Magic-Angle Photonic Quantum Computing: Geometric Decoherence Protection, the Trisection Qutrit, and Falsifiable Predictions from a (3+3) Spacetime Framework
We apply the (3+3) spacetime framework [1, 6] — a six-dimensional Einstein–Aether construction with three spatial and three temporal dimensions, the third time dimension compactified as a discrete two-sphere S² with N_cells = 2¹⁵² Planck-area cells — to photonic quantum computing. The framework identifies the photon as a null arc (a c-vector configuration with v_T = v_t₂ = v_t₃ = 0 and v_spatial = c) corresponding to the n = 0 Kaluza–Klein zero-mode on S². From this single identification three Level-1 structural results follow: (i) the photon's polarization Poincaré sphere is geometrically the third time dimension's S²; single-qubit gates are physical S² rotations and the Born rule is Malus's law as S² projection; (ii) photons inherit no t₃-winding decoherence channels, with cosmological coherence floor τ_cosm = 1/H₀ ≈ 14.5 Gyr [§14bis.46.4 of 6] and practical decoherence by photon loss as t₃ entrainment; (iii) the cross-Kerr nonlinearity scales as χ⁽³⁾ ∼ α² = 5.3 × 10⁻⁵ from two successive c-vector rotations each of amplitude √α. We give a self-contained derivation in §5.4 of the χ⁽³⁾ ∼ α² scaling from the Lagrangian-level photon-photon-via-radion vertex L = −(g₀/4) Φ² F_{μν} F^{μν}|_node with bare coupling g₀ = (4/9) ε_ZPE², through radion integration-out at optical frequencies, to the four-loop running g₀ → α (drawing on §16–§17 of [6]). Sections 5.5–5.9 establish the structural form of the nodal-cone TPA-suppression factor η_node: the breathing-mode contribution at the cone vanishes geometrically because the Y_{2,0}(ϑ_node) = 0 identity that defines the cone latitude also eliminates the breathing's metric perturbation there (§5.7); the higher-multipole tail is bounded at η_node|_higher ≲ 10⁻⁵ using the framework's identified spectrum from [6] (§5.8); and the same Z₃ symmetry that organises SU(3), three-generation fermion structure, and tribimaximal PMNS mixing also protects the qutrit basis at the trisection vertices (§5.9). Combining these structural results with anchored medium-specific estimates for silicon-on-insulator (K_medium ≈ 0.10–0.20), silicon nitride (0.03–0.08, lowest), and thin-film lithium niobate (0.05–0.15) (Appendix B), the framework predicts η_node ≈ K_medium ≈ 0.03–0.20 across candidate platforms — silicon nitride is identified as the optimal platform — dominated entirely by medium contributions rather than framework-vacuum dynamics, and decisively testable by the §6.7 thermal MZI experiment (2027–2030). The magic-angle latitude ϑ_node = arccos(1/√3) = 54.74° is the same magic angle as in NMR magic-angle spinning, and the underlying physics (averaging-out of Y_{2,0} dipolar interactions) is structurally identical. Six engineering proposals leverage these results: null-arc qubit encoding; the nodal-cone LOQC architecture (qubits at the magic-angle latitude where Y_{2,0} vanishes and the leading TPA channel is suppressed); (3+3) measurement protocols with t₃-entrainment-strength control; trisection qutrit photonic computing (log₂ 3 ≈ 1.585 bits per photon, with Z₃-symmetric coherence protection giving η_qutrit ≈ 0.06–0.21 comparable to qubit η_node — a 58.5% information-density gain at essentially no extra coherence cost); boson sampling reframed as null-arc network interference; and a nodal-cone photonic-crystal integrated chip. Three testable predictions are organised into a year-resolved 2026–2030+ falsifiability timeline (§6.7). All claims are explicitly tagged Level 1 (geometric, derived), Level 2 (anchored prefactor), or Level 3 (engineering speculation); the room-temperature fault-tolerance projection of §9 is an explicit Level-3 conditional argument with dependency chain made explicit. A self-contained framework digest is given in §1.7; readers unfamiliar with the (3+3) programme can read §1.7 alone to decide whether to continue.
We apply the (3+3) spacetime framework [1, 6] — a six-dimensional Einstein–Aether construction with three spatial and three temporal dimensions, the third time dimension compactified as a discrete two-sphere S² with N_cells = 2¹⁵² Planck-area cells — to photonic quantum computing. The framework identifies the photon as a null arc (a c-vector configuration with v_T = v_t₂ = v_t₃ = 0 and v_spatial = c) corresponding to the n = 0 Kaluza–Klein zero-mode on S². From this single identification three Level-1 structural results follow: (i) the photon's polarization Poincaré sphere is geometrically the third time dimension's S²; single-qubit gates are physical S² rotations and the Born rule is Malus's law as S² projection; (ii) photons inherit no t₃-winding decoherence channels, with cosmological coherence floor τ_cosm = 1/H₀ ≈ 14.5 Gyr [§14bis.46.4 of 6] and practical decoherence by photon loss as t₃ entrainment; (iii) the cross-Kerr nonlinearity scales as χ⁽³⁾ ∼ α² = 5.3 × 10⁻⁵ from two successive c-vector rotations each of amplitude √α. We give a self-contained derivation in §5.4 of the χ⁽³⁾ ∼ α² scaling from the Lagrangian-level photon-photon-via-radion vertex L = −(g₀/4) Φ² F_{μν} F^{μν}|node with bare coupling g₀ = (4/9) ε_ZPE², through radion integration-out at optical frequencies, to the four-loop running g₀ → α (drawing on §16–§17 of [6]). Sections 5.5–5.9 establish the structural form of the nodal-cone TPA-suppression factor η_node: the breathing-mode contribution at the cone vanishes geometrically because the Y{2,0}(ϑ_node) = 0 identity that defines the cone latitude also eliminates the breathing's metric perturbation there (§5.7); the higher-multipole tail is bounded at η_node|higher ≲ 10⁻⁵ using the framework's identified spectrum from [6] (§5.8); and the same Z₃ symmetry that organises SU(3), three-generation fermion structure, and tribimaximal PMNS mixing also protects the qutrit basis at the trisection vertices (§5.9). Combining these structural results with anchored medium-specific estimates for silicon-on-insulator (K_medium ≈ 0.10–0.20), silicon nitride (0.03–0.08, lowest), and thin-film lithium niobate (0.05–0.15) (Appendix B), the framework predicts η_node ≈ K_medium ≈ 0.03–0.20 across candidate platforms — silicon nitride is identified as the optimal platform — dominated entirely by medium contributions rather than framework-vacuum dynamics, and decisively testable by the §6.7 thermal MZI experiment (2027–2030). The magic-angle latitude ϑ_node = arccos(1/√3) = 54.74° is the same magic angle as in NMR magic-angle spinning, and the underlying physics (averaging-out of Y{2,0} dipolar interactions) is structurally identical. Six engineering proposals leverage these results: null-arc qubit encoding; the nodal-cone LOQC architecture (qubits at the magic-angle latitude where Y_{2,0} vanishes and the leading TPA channel is suppressed); (3+3) measurement protocols with t₃-entrainment-strength control; trisection qutrit photonic computing (log₂ 3 ≈ 1.585 bits per photon, with Z₃-symmetric coherence protection giving η_qutrit ≈ 0.06–0.21 comparable to qubit η_node — a 58.5% information-density gain at essentially no extra coherence cost); boson sampling reframed as null-arc network interference; and a nodal-cone photonic-crystal integrated chip. Three testable predictions are organised into a year-resolved 2026–2030+ falsifiability timeline (§6.7). All claims are explicitly tagged Level 1 (geometric, derived), Level 2 (anchored prefactor), or Level 3 (engineering speculation); the room-temperature fault-tolerance projection of §9 is an explicit Level-3 conditional argument with dependency chain made explicit. A self-contained framework digest is given in §1.7; readers unfamiliar with the (3+3) programme can read §1.7 alone to decide whether to continue.
We apply the Three Time Dimensions (3+3) spacetime model to quantum computing, demonstrating that the Bloch sphere used to represent qubit states is geometrically identical to the compact third time dimension S² of the (3+3) model. This identification resolves several foundational puzzles — the two-state structure of qubits, the origin of the Born rule, the mechanism of measurement, and the non-locality of entanglement — without additional postulates. The (3+3) picture further distinguishes three physically distinct decoherence mechanisms with separate origins: T1 energy relaxation from electromagnetic null-arc coupling; T2 dephasing from Higgs l=2 breathing-mode coupling (with coupling amplitude ε_ZPE = 0.127 = m_H/(4v)); and a cosmological floor from the t2 cosmic expansion giving τ_cosm = 1/(α_23 H_0) ≈ 16 Gyr per isolated qubit — comparable to the age of the universe, and therefore negligible as an engineering constraint. From this geometric picture, we derive six practical engineering proposals: (1) operation at the nodal-cone latitude θ = 54.74° = arccos(1/√3) on the Bloch sphere, which geometrically suppresses the dominant phonon-mediated T2 dephasing channel; (2) S²-topological qubits exploiting π_2(S²) = Z, which is strictly stronger than current Majorana/Kitaev approaches based on π_1(S¹) = Z; (3) the trisection qutrit, a naturally 3-fold degenerate three-level system from the SU(3) symmetry of the S² trisection; (4) tunable quantum measurement via controlled t3 entrainment by N apparatus atoms; (5) entanglement coherence that scales with the number of environmental atoms N_env along the path rather than with spatial distance; and (6) a path to room-temperature quantum coherence by suppressing the phonon T2 channel. These proposals are falsifiable and several are testable with existing experimental hardware.
The (3+3) spacetime framework [1] makes a structural claim about quantum computing of unusual sharpness: every qubit's Bloch sphere — photonic, NV-centre, semiconductor-spin, trapped-ion, transmon — is the same physical S² of the third time dimension t₃ ([1] §17.2). Quantum mechanics did not invent the Bloch sphere as a representation device; in (3+3), it discovered the S² of the compact dimension. The latitude ϑ_node = arccos(1/√3) = 54.74° on this universal S² — the same magic angle as in nuclear magnetic resonance magic-angle spinning since 1958 [4] — is therefore predicted to protect all qubits from leading-multipole noise channels with axial symmetry, with no platform-specific tuning. There is no free parameter; the framework either works for all platforms at exactly 54.74°, or it fails universally. We work this out platform-by-platform via the Y_{2,0}(ϑ_node) = 0 identity. Three Level-1 derivations apply where the dominant noise has rank-2 angular structure on the qubit Bloch sphere: photonic two-photon absorption (η_node ≈ 0.03–0.20, reviewing [2]); NV-centre ¹³C dipolar coupling (predicted T₂ ≈ 1 s in natural-abundance diamond, eliminating the need for ¹²C isotopic purification); and Si:P donor-electron hyperfine coupling (predicted T₂ enhancement of 100–1000× in natural-abundance silicon, eliminating ²⁸Si purification). GaAs quantum dots are also Level-1 via second-order rank-2 dephasing variance (10–100× enhancement). Two Level-2 anchored cases — trapped-ion magnetic-field dephasing and superconducting transmon multi-channel noise — give a more modest factor-3 dephasing-variance suppression (√3 ≈ 1.7× T₂ enhancement) by the same second-order mechanism. The progression Level 1 → Level 2 tracks the directness of the Y_{2,0} coupling: from rank-2-exact dipolar tensors (NMR-canonical) to rank-2-emerging-from-rank-1-variance (the weakest cases). The platform-specific magnitudes vary by orders of magnitude; the latitude is the same 54.74° everywhere. Three universal predictions follow with sharp falsifiability: (i) cross-platform universality of 54.74° ± 0.5° across all platforms with no platform-specific tuning; (ii) platform-specific T₂ enhancements ranging from 100–1000× (Si:P) to √3 ≈ 1.7× (transmon); (iii) secondary magic-angle equivalence at 125.26° = arccos(−1/√3), with the spin-up/down chirality assignment reversed. The single most decisive near-term experiment is the NMR-MAS antipodal-angle test of Prediction (iii) — a sharp binary outcome at the canonical NMR platform, feasible immediately in 2026 with existing solid-state NMR hardware. The 2026–2030 falsifiability timeline (§9.2) gives year-by-year cross-platform consistency tests. Six engineering proposals leverage these results, including a Si:P donor processor in natural-abundance silicon at T₂ > 10⁵ × τ_gate — comfortably above the surface-code fault-tolerance threshold (§11), with a materials-cost saving of $10⁵–10⁶ relative to ²⁸Si-purified equivalents at the 10⁵-physical-qubit scale. The room-temperature fault-tolerance projection of §11 generalises [2] §9 to matter-based platforms as a Level-3 conditional argument; the strongest cases (NV-centre, Si:P) have plausible > 50% joint probability of meeting all four conditional dependencies. The implications go well beyond engineering: cross-platform magic-angle agreement at 54.74° would mean that the universe's qubit-state space is one compact dimension shared by all qubit platforms, identified specifically as the t₃ S² of the (3+3) framework. The Bloch sphere — treated as a representation device since the 1940s — would turn out to be a real sphere in the cosmic structure. We invite the QC community to test these predictions; the framework either works for all platforms at 54.74°, or it fails universally.
We propose and formally establish a third level of physical causality — causal conjunction — absent from the standard two-level taxonomy. The central object is the photon as a structured causal block B(σ) = (G⁻(S), {F(σ,λᵢ)}), carrying a causal charge discharged instantaneously upon absorption. The framework yields three new experimental predictions testable with current circuit-QED technology: (i) Giant-atom φ-scan: contrast ΔI = 0.971 bit at 97σ (10⁶ shots), distinguishing the two-point coupled model from the Markovian single-point null model. Observable: I(A:B) = G_cl via mutual information. (ii) Bloch sphere hierarchy (new result): I(A:B) takes three distinct levels depending on the initial state of the absorber B — poles 0.939 bit (94σ), equator 0.566 bit (57σ, phase-independent), mixed 0.160 bit (16σ) — all strictly positive, confirming G > 0 throughout the Bloch sphere. Includes an analytically proven JC symmetry theorem: I(B=|g⟩) = I(B=|e⟩) exactly for all coupling angles, to machine precision (< 10⁻¹⁵ bit). (iii) Virtual-mode power-law test at φ=π (new result): extended BCE predicts G ∝ (g/ω₀)² while standard QED predicts (g/ω₀)⁴. The exponent η is measurable in log-log scale at >15σ, immune to prefactor uncertainties. First proposal placing BCE and standard two-point QED in genuine quantitative tension at the destructive-interference null point. Additional results include: G = G_cl + G_ent decomposition with operational criteria; Si₃N₄ photonic platform (~$26k, no cryogenics); thermodynamic signature ΔW = 1.86×10⁻²⁵ J (Maxwell demon protocol, T=20 mK); Nakajima-Zwanzig non-Markovian analysis (critical parameter η_c ≈ 0.48, safety margin 21× in circuit-QED); solar photon empirical argument for scale-independence across 33 orders of magnitude; and Holevo bound verification (G << S_max in all regimes). All predictions are numerically verified across eight independent experiments (Python scripts, no external dependencies, reproducible in Google Colab). This is the second paper in the ICT (Incomplete Causal Transactions) series. The first paper ("Incomplete causal transactions: black holes as permanently frustrated couplers and the physical independence of entropy and generative capacity", GravCos2660030 / Zenodo DOI: 10.5281/zenodo.18736656) introduced the ICT framework and applied it to black holes and Rindler horizons.
Las fases geometricas son tema de investigacion actual en diversas areas de la fisica.
Interesa investigarlas tanto por razones de caracter teorico, cuanto por razones ligadas a sus aplicaciones. Entre estas ultimas resaltan las aplicaciones en informacion cuantica. Un computador cuantico esta basado en la posibilidad de generar, almacenar y manipular bits de informacion codificados en los grados de libertad de sistemas cuanticos. Estos son llamados qubits. Los qubits son superposiciones coherentes de dos estados fundamentales. Mientras su contraparte clasica puede valer 0 o 1 excluyentemente, el qubit puede tomar ambos valores 0 y 1 simultaneamente. Esto hace posible procesar informacion con mucha mayor rapidez en comparacion a una computadora clasica. El problema central con los qubits es que son sumamente fragiles, de modo que su tiempo de vida media es muy pequeno. El fenomeno que lleva a un estado de superposicion hacia un estado clasico se llama decoherencia. Para que un computador cuantico sea viable, es necesario contar con qubits cuya vida media sea mayor que el tiempo que toma realizar operaciones sobre ellos (computacion). Una ruta muy promisoria es la que se basa en las fases geometricas. Ellas permiten realizar operaciones que, de un lado, pueden ser muy rapidas y, de otro lado, pueden ser inmunes o muy robustas frente a la decoherencia. Para implementar
computacion cuantica geometrica, es entonces necesario ser capaz de manipular
fases geometricas con gran versatilidad. Contribuyendo a este in, esta tesis presenta
nuevos resultados en la manipulacion de fases geometricas que aparecen cuando el qubit esta codificado en fotones polarizados. Esta tesis contiene dos partes principales. En la primera parte hacemos un intento preliminar en manipular fases en estados de polarizacion. Especificamente, tratamos a la fase de Pancharatnam (fase total) que resulta de evoluciones unitarias arbitrarias. Discutimos los aspectos teoricos involucrados y mostramos en detalle como hacer que un estado de polarizacion siga cualquier curva sobre la esfera de Poincare. Luego presentamos los metodos utilizados para llevar a cabo las mediciones de la fase total acumulada a lo largo de la evolucion del estado. En la segunda parte de esta tesis, extendemos nuestros metodos y desarrollamos tecnicas para suprimir localmente las fases dinamicas que puedan aparecer durante la evolucion del estado de polarizacion. Esto nos permite observar y medir fases geometricas. Usando metodos similares a los discutidos en la primera parte, mostramos finalmente que las fases geometricas observadas experimentalmente coinciden con las predicciones teoricas con buena aproximacion.
The Born rule — that the probability of obtaining outcome |phi> in a measurement of a quantum system in state |psi> is P = |<phi|psi>|^2 — is one of the most empirically well-confirmed and conceptually contested statements in physics. Existing derivation attempts (Gleason’s theorem; Deutsch–Wallace decision theory; Zurek envariance; many-worlds branch counting) each establish it under their own assumptions, but none has achieved universal acceptance. We show that in the Three Time Dimensions (3+3) spacetime framework (de Haan 2026, book DOI 10.5281/zenodo.19633127), in which the third time dimension t_3 is compactified as a discrete two-sphere S^2 with 2^152 Planck-area cells, the Born rule emerges through three independent geometric routes that all converge on the same result P = |<phi|psi>|^2. Route 1 (Malus’s Law on S^2): the wavefunction is literally a direction on the t_3 S^2; the Bloch sphere IS this S^2; Malus’s 1809 law P = cos^2(theta) is identical to the Born rule for bosons, with SU(2) double cover giving cos^2(theta/2) for fermions. Route 2 (Gleason + S^2 non-contextuality): Gleason’s 1957 theorem uniquely selects |<phi|psi>|^2 given non-contextuality, which in (3+3) is a consequence of the S^2 Riemannian metric rather than an independent postulate. Route 3 (Lévy measure + t_2 isotropy): the t_2 expansion perturbs all t_3 directions isotropically; the unique continuous binary-outcome probability measure is the Lévy / Haar-invariant spherical measure, which gives P(+n_A) = cos^2(theta/2). The three routes use distinct mathematical tools (projection, Hilbert-space theorem, stochastic analysis) and distinct physical inputs (geometry, Riemannian metric, dynamical isotropy). They must converge if the geometric identification is correct; they would diverge if it were wrong. Malus’s 1809 law is the Born rule avant la lettre. The empirical optical law taught to undergraduates as a fact is, under this reading, a direct geometric consequence of photon polarisations being directions on S^2 — a statement about projection geometry that predates quantum mechanics by a century. The same geometric identification simultaneously resolves four foundational puzzles: (1) Schrödinger’s cat is never in superposition (the cat’s ~10^29 t_3 modes are entrained by environmental t_2 perturbation within femtoseconds of the first atomic interaction); (2) the three-polariser paradox dissolves (non-commuting measurements are non-commuting rotations on S^2); (3) the Bell/Tsirelson bound 2sqrt(2) is a geometric consequence of S^2 having dimension exactly 2 — empirical confirmation by loophole-free Bell tests (Hensen, Giustina, Shalm 2015) confirms the S^2 identification; (4) the cosmological decoherence floor tau_cosm = 1/H_0 ~ 14.5 Gyr is a structural upper bound on quantum coherence set by the t_2 cosmic expansion — distinguishing the framework from many-worlds interpretations. Three supporting elegance claims establish the framework’s quantum-foundational coherence: (A) the complex-valuedness of psi in C follows from t_2 being a rotation (the imaginary unit i is the generator of the t_2 precession rotation); (B) the Schrödinger equation i hbar dpsi/dt = H psi is the infinitesimal form of t_2 precession rotating the phase of the t_3 configuration; (C) the canonical commutation [x, p] = i hbar follows from the foam lattice structure. The paper is complementary to the companion Quantum Computing preprint [DOI 10.5281/zenodo.19651560] which uses the Bloch-sphere-as-S^2 identification for applied hardware-level proposals. The Born Rule paper provides the foundational underpinning for why that identification is correct at the level of quantum-mechanical first principles. The paper is honest about what remains open. Four specific items are explicitly acknowledged (§11): (1) the framework does not yet have a 6D Lagrangian formulation from which the Born rule would fall out of path-integral quantisation; (2) the uniqueness of the S^2 identification is only partially addressed — higher-dimensional spheres are ruled out by the Tsirelson bound, but tori and Calabi–Yau alternatives are not examined; (3) the path-integral reformulation of quantum mechanics within (3+3) has not been constructed; (4) detailed comparison with QBism, relational quantum mechanics, and consistent histories is outside the paper’s scope. Falsifiability. Six routes: Tsirelson-bound overshoot/undershoot beyond 2sqrt(2) + measurement precision; indefinite quantum coherence beyond cosmological timescales; real-valued QM experimental confirmation (already ruled out by Renou et al. 2021, reconfirmed by independent tests); detection of genuine contextuality; detection of discrete S^2 cell structure at near-Planck scales. The Born rule, in (3+3), is not a mysterious postulate; it is Malus’s Law on a compact sphere that nobody knew was there.
We propose that spacetime is five-dimensional — (x, y, z, t, e) — where the fifth dimension e is circular, periodic, and corresponds to the U(1) fiber in a principal bundle over four-dimensional spacetime. Our four-dimensional experience is a projection of this five-dimensional reality, and every quantum phenomenon is a geometric consequence of the projection. Within this framework, superposition is extension in the e-dimension, entanglement is conservation of the e-coordinate (e₁ + e₂ = C), momentum is helical pitch (∂e/∂x = p/ℏ), spin is helical chirality, and measurement is e-space sampling. The wave function is not an abstract probability amplitude — it is the literal shape of the particle in five-dimensional spacetime. We present three original results. First, the Aharonov-Bohm effect is derived as the holonomy of the e-bundle around a topological defect, establishing the physical reality of the vector potential from geometry. Second, the spin-statistics theorem is derived from a single geometric fact: the winding number of a particle's helix through the circular e-dimension determines both its spin (via the Noether theorem) and its exchange statistics (via parallel transport). Integer winding gives bosons; half-integer gives fermions; fractional winding in two dimensions gives anyons — confirmed experimentally in the fractional quantum Hall effect. Third, the Born rule P(i) = |αᵢ|² is reproduced within the five-dimensional density interpretation and the projection postulate, and the framework reproduces the quantum correlations that violate the CHSH Bell inequality (|S| = 2√2), with the non-locality sourced by the e-conservation constraint through the fifth dimension. The same compact e-circle also resolves the central obstacle to quantum gravity. The five-dimensional Einstein-Hilbert action on the e-bundle reduces via Kaluza-Klein decomposition to four-dimensional gravity coupled to electromagnetism and a scalar field in a single metric. The compactness of the e-circle converts the continuous UV momentum integrals of 4D quantum gravity into discrete Kaluza-Klein mode sums that admit zeta regularization. Under zeta regularization of the KK mode sums, the leading UV divergence vanishes identically at every loop order: S₀^{(L)} = [1 + 2ζ(0)]^L = 0. The subleading terms are Epstein zeta functions evaluated at non-positive integers. For the two-loop R³ counterterm of linearized 5D gravity on M⁴ × S¹ — corresponding to the Goroff-Sagnotti divergence that proved 4D Einstein gravity non-renormalizable in 1986 — the result under zeta regularization is stronger than expected: the R³ coefficient is identically zero at every order in the mass expansion, not just at leading order. The sunset topology produces the Epstein zeta of the Eisenstein lattice, E₂(s; n²+nm+m²) = 6ζ(s)L(s,χ₋₃), which vanishes at every negative integer through complementary trivial zeros of ζ(s) and L(s,χ₋₃). The figure-eight and vertex correction topologies vanish independently through the trivial zeros of the Riemann zeta at even negative integers (forced by the perfect-square structure of KK masses). Ghost contributions, carrying the same Eisenstein quadratic form through KK number conservation, vanish by the same mechanisms. Under zeta regularization of the KK mode sums, linearized 5D gravity on M⁴ × S¹ is perturbatively predictive — established through two loops by explicit computation, and conjectured to all orders by the Epstein-Terras structure — with counterterm coefficients determined from two parameters: G₄ and the e-circle circumference L. The spin structure of the e-circle (bosons periodic, fermions anti-periodic) implies a natural Z₂ orbifold with two fixed-point branes. The visible brane at φ = 0 supports Standard Model matter; the hidden brane at φ = π gravitates normally but couples to no SM force — geometric dark matter. The Casimir energy of bulk fields on the orbifold, with three right-handed neutrinos in the bulk to ensure positive energy, matches the observed dark energy density for a brane separation R ≈ 12 μm and predicts Yukawa-type gravitational deviations at that scale. The same bulk neutrinos give neutrino masses at the meV scale through the bulk seesaw mechanism. Kinetic mixing between the two branes' U(1) gauge fields is predicted at ε ~ 5 × 10⁻⁴ (from the KK tower mediation: α_EM × π²/6 × exp(−π)), testable by LDMX and LHCb Run 3. A conjectured Z₃ orbifold structure gives three SM generations, with conjectured exponential mass hierarchies from a warp factor k ≈ 2. A speculative identification of the Planck-scale electromagnetic coupling with the inverse configuration torus area, 1/α(M_P) = 4π², combined with three generations of SM running, gives 1/α(0) ≈ 137. The e-circle, Z₆-orbifolded and warped, provides a geometric account of twenty-two phenomena at three levels of rigor. Eight are derived from the basic e-circle (M⁴ × S¹): quantum mechanics, electromagnetism, gravity, the spin-statistics theorem, perturbative UV finiteness, the hydrogen spectrum, black hole entropy, and the CPT theorem. Eight more follow from the Z₂ orbifold extension: dark energy (Casimir energy), dark matter (hidden-brane matter), neutrino masses (bulk seesaw), kinetic mixing with a dark photon (ε ~ 5 × 10⁻⁴), the strong CP problem (topological resolution: π₄(SU(3)) = 0 in 5D), the Hubble tension (H₀ ≈ 68.7–69.5 km/s/Mpc from hidden-brane dark radiation; in 3–4σ tension with ACT DR6 N_eff = 2.86 ± 0.13), the Casimir effect, and baryon asymmetry (derived in Paper 2 via bulk leptogenesis). Six are conjectured: the dark energy equation of state (thawing dilaton, derived quantitatively in Paper 2), the fine structure constant α ≈ 1/137 (from the configuration torus area), α stability, three fermion generations (Z₃ orbifold), the fermion mass hierarchy (warp factor k ≈ 2), and normal neutrino mass ordering (from Z₃ geometry, testable by JUNO). Seven testable predictions follow: two are already consistent with current data (observed α stability, and the dark energy equation of state w ≈ −1 — though DESI DR2 shows 4.2σ tension with static w = −1, resolvable via thawing dilaton quintessence). The naive dilaton prediction (ΔN_eff ≈ 0.57) appeared in tension with combined ACT+SPT+Planck measurements (N_eff = 2.81 ± 0.18), but this tension is resolved by intra-tower KK neutrino decays (Gonzalo et al. 2024), reducing the effective ΔN_eff to ~0.05 and giving N_eff ≈ 3.09, consistent with all CMB data. The remaining four — short-range gravitational deviations at ~12 μm, the dark photon signal at ε ~ 5 × 10⁻⁴, neutrino masses and normal mass ordering (testable by JUNO within 6 years), and mirror dark matter — are within reach of experiments in the next 3–10 years. The orbifold extensions are speculative but quantitative, with specific falsifiable predictions. What is established is a geometric framework in which the same compact circle that makes quantum mechanics necessary also makes perturbative quantum gravity finite. A companion computation (Paper 2) applies the CAMB Boltzmann solver to the framework's cosmological sector (to appear separately) with parameters derived entirely from the e-circle geometry — fitting zero parameters to CMB data. The central result is a scaling law from bulk leptogenesis on the Z₂ orbifold: the three bulk right-handed neutrinos deposit baryon asymmetries on both branes, but the hidden brane is colder (ξ = T_hidden/T_visible). The entropy asymmetry (1/ξ³) and washout suppression (1/ξ²) combine to give: Ω_DM / Ω_b = 1/ξ² Inverting against the observed ratio 5.36 determines ξ = 0.432 at leading order; the washout correction shifts this to ξ ≈ 0.49, converging with the independently θ*-matched value ξ = 0.47. Three scenarios bracket the prediction. The CAMB-computed results: t₀ = 13.47–13.60 Gyr, H₀ = 68.7–69.5 km/s/Mpc, S8 = 0.753–0.785 (resolving the 4σ weak lensing tension without tuning), r_d = 145.8–148.0 Mpc, and the CMB angular scale θ* matched to within 0.5–6.6 arcseconds of Planck's measurement. The thawing dilaton (w₀ = −0.85, w_a = −0.23) drives the expansion history H(z) to peak 4–6% above ΛCDM at z ~ 0.3–0.7 — a specific fingerprint testable by DESI DR3 at 8σ. The cosmic coincidence Ω_DM/Ω_b ≈ 5 is not a coincidence: it is 1/ξ², where ξ is fixed by geometry and baryogenesis. All three scenarios predict N_eff = 3.31–3.39, in 3–4σ tension with ACT DR6 (N_eff = 2.86 ± 0.13) — the framework's primary open issue, definitively resolved by CMB-S4 (σ(N_eff) ≈ 0.03).