapidus–Maier state RH as an inverse spectral problem for fractal strings. Check this one twice; it's the kind of statement that is either a genuine edge or a restatement of a resemblance, and I can't tell from the dictionary entry alone.

Status of its own content: honest and mixed, which is why I'd take it. One prediction failed outright (anyons on fractals interpolating by cut-point structure — Manna–Pal–Wang–Nielsen refuted it, corrected entry recorded), one result was downgraded by its own follow-up (§26 → §27, the cylinder pump null result), and CASE-01 exists precisely because two diagnostics disagree. R1 settles it and is a real overnight computation.

N4. DIVALG — Division algebras and the Standard Model

Sources: CASE-02, CASE-03, CASE-06, dictionary §39, §11, §20.

Three of the seven cases are here, which by itself argues for a node. The construction: $T = \mathbb{R}\otimes\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$ acting on itself, particles as the algebra's own module, gauge group as the symmetries of that action, $SU(3)$ as the stabilizer in $G_2$ of an imaginary unit, $U(1)$ as the number operator.

Mechanism to SYMM: SYMM's registry line already includes the Higgs mechanism, electroweak mixing and GUT running; CASE-06 is the same question one rung earlier (is chirality a vacuum choice, an algebraic necessity, or brute?), and left–right symmetric models are the explicit-becomes-spontaneous move SYMM is about.

Mechanism to RAMIF (the interesting one): triality is the outer automorphism $S_3$ of $D_4$; charge conjugation is the diagram automorphism of $SU(N)$, $N\ge3$; and the depth-shift story is governed by the outer action of Galois on the dual group. Three appearances of the same object — an outer automorphism of a Dynkin diagram acting on a group — in settings that don't otherwise touch. The accidental-symmetries note already makes half of this explicit ("charge conjugation… literally is a rung-1 outer action; the table says it is a tame one"). If any edge in this harvest is going to force a revision rather than just place cleanly, I'd bet on this one.

Caveat, from the material itself: CASE-02 records a bet that two of the three "3"s were the same failing cleanly, and the $S_3$ flavour structure fails on $|V_{ub}|$ (predicted ≈ 0.06 against 0.0037 observed). The node comes with its own falsifications attached, which is the right shape.

N5. AZUMAYA — Rational flux, Frobenius twists, and the Brauer invariant

Source: note-hofstadter-reading-of-char-p-langlands.md, dictionary §16–17.

The most mechanism-dense thing in the whole sweep, and the one I'd promote fastest after B1.

The shared object, stated once: an algebra with two generators whose $n$-th powers are central, hence a bundle of $n\times n$ matrix algebras over the spectrum of that centre. The rotation algebra at flux $p/q$ ($U^qV = VU^q$, centre generated by $q$-th powers, fibre $M_q$ over the magnetic Brillouin zone); the Weyl algebra in characteristic $p$ ($[\partial,x^p]=px^{p-1}=0$, centre $k[x^p,\partial^p]$, Azumaya over the Frobenius twist, BMR); the quantum group at $q^\ell=1$. Frobenius twist = magnetic unit cell.

Mechanism to CFT: $\mathrm{Br}(F)[n] = H^2(F,\mu_n) \cong \mathbb{Z}/n$ is the invariant map of local class field theory; the rotation algebra at $p/q$ is Azumaya of degree $q$ over $T^2$ with class $p \in H^2(T^2,\mu_q)$, the first Chern class reduced mod $q$. The Brauer invariant is flux per cell mod $n$ — the same cyclic-algebra presentation $(F_n/F,\mathrm{Frob},\varpi^k)$ in both.

Mechanism to RAMIF: for Artin–Schreier $y^p - y = t^{-m}$, the $p$-curvature $\psi = f^p + \partial^{p-1}f$ evaluates to $(-m)^p t^{-p(m+1)}$ (the $\partial^{p-1}$ term vanishes because its coefficient runs over every residue class mod $p$ except $-m$, hence contains 0). Swan conductor = irregularity of the $p$-curvature. That's Deligne's comparison, computed, and it is the bridge between the two wildnesses of N1.

Mechanism to LANG: Braverman–Bezrukavnikov and Chen–Zhu prove characteristic-$p$ geometric Langlands as a twisted Fourier–Mukai transform over the Frobenius-twisted Hitchin base, using exactly the Azumaya structure; the Hitchin map in characteristic $p$ is the $p$-curvature.

Mechanism to FRACTAL: the integer lift of the mod-$n$ class is fixed in Hofstadter by the TKNN Diophantine equation — which is what R5_fractal_streda.py goes and tests on fractal Chern gaps.

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3. Section E — edges to existing nodes, no new node needed

E1. FRAC → PA. Computed, 30 digits, and the strongest single result in the dictionary: the Riesz constant is $-\zeta_\mathbb{R}(n+2s)/\zeta_\mathbb{R}(-2s)$ and Vladimirov's constant is $-\zeta_p(1+\alpha)/\zeta_p(-\alpha)$ — one uniform formula $-\zeta_v(1+\alpha)/\zeta_v(-\alpha)$ at every place, with the product over all places equal to 1 by $\xi(s)=\xi(1-s)$. Fractional calculus is not an add-on at $p$; on $\mathbb{Q}_p$ integer-order calculus is degenerate (zero derivative doesn't mean constant) and Vladimirov's nonlocal operator is the calculus. Also: the Vladimirov operator is scattering on the Bruhat–Tits tree, i.e. $p$-adic AdS/CFT.

E2. FRAC → RAMIF. Prediction P5: the conductor $a(\chi)$ is the localization scale $p^{-a}$ of character-twisted fractional kernels, via Tate's $\varepsilon$-factors. The depth filtration read by fractional calculus.

E3. RMT → PA. Dictionary §58–60 assembles the explicit formula place by place on Connes' adele class space (pole 2.266297, archimedean −1.196347, primes 1.069950, residual −8.8e−8) and computes the Ollivier curvature of the Bruhat–Tits trees ($\kappa = -2/(p+1)$, exact for $p=2,3,5$). The mechanism: the periods of the scaling flow are $\log p$ because valuations are integers — nothing put in by hand.

E4. RMT — the missing-rung diagnosis. The sharpest statement in the sweep, and it belongs in the RMT note whether or not any edge is drawn: over $\mathbb{F}_q$ the fixed-point rung is Frobenius eigenvalues on cohomology with $|\alpha| = \sqrt q$ from duality plus positivity; over $\mathbb{Q}$ that rung is empty — no curve, no cohomology, no Frobenius. The recorded failure point is structural and specific: the adele class space gives the trace formula but no surface, no cycles, no intersection theory, because there is no $\mathbb{Q}\times_{\mathbb{F}_1}\mathbb{Q}$. Plus the type-error observation that over $\mathbb{F}_q$ the Frobenius graph is a single cycle while over $\mathbb{Q}$ the scaling flow is a one-parameter family, so the explicit formula is the intersection number, smeared. The Weil determinant check at $q=7,g=1$ ($a=2,-4,5$ pass with det 24, 12, 3; $a=\pm6$ fail with det −8, matching $|a|\le 2\sqrt7$) is a verified worked example sitting there unused.

E5. RMT → INTEG / §24's Mixmaster computation. Periodic BKL sequences = periodic continued fractions = closed geodesics, with multiplicity = the number of $SL_2(\mathbb{Z})$-classes of forms of discriminant $t^2-4$, i.e. class numbers; mean multiplicities 4.3/8.1/17.0/24.9 against $e^{L/2}/L$ = 3.4/6.8/14.8/22.2. R3_artin_billiard_maass.py tests whether the resulting spectrum is Poisson. This touches INTEG, which the map records as genuinely disconnected — worth checking whether it stays that way.

E6. SYMM gains Feigenbaum content it doesn't have. Dictionary §57 computes $\delta$ and $\alpha$ from scratch by bisection (4.669061 and 2.502901 against 4.669201609 and 2.502907875), finds the attractor's dimension 0.5245 from the exact $\alpha$, and — the part I'd flag — verifies that the attractor's dynamics is the 2-adic odometer $x \mapsto x+1$ on the period-256 orbit, with chaos past the accumulation point being the shift turning on. If that holds up it's an unforced SYMM → PA edge via a computed identification, not a vocabulary match. The existing §3 note already cross-references Feigenbaum's $\delta$ between SYMM and FRAC; this would change its status.

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4. Section F — false friends (§4 additions)

F1. "Ramification" — arithmetic vs fractal. CASE-04's wild/tame ramification filtration of a Galois action, and R7/§23's finite ramification of the Sierpinski gasket (removing finitely many cut points disconnects it), are different objects with no shared theorem. Both are live in this project right now and both are load-bearing in their own cluster, which makes this the highest-collision-risk pair in the sweep. Log before it costs anything.

F2. "Fractional" / "fractal" / "fractional quantum Hall." Three senses, one root. The real edge is the specific one in N3 (fractional order = fractal/spectral dimension, by the Caffarelli–Silvestre and Laskin theorems); FQH filling fractions are not in that family. Log the distinction alongside the edge so the edge doesn't get read as licensing the others.

F3. The three "3"s. CASE-02 is a ready-made §4 entry and a rare one — a tested and failed identification, not a suspected one. The centre of colour $SU(3)$ (a gauge element, discrete anomalies vanish for every generation count, cannot count anything), baryon triality (not in the gauge group, anomaly $2N_\text{gen} \bmod 3$, counts but only in multiples of 3, presupposes a SUSY-type discrete gauge symmetry), and $D_4$ triality (the $S_3$ outer automorphism permuting three 8-dimensional reps). The bet that two were the same failed cleanly. The gauge-centre candidate is ruled out, which is a stronger statement than "no mechanism found."

F4. Lyapunov, fifth instance. As in B4.

F5. The chromatic/anonymous criterion — this one isn't an entry, it's a test. The note's central claim is that "consistent on every local view, inconsistent globally" is a symptom with two distinct mechanisms, and it gives the discriminator: ask whether the local observers act on the coefficients. If not, the obstruction is a cycle in a labelled nerve (chromatic: FLP, $k$-set agreement, Fekete's read-only anomaly, Borromean rings, Massey products, Rédei symbols). If so, it's a fixed-point failure of a group action (anonymous: Angluin, Yamashita–Kameda, Grunwald–Wang, Tate's $H^3$).

That is exactly what §4 is for, one level up. Right now §4 is a list of rejected pairs; this converts part of it into a decision procedure that tells you which kind of local-to-global failure you're looking at before you go looking for a mechanism. If you take one structural change from this sweep, I'd take this one — it's the "forces a revision" cell of your own triage grid.

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5. The probe registry question looks answered

The open item was whether probes live as a section in map-index.md or as probes.md. The project already contains a third answer that's better than both: README.md plus one file per probe. The README table has exactly the schema that was specified — question, prediction, runtime, and what a positive or negative result is worth — and each R*.py is a self-contained statement of its own case (purpose, prediction, method, install, what the result buys), runnable over SSH without reference to any map node. That's the self-containment requirement satisfied structurally rather than by discipline.

What's missing is only the link in the other direction: map-index.md §5 gaining the fourth triage outcome — (d) real friction, mechanism not yet in hand → open a probe, record the file — and §2b (B2) recording which node each probe attaches to. Probes and cases are the same lifecycle at different stages (CASE-01 → R1, CASE-07 → R6), which the numbering already reflects.

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6. What I parked rather than placed

• Dictionary §55–56 (Monster, sporadics, moonshine). Deep, and it doesn't place — no mechanism to anything currently on the graph. Two of three bets failed with computation behind the failures (O'Nan is not the Miyamoto group of any Monster-type axial algebra, via PSL(2,31) involution products of order 15 and 16; string cosmological billiards are chaotic only for $D\le10$ so the Monster Lie algebra lives past the end of chaos). The surviving bet — umbral moonshine = the walls rung, Monster = the root-free limit, Mathieu = 24 walls — is held firmly and checked via the Golay weight distribution. This is the "park it, deliberately" cell, and I'd park it with that reason recorded rather than inventing a home.
• Dictionary §49–52 (information, synthetic curvature, symplectic). These act as machinery used by §58–60 rather than as subject matter of their own; I'd let them ride inside E3/E4 until something needs them standing alone.
• CASE-03 (global form of the SM gauge group). Placed under N4, but its actual diagnosis — that the two camps answer different questions, and "the room is closed" is an added axiom deciding an independent statement the way $V=L$ decides CH — is a logic move with no node anywhere near it. Flagged, not placed.

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7. Your calls

Ranked by what I'd do first:

1. B1 — file the p-adic node. Not a judgment call; four existing edge labels already depend on it.
2. F5 — decide whether the chromatic/anonymous criterion restructures §4. This is the one that could force a revision.
3. N5 (AZUMAYA) and N1 (RAMIF) — I'd take both; they're mutually load-bearing via Swan = irregularity of $p$-curvature, and both attach to nodes you already have.
4. N2, N3, N4 — real, but each has an edge I'd reject inside it (LOCGLOB→SYMM; the Lapidus/RMT row; the physics half of the accidental-symmetries identification). Worth going through one at a time rather than as a batch.
5. F1–F4 — cheap, do them whenever.
6. B3 — find out whether R2 and R6 exist.