Quantinion Simplex Cubic Decomposition
Global Recovery over 2-Power Rings
Muhammad Arshad · September 2026 · Symmetric cubic tensors, binary simplex frames, global parity identifiability, and exact bit lifting
Canonical paper (v1, CC BY 4.0): 10.5281/zenodo.22850767. All five papers and DOI links →
An Overcomplete Cubic with Exact Recovery
This chapter records the Quantinion Simplex Cubic Decomposition Theorem. Let Rₘ=Z/(2ᵐ) and let n≥4 be even. Consider a symmetric cubic tensor
If the parity reductions s_q=v_q mod 2 form a binary simplex frame — their sum is zero and any n of the n+1 vectors are independent — then the entire n+1 term decomposition is globally identifiable up to permutation. The recovery is constructive: first recover the parity frame from contraction ranks over F₂, then lift the atoms one bit at a time through Z₄,Z₈,…,Z_{2ᵐ}. For the Quantinion instance (n,m)=(8,8), this gives an exact 9-component decomposition in an 8-dimensional state space over Z₂₅₆.
Scope. The theorem is not a claim about arbitrary tensor decompositions. It applies to the stated simplex-frame cubic family over 2-power rings. The boundaries — even coefficients, odd n, and non-simplex overcomplete families — remain separate problems.
The Ring and the Cubic
The decomposition is already overcomplete because r=n+1 exceeds the vector dimension n. In the MPRC/Quantinion specialization, n=8, m=8, r=9 and the observable cubic has 8³=512 ordered entries.
The Parity Skeleton
Reduce every atom modulo two:
The required frame satisfies
Every such frame is GLₙ(F₂)-equivalent to the canonical form
The theorem is proved in canonical coordinates and transported through the invertible binary change of basis.
The Tensor Reveals Its Dual Frame
Let T̄=T mod 2. For each nonzero observer x∈F₂ⁿ, contract the first mode:
In canonical coordinates, with w=wt(x),
Because n is even, exactly n+1 observers have full rank n: the all-one vector and the n vectors obtained by clearing one coordinate. These form the dual simplex frame.
No Second n+1-Term Parity Decomposition
Suppose the same binary tensor had another n+1-term decomposition T̄=Σ t_q^{⊗3}. Tensor diagonal entries imply Σt_q=0. For every full-rank observer d, the contraction
has rank n, so at least n atoms are active. The active count is even because dᵀΣt_q=0. Since n is even and there are only n+1 atoms, exactly n are active: every full-rank observer annihilates exactly one atom and the remaining n atoms form a basis.
Distinct full-rank observers must annihilate distinct atoms. After ordering the observers d₀,…,d_n and stacking them as D, every hidden parity atom obeys
The matrix D has column rank n, so each system has one solution. Those solutions are exactly the original simplex atoms. Therefore the n+1-term parity decomposition is globally unique up to permutation.
Cubic Recovery Becomes Linear
Assume the atoms are known modulo 2ᵇ. Write the next lift as
Modulo 2ᵇ⁺¹, all terms containing two factors of 2ᵇ vanish. Therefore
The next bit is therefore obtained from a binary linear system J_S δ = R_b rather than from nonlinear optimization.
The Lift Has No Hidden Direction
For the canonical simplex, suppose J_Sδ=0. On a tensor coordinate (i,j,k) with all indices distinct, only the all-one atom contributes, giving
Because n≥4, comparing triples forces all entries of δ₀ to be equal; substituting back forces that common bit to be zero. Then coordinate (i,i,j) gives δ_i,j=0 for i≠j, and (i,i,i) gives δ_i,i=0. Hence every perturbation vanishes:
Each lift is unique.
Quantinion Simplex Cubic Decomposition Theorem
Theorem. Let n≥4 be even and Rₘ=Z/(2ᵐ). If T=Σ_{q=0}^{n}v_q^{⊗3} and the parity reductions {v_q mod 2} form a binary simplex frame, then any other n+1-term symmetric cubic decomposition of T consists of exactly the same atoms, up to permutation.
Proof structure:
The 8 → 9 → 512 Case
The observer search contains only 2⁸−1=255 nonzero binary observers. Once the nine parity atoms are recovered, seven bit lifts reconstruct all 72 atom coordinates:
Weighted Cubics Stay in the Same Class
Consider T=Σ λ_q v_q^{⊗3} with every λ_q odd. The units of Z/(2ᵐ) form a finite 2-group, so the cube map is an automorphism. Each odd λ_q therefore has a unique unit cube root μ_q. Thus
The parity simplex geometry is unchanged. This matches the MPRC scalar BIND restriction to odd multiplicative units.
The Theorem Was Reached Through Failures
| Attempt | Result |
|---|---|
| Second-order tensor should retain orientation | False: v⊗²=(−v)⊗² |
| Third order always distinguishes v from −v on Z₂₅₆ | False for states divisible by 8 |
| Primitive/odd atoms alone make overcomplete mixtures identifiable | False: exact rank-2 collisions exist on one direction |
| Simplex parity frame + cubic + bit lift | Survives proof and regression tests |
The failed statements are retained because they define the boundary of the theorem rather than being edited away after the result was known.
Independent Regression Checks
A separate exact-integer verifier reconstructs random hidden simplex frames after random GLₙ(F₂) changes of basis and random Z₂₅₆ lifts. The current regression set covers n=4,6,8 and checks the contraction-rank criterion, parity support recovery, full Jacobian rank, and all seven byte-lifting stages. The verification code is published with the formal paper.
The First Solvable r = n+2 Class
The main theorem stops at r=n+1, but a structured n+2 family can also be recovered exactly. Let the parity atoms consist of a binary simplex frame {s₀,…,s_n} plus one residual atom u. Because the simplex frame sums to zero, the diagonal of the parity tensor exposes the residual immediately:
Subtract u^{⊗3} modulo two and the remaining parity tensor is exactly the simplex cubic of the main theorem. The only remaining issue is whether the n+2 atoms lift uniquely through the 2-power ring.
Residual-Simplex Lift Theorem. In canonical coordinates let U={i:u_i=1}. If |U|≥4 and |Uᶜ|≥3, then the combined one-bit lifting Jacobian for the simplex frame plus u has trivial kernel. Hence the residual-simplex decomposition lifts uniquely through Z₄,Z₈,…,Z_{2ᵐ}.
For the Quantinion case n=8, this leaves exactly Hamming weights 4 and 5. In simplex language the tenth atom is a balanced 4/5 split of the nine simplex vertices. Random hidden-frame Z₂₅₆ regressions recover all 80 byte coordinates exactly.
The deficient cases are genuinely non-identifiable inside this family: nonzero Jacobian directions already create distinct decompositions modulo four. This is not repaired by later lifting.
Boundary. This extension proves uniqueness within the simplex-plus-one-residual family. It does not prove that an arbitrary unrelated 10-atom family cannot represent the same tensor.
The Theorem's Boundary
| Question | Status |
|---|---|
| Even n≥4, r=n+1 simplex cubic over Z_{2ᵐ} | Proven |
| Odd unit weights | Reduced to unweighted theorem by unique cube roots |
| Odd n | Open: full-rank contraction pattern changes |
| Even/non-unit coefficients | Not covered; 2-adic torsion can destroy identifiability |
| General arbitrary r=n+2 | Open |
| Residual-simplex r=n+2 with |supp(u)|≥4 and |supp(u)ᶜ|≥3 | Proven within the residual-simplex family |
| Quantinion n=8 balanced r=10 residual class | Proven within family; weights 4/5 |
| General overcomplete cubic decomposition | Not claimed |
A Theorem Generated by the 8-State Geometry
The route to the theorem began from the Quantinion/MPRC operator structure: [1,2,4], its 8 binary subset states, the 8×8=64 operator space, and the 8³=512 cubic composition space. The final theorem is stated independently over Z/(2ᵐ), so it can be tested and used without accepting any broader physical interpretation of MPRC.
What Is Earned
| Statement | Status |
|---|---|
| Canonical binary simplex form | Exact linear-algebra reduction |
| Full-rank contraction characterization for even n | Proven |
| Global uniqueness of the n+1-term parity decomposition | Proven |
| One-bit cubic lift is linear | Exact modulo-2 expansion |
| Jacobian injectivity for n≥4 | Proven |
| Global uniqueness over Z/(2ᵐ) | Proven by induction |
| Quantinion n=8,m=8,r=9 specialization | Immediate corollary |
| General arbitrary overcomplete tensor decomposition | Not claimed |
Formal Manuscript and Exact Verifier
The theorem chapter has a separate formal LaTeX manuscript and an exact-integer reference verifier. The verifier uses no floating point, NumPy, or third-party numerical package.
The published regression seed is 20260918. It verifies the contraction-rank law for n=4,6,8,10, the lifting-Jacobian ranks, and blind complete Z₂₅₆ recoveries for random hidden simplex frames at n=4,6,8.