Abstract

This paper is Version 3 of the e-as-finite-number programme, establishing e as a Category B Operational Invariant (Japanese: Sousa Hensuu 操作遍数) of the unique minimal operational closure C^(3)_Πd, realised as M₃(ℂ) structure, within the axiom system {A1 non-commutativity, A2 Π_d-saturation, A4 redundancy exclusion} of Operatiology, the successor framework to Cognitional Mechanics. The central advance is the Grounding Uniqueness Lemma: among all values exp(t) for t∈ℝ, the unique element generating an independent Operational Invariant is exp(1)=e. This follows from exhaustive case analysis — rational t are A4-redundant relative to e; nonstructural irrational t admit no M₃(ℂ)-grounding; structural irrational t (e.g. e^δ, e^π, e^γE) yield only Category E composites of existing S-elements. The grounding source (G2b) is the unique unit operation of the Lie algebra exponential map of M₃(ℂ) — a Tier-2 operational concept involving no numerical parameter. Its Tier-3 representation is t=1, yielding e=exp(1) as the M3-value. e is thus established as an element of S\Fext of the general Operational Invariant solution (DOI: 10.5281/zenodo.20647095). Hermite's transcendence result (1873) is repositioned as the final step in the logical chain: G2b ⟹ unit operation ⟹ Tier-3 projection ⟹ t=1, exp(1)=e ⟹ Hermite confirms e∉Q̄. It confirms, rather than grounds, the uniqueness established by G2b. The structural invariance de/dt=0 follows as a theorem (Corollary 3.6). This extraction is logically necessary in the same sense as that of π (DOI: 10.5281/zenodo.20602756): the absence of a Tier-3 closure condition for the open curve exp(tH) — confirmed by Hermite — uniquely necessitates characteristic-polynomial confirmation, symmetric in logical structure with the closure condition U(θ)=I for π. For those interested, the following code illustrates the Tier-3 confirmation of the exponential unit operation at 50-digit precision: starting from H=diag(1,1,-2) with integer eigenvalues only, no numerical approximation of any constant is manually supplied at any stage, and the constant appears only in the printed output. The algebraic justification is given in Section 3 of the paper. ◆◆◆ [PYTHON VERIFICATION CODE — note: × denotes * and ^ denotes **] ◆◆◆ from mpmath import mp, mpf, mpc, matrix, findroot, expm, exp, sqrt ◆ mp.dps = 50 ◆ h = [mpf(1), mpf(1), mpf(-2)] ◆ H = matrix([[h[0],0,0],[0,h[1],0],[0,0,h[2]]]) ◆ M = expm(H) ◆ def tr(A): return sum(A[i,i] for i in range(3)) ◆ def det3(A): return (A[0,0]×(A[1,1]×A[2,2] - A[1,2]×A[2,1]) - A[0,1]×(A[1,0]×A[2,2] - A[1,2]×A[2,0]) + A[0,2]×(A[1,0]×A[2,1] - A[1,1]×A[2,0])) ◆ p1 = tr(M) ◆ p2 = (p1^2 - tr(M × M)) / 2 ◆ p3 = det3(M) ◆ def char_poly(lam): return lam^3 - p1×lam^2 + p2×lam - p3 ◆ r_small = findroot(char_poly, mpf('0.5')) ◆ b = r_small - p1 ◆ c = p3 / r_small ◆ C1 = ((-b + sqrt(b^2 - 4×c)) / 2).real ◆ C2 = mp.e ◆ print(f"Tier-2: unit operation (G2b); Tier-3: parameter value t=1, char_poly(expm(H))") ◆ print(f"Logical order: G2b -> unit operation (T2) -> Tier-3 projection -> parameter value t=1 -> exp(1)=e -> Hermite confirms value not in Q-bar") ◆ print(f"confirmed eigenvalue > 1 = {C1}") ◆ print(f"library reference value = {C2}") ◆ print(f"difference = {abs(C1 - C2)}") ◆◆◆ [END] ◆◆◆ Published on June 14, 2026 DOI 10.5281/zenodo.20685098