e as the Finite Number: Algebraic Derivation of the Natural Logarithm Base

The present version reframes the derivation within 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 a Category B Operational Invariant (e∈S\Fext, 操作遍数) of the general solution (DOI: 10.5281/zenodo.20647095). Hermite's transcendence result (1873) is repositioned as the final step: 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. 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; the constant appears only in the printed output. The algebraic justification is given in Section 3 of the paper. from mpmath import mp, mpf, mpc, matrixfrom mpmath import findroot, expmfrom mpmath import 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)) / 2p3 = 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 - p1c = p3 / r_smallC1 = ((-b + sqrt(b**2 - 4*c)) / 2).realC2 = mp.e print("=== Part A-1: Tier-3 confirmation" " (exponential unit) ===")print(f"Tier-2: unit operation (G2b)")print(f"Tier-3: parameter value t=1,")print(f" char_poly(expm(H))")print(f"Logical order:")print(f" G2b -> unit operation (T2)")print(f" -> Tier-3 projection")print(f" -> parameter value t=1")print(f" -> exp(1)=e")print(f" -> Hermite confirms")print(f" value not in Q-bar")print(f"confirmed eigenvalue > 1 = {C1}")print(f"library reference value = {C2}")print(f"difference = " f"{abs(C1 - C2)}")