The Two-Tier pi-Locked Unit Architecture (l_0, L) and Discrete Vacuum Geometry on l^2(N >= 2): From Sakharov Throat Gravity (G = l_0^2 / 16*pi) and the s = 5 <-> k_max = 5 Dual Valve to Unaligned Hamiltonian Collapse (18.259%), I_0 Metrology, Zero-Da

This repository presents the complete capstone monograph suite, intuitive pocket-calculator companion guide, 15-suite master bibliographic reading index, and standalone numerical verification engine of K-PROTOCOL Series 00-G (Core Monograph VII and Companion Guide VII; Permanent Concept DOI: 10.5281/zenodo.24074080; Version v1 DOI: 10.5281/zenodo.24074081). Uniting K-PROTOCOL Series 00-A through 00-F (DOIs: 10.5281/zenodo.23117791, 10.5281/zenodo.23355922, 10.5281/zenodo.18797547, 10.5281/zenodo.23911703, 10.5281/zenodo.23913158, 10.5281/zenodo.23972445), we demonstrate that the apparent decimal randomness of physical constants in reference tables (g_0 = 9.80665 m/s^2, G = 6.67430e-11 m^3 kg^-1 s^-2, m_e = 9.10938e-31 kg) is not an intrinsic feature of physical law, but a coordinate veil created by two historical conventions: (i) retaining the singular boundary node k = 1 in continuum mode integrals, and (ii) measuring a curved, pi-governed manifold with an historical 1790 terrestrial meter stick (1 m) locked to a single-laboratory proper speed of light (c_0 = 299,792,458 m/s). Formulating the dynamics on the shifted discrete Hilbert space H_lat = l^2(N >= 2) equipped with the normalized 3-point harmonic measure w(k) = 4 / [(k-1)k(k+1)] (Sum_{k=2}^inf w(k) = 1) under a two-tier pi-locked unit system (l_0, L) unifies micro-to-macro physics with zero adjustable parameters: ========================================================================REPOSITORY CONTENTS & FILE ARCHITECTURE======================================================================== 1. Core Monograph VII (00-G_Core_Monograph_VII_Pi_Units_and_Gravity.pdf):Provides the formal definition-theorem-proof derivations, exact algebraic identities, and multi-scale unification across five foundational pillars:- Pillar 1 (Operator Foundations & Mellin Dual Closure s = 5 <-> k_max = 5): Bilateral central differences select k_min = 2, while Riemann critical-line unitarity (Re(s) = 1/2) fixes the self-energy denominator 4k in [W_w]_kk = w(k)^2 / (4k). Stride-2 partial-fraction telescoping cancels Euler's planar wave constant zeta(2) = pi^2 / 6 from the diagonal rest-mass trace (Z_diag = 4*zeta(3) - 19/4 = 0.0582276126) while preserving pi^2 in off-diagonal wave hopping (J_total^2 = (238 - 24*pi^2)/9 = 0.1254993749). Across the 2D Euler-Goldbach power grid G_{k,n} = k^-n, Mellin row-column duality links the spectral vanishing of 4(2-2)[zeta(5)-1] = 0 at s = 5 to the hyperelliptic genus-2 rupture at k_max = 5, locking the two-way channel count 2*k_max = 10, the 5-body EFT factor F_EFT = 21/4 = 5.25, the 4D loop torus measure (2*k_max)^-4 = 10^-4, and the 6-mode throat barrier Pi_6 = (2^14 * 3^9 * 5^4)^-1 = 1 / 201,553,920,000.- Pillar 2 (Codimension Projectors & Unaligned 3x3 Hamiltonian Crossing): Acting with the horizon surface operator on orthogonal codimension subspaces P_q (q in {0, 1, 2}) fixes H_00(0) = 0, H_11(0) = Z_loop = 9.129635%, and H_22(0) = Z_loop + |gamma_vac| with zero pre-aligned offsets. Feshbach-Lowdin level repulsion (-g_01^2 / E_0 = +0.07412%) delays the 50:50 particle collapse from the bare tension |gamma_vac| = 18.18480% to the spontaneous saddle eta_c2^dressed = 18.25906% (18.26012% 1st-order).- Pillar 3 (Two-Tier pi-Locked Units (l_0, L), I_0 Metrology & Subatomic Spectrum): In Tier-1 microscopic throat units (M_Pl = l_0^-1), Sakharov induced gravity locks G = l_0^2 / (16*pi) and Einstein's coupling kappa_E = 8*pi*G = 1/2 l_0^2 (equal 50% temporal g_00 and 50% spatial g_rr partition), reducing subatomic quantities to pure pi-exponentials e^{+16*pi(b-a)} (bare electron mass m_e = 2*e^{-16*pi} M_Pl, effective action S_eff = 49.573676 [0.24 ppm], Koide ratio m_mu / m_e = 206.768128 [0.75 ppm], PMNS mixing angles sin^2(theta_13) = 0.02205, sin^2(theta_23) = 0.5454, sin^2(theta_12) = 0.3029, Higgs VEV v_lat = 246.27 GeV, and Higgs boson mass m_H = 125.29 GeV [0.22-sigma]). In Tier-2 macroscopic units at the quadrupole-free origin I_0 (P_2(sin phi_0) = 0 <=> sin^2(phi_0) = 1/3, cos^2(phi_0) = 2/3 = w(2)), defining 1 L = 9.80665 / pi^2 m = 0.99362139 m locks reference gravity to g_0 = pi^2 L/s^2, reconciles the 5.4-sigma Paris-Harvard-Berkeley alpha^-1 spread to 0.07-sigma, proves the +2.7890 mm (+2.7863 mm effective) GNSS vertical bounce via the 50% Spatial Metric Omission Theorem, and predicts the -4.914e-11 BNL-to-FNAL (+111 mGal) muon g-2 shift.- Pillar 4 (Multi-Scale Pade Gravity & Zero-Dark-Sector Cosmology): Monotonic [1/1] Pade-regularized flux susceptibility S_g^reg(x) explains the +45 to +145 ppm crustal Bouguer Big G scatter on Earth and saturates smoothly at G_eff / G = (1 - |gamma_vac|)^-1 = 1.2223 in deep space. Below a_0 = c * H_0^bare * |gamma_vac| = 1.190e-10 m/s^2, Kaluza-Klein 2D dilaton reduction (M^4 -> M^2 x S^2) onto the Apery floor e^{2*phi_0} = Z_diag yields flat galactic rotation curves (v_flat^4 = G_eff * M_baryon * a_0, exact slope 4.0000), the 12.00x Bullet Cluster Hessian lensing peak ratio, early JWST galaxy formation (z_coll ~ 17.12), and void metric-slip Hubble expansion (H_0^local = 73.56 km/s/Mpc, 0.5-sigma SH0ES match) without dark matter or dark energy.- Pillar 5 (Quantum Botany & gcd(N_c, 3) Bio-Motor Dichotomy): The identical tri-sector Hamiltonian governs all 8 core processes of photosynthesis (0.168% mean error across 14 observables) and partitions all biological F_o*F_1-ATP synthase c_N-rings (N_c in [8, 15]) against the 3-fold alpha_3*beta_3 stator into Class I Continuous Cogging-Free Rotary Turbines (gcd(N_c, 3) = 1 => {8, 10, 11, 13, 14}) and Class II Anti-Backflow Mechanical Ratchets (gcd(N_c, 3) = 3 => {9, 12, 15}). 2. Companion Guide VII (00-G_Companion_Guide_VII_Pi_Units_and_Gravity.pdf):"Why Nature Speaks in pi While Tables Speak in Decimals" -- Provides a 1-to-1 translation dictionary, five intuitive physical mental models (the Leapfrog Domino Eraser, the Free Bird / One-Rope Tether / Two-Rope Tent, the Open Umbrella in GNSS, the Balloon Between Glass Plates, and the 3-Pin Bicycle Gear Law), unabridged hand derivations with zero steps skipped, and a 12-step pocket-calculator audit sheet verifiable in five minutes. 3. Master Pedagogical Catalog & Reading Guide (00-G_Master_Series_Catalog_and_DOI_Index.pdf):Provides the complete bibliographic index and logical reading order across all 15 official K-PROTOCOL Zenodo suites (Suites 01 through 15) with Permanent Concept DOIs. 4. Standalone Python Verification Suite & 2-Page Report (verify_series_00_G.py & 00-G_Numerical_Verification_Report.pdf):A dependency-free, pure-Python verification engine using exact rational arithmetic (fractions.Fraction) and a custom Jacobi eigenvalue solver. Dynamically executes all 5 parts with zero free parameters and zero hardcoded output strings, generating the widescreen 2-page numerical verification summary and 4-panel analytical plots.