Beyond Octonions: Universal Symmetry and Instant Inversion in Algebras with Anticommuting Generators

**Beyond Octonions: Universal Symmetry and Instant Inversion in Algebras with Anticommuting Generators** M. V. Kostiuk February 2026 --- **ABSTRACT** The classical division algebras (ℝ, ℂ, ℍ, 𝕆) possess a remarkable property: an element \(a(1+e_1+\ldots+e_{d-1})\) is inverted by simply changing the signs of all imaginary units and dividing by \(a\cdot d\). It turns out that this property is **not** connected with associativity, nor with the existence of a norm, nor even with dimensions that are powers of two. It suffices that the imaginary units anticommute and that their square equals \(-1\). **Main discovery:** the formula works for **ANY integer dimension \(d\)**. One may take \(d = 3,5,6,7,9,10,\ldots,100,\ldots\) — and for all such “algebras” the diagonal element with arbitrary signs \(\pm1\) is inverted in \(O(1)\) time with coefficient \(1/(a\cdot d)\). The parameter \(a\) can be **any nonzero real number**: be it \(\pi\) or \(-\sqrt{2}\). **Symmetric diagonal elements** are expressions of the form \(a(\varepsilon_0+\varepsilon_1 e_1+\ldots+\varepsilon_{d-1}e_{d-1})\) where all coefficients of the basis vectors equal \(\pm a\). For fixed \(a,d\) there are exactly \(2^d\) such elements — the vertices of a \(d\)-dimensional hypercube. And **EACH** of them is inverted instantly by a single pattern: the real unit keeps its sign, all imaginary units change sign, and the whole expression is multiplied by \(1/(a\cdot d)\). **Table MI 1** (Instant Inversion No. 1) is an \(8\times8\) table of the values of the coefficient \(1/(a\cdot d)\). Its diagonal (\(a=d\)) contains the reciprocal squares \(1/1^2, 1/2^2, \ldots\) whose sum is the famous \(\pi^2/6\) (the Basel problem). Every entry of the table is a key to **all** \(2^d\) diagonal elements of that dimension. The table becomes an explicit reference: for \(d=4,\;a=1\) take \(1/4\) — and you are done. **Purely imaginary algebras** (without a real unit) obey the same law: for an element \(a(\varepsilon_1 e_1+\ldots+\varepsilon_d e_d)\) the inverse is obtained by changing the signs of all \(e_i\) with the same coefficient \(1/(a\cdot d)\). Octonions without real part (\(d=7\)), quaternions without unit (\(d=3\)) — all fit into the same table. **CONCLUSION:** beyond octonions lies an infinite world of “hypercubic numbers” for which inversion is not a computational problem but an instantaneous transformation. The work contains a rigorous proof for any dimension, combinatorial consequences, a table of coefficients, and a conjecture on fractional dimensions. --- **1. INTRODUCTION** In [1] the formula for instant inversion was discovered for elements \(S_d(a)=a(1+e_1+\ldots+e_{d-1})\) in the four classical division algebras: real numbers (\(d=1\)), complex numbers (\(d=2\)), quaternions (\(d=4\)) and octonions (\(d=8\)). The proof relied only on two simple properties: * \(e_i^2 = -1\); * \(e_i e_j = -e_j e_i\) for \(i\neq j\). Neither associativity, nor the existence of a norm, nor closure under multiplication are needed for this. It is enough that the imaginary units anticommute and that their squares are negative. A natural question arises: does this magic work for other dimensions? For example, can one take 3 imaginary units \(i,j,k\) with \(ij=-ji,\;ik=-ki,\;jk=-kj\) and obtain the inverse of \(a(1+i+j)\) simply by changing signs? The answer is yes. Moreover, **ANY** natural dimension \(d\) admits such a construction. It does not matter whether the algebra has a multiplication defined for all pairs — the inversion formula only requires that the product \((\varepsilon_0+v)(\varepsilon_0-v)\) be defined for the particular vector \(v=\sum \varepsilon_i e_i\). We also discover that **ALL** alternating diagonal elements — those for which the coefficients of all basis vectors are \(\pm a\) — behave identically: the inverse always carries the factor \(1/(a\cdot d)\) and opposite signs for the imaginary units. This yields an exponential family (\(2^d\) elements) with a uniform inversion algorithm. Finally, **purely imaginary algebras** (without a distinguished real unit) obey the same law up to an overall sign. This extends the applicability to subspaces of classical algebras and to abstract constructions of arbitrary dimension. --- **2. UNIVERSAL FORMULA FOR ANY DIMENSION** **2.1 Algebra with anticommuting generators** Let \(d\) be an arbitrary natural number. Consider an algebra over \(\mathbb{R}\) with basis \(\{1, e_1, e_2, \ldots, e_{d-1}\}\) subject to the relations \[ e_i^2 = -1 \qquad (i=1,\ldots,d-1), \] \[ e_i e_j = -e_j e_i \qquad (i\neq j). \] No further conditions (associativity, alternativity, normed property) are imposed. **2.2 Fundamental identity** **THEOREM 1.** For any choice of signs \(\varepsilon_0,\varepsilon_1,\ldots,\varepsilon_{d-1}\in\{+1,-1\}\) and the vector \(v = \sum_{i=1}^{d-1} \varepsilon_i e_i\), \[ (\varepsilon_0+v)(\varepsilon_0-v)=d. \] *Proof.* Expanding the product gives \(\varepsilon_0^2 - v^2 = 1 - v^2\). Compute \(v^2\): \[ v^2 = \Bigl(\sum_{i=1}^{d-1}\varepsilon_i e_i\Bigr)^2 = \sum_{i=1}^{d-1} (\varepsilon_i e_i)^2 + \sum_{i\neq j} (\varepsilon_i e_i)(\varepsilon_j e_j) = \sum_{i=1}^{d-1}(-1) + \sum_{i\neq j} \varepsilon_i\varepsilon_j e_i e_j. \] Because of anticommutation \(e_i e_j = -e_j e_i\), for each pair \(i\neq j\) the terms \(\varepsilon_i\varepsilon_j e_i e_j\) and \(\varepsilon_j\varepsilon_i e_j e_i = -\varepsilon_i\varepsilon_j e_i e_j\) cancel. Hence the double sum is zero, and \(v^2 = -(d-1)\). Therefore \[ (\varepsilon_0+v)(\varepsilon_0-v)=1-(-(d-1))=d. \quad \blacksquare \] **IMPORTANT:** the identity holds for **EVERY** natural \(d\). The classical dimensions \(1,2,4,8\) are only special cases. **2.3 Instant inversion formula (Class I)** **THEOREM 2.** For any nonzero real number \(a\) (positive, negative, rational, irrational) and any element \(S = a(\varepsilon_0+v)\), the inverse exists and is given by \[ S^{-1} = \frac{1}{a\cdot d}\,(\varepsilon_0 - v). \] *Proof.* \(S\cdot\frac{1}{a d}(\varepsilon_0-v) = \frac{1}{d}(\varepsilon_0+v)(\varepsilon_0-v) = \frac{d}{d}=1.\) ∎ **REMARK.** The coefficient \(\frac{1}{a d}\) may be negative (if \(a