The 3rd-Order Interaction Tensor
A 3rd-order tensor is a three-dimensional generalization of a matrix. Where a matrix captures pairwise relationships (row × column), a rank-3 tensor captures triadic interactions — three dimensions simultaneously. In the context of the Guenhwyvar deck, we define:
Where ci is the card quantity, T is the tag incidence tensor (5 modes), R is the role tensor (20 functional roles), and B is the behavior tensor (7 archetypes). The result: a 5 × 20 × 7 = 700-cell interaction space capturing the weighted mass of cards existing at every possible (tag, role, behavior) triple simultaneously.
This is not a toy exercise. The same tensor decomposition used here is applied in signal processing, neuroscience, recommender systems, and — critically — cybersecurity threat modeling, where attack vectors exist at the intersection of multiple independent dimensions.
Incidence Tensors & Mass Vectors
Before constructing the 3rd-order tensor, we extract three binary incidence matrices from the deck's 22 unique nonland cards (43 total copies):
Tag Tensor T
T ∈ {0,1}22×5
Five mode tags classifying each card's strategic role in the deck architecture:
| Tag | Mass |
|---|---|
| STORY | 27 |
| APEX | 26 |
| HUNT | 24 |
| CORE | 23 |
| ACCEL | 2 |
Role Tensor R
R ∈ {0,1}22×20
Twenty functional roles — what each card does mechanically. Top 5 by weighted mass:
| Role | Mass |
|---|---|
| Combat Protection | 6 |
| Anthem | 5 |
| Early Threat | 4 |
| Equipment | 4 |
| Recovery | 4 |
Behavior Tensor B
B ∈ {0,1}22×7
Seven behavioral archetypes — how the card behaves in the game ecosystem:
| Behavior | Mass |
|---|---|
| PREDATOR | 13 |
| DIVINE | 11 |
| RELIC | 6 |
| PACK | 5 |
| Control | 3 |
| TERRITORY | 3 |
| SCOUT | 2 |
Weighted Co-Occurrence & Jaccard Similarity
Before the rank-3 tensor, we compute pairwise coupling. The quantity-weighted co-occurrence matrix measures how often two tags appear on the same card, weighted by copy count:
We then normalize via weighted Jaccard similarity — the gold standard for measuring overlap in binary vectors with unequal support:
Tag-Tag Weighted Jaccard Similarity
| ACCEL | APEX | CORE | HUNT | STORY | |
|---|---|---|---|---|---|
| ACCEL | 1.000 | 0.077 | 0.087 | 0.083 | 0.000 |
| APEX | 0.077 | 1.000 | 0.633 | 0.613 | 0.262 |
| CORE | 0.087 | 0.633 | 1.000 | 0.958 | 0.429 |
| HUNT | 0.083 | 0.613 | 0.958 | 1.000 | 0.417 |
| STORY | 0.000 | 0.262 | 0.429 | 0.417 | 1.000 |
Key insight: CORE–HUNT coupling at J=0.958 means 23 of 24 HUNT-tagged copies also carry CORE — these modes are nearly fused. STORY operates more independently (J ≤ 0.429), providing strategic diversification. ACCEL is surgically isolated (2 copies, zero STORY overlap).
Top Weighted Triples: X[tag, role, behavior]
The full 3rd-order tensor X contains 700 cells. Most are zero — only specific (tag, role, behavior) combinations carry weight. The top triples reveal the deck's structural spine:
| Tag | Role | Behavior | Weight | Interpretation |
|---|---|---|---|---|
| APEX | Combat Protection | PREDATOR | 6 | Primary defensive shell — Kutzil at 6 copies |
| CORE | Combat Protection | PREDATOR | 6 | Same cards, multi-tagged: system redundancy |
| HUNT | Combat Protection | PREDATOR | 6 | Triple-mode coverage ensures draw reliability |
| APEX | Anthem | DIVINE | 4 | Board-wide power scaling through divine buffs |
| STORY | Equipment | RELIC | 4 | Dancing Sword — narrative + mechanical payload |
| STORY | Recovery | DIVINE | 4 | Oketra's Last Mercy — reset to full life |
| APEX | Scaling Finisher | PREDATOR | 2 | Drizzt + Guenhwyvar as apex predator pair |
| STORY | Midrange Body | TERRITORY | 2 | Kudo, King Among Bears — territorial control |
Structural observation: The weight-6 triples all share the same (Combat Protection, PREDATOR) pair across three different tags. This is deliberate engineering: Kutzil, Malamet Exemplar at 6 copies creates a triple-mode redundant protective shell — the card is simultaneously APEX, CORE, and HUNT, ensuring it appears in nearly every opening hand.
Mode-Wise SVD of Unfolded Tensor
We unfold X along each mode and compute Singular Value Decomposition (SVD). The energy share of each singular value reveals how much structural information is concentrated in each principal component — a measure of dimensional coupling.
Energyj = Sj² / Σ S²
Tag-Mode Spectrum
σ₁ captures 80.0% — near-rank-1 structure
Interpretation: The tag dimension is highly coherent — one principal direction explains 80% of all tag-mediated variation. The deck's tagging system is well-aligned, not scattered.
Role-Mode Spectrum
More distributed — 20 roles across 22 cards
Interpretation: Roles are more diverse — two principal components explain 75%, with meaningful structure in the tail. The deck has a clear primary strategy with deliberate secondary layers.
Behavior-Mode Spectrum
PREDATOR + DIVINE dominate — aggressive divine deck
Interpretation: The behavioral signature is dual-axis dominant — PREDATOR (aggressive) and DIVINE (protective) together explain 82.6% of behavioral variation. This is a deck that attacks and shields simultaneously.
Marginal Projections
Contracting (summing over) one or more dimensions of X yields lower-order views of the data — the tensor equivalent of marginal distributions in probability:
X_role[r] = Σt,b X[t,r,b] = w_role[r]
X_beh[b] = Σt,r X[t,r,b] = w_beh[b]
X_tr[t,r] = Σb X[t,r,b] ← tag-role coupling
X_tb[t,b] = Σr X[t,r,b] ← tag-behavior coupling
X_rb[r,b] = Σt X[t,r,b] ← role-behavior coupling
The marginals recover the 1st-order mass vectors exactly — a consistency check proving the tensor was constructed correctly. The 2nd-order slices (X_tr, X_tb, X_rb) reveal which pairwise couplings dominate, informing deck-building decisions analogous to dependency analysis in system architecture.
"The same tensor decomposition used to analyze a 69-card deck is applied to production systems with millions of users. The scale changes. The math doesn't." — ArchDaemon™ · Quality Engineering Framework
ArchDaemon™ (US Serial 98940257) · GoldHat™ (US Serial 98925168) · All mathematical analyses are original IP of David Leo Sylvester.