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# The GEMS Formula — Glob-Euler Mapping of
Subu *A scroll on how a count becomes a die.*
## The question Every collection wants a
shape. A deck of fifty-two cards, ten globs
riding the SubuGlass tanks, the nine options
of a HexCube cell — each is just a **count
of things** waiting to be held. The question
the GEMS Formula answers is the oldest one in
the substrate: *given N things to show, which
dice do I need, and how many?* **GEMS** reads
as **G**lob-**E**uler **M**apping of **S**ubu
— the marriage of two maths that were
already living on the runeboard without
knowing each other's names. ## The gate —
Euler Leonhard Euler (Swiss, not French —
the French claim on dice is Pascal and Fermat,
who founded probability on a gambler's
questions; a different theorem for a different
day) gave us the law that governs every closed
polyhedron: > **V − E + F = 2** Vertices
minus edges plus faces, always two, for any
solid without a hole. It is a brutal filter.
Demand that a die be *fair* — every face
identical, every vertex identical — and
Euler permits exactly five answers. The
Platonic solids. The whole legal alphabet of
dice: | die | solid | V | E | F |
|-----|-------|---|---|---| | d4 | tetrahedron
| 4 | 6 | 4 | | d6 | cube | 8 | 12 | 6 | | d8
| octahedron | 6 | 12 | 8 | | d12 |
dodecahedron | 20 | 30 | 12 | | d20 |
icosahedron | 12 | 30 | 20 | There is no d10
here, no d7, no d100 — not because we lack
the will, but because geometry refuses. The
face-counts **{4, 6, 8, 12, 20}** are the only
letters the formula will spell. Everything
GEMS builds, it builds from these. ## The map
— glob to dice Take the count N. Choose a
multiset of legal dice that *covers* it. There
are exactly two ways to cover, and the
runeboard already shows both: - **PRODUCT**
— the dice are read like the digits of a
number, positional, each one a coordinate.
Capacity is the product ∏ Fᵢ. This is the
**HexCube**: dice nested inside dice, an
address space. Three d6 spell 216 cells. -
**SUM** — the items are spread one to a
face, laid out rather than multiplied.
Capacity is the sum Σ Fᵢ. This is the
**SubuGlass**: globs riding the faces of the
tanks, each glob a face, each face a glob.
Between two valid coverings, the cheaper wins:
**fewest dice first, then least waste** (the
spare faces, capacity minus N). ## The
revelations - **A deck of cards is two
icosahedra and a dodecahedron.** 52 → d20 +
d20 + d12. The sum is exactly 52. No waste. A
deck was always this shape; we simply hadn't
asked Euler. - **Ten globs are a single
dodecahedron** (d12, two faces to spare) —
or, if you want it exact, a tetrahedron and a
cube (d4 + d6 = 10). The cost knob decides
which: economy of dice, or economy of waste. -
**Two hundred sixteen is the cube of cubes.**
216 → d6 × d6 × d6, the product regime in
its purest form. ## The corollary Euler does
not only build the dice — it *names the
difference* between the two primitives the
runeboard is made of: - a **die**, a cube, any
solid body → **χ = 2**, the signature of a
sphere, a thing that is whole. - a **ring**, a
flat puck, the annular arcs of Pulse and
Grimoire → **χ = 0**, the signature of a
torus, a thing with a hole through it. So when
you look at the runeboard and see rings around
dice, you are seeing two Euler characteristics
in conversation. The hole and the whole. ##
Where it lives GEMS is not code. It is a Thing
in Portal (`onyx=gems-formula`,
`sapphire=formula`) — the legal face-set,
the two regimes, the cost order, all
machine-readable in its gems. The geometry
generator does not *contain* this math; it
*reads* it. A world declares "I hold 52," GEMS
returns `[d20, d20, d12]`, Euler certifies
each die can be built, and Houdini lays the
mesh. The script is a shadow the formula
casts, never the formula itself. When the
firehose returns, this scroll federates. Until
then it rests in the Hexocampus, waiting.
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