The Infinitum as an Energy Economy Protocol Why Any Cognitive System Striving to Minimize Costs While Maximizing Predictive Power Inevitably Arrives at △₁ₓ₁ and Only at △₁ₓ₁

We present a treatise demonstrating that the right isosceles triangle △₁ₓ₁ — the Infinitum — is the unique attractor for any energy‑efficient cognitive system. Starting from the physical principle that information processing always carries a thermodynamic cost (Landauer's principle), we compare the energy expenditures of various geometric primitives and prove that △₁ₓ₁ minimizes storage and computational costs simultaneously. Three key properties — orthogonality, fractal self‑similarity, and a spectral gap — form an optimal compression protocol. We show that the brain's hexagonal grid cells are an isotropic adaptation of this protocol, derived from △₁ₓ₁ through the chain √2 → √3 → 2. We further demonstrate that the primary structure △₁ₓ₁ arises from the need to compute minimal shifts and compensate for gravity in order to maintain stability, while the equilateral triangle is a secondary structure generated from △₁ₓ₁. We predict that optimal AI architectures will converge to △‑mosaics under energy constraints, and we link the spectral gap λ₁ = 1 – ½√2 to the thresholds of neural predictive coding. The treatise concludes with a theorem: △‑ontology is the sole attractor for any energy‑efficient cognitive system, making it as inevitable as the second law of thermodynamics.