h¹¹=4 · h²¹=68 · χ=-128

exact counts · proxy meaning Exact counts from Hodge numbers for a CY3: dim of Kähler moduli space is h¹¹; complex-structure moduli space dimension is h²¹ (before quotienting by discrete symmetries / flux stabilization). Stabilization map is mechanism taxonomy, not a solved potential.

Mirror swaps h¹¹ ↔ h²¹ and sends χ → −χ at the Hodge level.

Mirror compare: (4,68) ↔ (68,4) · CS-heavy

Counts of unknown intersection numbers if the ring were fully generic. Real CY3 intersection rings are highly constrained; these are size estimates for the computational problem, not computed κ_ijk.

Hodge mirror: h¹¹=68 · h²¹=4 · χ=128 — not yet in the Hall of Fame

• h¹¹ counts independent Kähler (size) deformations of even cycles.
• h²¹ counts complex-structure (shape) deformations.
• Period / Picard–Fuchs problem size proxy: K = h²¹+1 = 69.
• G₃ fluxes can fix complex structure in principle; Kähler moduli need non-perturbative effects — see stabilization map below.

Stabilization map

• Complex structure 68 in_principle
  G₃ flux (tree-level GVW superpotential)
  68 moduli; flux lattice dimension proxy K = h²¹+1
• Kähler 4 needs_extra
  Non-perturbative (gaugino condensation / instantons)
  4 moduli; fluxes alone do not stabilize volume moduli
• Dilaton / axio-dilaton 1 needs_extra
  Fluxes + non-perturbative corrections
  Universal modulus; needs a concrete flux+np setup

exact budget · Stirling proxy Exact: tadpole budget L = |χ|/24 from the Euler characteristic. Computed proxy: Stirling / Bousso–Polchinski log-density and asymptotic N_flux estimate. Not a vacuum scan over a real flux lattice.

Periods / special geometry

Period integrals are not evaluated numerically here. We give the exact dimensions and the standard special-geometry identities.

Orientifold / O3–O7 tadpole

• exact
  Geometric tadpole χ(X)/24

  value=5.333333
• needs_involution
  O7 contribution χ(D_O7)/12

  unknown
• needs_involution
  O3 plane charges

  unknown
• free_after_fluxes
  Mobile D3 count N_D3

  unknown

Orientifold involution and O-plane loci are not stored. We expose the exact χ/24 piece and leave O3/O7 as explicit unknowns.

Toy flux-lattice enumeration

K=69 too large for in-process enumeration (max_k=5)

Flux-scan readiness (7/8 · 87.5%)

Readiness for a *real* flux vacuum scan. Green items are available from Hodge/curated metadata; red items need an external geometry pipeline.

• HAVE
  Hodge invariants locked

  (h¹¹,h²¹,χ)=(4,68,-128)
• HAVE
  Tadpole budget L=|χ|/24

  L=5.3333
• HAVE
  Ambient / curated construction class

  Textbook class or stored ambient
• HAVE
  Hypersurface / CICY equation

  Present
• HAVE
  Polytope vertex matrix

  Present
• HAVE
  Fine regular star triangulation

  Present
• HAVE
  Periods / prepotential

  Symbolic PF structure available (K=h²¹+1=69); numerical periods not evaluated
• NEED
  Orientifold O3/O7 data

  χ/24 term known; O3/O7 charges need involution

Constraints

• D3 tadpole (IIB sketch) exact_budget
  N_{D3} + N_{\mathrm{flux}} = L = |\chi|/24
  L = 5.3333; headroom proxy = 0.0
• Orientifold-extended tadpole partial
  N_{D3} + \tfrac{1}{2}N_{\mathrm{flux}}= \tfrac{\chi(X)}{24} + \tfrac{\chi(D_{O7})}{12} + \cdots
  Orientifold involution and O-plane loci are not stored. We expose the exact χ/24 piece and leave O3/O7 as explicit unknowns.
• Period / flux monomial count exact_count
  K = h^{2,1}+1,\quad b_3 = 2(h^{2,1}+1)
  K=69, b₃=138 (CY3 Betti)
• Flux vacua asymptotic estimate proxy
  \log N_{\mathrm{flux}} \sim 2K\log(2\pi L) - \log K!
  K = 69; log N ≈ 258.4456; N ~ 1.744e+112; ranking sigmoid = 0.9318
• Toy flux-lattice count skipped
  \sum_{i=1}^{K} n_i^2 \le \lfloor 2L\rfloor
  counted=None · K=69 too large for in-process enumeration (max_k=5)

For a full flux scan we would need

• Numerical periods / prepotential on the complex-structure moduli space
• Orientifold involution with explicit O3/O7 charges
• Physical H³ flux lattice (beyond the toy ∑n² bound)
• A concrete triangulation / hypersurface when not in the geometry pack

indices + proxies Topological indices are exact formulas; soft terms and Yukawas are symbolic / combinatorial until moduli vevs and bundles/branes exist.

Soft terms (symbolic)

No soft spectrum is computed: that needs stabilized moduli vevs, gauge kinetic functions, and a mediation model. Formulas are shown so the missing inputs are explicit.

Toy soft-parameter card

illustrative · not from this CY Illustrative MSSM soft pattern from user-chosen A0, m1/2, tanβ, m0. NOT derived from this CY, NOT RGE-evolved, NOT physical GeV predictions.

Yukawa texture shown as generation-index placeholders Y_{ij}; no numerical SM Yukawas are claimed.

Yukawa structure

Closed-string triple intersections need the Kähler ring; open-string/MSSM Yukawas need D-brane/bundle data.

Gauge embedding roadmap

Gauge embeddings are not unique at the Hodge level; this is a roadmap.

• needs_geometry
  IIB / F-theory gauge from 7-branes

  G\subset E_8\ \text{(local F-theory)}
• needs_geometry
  Intersecting brane / quiver SM

  [D_a]\cdot[D_b]\cdot[D_c]
• exact_index
  Generation index from topology

  |\chi|/2=64\ \text{(heterotic-like index; IIB uses intersections)}

Computed from topology

• Net generation index exact_formula → 64
  For heterotic compactifications the net number of chiral generations is |χ|/2 when the gauge bundle index equals the Euler characteristic index. This is a topological necessary condition, not a full model.
• Three-generation necessary condition exact_check → no
  |χ| = 6 ⇔ |χ|/2 = 3 generations (heterotic index sketch)
• c₂·J scale proxy proxy → 456
  Linear stand-in used in ranking features; a real c₂·J needs the resolved toric geometry and a chosen Kähler class.
• Symmetric triple-intersection unknowns (generic) combinatorial → 20
  Counts of unknown intersection numbers if the ring were fully generic. Real CY3 intersection rings are highly constrained; these are size estimates for the computational problem, not computed κ_ijk.
• Soft-term skeleton symbolic → gravity / modulus mediation sketch
  No soft spectrum is computed: that needs stabilized moduli vevs, gauge kinetic functions, and a mediation model. Formulas are shown so the missing inputs are explicit.
• Yukawa structure combinatorial → combinatorial
  Closed-string triple intersections need the Kähler ring; open-string/MSSM Yukawas need D-brane/bundle data.
• Gauge embedding roadmap checklist → checklist
  Gauge embeddings are not unique at the Hodge level; this is a roadmap.

textbook curated cicy featured-seed verified

Still needs vevs / branes / bundles

• Numerical soft masses (need moduli vevs + mediation)
• Explicit SM Yukawa matrices (need branes/bundles + wavefunctions)
• Unique gauge embedding (need concrete geometry)

geometry · specimen Pure-mathematics framing of stored Hodge data and construction metadata. No invented theorems, intersection numbers, or curve counts.

Invariants for geometers

Hodge mirror: h¹¹=68 · h²¹=4 · χ=128 — not yet in the Hall of Fame

Mirror symmetry

Mirror symmetry exchanges Kähler and complex-structure deformations at the level of Hodge numbers for Calabi–Yau threefolds. A full mirror map identifies period data / quantum-corrected Kähler geometry with the complex-structure VHS of the mirror — that map is not computed here.

hodge_swap_only Only the Hodge-level swap (h¹¹,h²¹,χ) ↔ (h²¹,h¹¹,−χ) is known here. No explicit Greene–Plesser / toric mirror construction is claimed unless curated construction notes say so.

Mirror swaps h¹¹ ↔ h²¹ and sends χ → −χ at the Hodge level.

Enumerative geometry / periods

Periods pending — hand off to CYTools/Sage. Pipeline stage reflects stored geometry only; no invented curve counts or Yukawa numbers.

filled: vertices; pending: triangulated, intersections, periods

Gromov–Witten / enumerative invariants would use the intersection ring and (quantum-corrected) periods. This site does not enumerate curves.

• have
  Vertices

  Curated / stored polytope vertices present.
• pending
  Triangulation / ambient

  Triangulation or ambient resolution note present.
• pending
  Intersections

  Combinatorial / symbolic only unless literature or offline dump is attached — Hodge numbers alone do not fix intersections.
• pending
  Periods

  Numerical periods pending offline worker — not invented here.

Periods pending — hand off to CYTools/Sage.

Toric / combinatorial classification

representative Toric / combinatorial data are classification input (KS reflexive polytopes, weights, configuration matrices). Status “representative” means Hodge numbers alone do not uniquify a polytope unless a curated construction says otherwise.

Multidegree (2,2,2,2) hypersurface

Vertices / vertex matrix

[
  [
    1,
    0,
    0,
    0
  ],
  [
    1,
    2,
    0,
    0
  ],
  [
    1,
    0,
    2,
    0
  ],
  [
    0,
    0,
    0,
    1
  ],
  [
    0,
    -4,
    4,
    1
  ],
  [
    0,
    -4,
    0,
    1
  ],
  [
    1,
    6,
    -2,
    -2
  ],
  [
    -4,
    16,
    -12,
    -7
  ]
]

Configuration matrix

[
  [
    2,
    2,
    2,
    2
  ]
]

• Kreuzer–Skarke CY data (TU Wien) — Search polytopes with h11=4, h12=68 in the online forms / dumps.
• CYTools (geometry toolkit) — Offline: polytopes, triangulations, intersection numbers.

Hodge theory / moduli

For a Calabi–Yau threefold the intermediate Jacobian / Hodge filtration on H³ gives a variation of Hodge structure (VHS) over the complex-structure moduli space. Exact moduli dimensions follow from Hodge numbers alone.

Period domain / PF problem size proxy K = h²¹+1 = 69 (symbolic; not a solved VHS).

Cite this specimen

Citeable geometric specimen: Hodge class plus any stored construction metadata (toric data, literature periods, mirror link when available).

Deep links: #mathematics · #certificates

Handoff checklist

• READY
  Download / export geometry JSON

  GET /api/geometry/kreuzer-skarke-af211369e6bf or /api/geometry/lookup for this Hodge class; analysis bundle at /api/analysis/kreuzer-skarke-af211369e6bf/bundle.
• READY
  Open in CYTools / PALP

  Import vertices / weight system when present; triangulate favourable hypersurface and compute intersection ring offline.
• READY
  Macaulay2 / Sage algebraic checks

  Use ambient / configuration matrix when present; otherwise reconstruct from KS reflexive polytope data.
• TODO
  Periods / Gromov–Witten handoff

  Periods pending — hand off to CYTools/Sage.
• TODO
  Still missing on this site

  intersection numbers; numerical periods

metadata · curated · workplan Exact: stored metadata keys and Hodge invariants. Curated: textbook constructions when (h¹¹,h²¹) matches a known class. Geometry DB: offline / seeded SQLite hits preferred when richer (vertices). Unavailable: unique polytope vertices / triangulation unless stored.

Tetraquadric class in (P¹)⁴.

Citations

• Classic CICY / toric example — curated
• Classic CICY / mirror symmetry pedagogy; Kreuzer CY site — geometry-pack
• Kreuzer–Skarke arXiv:hep-th/0002240; HF calabi-yau-data/polytopes-4d — geometry-db
• Dataset / catalog source — dataset

Pipeline stage checklist

Honest progress: vertices → triangulation / ambient → intersections → periods. Literature / combinatorial counts are labeled; numerical periods are never invented.

• HAVE
  Vertices

  Curated / stored polytope vertices present.
• TODO
  Triangulation / ambient

  Triangulation or ambient resolution note present.
• TODO
  Intersections

  Combinatorial / symbolic only unless literature or offline dump is attached — Hodge numbers alone do not fix intersections.
• TODO
  Periods

  Numerical periods pending offline worker — not invented here.

Stored features

Construction data present

Vertex matrix

[
  [
    1,
    0,
    0,
    0
  ],
  [
    1,
    2,
    0,
    0
  ],
  [
    1,
    0,
    2,
    0
  ],
  [
    0,
    0,
    0,
    1
  ],
  [
    0,
    -4,
    4,
    1
  ],
  [
    0,
    -4,
    0,
    1
  ],
  [
    1,
    6,
    -2,
    -2
  ],
  [
    -4,
    16,
    -12,
    -7
  ]
]

Construction workplan

1. Lock topological invariants
   Use (h¹¹,h²¹,χ)=(4,68,-128) as search keys.
2. Find reflexive 4-polytopes with these Hodge numbers
   PALP / KS database / CYTools: filter polytopes whose favourable hypersurfaces give h¹¹=4, h²¹=68.
3. Choose a fine regular star triangulation
   Fixes the toric ambient and the resolved CY hypersurface.
4. Compute periods / prepotential
   Needed for a real flux scan on the 68-dimensional complex-structure moduli space.
5. Impose D3 tadpole
   Target budget L = |χ|/24 = 5.3333.

Look up geometry externally

• Kreuzer–Skarke CY data (TU Wien) — Search polytopes with h11=4, h12=68 in the online forms / dumps.
• KS list paper (arXiv) — Original reflexive polytope enumeration.
• CYTools (geometry toolkit) — Offline: polytopes, triangulations, intersection numbers.

How one would reconstruct

Kreuzer–Skarke: look up a reflexive 4-polytope with these Hodge numbers (PALP / KS database), choose a triangulation, then form the anticanonical hypersurface. Many polytopes can share the same (h¹¹, h²¹, χ) — the content-addressed id here is not a KS polytope id.

Dataset source: http://hep.itp.tuwien.ac.at/~kreuzer/CY/

model-building · necessary only Model-building aids: topological exclusions and literature cards. Spectra only when cited from published references.

Geometry pipeline stage

filled: vertices; pending: triangulated, intersections, periods

Pipeline: vertices → triangulation (ambient) → intersections (combinatorial / literature unless offline dump) → periods (CdOGP literature formulas when applicable — never invented numerics).

• HAVE
  Vertices

  current offline stage

  Curated / stored polytope vertices present.
• TODO
  Triangulation / ambient

  Triangulation or ambient resolution note present.
• TODO
  Intersections

  Combinatorial / symbolic only unless literature or offline dump is attached — Hodge numbers alone do not fix intersections.
• TODO
  Periods

  Numerical periods pending offline worker — not invented here.

Exclusion certificates

PASS = does not rule out under the stated assumptions; FAIL = rules out that model class under Y. Never sufficiency.

• FAIL
  Heterotic standard-embedding 3-generation index exclusion

  Rules out: Standard-embedding heterotic models with net chirality n_gen = |χ|/2 = 3 generations

  χ=-128, |χ|=128, n_gen=|χ|/2=64. |χ|≠6 so standard-embedding 3-generation models are ruled out (would give n_gen=64, not 3).

  • Heterotic E₈×E₈ (or Spin(32)/ℤ₂) on a CY3
  • Standard embedding: gauge connection = spin connection (or index equals the Euler characteristic index)
  • Net chiral generation count n_gen = |χ|/2
  • Target phenomenology: exactly three net generations
• PASS
  IIB/KS D3 tadpole budget L = |χ|/24 exclusion

  Rules out: Flux + D3 constructions that require tadpole budget L ≥ L_min=1

  L=|χ|/24=5.33333 with |χ|=128. L ≥ L_min=1: does not rule out models needing that budget.

  • Type IIB / O3–O7 orientifold sketch on a CY3
  • Tadpole identity L = |χ|/24 (Euler characteristic budget)
  • Stated minimum budget L_min=1 for the model class under study
  • No additional localized sources that enlarge the effective budget

Literature model cards

No literature model card is seeded for this Hodge class yet.

certificates · not theorems Machine-checkable identities and necessary conditions on the stored invariants. These are certificates, not uniqueness theorems or existence proofs for string vacua.

Necessary checks

• PASS
  Euler identity

  \chi = 2(h^{1,1} - h^{2,1})

  derived χ=-128, supplied χ=-128
• FAIL
  Dataset target rule

  |\chi| < 100

  |χ|=128
• PASS
  Tadpole charge defined

  L = |\chi|/24 \ge 0

  L=5.3333
• PASS
  Positive Hodge numbers

  h^{1,1} \ge 1,\quad h^{2,1} \ge 1

  h¹¹=4, h²¹=68
• PASS
  Search verification flag

  verified_target from ranking run

  yes
• PASS
  Generation index defined

  n_{\mathrm{gen}} = |\chi|/2 \ge 0

  n_gen=64
• PASS
  Euler parity (CY3)

  \chi = 2(h^{1,1}-h^{2,1})\ \text{even}

  χ=-128
• PASS
  Period count identity

  K = h^{2,1}+1,\quad b_3 = 2(h^{2,1}+1)

  K=69, b₃=138
• PASS
  Flux-scan readiness

  \text{readiness } 7/8

  87.5% — see Fluxes / Construction tabs

Identity certificates