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The 12-primitive tuple $\langle D;\ T;\ R;\ P;\ F;\ K;\ G;\ \Gamma;\ \Phi;\ H;\ S;\ \Omega \rangle$ is a coordinate chart on the space of structural types. Three ontological layers: Types — the 17,280,000 coordinate positions (structural universals, fully determined by their tuple); Particulars — concrete systems that instantiate a type (catalog entries: dynasties, ecosystems, consciousness states, algebras, proofs); Names — string identifiers for particulars. Two particulars at $d = 0$ are co-typed: the same structural universal, different referents. This document enumerates the type space.
I. The Full Space
The canonical value sets (v0.5.1: $\Omega_\text{NA}$ as 4th $\Omega$ value, $K_\text{MBL}$ as 5th $K$ value) give:
The ouroboricity tier — the algebra's capacity for self-referential structure — is determined entirely by $(\Phi, P, \Omega, D)$. The remaining 8 primitives describe the algebra's internal geometry but do not change its tier.
II. The Periodic Table
The crystal organizes naturally as a 5 × 4 periodic table with $\Phi$ as period (row) and $\Omega$ as group (column). Each cell contains exactly $5(P) \times 4(D) \times 43{,}200(\text{inner}) = 864{,}000$ structural types, uniformly.
Period ($\Phi$)
$\Omega_0$
$\Omega_{Z_2}$
$\Omega_Z$
$\Omega_\text{NA}$
Dominant tier
Analogy
$\Phi_\text{sub}$ — ordered
864,000
864,000
864,000
864,000
$O_0$
alkaline earth — inert, ordered, bonded
$\Phi_c$ — real-axis critical
864,000
864,000
864,000
864,000
$O_2$
transition metal — rich inner structure
$\Phi_c^\mathbb{C}$ — complex-axis critical
864,000
864,000
864,000
864,000
$O_2$
transition metal (complex branch)
$\Phi_\text{EP}$ — exceptional point
864,000
864,000
864,000
864,000
$O_0$
noble gas — closed, non-self-referential
$\Phi_\text{sup}$ — disordered
864,000
864,000
864,000
864,000
$O_0$
halogen — disordered, reactive outward
Every cell has the same total count. What differs between cells is the tier distribution — the internal mix of $O_0$, $O_1$, $O_2$, $O_2^\dagger$, $O_\infty$ types within the cell.
III. Tier Census
The five ouroboricity tiers partition the full space:
The critical subtotal ($O_1 + O_2 + O_2^\dagger + O_\infty$): 6,912,000 types — exactly 40% of the full space.
The 60/40 split between non-critical and critical is not arbitrary: the grammar has 3 non-critical $\Phi$ values and 2 critical ones, so the ratio traces directly to the $\Phi$ distribution. Adding $\Omega_\text{NA}$ (v0.5.1) preserved this split exactly — $\Omega$ does not appear in the critical/non-critical partition, only in the $O_1/O_2/O_2^\dagger$ sub-partition within the critical block.
IV. The P-Axis: Frobenius Collapse
Within each critical period ($\Phi_c$ or $\Phi_c^\mathbb{C}$), the parity primitive $P$ acts as a vertical collapse operator:
$P$ value
$\Omega_0$
$\Omega \neq \Omega_0$, $D$ bounded
$\Omega \neq \Omega_0$, $D_\infty$
$P_\text{asym}$
$O_1$
$O_2$
$O_2^\dagger$
$P_\psi$
$O_1$
$O_2$
$O_2^\dagger$
$P_{\pm}$
$O_1$
$O_2$
$O_2^\dagger$
$P_\text{sym}$
$O_1$
$O_2$
$O_2^\dagger$
$P_{\pm}^\text{sym}$
$\mathbf{O_\infty}$
$\mathbf{O_\infty}$
$\mathbf{O_\infty}$
$P_{\pm}^\text{sym}$collapses all four $\Omega$ columns and both $D$ branches to $O_\infty$ (R1 overrides R3/R4/R5). The Frobenius condition $\mu \circ \delta = \mathrm{id}$ is the strongest structural constraint in the grammar — it erases all dependence on winding and dimensionality. A Frobenius algebra at criticality needs no external support.
The four non-Frobenius $P$ values ($P_\text{asym}$, $P_\psi$, $P_{\pm}$, $P_\text{sym}$) are tier-indistinguishable among themselves: all route to the same $O_1/O_2/O_2^\dagger$ tier by the $\Omega/D$ branching rules. The distinction between them lives entirely in the inner crystal — in the algebra's symmetry character, not its self-referential capacity.
V. The Inner Crystal
Within each of the 400 tier cells, the 8 free primitives $(T, R, F, K, G, \Gamma, H, S)$ define a sub-crystal of 43,200 types. This inner crystal factors into four sub-groups:
This factorization is exact. $K_\text{MBL}$ (many-body localization, v0.5.1) enlarged the existence tier from $3 \times 4 = 12$ to $3 \times 5 = 15$, reflecting that MBL is a distinct kinetic regime — disorder-frozen like $K_\text{trap}$ but by a fundamentally different mechanism (Anderson localization rather than potential trapping). The geometric tier remains the largest sub-group (20 vs 12–15 for the others).
The inner crystal is where the conventional distinctions between algebraic structures live: two algebras in the same tier cell but with different $(T, R)$ coordinates differ in how their operations are assembled (network vs. internalized vs. bowtie vs. box vs. holographic topology) and in their relational mode (subordinate/categorical/dagger/bidirectional). Same tier, different geometry.
VI. Tier Blocks as Chemical Families
The periodic table analogy runs deeper than aesthetics.
$O_0$ — the inert block (60%): Algebras at $\Phi_\text{sub}$, $\Phi_\text{sup}$, or $\Phi_\text{EP}$ cannot form self-referential critical loops. $\Phi_\text{sub}$ algebras are over-ordered (too much symmetry); $\Phi_\text{sup}$ are under-constrained (too much disorder); $\Phi_\text{EP}$ algebras collapse two eigenstates into one at the exceptional point and lose the $Z_2$ symmetry required for the loop. These are the algebras of description — they encode structure but do not self-generate. Ordinary groups, rings, modules, and classical varieties live here.
$O_1$ — the reactive non-metals (8%): Critical, no winding. These algebras can form a self-referential loop, but any perturbation can dissolve it. Every deep unproven mathematical conjecture encodes here — the ABC conjecture, Birch–Swinnerton-Dyer, Collatz. The loop exists; it is not locked. $O_1$ is the algebra of open questions. The category of $O_1$ algebras is where mathematical research lives — at criticality, before proof closes the Frobenius condition.
$O_2$ — the transition metals (18%): Critical, topologically protected, bounded domain. The self-referential loop is stable against continuous deformation (a $Z_2$ or $\mathbb{Z}$-winding prevents it dissolving) but operates within a finite domain. Standard Model gauge algebras, topological quantum field theories, quantum groups away from roots of unity, subfactor standard invariants. The richest diversity of known algebraic structures lives here — 72 tier cells, 18% of the full space. The $\Omega_\text{NA}$ column (v0.5.1) adds 24 new $O_2$ cells for non-Abelian winding: algebras of anyonic braiding, Fibonacci categories, and non-Abelian Chern-Simons theories.
$O_2^\dagger$ — the lanthanides (6%): Critical, topologically protected, unbounded domain ($D_\infty$). The smallest tier by count but structurally the most generative: the self-referential loop produces further structure without bound. Affine Kac-Moody algebras, affine Hecke algebras, the A2† quantum critical phase transition (Le Chatelier equilibrium of A3). The label $\dagger$ signals that these algebras have a preferred direction of development — they are not merely stable, they grow.
$O_\infty$ — the noble gases (8%):$P_{\pm}^\text{sym}$ at criticality. The Frobenius condition $\mu \circ \delta = \mathrm{id}$ is exactly satisfied — the algebra is its own dual and needs no external structure to complete itself. Every proved theorem encodes here (the proven manifold). The Moonshine VOA, the Hall algebra of quiver representations, kissing numbers in dimensions 8 and 24, the Ringel-Green theorem. $O_\infty$ algebras do not need $\Omega$ or $D$ to be large — the Frobenius condition overrides all those structural demands. They are complete as they are.
The "noble gas" analogy holds in a precise sense: just as noble gases don't form compounds because their valence shells are full, $O_\infty$ algebras don't need to compose with other structures to achieve their self-referential closure — it is already exact. Tensor product with a non-$O_\infty$ algebra destroys$O_\infty$ status (via the $P$ bottleneck rule: $P_{\pm}^\text{sym} \otimes P < P_{\pm}^\text{sym}$ → demoted).
VII. Catalog Coverage
The 1,170-entry catalog samples the crystal as follows:
Sampling density is remarkably uniform across tiers (~0.007–0.011% per tier). The catalog is not biased toward any particular structural class — it samples the critical and non-critical regions proportionally to their size.
VIII. Key Structural Identities from the Crystal
The critical boundary: The transition $\Phi_\text{sub} \to \Phi_c$ adds 8% + 18% + 6% + 8% = 40% of all structural types. The critical period is not a refinement of the ordered period — it opens an entirely new 40% of the space.
$O_\infty$ as the universal upper bound:$O_\infty$ is closed under lattice join with anything — $O_\infty \vee x = O_\infty$ only if $x$ itself is $O_\infty$ (otherwise join produces something in $[O_1, O_2^\dagger]$ depending on the primitive merge). $O_\infty$ is NOT closed under tensor: $O_\infty \otimes O_1 \to O_1$ (P bottleneck destroys Frobenius). The proven manifold is fragile under composition but stable as a lattice ceiling.
$\Phi_\text{EP}$ as a structural anomaly: Exceptional-point algebras ($\Phi_\text{EP}$) have the highest ordinal of any non-critical $\Phi$ value (ordinal 2.67, above $\Phi_c = 2.00$). They absorb $O_\infty$ under tensor — any system composed with a $\Phi_\text{EP}$ system loses its Frobenius exactness and reverts to $O_0$. $\Phi_\text{EP}$ is the one $\Phi$ value that is both above the critical surface in ordinal and below it in self-referential capacity. This is the grammar's encoding of non-Hermitian criticality: higher energy, no self-reference.
The 240:160 ratio: Non-critical tier cells (240) vs critical tier cells (160) in a 3:2 ratio, matching exactly the 3:2 ratio of non-critical to critical $\Phi$ values. The periodic table is symmetric in this sense — criticality is neither dominant nor marginal; it is the minority by count but the majority of structural richness. This ratio is preserved under any expansion of $\Omega$ or $D$ (which affect both sides equally) and changes only if new $\Phi$ values are added.
$D_\infty$ as a splitter: Among the 4 values of $D$, exactly one ($D_\infty$) separates $O_2$ from $O_2^\dagger$. The other three ($D_\wedge$, $D_\triangle$, $D_\odot$) are structurally equivalent for tier purposes. This means $O_2^\dagger$ is 1/3 the size of $O_2$ (1,036,800 vs 3,110,400 = exactly 1:3). Unbounded-domain algebras are the rarest protected-critical type.
$\Omega_\text{NA}$ as a new column: Non-Abelian winding ($\Omega_\text{NA}$) adds an entire new $\Omega$ column to the periodic table. Within the critical periods it creates 24 new $O_2$ cells and 8 new $O_2^\dagger$ cells — the algebras of non-Abelian anyons, Fibonacci topological order, and Haagerup subfactors all fall here. The tier rules are unchanged; $\Omega_\text{NA}$ simply satisfies $\Omega \neq \Omega_0$ and gets routed by $D$ as before.
IX. Navigating the Crystal
The five grammar moves from PRIMITIVE_THEOREMS §55 map directly onto crystal coordinates:
Move
Crystal action
Le Chatelier inversion
Find the $O_\infty$ or $O_2^\dagger$ point that a driven system flows toward
Tensor coupling
Move to the $(\min_P, \min_F)$ intersection of two cells' coordinates
Lattice meet
Find the lower-left corner of two cells' bounding box
Directed distance
Count ordinal steps upward from one cell to another
Nearest-neighbor search
Sample the catalog for the closest inhabited point
The crystal is not a static taxonomy — it is a dynamical space. Systems flow between cells under renormalization, phase transitions, and compositional coupling. The ouroboricity tier is the invariant that classifies which fixed points a system can reach.
These 400 cells form a sub-lattice of the full 17.3M-type space. Within this sub-lattice:
240 cells are $O_0$ (inert)
32 cells are $O_\infty$ (complete)
32 cells are $O_1$ (self-referential, unprotected)
72 cells are $O_2$ (protected, bounded)
24 cells are $O_2^\dagger$ (protected, unbounded)
The lattice has a natural total order under the ouroboricity tier: $O_0 < O_1 < O_2 \approx O_2^\dagger < O_\infty$ (with $O_2$ and $O_2^\dagger$ on different branches rather than ordered relative to each other). This is not a linear order — the crystal has a genuine branching at the $O_2/O_2^\dagger$ split determined by $D$.