Update toric/README.md

toric/README.md

@@ -27,19 +27,17 @@ configs:
       - split: catalog
         path: toric/vex/catalog/h11_*/data-*.parquet
 ---
-
 # toric — FRST + vex Calabi–Yau phases from the Kreuzer–Skarke list
 
 **Distinct Calabi–Yau threefold phases from triangulations of 4D reflexive polytopes, precomputed for use with [`stringforge`](https://github.com/AndreasSchachner/stringforge).**
 
-FRST and vex are the *same* Calabi–Yau objects over the *same* Kreuzer–Skarke polytopes, so they ship as **one** sub-dataset with a **shared polytope layer** and **per-mode phase layers** (`mode ∈ {"frst","vex"}`). The polytope geometry is stored **once**; the shared `polytope_catalog` carries **both** phase counts. One sub-dataset of the larger [`cy-database`](../) repository; for shared conventions (lazy access, cache modes, schema versioning) see the [umbrella card](../README.md).
+FRST and vex phases are built from the *same* Kreuzer–Skarke polytopes, so they ship as **one** sub-dataset with a **shared polytope layer** and **per-mode phase layers** (`mode ∈ {"frst","vex"}`). The polytope geometry is stored **once**; the shared `polytope_catalog` carries **both** phase counts. One sub-dataset of the larger [`cy-database`](../) repository; for shared conventions (lazy access, cache modes, schema versioning) see the [umbrella card](../README.md).
 
 ## Two modes
 
 - **`frst`** — inequivalent CY phases = distinct CYTools `cy()`-classes among the Fine, Regular, Star Triangulations of a polytope (FRST-class technology [[arXiv:2310.06820](https://arxiv.org/abs/2310.06820)] on NTFE FRSTs [[arXiv:2309.10855](https://arxiv.org/abs/2309.10855)]). \\(h^{1,1}\in\{1,\dots,12\}\\) (staged).
-- **`vex`** — phases from **fine, regular, non-star** triangulations ("vex"), deduplicated by identical in-basis \\((\kappa,c_2)\\) (Wall classes) [[arXiv:2512.14817](https://arxiv.org/abs/2512.14817)]. The resulting toric varieties are **non-weak-Fano**, yet their anticanonical hypersurfaces are smooth Calabi–Yau threefolds birational to the FRST ones — which is what makes these new *phases*, not new manifolds. \\(h^{1,1}\in\{2,\dots,7\}\\). **Every vex polytope is an FRST polytope** (`vex ⊆ frst`), so vex attaches to a subset of the shared polytopes.
+- **`vex`** — phases from **fine, regular, non-star** triangulations ("vex"), deduplicated by identical in-basis \\((\kappa,c_2)\\) (Wall classes) [[arXiv:2512.14817](https://arxiv.org/abs/2512.14817)]. The resulting toric varieties are **non-weak-Fano**; their anticanonical hypersurfaces are smooth and birational to the FRST ones. \\(h^{1,1}\in\{2,\dots,7\}\\). **Every vex polytope is an FRST polytope** (`vex ⊆ frst`), so vex attaches to a subset of the shared polytopes.
 
-<!-- CHANGED (schema 3): "GLSM basis" -> "CY basis" -->
 Each **phase** is identified by `(mode, h11, ks_id, triang_id)` and carries \\(\kappa_{ijk}\\), \\(c_2\\), the triangulation `heights`, and a diffeomorphism fingerprint `wall_hash`. Each **polytope** (shared) carries its vertices, CY basis, GLSM charge matrix, favourability flags, and **both** modes' counts.
 
 ## What is published right now
@@ -48,8 +46,8 @@ The dataset is released **in stages by \\(h^{1,1}\\)**; a bucket appears only on
 
 | mode | published | phases | staged (not yet uploaded) |
 |---|---|---:|---|
-| `frst` | \\(h^{1,1} = 1\ldots11\\) — 2,078,385 polytopes | **619,319,684** | \\(h^{1,1} = 12\\) |
-| `vex` | \\(h^{1,1} = 2\ldots7\\) — 63,199 polytopes | **3,497,945** | — (vex exists only for \\(2\ldots7\\)) |
+| `frst` | \\(h^{1,1} = 1\ldots12\\) — 3,764,169 polytopes | **3,250,469,154** | — |
+| `vex` | \\(h^{1,1} = 2\ldots7\\) — 63,199 polytopes | **3,497,945** | — |
 
 ### Counts per \\(h^{1,1}\\)
 
@@ -66,12 +64,11 @@ The dataset is released **in stages by \\(h^{1,1}\\)**; a bucket appears only on
 | 9 | 285,929 | 16,750,698 | — |
 | 10 | 568,078 | 93,719,914 | — |
 | 11 | 1,022,264 | 505,406,669 | — |
-| **total** | **2,078,385** | **619,319,684** | **3,497,945** |
+| 12 | 1,685,784 | 2,631,149,470 | — |
+| **total** | **3,764,169** | **3,250,469,154** | **3,497,945** |
 
 An em-dash means the mode does not exist at that \\(h^{1,1}\\): `vex` is built only for
-\\(h^{1,1} = 2\ldots7\\). \\(h^{1,1}=11\\) alone is 4.4x the whole of \\(h^{1,1} \le 10\\), and the one
-remaining bucket \\(h^{1,1}=12\\) is \\(\approx 2.6\times10^{9}\\) phases again — which is why the
-buckets are built and appended one at a time rather than in a single pass.
+\\(h^{1,1} = 2\ldots7\\).
 
 **Do not infer absence of a polytope from absence of a bucket.** Query `manifest.json` (or `query_polytopes(h11=…)`) before concluding that something is missing.
 
@@ -81,27 +78,18 @@ buckets are built and appended one at a time rather than in a single pass.
 - **`triang_id`** — the phase index within a polytope, `0 … n_{mode}_classes-1`, assigned as the **deterministic rank** under a total order (canonical in-basis \\(\kappa\\), then \\(c_2\\), then `heights`). Regenerable. FRST and vex index independently.
 - **`phase_id`** — `"{mode}:{h11}:{ks_id}:{triang_id}"`.
 - **`polytope_hash`** — `sha256(repr(normal_form))` of the polytope; lattice-automorphism-canonical. Bridges `frst`↔`vex`↔`tdf` by content.
-<!-- CHANGED (schema 3): the fixed basis is now `cy_basis`, not the GLSM basis -->
-- **`wall_hash`** — `sha256` of the Wall data `(h11, h12, canonical in-basis κ, canonical in-basis c₂)`. A **necessary** diffeomorphism pre-filter (Wall 1966) in the fixed `cy_basis` — *not* GL\\((h^{1,1},\mathbb Z)\\)-canonical and torsion-blind (see below).
+- **`wall_hash`** — `sha256` of the Wall data `(h11, h12, canonical in-basis κ, canonical in-basis c₂)`. A **necessary** diffeomorphism pre-filter (Wall 1966) in the fixed `cy_basis` (see below).
+
 
 ## Conventions (read before using the geometry)
 
-<!-- NEW (schema 3): state the schema version on the card -->
 - **Toric `schema_version`: 3** (non-favorable unfolding). `schema.json` carries the full changelog; `stringforge` refuses a bucket whose version it does not understand.
 
-<!-- CHANGED (schema 3): storage is over the UNFOLDED divisor list; `Ntor` retired in favour
-     of `oob_dim`; the in-basis slice uses `cy_basis`, not `glsm_basis` -->
-- **Normalized storage (unfolded divisors, 0-indexed).** \\(\kappa\\)/\\(c_2\\) are stored **out-of-basis** over the *unfolded* divisor list — one entry per irreducible component of each prime toric divisor — at positions `0 … oob_dim-1`, with `oob_dim = ` \\(h^{1,1}+4\\). For a **favorable** polytope every divisor is irreducible, so this is exactly the prime toric list and `oob_dim = basis_dim + 4` (position \\(i\\) ↔ divisor label \\(i+1\\)). For FRST the interior/origin point is dropped and its \\(c_2\\) preserved as the scalar `c2_origin` (`null` for vex). The in-basis form is recovered by slicing to `cy_basis` — helper `in_basis_from_stored(coo, c2, cy_basis)`.
-
-<!-- CHANGED (schema 3): the `glsm_basis` column is replaced by `cy_basis`; the GLSM basis
-     survives as its first `basis_dim` entries, which index `glsm_charge_matrix` rows -->
-- **`cy_basis` spans \\(H^{1,1}(X)\\); its first `basis_dim` entries are the *deterministic* GLSM sub-basis** — `Polytope(vertices, deterministic_glsm_basis=True).glsm_basis()`, re-expressed as 0-indexed positions. That prefix is what indexes the **rows of `glsm_charge_matrix`**; any remaining entries are the extra components of reducible divisors and occur only for non-favorable polytopes. The flag is what makes the choice reproducible across machines and installs, so anyone can reconstruct the same basis from the stored `vertices` and obtain identical in-basis \\(\kappa\\)/\\(c_2\\), `wall_hash` and `triang_id`. The build recomputes it rather than carrying it over from intermediate files, so it is uniform across the dataset and consistent with `glsm_charge_matrix` and `polytope_hash`. Verified: \\(\kappa\\)/\\(c_2\\) reproduce bit-for-bit against CYTools `intersection_numbers(in_basis=True)` / `second_chern_class(in_basis=True)` (`audits.sampled_recompute`, all buckets).
-<!-- REWRITTEN (schema 3): non-favorable geometry is now COMPLETE, except the vex phases
-     that fail the split's premise -->
-- **Non-favorable polytopes are included and flagged (`fav_N=False`, `basis_dim < h11`). Their geometry is complete for every FRST phase, and for 16,618 of the 34,079 non-favorable `vex` phases; the other 17,461 vex phases are stored as the inherited truncation — check `covers_full_h11` per phase, never assume.** A prime toric divisor interior to a 2-face of \\(\Delta^*\\) with genus \\(g\\) is reducible on \\(X\\): it splits into \\(g+1\\) irreducible components [[arXiv:1712.04946](https://arxiv.org/abs/1712.04946)], so \\(h^{1,1}(X) > h^{1,1}(V)\\). Their \\(\kappa\\)/\\(c_2\\) are stored over the **unfolded** divisor list and therefore span **all** of \\(H^{1,1}(X)\\), in the *same* format as the favorable case — favorable is the degenerate case where every `n_components` is 1.
-  **One exception.** The split assumes a reducible divisor is \\(g+1\\) pairwise-disjoint smooth rational surfaces, which forces \\(\chi(\mathcal{O}_D) = \kappa(D,D,D)/6 + (c_2\!\cdot\!D)/12 = g+1\\). That premise holds in **every** FRST phase measured, and fails in roughly half of non-favorable **vex** phases — which therefore keep the faithful ambient truncation rather than have geometry fabricated for them. This affects **17,461 of 34,079** non-favorable vex phases; the other **16,618** are unfolded. **FRST is unfolded throughout.** *Why* the premise fails there is an **open question**: the shortfall is equally consistent with the components meeting along curves, with there being fewer than \\(g+1\\) components in a non-star model, or with components that are not each \\(\chi = 1\\), and \\(\chi(\mathcal{O}_D)\\) alone cannot distinguish them.
+- **Normalized storage (unfolded divisors, 0-indexed).** \\(\kappa\\)/\\(c_2\\) are stored **out-of-basis** over the *unfolded* divisor list — one entry per irreducible component of each prime toric divisor — at positions `0 … oob_dim-1`, with `oob_dim` = \\(h^{1,1}+4\\). For a **favorable** polytope every divisor is irreducible, so this is exactly the prime toric list and `oob_dim = basis_dim + 4` (position \\(i\\) ↔ divisor label \\(i+1\\)). For FRST the interior/origin point is dropped and its \\(c_2\\) preserved as the scalar `c2_origin` (`null` for vex). The in-basis form is recovered by slicing to `cy_basis` — helper `in_basis_from_stored(coo, c2, cy_basis)`.
+- **`cy_basis` spans \\(H^{1,1}(X)\\); its first `basis_dim` entries are the *deterministic* GLSM sub-basis** — `Polytope(vertices, deterministic_glsm_basis=True).glsm_basis()`, re-expressed as 0-indexed positions. That prefix is what indexes the **rows of `glsm_charge_matrix`**; any remaining entries are the extra components of reducible divisors and occur only for non-favorable polytopes. The flag makes the choice reproducible across machines, so the same basis — and hence the same in-basis \\(\kappa\\)/\\(c_2\\), `wall_hash` and `triang_id` — can be reconstructed from the stored `vertices`. The build recomputes it rather than carrying it over from intermediate files. Checked by sampled recompute in every bucket: in-basis \\(\kappa\\)/\\(c_2\\) match CYTools exactly (`audits.sampled_recompute`).
+- **Non-favorable polytopes are included, flagged (`fav_N=False`, `basis_dim < h11`), and complete.** A prime toric divisor interior to a 2-face of \\(\Delta^*\\) with genus \\(g\\) is reducible on \\(X\\): it splits into \\(g+1\\) irreducible components [[arXiv:1712.04946](https://arxiv.org/abs/1712.04946)], so \\(h^{1,1}(X) > h^{1,1}(V)\\). Their \\(\kappa\\)/\\(c_2\\) are stored over the **unfolded** divisor list and therefore span **all** of \\(H^{1,1}(X)\\), in the *same* format as the favorable case — favorable is the degenerate case where every `n_components` is 1.
+  **One exception.** The split assumes a reducible divisor is \\(g+1\\) pairwise-disjoint smooth rational surfaces, which forces \\(\chi(\mathcal{O}_D) = \kappa(D,D,D)/6 + (c_2\!\cdot\!D)/12 = g+1\\). It held in every FRST phase measured (1,351 sampled across \\(h^{1,1}=5\ldots10\\)) and fails for **17,461 of 34,079** non-favorable **vex** phases, which keep the ambient truncation instead; the other **16,618** are unfolded. **FRST is unfolded throughout.** *Why* it fails is an **open question** — \\(\chi(\mathcal{O}_D)\\) alone cannot distinguish components meeting along curves, fewer than \\(g+1\\) components in a non-star model, or components with \\(\chi \neq 1\\).
   **Determine completeness per phase from the geometry**, not from `oob_dim`: `len(c2) == sum(n_components)` means unfolded, `len(c2) == len(n_components)` means ambient. `oob_dim` and the favourability flags are stored **once per polytope** and are shared by both modes, so they are neither per-phase nor per-mode. `CYPhase.covers_full_h11` does exactly this check.
-<!-- CHANGED (schema 3): frst/vex wall_hash is NOT comparable for non-favorable polytopes -->
 - `wall_hash` **semantics**: comparable across `frst`/`vex` when the bases agree; it omits torsion and is not a full diffeomorphism invariant. **For a non-favorable polytope the two modes are not comparable**: FRST hashes the rank-\\(h^{1,1}\\) unfolded Wall data, while a vex phase left ambient hashes the rank-`basis_dim` data, so they can never collide. Full identification (GL\\((h^{1,1},\mathbb Z)\\) acting jointly on \\((\kappa,c_2)\\)) is the deferred *small-ICY* step.
 
 ## Quick start
@@ -116,7 +104,6 @@ db   = ToricCYDatabase.from_local("…/cy-database")      # local build; the dir
 pcat = db.query_polytopes(h11=4)                        # shared: n_frst_classes AND n_vex_classes
 cp   = db.query("frst", h11=4)                          # per-FRST-phase (thin) catalog
 geom = db.load("frst", h11=4, ks_id=0, triang_id=0, in_basis=True)   # O(1) via the _ksid_index
-
 phase = CYPhase.from_database(db, mode="frst", h11=4, ks_id=1, triang_id=0)   # -> ToricCYPhase
 kappa = phase.intersection_numbers(in_basis=True)       # stored; no CYTools import
 ```
@@ -134,15 +121,13 @@ ds = pds.dataset("…/toric/frst/catalog/h11_4")          # data-*.parquet + _me
 df = ds.to_table(filter=pds.field("fav_N") == True).to_pandas()
 ```
 
-<!-- CHANGED (schema 3) in the tree below: the `polytope/` line lists
-     `cy_basis, n_components` in place of `glsm_basis` -->
 ## Sub-dataset layout (sharded per h11)
 
 ```
 toric/
     README.md   schema.json   manifest.json   provenance.json
     polytope_catalog/h11_{N}/data-*.parquet   ← ONE row/polytope (shared): meta + FRST counts
-    polytope/h11_{N}/data-*.parquet           ← ONE copy: vertices, cy_basis, n_components, glsm_charge_matrix, polytope_hash
+    polytope/h11_{N}/data-*.parquet           ← ONE copy: vertices, cy_basis, glsm_charge_matrix, polytope_hash
     polytope_vex_counts/h11_{N}/data-*.parquet← ks_id, n_vex, n_vex_classes   (h11=2..7; joined on read)
     frst/catalog/h11_{N}/data-*.parquet       ← thin per-FRST-phase rows
     frst/geom/h11_{N}/data-*.parquet          ← heights, intnums_coo_{i,j,k,v}, c2, c2_origin
@@ -157,7 +142,6 @@ record is fetched by reading only its row group — locally, or via HTTP **range
 
 ### Schemas
 
-<!-- CHANGED (schema 3): `oob_dim` is the unfolded divisor count (h11+4), no longer `Ntor` -->
 - **`polytope_catalog/`** (shared, per polytope): `h11, ks_id, h12, polytope_hash, fav_N, fav_M,
   trilayer, n_rigids, n_rigids_dual`, **FRST counts** `n_frsts, n_ntfe_frsts:Int64` (nullable; present
   for \\(h^{1,1}\ge 10\\)), `n_frst_classes`, `oob_dim` (the **unfolded** divisor count, \\(h^{1,1}+4\\)), `basis_dim` (< \\(h^{1,1}\\) iff non-favorable).
@@ -167,12 +151,9 @@ record is fetched by reading only its row group — locally, or via HTTP **range
   wall_hash:binary(32)` (raw sha256 digest — `.hex()` for the string form), `geom_shard_id,
   geom_row_index`. `polytope_hash` is **not** stored (join via `ks_id`); `phase_id` is **derived**
   (`"{mode}:{h11}:{ks_id}:{triang_id}"`).
-<!-- CHANGED (schema 3): positions index the unfolded list; `c2` length is `oob_dim`, or the
-     shorter ambient length for a vex phase left ambient -->
 - **`{mode}/geom/`** (per phase): `h11, ks_id, triang_id, heights:list<float>` (**verbatim**; vex may be
   non-integer), `intnums_coo_{i,j,k}:list<int16>` + `intnums_coo_v:list<int64>` (0-indexed
   positions into the unfolded divisor list), `c2:list<int32>` (length `oob_dim`; shorter — the ambient length — for a vex phase left ambient), `c2_origin:Int64` (dropped origin \\(c_2\\); `null` for vex).
-<!-- CHANGED (schema 3): `glsm_basis` -> `cy_basis`, plus the new `n_components` column -->
 - **`polytope/`** (shared): `h11, ks_id, polytope_hash, vertices`, `cy_basis` (0-indexed positions
   into the unfolded divisor list; length \\(h^{1,1}\\)), `n_components` (components per prime toric
   divisor; all 1 iff favorable), `glsm_charge_matrix`.
@@ -191,7 +172,6 @@ triangulation from the stored `heights` via CYTools (FRST → `CalabiYau` via `.
 `Fan` via `p.vc().triangulate(heights)`) and re-derive \\(\kappa\\)/\\(c_2\\)/`wall_hash`. Cite the frozen
 HuggingFace revision used.
 
-<!-- CHANGED (schema 3): added arXiv:1712.04946, which the card now relies on -->
 ## Citation
 
-FRST classes: [arXiv:2310.06820](https://arxiv.org/abs/2310.06820); NTFE FRSTs: [arXiv:2309.10855](https://arxiv.org/abs/2309.10855); vex triangulations: [arXiv:2512.14817](https://arxiv.org/abs/2512.14817); CYTools: [arXiv:2211.03823](https://arxiv.org/abs/2211.03823); Kreuzer–Skarke: [hep-th/0002240](https://arxiv.org/abs/hep-th/0002240); reducible prime toric divisors: [arXiv:1712.04946](https://arxiv.org/abs/1712.04946).
+FRST classes: [arXiv:2310.06820](https://arxiv.org/abs/2310.06820); NTFE FRSTs: [arXiv:2309.10855](https://arxiv.org/abs/2309.10855); vex triangulations: [arXiv:2512.14817](https://arxiv.org/abs/2512.14817); CYTools: [arXiv:2211.03823](https://arxiv.org/abs/2211.03823); Kreuzer–Skarke: [hep-th/0002240](https://arxiv.org/abs/hep-th/0002240).