Ontology asks what the three states mean.
Mathematics asks what happens when we forget their meanings for a moment and treat them simply as elements of a formal system.
What emerges is surprisingly compact.
Tetriamancy consists of a three-element arithmetic, a set of 81 four-dimensional figures, and a tetrahedral structure containing every non-empty combination of four Sources.
Three States as a Finite Field
Tetriamancy ordinarily numbers its states:
- 1 - Manifest
- 2 - Liminal
- 3 - Latent
For mathematical purposes, it is more useful to recenter these values around zero:
| State | Figure value | Mathematical value |
|---|---|---|
| Manifest | 1 | -1 |
| Liminal | 2 | 0 |
| Latent | 3 | +1 |
The mathematical values are not measurements or rankings. They are representatives of the three elements of arithmetic modulo 3.
Mathematically, this system is usually written:
F₃ = {0, 1, 2}
but because 2 ≡ -1 (mod 3), Tetriamancy can write the same system more symmetrically as:
F₃ = {-1, 0, +1}
This makes the relationship between the two non-zero states immediately visible.
Composition is simply addition modulo 3:
| ⊕ | -1 | 0 | +1 |
|---|---|---|---|
| -1 | +1 | -1 | 0 |
| 0 | -1 | 0 | +1 |
| +1 | 0 | +1 | -1 |
Because 3 is prime, this is not merely a cyclic set. It is a finite field.
That allows the larger figure system to behave like ordinary linear algebra, except that all arithmetic takes place among only three values.
Why -1 + 1 = 0 Matters
The equation
-1 + 1 = 0
is mathematically important because -1 and +1 are additive inverses.
Each cancels the other.
The value 0 is the additive identity:
x + 0 = x
This gives the three values a very specific algebraic relationship:
+1 = -(-1)
-1 = -(+1)
-0 = 0
There is an important subtlety here.
Although we write the values as:
-1, 0, +1
they do not form an ordinary number line.
There is no meaningful mathematical claim here that:
-1 < 0 < +1
The field F₃ cannot be ordered in a way compatible with its arithmetic.
Instead, it is cyclic.
Repeatedly adding +1 gives:
0 → +1 → -1 → 0
Repeatedly adding -1 gives the same cycle in the opposite direction:
0 → -1 → +1 → 0
There is no endpoint.
Three Returns to Zero
One of the most characteristic facts about F₃ is:
x + x + x = 0
for every value x.
This is called characteristic 3.
For example:
+1 + +1 + +1 = 0
and:
-1 + -1 + -1 = 0
From this follows another useful identity:
x + x = -x
In other words, doubling any non-zero state produces its additive inverse.
This is not a separate rule added to Tetriamancy. It follows automatically from arithmetic modulo 3.
The same fact remains true when entire figures are added:
F ⊕ F = -F
and therefore:
F ⊕ F ⊕ F = 0
Every figure added to itself twice returns the identity figure.
The 81 Figures
A Tetriamantic figure has four coordinates:
(Fire, Air, Water, Earth)
Each coordinate can contain any element of F₃.
A figure is therefore mathematically a vector:
(x₁, x₂, x₃, x₄)
where each x is -1, 0, or +1.
There are:
3⁴ = 81
possible vectors.
So the complete collection of Tetriamantic figures is:
F₃⁴
This is a four-dimensional vector space over the three-element field F₃.
The phrase sounds much larger than the thing itself.
It simply means:
- four coordinates;
- three possible values in each;
- coordinate-by-coordinate arithmetic modulo 3.
For example:
(-1, 0, +1, -1)
is one point in this 81-point space.
Composition of figures is vector addition:
(a₁,a₂,a₃,a₄) ⊕ (b₁,b₂,b₃,b₄)
= (a₁+b₁, a₂+b₂, a₃+b₃, a₄+b₄)
with every coordinate calculated modulo 3.
The Zero Figure and Complement Pairs
The zero vector is:
(0, 0, 0, 0)
or, in Tetriamantic notation:
2-2-2-2
This is the Hinge, the identity figure.
Every other figure has exactly one additive inverse.
If:
F = (a,b,c,d)
then:
-F = (-a,-b,-c,-d)
The 81 figures therefore have a simple internal structure:
1 identity figure + 40 complementary pairs = 81 figures
The Hinge is the only figure that is its own complement.
All other figures occur in pairs F and -F.
This is a direct consequence of the geometry of F₃⁴, not an additional property assigned to the figures.
Four Sources
A complete chart begins with four Source figures:
A, B, C, D
Each Source can independently be any of the 81 figures.
The number of possible sets of four Sources is therefore:
81⁴
which is:
3¹⁶ = 43,046,721
possible Source configurations.
That number also reveals something useful about the information contained in a chart.
Each Source contains four independently generated ternary values.
Four Sources therefore contain:
4 × 4 = 16
independent ternary values, or 16 trits.
The completed chart displays fifteen figures, containing sixty individual state positions.
But only sixteen of those sixty values are independently generated.
The remaining forty-four are mathematically derived from them.
A completed chart is therefore highly structured rather than a collection of fifteen independent figures.
The Tetrahedron
The tetrahedron is not merely a convenient diagram for arranging fifteen positions.
Its combinatorics produce exactly the structure Tetriamancy requires.
A tetrahedron is a 3-simplex.
It contains:
- 4 vertices
- 6 edges
- 4 triangular faces
- 1 complete tetrahedron
Tetriamancy maps these directly:
| Number of Sources | Tetriamancy | Geometry | Count |
|---|---|---|---|
| 1 | Sources | vertices | 4 |
| 2 | Bonds | edges | 6 |
| 3 | Fields | faces | 4 |
| 4 | Whole | tetrahedron | 1 |
These numbers are not arbitrary.
They are binomial coefficients:
C(4,1) = 4
C(4,2) = 6
C(4,3) = 4
C(4,4) = 1
Together:
4 + 6 + 4 + 1 = 15
There are exactly fifteen non-empty ways to select members from a set of four things.
Equivalently:
2⁴ - 1 = 15
So every Tetriamantic position corresponds to one non-empty subset of:
{A, B, C, D}
A is {A}.
AB is {A,B}.
ABC is {A,B,C}.
ABCD is {A,B,C,D}.
Nothing is omitted and nothing is duplicated.
The entire chart is the collection of every possible non-empty combination of its four Sources.
The Missing Sixteenth Combination
The full power set of four Sources contains not fifteen but sixteen subsets:
2⁴ = 16
The missing member is:
∅
the empty set.
This gives the Null outside the chart a particularly clean mathematical analogue.
The fifteen positions inside the Whole correspond to every non-empty subset of {A,B,C,D}.
The empty subset corresponds to selecting no Source at all.
This also demonstrates why ∅ and 0 are not the same thing.
They belong to different mathematical structures.
∅ is the empty subset:
no Sources selected
while 0 is the zero vector:
(0,0,0,0)
which is a perfectly valid member of F₃⁴.
Thus:
∅ ≠ 0
The Null is not the Hinge.
One is outside the selection of Sources entirely.
The other is an actual figure within the 81-figure space.
Geometry Selects; Algebra Combines
The tetrahedron tells us which Sources belong to a position.
The ternary algebra tells us how their figures combine.
If S is any non-empty subset of the four Sources, its figure is simply:
T(S) = Σ X
for every Source X contained in S, with addition performed in F₃⁴.
Thus:
T({A,B}) = A ⊕ B
T({A,B,C}) = A ⊕ B ⊕ C
T({A,B,C,D}) = A ⊕ B ⊕ C ⊕ D
A single rule generates every derived position.
Tetrahedral Identities
Because the chart combines tetrahedral geometry with arithmetic of characteristic 3, several relationships follow automatically.
The three pairs of opposite edges all produce the Whole when combined:
AB ⊕ CD = ABCD
AC ⊕ BD = ABCD
AD ⊕ BC = ABCD
Likewise, the three edges meeting at any vertex sum to the Whole.
For A:
AB ⊕ AC ⊕ AD = ABCD
because:
(A+B) + (A+C) + (A+D)
= 3A + B + C + D
and in F₃:
3A = 0
leaving:
A + B + C + D
The same is true at B, C, and D.
There are other automatic identities.
All six Bonds together sum to the zero figure:
AB ⊕ AC ⊕ AD ⊕ BC ⊕ BD ⊕ CD = 0
because each Source occurs exactly three times.
Likewise:
ABC ⊕ ABD ⊕ ACD ⊕ BCD = 0
because each Source again occurs exactly three times.
These are not additional rules of interpretation.
They are mathematical invariants of the chart.
What Kind of Object Is Tetriamancy?
At its most abstract, the mathematics of Tetriamancy can be summarized surprisingly briefly.
The states form the finite field:
F₃
The figures form the four-dimensional vector space:
F₃⁴
The four Sources are four vectors in that space.
The fifteen chart positions correspond to the fifteen non-empty subsets of four Sources, equivalently the non-empty faces of a 3-simplex.
Each position receives the sum of the Source vectors belonging to its subset.
The external Null corresponds naturally to the one subset not represented internally:
∅
So beneath the vocabulary, figures, and correspondences is a compact mathematical object:
ternary linear algebra arranged on the combinatorics of a tetrahedron.
That is the mathematical skeleton of Tetriamancy.