Euler Brick (Perfect Cuboid) - Complete Proof

This work presents a complete proof that a perfect cuboid does not exist. A perfect cuboid is a rectangular box with positive rational edges W1,W2,W3W₁, W₂, W₃W1,W2,W3, positive rational face diagonals Z1,Z2,Z3Z₁, Z₂, Z₃Z1,Z2,Z3, and positive rational space diagonal CCC, satisfying the standard cuboid system W₁² + W₂² = Z₃²W₁² + W₃² = Z₂²W₂² + W₃² = Z₁²W₁² + W₂² + W₃² = C² Equivalently, the proof works in the explicit cuboid ring B := ℚW₁, W₂, W₃, Z₁, Z₂, Z₃, C / I with III generated by the four displayed relations. The main idea of the paper is to compress the perfect cuboid problem into a local algebraic–differential contradiction pipeline. The proof is organized around the following structural fact: every local face can be rewritten in triangular-remainder coordinates a = r + x, b = r + y, c = r + x + y and the Pythagorean face equation becomes exactly a² + b² = c² ⇔ r² = 2xy This identity is the unique nonlinear local source of the problem. All three-face gluing conditions are affine-linear. Thus the entire nonlinear local mechanism is reduced to a single quadratic face law together with affine compatibility. The proof then shows that, after normalization, the transported local shell is governed up to degree 3 by exactly two visible equations: R₁² = 2uv and uV₂ + vU₂ = R₁R₂ Equivalently, the exact transported local ideal truncated in degree ≤ 3 is J≤3 = ⟨f₂, f₃⟩≤3 with f₂ = R₁² − 2uvf₃ = uV₂ + vU₂ − R₁R₂ So there is no hidden quadratic or cubic rescue mechanism outside the visible normalized shell. A second key step is the first-order readout theorem. The local readout morphism has the form u = s + q₁v = t + q₂w = q₃ with qᵢ ∈ ⟨s², st, t²⟩ Hence the first-order part of the readout is completely linear in the packet variables (s,t)(s,t)(s,t); all nonlinear corrections begin in degree ≥ 2. From this one obtains the faithful edge-readout statement: on the de-gauged strict-core nonsingular tangent space, the edge readout coincides with the genuine first-order edge differential and has rank 2. In particular, any nonzero tangent direction forces motion of at least one edge: (dW₁, dW₂, dW₃) ≠ (0, 0, 0) This yields admission completeness: every hypothetical perfect cuboid necessarily produces a detected nontrivial local branch at some edge. The rest of the argument is eliminative. The manuscript excludes every possible survival mode for such a branch: no hidden first-order branch may survive outside the packet-visible layer; no hidden degree-≤3 branch may survive outside the visible shell J≤3=⟨f2,f3⟩≤3J≤3 = ⟨f₂, f₃⟩≤3J≤3=⟨f2,f3⟩≤3; no branch may survive in the face-trap locus; no singular or degenerate decoration may create a new reduced support-carrying branch after blowdown and reduction. The singular/degenerate sector is treated by a finite pathology atlas consisting of three strata: corank-one rank-defect, kernel-type degeneracy, higher-corank degeneracy. For each stratum the reduced downstairs support is shown to coincide with that of the associated strict core. Thus singular or degenerate decoration may alter multiplicity, nilpotent thickness, or scheme-theoretic shadow, but it does not create a new support-bearing branch. At that point the remaining local alternatives are exhausted. Every detected branch either collapses or yields an unsupported reduced branch, and the singular/degenerate route gives no new surviving support. Therefore no local branch compatible with a hypothetical perfect cuboid remains. The contradiction is global and final: A hypothetical perfect cuboid would force a nontrivial local branch.The local branch analysis proves that no such branch can survive. Therefore There does not exist a perfect cuboid. The manuscript is written in a theorem-driven style and is intended to be read both as a human-verifiable mathematical proof and as a structurally transparent proof object suitable for further formalization, symbolic checking, and AI-assisted parsing. In particular, the proof spine is deliberately compressed to the following sequence: cuboid ring→ triangular-remainder local reduction→ affine gluing / single nonlinear source→ exact shell closure up to degree 3→ faithful first-order edge readout→ detected local branch→ exclusion of all surviving branch types→ contradiction This publication provides the full manuscript of the proof in journal-style form.