超越数坐标系三区构造与π刻度结构_判据方向表·母Z₂谱系·弱电角闭环_CMG-C3

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Abstract (English)

We develop the criterion structure of the transcendental coordinate geometry z = e^a·(-1)^b (a in C, b in C) — built on the two branches of the logarithm, the real branch ln e and the imaginary branch iπ = ln(−1) — and derive from its region table a structural mapping to transcendence classes. Contribution declaration: the new contributions of this paper lie in the framework organization, the classification theorems (HV-1 integer classification / HV-6 boundary-number structure / HV-11 criterion completeness) and the structural observations (Z2 reflection fixed point, force-spectrum signature, weak-mixing-angle coordinate reading); the identity family N1–N14 is a systematized compilation of classical results (Wallis product, Legendre duplication, Beta integral, Γ reflection) and no novelty is claimed for it. Main results (evidence grading: ✅ = rigorous mathematical fact, 🟠 = observational-level correspondence with structural support; conjectural-level and open-problem content placed respectively in Appendices A/B):

1. Criterion direction table (5x3 = 15 cells) and the P-group action (✅): with the representations of the coordinate decomposition z = e^a·(−1)^b as the objects of classification (not z itself — the same z has infinitely many equivalent representations, the invariant being only w = a+iπb mod 2πi), the representation space is partitioned into the intersection region / conflict band / difference region / complement region, each refined by the criterion directions of the two ends, giving 15 cells in total (a-side classes C'/LW/special/G/open five types x b-side classes H/H'/open three types, the Gelfond anchor G formally admitted); the locked↔open exchange involution P is defined (P²=I, component involution re-verified 15/15 cells, true bijection independently confirmed), whose fixed points are exactly the three cells {LW, special, G}xH' (follows directly from the definition of P); and the determinacy mapping is given (🟠 definition corollary): P pairs the criterion-decisive side with the zero-criterion side (intersection-region rigorous cell C'xH ↔ complement-region zero-criterion cell).

2. Integer classification theorem for the measure function and the Π_2(d) scale (✅): C(d) = 2^k ⟺ (d,k) ∈ {(1,1), (3,2)}; C(d) ∈ Z ⟺ d ∈ {1,3} (Bertrand theorem proof, upgrading "d=3 unique" from observation to theorem); together with the strict decrease of Π2 (Bohr–Mollerup three-point log-convexity) this gives real-domain uniqueness: Π_2(d)=π ⟺ d=1, C(d)=4 ⟺ d=3. Two-level measure convention: representation-space convention — the decidable skeleton (representations with ≥2 criteria) is Lebesgue null (finite-grid sampling share 2.58×10⁻⁴, tending to 0 as the window refines), its complement of full measure; z-assignment convention (criterion-priority gauge) — any z≠0 always has the representation (ln z, 0) with efficacy ≥ single-criterion decisive, hence the complement region (zero-criterion cell) is empty at the z-assignment layer; a general z lands in the difference region with single criterion, of full measure — "fully locked (dual criterion) is the exception; single criterion is the norm". The spherical measure ratio Π_2(d) = |S^d|/|S^(d−1)| = √π·Γ(d/2)/Γ((d+1)/2) satisfies the closed forms Π_2(2m+1) = π·C(2m,m)/4^m and the even family Π_2(2m) = 4^m/(m·C(2m,m)) — the rigorous geometric endorsement of "π-content = dimension": dimension one is the unique dimension giving the complete π (Π_2(1)=π); odd dimensions retain π, even dimensions cancel π.

3'. The Z2 reflection and half-integer self-duality of Π2 (✅ new, HV-9): Π2 satisfies the reflection functional equation Π_2(1−d) = tan(πd/2)·Π_2(d) (two-line proof from Γ reflection); the reflection σ:d↦1−d is a Z2 action; the multiplier m(d)=tan(πd/2) satisfies the cocycle m(d)·m(1−d)=1; the unique fixed point d=1/2 is self-dual (m(1/2)=1), with the self-dual value Π_2(1/2) = Γ(1/4)²/√(2π) = 2ϖ = 2·Π_4(2), where ϖ = 2∫₀¹(1−t⁴)^{−1/2}dt = Γ(1/4)²/(2√(2π)) = 2.622057554292… is the Gauss lemniscate constant — the half-integer self-dual point coincides with the Z2 fixed point, and the self-dual value 2ϖ is exactly twice the M4 closed-form value Π_4(2) (AGM connection: Π_2(1/2) = 2π/AGM(1,√2), Π_2(3/2) = 2·AGM(1,√2), and Π_2(1/2)·Π_2(3/2) = 4π = |S²|, the fifth occurrence of 4π, now as an exact identity).

3. Master Z2 family and quotient structure (✅): from the solution-lattice quotient L/K = πiZ/2πiZ ≅ Z/2Z of the exp period kernel, "odd iπ on the negative axis" is upgraded from a placement postulate to a group-theoretic theorem; the master Z2 appears in five places — number theory (π-cancel/π-retain), coordinate system (return/sign-change), interface (type I/type II), basis (e-system/π-system), force spectrum (positive π axis/negative π axis, 3:3:1) — with the topological-statistical layer π₁(SO(2))=Z → π₁(SO(d≥3))=Z/2 as the sixth carrier.

4. π-emergence family and the braid one-dimensional representation (✅): the odd-dimension closed forms are isomorphic to symmetric-random-walk return probabilities — a dimension-dependent π-power emergence spectrum: 1D return probability ~ 1/√(πm), 2D ~ 1/(πm), 3D ~ 2(3/4πm)^{3/2} (Pólya theorem: d=1,2 recurrent / d≥3 transient); the anyon translation operator T_m is explicit as the braid-group B_n one-dimensional representation σ_j ↦ e^{2πi/m}, with full twist Δ² ↦ e^{in(n−1)θ}.

5. Transcendence classification catalog (✅🟠): 20 classical constants, each coordinated into a cell (z = e^a·(−1)^b), five-part classification: dual-confirmed by coordinate-system criterion × number-theory theorem 5 numbers (e^π, (−1)^√2, (−1)^i, i^i, e^{π√3}); theorem-single-confirmed 3 numbers (2^√2, 2^i, ln 2, not covered by the coordinate-system criterion); criterion-attempted pending 2 numbers (e^e/lnπ); criterion-undecided-but-transcendence-proved 4 numbers (sin 1, π√2, √π, 1/π); genuinely-unknown 6 numbers (π^e, γ, Catalan G, π+e, π·e, ζ(3) — of which only ζ(3) has proved irrationality) — genuine-unknown rate 30%.

6. Weak-mixing-angle Ledger (🟠 observational-level coordinate reading): parameter-free π-scale coordinate reading sin²θ_W = 3π²/128 (MS-bar @ M_Z +0.72σ; sin²θ_eff^ℓ convention −5.78σ, sign varies with convention — 3π²/128 lies between the two observed values, dual-convention parallel disclosure), co-sourced with cos²θ_W, M_W/M_Z, α₂, g_W, g_V/g_A from the constant family {128, 3π², 32}, forming a "one parameter, six derived quantities" consistency residual ledger; the identity decomposition 3π²/128 = (3/8)·(π/4)² embeds it in the GUT tree-level Weinberg-angle family.

7. Conflict-band information-theoretic layer (✅): the composition map f: Z2xZ2→Z2 (condensation +1 identity element, leakage −1 generator) outputs −1 on all 4 conflict-band cells (constant image, Σ=−4); the level sets of f are exactly the cosets of the return/sign-change quotient; the criterion-entropy comparison gives ΔH = 1 bit = the information content of "multiplicative composition overriding additive voting" (H_vote=1.000 bit → H_out=0 bit).