Upload README.md

README.md

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+---
+license: cc-by-4.0
+language:
+- en
+tags:
+- physics
+- dimensional-analysis
+- buckingham-pi
+- vaschy-buckingham
+- symbolic-regression
+- science
+pretty_name: "Vashy: Dimensionally-Consistent Physics Equations"
+size_categories:
+- 1M<n<10M
+configs:
+- config_name: default
+  data_files: data/circuits.parquet
+---
+
+# PhysicsBabel — A Complete Catalogue of Dimensionally-Consistent Physics Equations
+
+**Vashy** is an exhaustive, machine-generated catalogue of every *irreducible
+dimensionless relation* that can be formed from a curated pool of 60 physical
+quantities spanning all of classical physics. It is built directly from the
+**Vaschy–Buckingham π theorem**: any physically meaningful equation must be
+dimensionally homogeneous, so the set of dimensionally-consistent monomials is
+exactly the set of dimensionless products the theorem describes.
+
+Each of the **2,325,320 rows** is one such relation — for example
+`F = k₀ · m·a` (Newton's second law), `ℓ = k₀ · v⁻¹·ρ⁻¹·μ` (the Reynolds
+number), or `m = k₀ · E·c₀⁻²` (mass–energy equivalence) — annotated with
+metrics that let you filter the physically meaningful laws out of the vast space
+of mere dimensional coincidences.
+
+> `k₀` denotes the dimensionless constant of order unity that dimensional
+> analysis *cannot* determine (e.g. the ½ in kinetic energy, or 6π in Stokes'
+> drag). This is an intrinsic limitation of the method, not of the dataset.
+
+---
+
+## What is in the dataset
+
+The catalogue is **complete**: it contains *every* minimal dimensionless
+relation of size 2 to 8 over the quantity pool. Size 8 is a hard mathematical
+ceiling — with 7 SI base dimensions, no irreducible relation can involve more
+than 8 quantities (a relation of `k` quantities needs rank `k−1 ≤ 7`).
+
+| # quantities | rows | note |
+|---|---|---|
+| 2 | 22 | pairs with identical dimensions (e.g. energy vs torque) |
+| 3 | 593 | e.g. `F = k₀·m·a`, `E = k₀·m·v²` |
+| 4 | 17,325 | e.g. Reynolds, Weber, Strouhal numbers |
+| 5 | 280,646 | |
+| 6 | 813,919 | |
+| 7 | 1,140,445 | |
+| 8 | 72,370 | must span all 7 base dimensions at once (rare) |
+| **Total** | **2,325,320** | |
+
+Of these, **46,720 are flagged `plausible`** (a conservative gate for genuine
+physical candidates) and **20 correspond to classically named laws / numbers**.
+
+---
+
+## Provenance & method
+
+Every quantity is represented as a vector of integer exponents over the seven SI
+base dimensions. A **dimensionless relation** among a set of quantities is an
+integer null vector of the matrix whose columns are those dimension vectors.
+
+A row in this dataset is a **circuit** of that "dimensional matroid": a
+*minimal* dependent set, where every quantity is essential (removing any one
+destroys the dimensionless product) and which is not a product of smaller
+relations. Formally, a set of `n` quantities is a circuit iff its dimension
+matrix has rank `n−1` and the (unique) null vector has full support. Each
+circuit corresponds to exactly **one** irreducible dimensionless number, and the
+catalogue enumerates all of them without duplication.
+
+Dimensional analysis fixes only the *form* of a law up to `k₀`; it does not
+prove physical existence. That is why the dataset ships with relevance metrics
+rather than claiming every row is a real law.
+
+---
+
+## Dataset structure
+
+A single split (`default`) backed by `data/circuits.parquet` (~108 MB, zstd).
+
+| column | type | description |
+|---|---|---|
+| `size` | int | number of quantities in the relation (2–8) |
+| `keys` | string | sorted, comma-separated quantity keys (the relation's identity) |
+| `symbols` | string | the quantities' symbols, `·`-joined |
+| `domains` | string | distinct physics domains involved, comma-separated |
+| `pi` | string | the dimensionless number (Π group), e.g. `ℓ·v·ρ·μ⁻¹` |
+| `equation` | string | the relation solved for one quantity, e.g. `ℓ = k₀ · v⁻¹·ρ⁻¹·μ` |
+| `name` | string \| null | recognized name of the law / dimensionless number, if any |
+| `n_const` | int | number of universal constants (c₀, G, h, k_B, e, N_A) involved |
+| `n_domains` | int | number of distinct physics domains bridged |
+| `max_exp` | int | largest absolute exponent in the monomial |
+| `exp_l1` | int | sum of absolute exponents (monomial "weight") |
+| `complexity` | float | composite complexity score (lower = simpler) |
+| `score` | float | composite physicality score (higher = more plausible) |
+| `plausible` | bool | passes the conservative physicality gate (see below) |
+| `exponents` | string (JSON) | `{quantity_key: integer_exponent}` for the dimensionless monomial |
+
+### The `plausible` gate
+
+A row is flagged `plausible = true` when it satisfies **all** of:
+
+- `size ≤ 5` — real laws rarely couple more than five quantities;
+- `n_const ≤ 1` — a genuine law uses at most one universal constant;
+- `n_domains ≤ 2` — at most one cross-domain bridge;
+- `max_exp ≤ 3` and `exp_l1 ≤ 8` — small integer exponents.
+
+This is intentionally conservative: it favours precision over recall. The
+remaining ~2.28 M rows are kept so that the catalogue stays complete and
+auditable, but the overwhelming majority are dimensional coincidences (e.g.
+`m = k₀ · c⁻¹·k_B`), not physical laws.
+
+### The `score`
+
+A heuristic that rewards simplicity (few quantities, small exponents, single
+domain, at most one constant). Sorting `plausible` rows by `score DESC` surfaces
+textbook laws and named dimensionless numbers first.
+
+---
+
+## Example rows
+
+| equation | pi | name | size | score | plausible |
+|---|---|---|---|---|---|
+| `m = k₀ · a⁻¹·F` | `m·a·F⁻¹` | Newton 2nd law | 3 | 9.0 | ✅ |
+| `m = k₀ · v⁻²·E` | `m·v²·E⁻¹` | kinetic energy | 3 | 7.8 | ✅ |
+| `m = k₀ · E·c₀⁻²` | `m·E⁻¹·c₀²` | mass-energy (E=mc²) | 3 | 6.7 | ✅ |
+| `ℓ = k₀ · v⁻¹·ρ⁻¹·μ` | `ℓ·v·ρ·μ⁻¹` | Reynolds number | 4 | 8.0 | ✅ |
+| `E = k₀ · f·h` | `E·f⁻¹·h⁻¹` | Planck relation (E=hf) | 3 | — | ✅ |
+| `i = k₀ · U·R⁻¹` | `i·U⁻¹·R` | Ohm's law | 3 | — | ✅ |
+
+---
+
+## The quantity pool (60 quantities, 7 base dimensions)
+
+Base dimensions: **M** (mass), **L** (length), **T** (time), **Θ** (temperature),
+**I** (electric current), **N** (amount of substance), **J** (luminous intensity).
+
+### Mechanics
+| symbol | quantity | dimensions |
+|---|---|---|
+| m | mass | M |
+| ℓ | length | L |
+| t | time | T |
+| v | velocity | L·T⁻¹ |
+| a | acceleration | L·T⁻² |
+| g | gravitational acceleration | L·T⁻² |
+| F | force | M·L·T⁻² |
+| E | energy | M·L²·T⁻² |
+| τ | torque | M·L²·T⁻² |
+| P | power | M·L²·T⁻³ |
+| p | momentum | M·L·T⁻¹ |
+| Lₐ | angular momentum | M·L²·T⁻¹ |
+| S | action | M·L²·T⁻¹ |
+| J | moment of inertia | M·L² |
+| P_p | pressure | M·L⁻¹·T⁻² |
+| ρ | mass density | M·L⁻³ |
+| μ | dynamic viscosity | M·L⁻¹·T⁻¹ |
+| ν | kinematic viscosity | L²·T⁻¹ |
+| ω | angular velocity | T⁻¹ |
+| f | frequency | T⁻¹ |
+| k | spring constant | M·T⁻² |
+| γ | surface tension | M·T⁻² |
+| A | area | L² |
+| V | volume | L³ |
+| Q | volumetric flow rate | L³·T⁻¹ |
+
+### Thermodynamics
+| symbol | quantity | dimensions |
+|---|---|---|
+| Θ | temperature | Θ |
+| S_e | entropy | M·L²·T⁻²·Θ⁻¹ |
+| C | heat capacity | M·L²·T⁻²·Θ⁻¹ |
+| c | specific heat capacity | L²·T⁻²·Θ⁻¹ |
+| κ | thermal conductivity | M·L·T⁻³·Θ⁻¹ |
+| α | thermal diffusivity | L²·T⁻¹ |
+| β | thermal expansion coefficient | Θ⁻¹ |
+
+### Electromagnetism
+| symbol | quantity | dimensions |
+|---|---|---|
+| q | electric charge | T·I |
+| i | electric current | I |
+| U | electric potential | M·L²·T⁻³·I⁻¹ |
+| R | electric resistance | M·L²·T⁻³·I⁻² |
+| C_e | capacitance | M⁻¹·L⁻²·T⁴·I² |
+| L_e | inductance | M·L²·T⁻²·I⁻² |
+| E_f | electric field | M·L·T⁻³·I⁻¹ |
+| B | magnetic flux density | M·T⁻²·I⁻¹ |
+| Φ | magnetic flux | M·L²·T⁻²·I⁻¹ |
+| ε | permittivity | M⁻¹·L⁻³·T⁴·I² |
+| μ₀ | permeability | M·L·T⁻²·I⁻² |
+| σ | electrical conductivity | M⁻¹·L⁻³·T³·I² |
+| ρ_e | electrical resistivity | M·L³·T⁻³·I⁻² |
+
+### Chemistry
+| symbol | quantity | dimensions |
+|---|---|---|
+| n | amount of substance | N |
+| M_m | molar mass | M·N⁻¹ |
+| c_n | molar concentration | L⁻³·N |
+| E_m | molar energy | M·L²·T⁻²·N⁻¹ |
+| R | molar gas constant | M·L²·T⁻²·Θ⁻¹·N⁻¹ |
+| z | catalytic activity | T⁻¹·N |
+
+### Photometry
+| symbol | quantity | dimensions |
+|---|---|---|
+| I_v | luminous intensity | J |
+| E_v | illuminance | L⁻²·J |
+| L_v | luminance | L⁻²·J |
+
+### Universal constants
+| symbol | quantity | dimensions |
+|---|---|---|
+| c₀ | speed of light | L·T⁻¹ |
+| G | gravitational constant | M⁻¹·L³·T⁻² |
+| h | Planck constant | M·L²·T⁻¹ |
+| k_B | Boltzmann constant | M·L²·T⁻²·Θ⁻¹ |
+| e | elementary charge | T·I |
+| N_A | Avogadro constant | N⁻¹ |
+
+---
+
+## Loading
+
+```python
+import pandas as pd
+df = pd.read_parquet("data/circuits.parquet")
+
+# the physically-plausible laws, best first
+laws = df[df.plausible].sort_values("score", ascending=False)
+
+# every relation involving viscosity, within mechanics
+df[df["keys"].str.contains("viscosity") & (df.domains == "mechanics")]
+```
+
+Or with the 🤗 `datasets` library:
+
+```python
+from datasets import load_dataset
+ds = load_dataset("<user>/vashy", split="train")
+```
+
+---
+
+## Intended uses
+
+- **Symbolic regression & scientific ML**: a dimensionally-valid hypothesis
+  space / prior over candidate equations.
+- **Physics education**: browsing dimensionless numbers and their structure.
+- **Benchmarking**: testing whether models can recover known laws from
+  dimensional constraints.
+
+## Limitations & caveats
+
+- **Dimensional consistency ≠ physical truth.** Most rows are coincidences; use
+  `plausible` and `score`, and validate against physics.
+- **The constant `k₀` is undetermined** by dimensional analysis.
+- **Only single dimensionless numbers are represented.** When a phenomenon needs
+  two or more independent Π groups, the true law is `Π₁ = f(Π₂, …)` with an
+  arbitrary function `f`; the dataset lists the individual irreducible Π groups
+  (the circuits), not the functional relation between them.
+- **Pool-dependent.** The catalogue is exhaustive *for this 60-quantity pool*;
+  a different pool yields a different catalogue.
+- Deliberate dimensional collisions in the pool (energy vs torque, action vs
+  angular momentum, entropy vs heat capacity, acceleration vs gravity) are real
+  and produce legitimate size-2 identities.
+
+## License
+
+Released under **CC-BY-4.0**. The underlying facts are mathematical
+consequences of SI dimensional definitions.
+
+## Citation
+
+```bibtex
+@misc{vashy_dimensional_equations,
+  title  = {Vashy: A Complete Catalogue of Dimensionally-Consistent Physics Equations},
+  note   = {Generated via circuits of the dimensional matroid (Vaschy--Buckingham pi theorem)},
+  year   = {2026}
+}
+```