UNIFIED DERIVATION: EMERGENT STRUCTURES FROM A SINGLE GEOMETRIC FIELD ================================================================================ This document establishes, by pure mathematics, the complete chain from a single scalar field on an n-dimensional manifold through dimensional reduction to a verification system on a finite-dimensional state space. All symbols denote pure geometric quantities. No physical interpretation is assigned. ================================================================================ SECTION 0: AXIOMS ================================================================================ Axiom A1. Let N be a connected, paracompact, Hausdorff, smooth manifold of dimension n >= 4, equipped with a smooth pseudo-Riemannian metric gamma of signature (-,+,+,...,+). Coordinates are denoted x^A with A in {0,1,...,n-1}. The metric determinant is gamma := det(gamma_AB). Axiom A2. The Levi-Civita connection nabla on N is uniquely determined by gamma via metric compatibility nabla_A gamma_BC = 0 and torsion freedom nabla_[A nabla_B] f = 0 for all smooth scalar functions f on N. The Christoffel symbols are Gamma^A_{BC} = (1/2) gamma^{AD} (partial_B gamma_{DC} + partial_C gamma_{DB} - partial_D gamma_{BC}). (E1) Axiom A3. There exists a smooth scalar field Phi : N -> R that is the sole fundamental object. No additional independent tensor fields are postulated. Axiom A4. The differential dPhi is non-vanishing on an open dense subset of N, ensuring that the level sets of Phi are regular embedded submanifolds of codimension 1. Axiom A5. Where applicable, gamma and Phi satisfy boundary conditions such that all integrals below are finite and surface terms from integration by parts vanish. Axiom A6. The geometric configuration (gamma, Phi) is determined by the variational principle delta S = 0 for arbitrary compactly supported variations delta gamma^{AB} and delta Phi. ================================================================================ SECTION 1: THE FIELD EQUATIONS ON N ================================================================================ Definition 1. The most general diffeomorphism-invariant functional of gamma_AB and Phi, involving no more than two derivatives, takes the form S[gamma, Phi] = integral_N d^n x sqrt{|gamma|} L, (E2) L = Z(Phi) R + G(Phi) gamma^{AB} (nabla_A Phi)(nabla_B Phi) + H(Phi) + W(Phi) box_gamma Phi + P(Phi) gamma^{AB} gamma^{CD} (nabla_A nabla_B Phi)(nabla_C nabla_D Phi) + Q(Phi) R^{AB} (nabla_A Phi)(nabla_B Phi) + T(Phi) R gamma^{AB} (nabla_A Phi)(nabla_B Phi) + U(Phi) (nabla_A Phi)(nabla_B Phi)(nabla^A Phi)(nabla^B Phi). (E3) Here R is the Ricci scalar of gamma, R^{AB} the Ricci tensor, box_gamma := gamma^{AB} nabla_A nabla_B, and Z, G, H, W, P, Q, T, U are smooth functions Phi -> R. The variational principle is delta S = 0. (E4) Lemma 1. Under delta gamma^{AB}, delta sqrt{|gamma|} = -(1/2) sqrt{|gamma|} gamma_{AB} delta gamma^{AB}, (E5) delta R = R_{AB} delta gamma^{AB} + nabla_A v^A, (E6) where v^A = gamma^{AB} (delta Gamma^C_{BC} - delta Gamma^C_{CB}). Using (E5), (E6), and discarding the divergence nabla_A(Z v^A) as a surface term (A5), the variation of (E2) with respect to gamma^{AB} yields a symmetric tensor E_{AB} defined by E_{AB} := Z(Phi) G_{AB} + T_{AB}[Phi, nabla Phi, nabla^2 Phi; gamma], (E7) where G_{AB} := R_{AB} - (1/2) gamma_{AB} R is the Einstein tensor of gamma, and T_{AB} collects all terms arising from the non-curvature sectors of L: T_{AB} = (1/2) gamma_{AB} L_{non-R} - G(Phi)(nabla_A Phi)(nabla_B Phi) - W(Phi)(nabla_A nabla_B Phi) + coupling terms from P,Q,T,U sectors. (E8) The vanishing of delta S / delta gamma^{AB} gives E_{AB} = 0. (E9) Varying (E2) with respect to delta Phi and integrating by parts (A5) gives the scalar equation D_Phi[gamma; Phi] = 0, (E10) where D_Phi denotes the differential operator obtained by collecting all terms from delta L / delta Phi. Equations (E9) and (E10) constitute the coupled system on N. Definition 2 (Fundamental n-Space Operator). The self-adjoint differential operator O_n acting on scalar densities on N is O_n := -(1/sqrt{|gamma|}) partial_A ( sqrt{|gamma|} F^{AB}(Phi,nabla Phi) partial_B ) + V(Phi, R, R_{AB}), (E11) where F^{AB}(Phi, nabla Phi) := G(Phi) gamma^{AB} + P(Phi) nabla^A nabla^B Phi + Q(Phi) R^{AB} + T(Phi) R gamma^{AB} + U(Phi) (nabla^A Phi)(nabla^B Phi), (E12) and V(Phi, R, R_{AB}) collects all non-derivative potential terms from the variation of the action. In compact form, O_n acts on a test scalar psi as O_n psi = - nabla_A ( F^{AB} nabla_B psi ) + V psi. (E13) Lemma 2. The field equation (E10) is equivalent to O_n Phi = 0, (E14) provided the higher-derivative terms P, Q, T, U are set to zero or absorbed into F^{AB}. In the general case, (E10) is a quasilinear fourth-order equation that extends (E14). Definition 3 (Spectral Decomposition). On a suitable complete slice of N, O_n admits a spectral decomposition with complete orthonormal eigenfunctions {phi_m} satisfying O_n phi_m = lambda_m phi_m, (E15) with respect to the L^2 inner product on (N, gamma). The associated heat kernel trace and zeta function are K(t) := Tr e^{-t O_n} = Sigma_m e^{-t lambda_m}, (E16) zeta_{O_n}(s) := Tr O_n^{-s} = Sigma_{lambda_m != 0} lambda_m^{-s}. (E17) The coefficients of the small-t expansion K(t) ~ Sigma_{j=0}^infty a_j(O_n) t^{(j-n)/2} are locally computable curvature invariants that depend polynomially on R_{ABCD}, nabla_A Phi, nabla_A nabla_B Phi, and gamma. ================================================================================ SECTION 2: EMERGENT SUBMANIFOLD VIA LEVEL SETS ================================================================================ Definition 4 (Level-Set Submanifold). Let c in R be a regular value of Phi (guaranteed on a dense set by A4). The codimension-1 submanifold is M_c := { p in N : Phi(p) = c }. (E18) By the regular value theorem, M_c is a smooth, closed, embedded (n-1)-dimensional submanifold of N. We denote its intrinsic coordinates by y^mu with mu in {0,1,...,n-2}. Definition 5 (Induced Metric). The inclusion map iota : M_c -> N induces the metric g_{mu nu}(y) := gamma_{AB}(iota(y)) e^A_mu(y) e^B_nu(y), (E19) where e^A_mu := partial x^A / partial y^mu are the n-1 tangent frame fields. Definition 6 (Unit Normal). The 1-form n_A := (nabla_A Phi) / |nabla Phi| with |nabla Phi| := sqrt{gamma^{BC} (nabla_B Phi)(nabla_C Phi)} is orthogonal to M_c by construction: n_A e^A_mu = 0. The normalization gamma^{AB} n_A n_B = +/- 1 fixes n as the unit conormal. Definition 7 (Extrinsic Curvature). The extrinsic curvature of M_c in N is the symmetric tensor K_{mu nu} := - gamma_{AB} e^A_mu nabla_A n_B e^B_nu = - e^A_mu e^B_nu nabla_A n_B. (E20) Equivalently, K_{mu nu} = -(1/2) L_n gamma_{mu nu}, (E21) where L_n denotes the Lie derivative along the normal. The mean curvature is K := g^{mu nu} K_{mu nu}. (E22) Theorem 1 (Gauss Equation). Let R^N_{ABCD} and R^M_{mu nu rho sigma} denote the Riemann tensors of (N, gamma) and (M_c, g) respectively. Then R^M_{mu nu rho sigma} = R^N_{ABCD} e^A_mu e^B_nu e^C_rho e^D_sigma + K_{mu rho} K_{nu sigma} - K_{mu sigma} K_{nu rho}. (E23) Theorem 2 (Codazzi Equation). With nabla-bar the Levi-Civita connection of g, nabla-bar_mu K_{nu rho} - nabla-bar_nu K_{mu rho} = R^N_{ABCD} n^A e^B_mu e^C_nu e^D_rho. (E24) Theorem 3 (Contracted Gauss Equation). R^N = R^M + K^2 - K^{mu nu} K_{mu nu} - 2 R^N_{AB} n^A n^B, (E25) and equivalently with the Einstein tensor G^N_{AB} = R^N_{AB} - (1/2) gamma_{AB} R^N, R^N_{AB} n^A n^B = -(1/2) G^N_{AB} n^A n^B - (1/2)(K^2 - K_{mu nu} K^{mu nu}). (E26) Lemma 3 (Nested Reduction). If n > 5, define a sequence of nested submanifolds M^{(n)} := N, M^{(k-1)} := { Phi_{k-1} = c_{k-1} } subset M^{(k)} for k = n, n-1, ..., 5. (E27) The final 4-dimensional submanifold is M := M^{(4)} with coordinates x^mu, mu in {0,1,2,3}, and induced metric g_{mu nu}. The extrinsic curvature of each step is denoted K^{(k)}_{mu nu} for the embedding M^{(k)} in M^{(k+1)}. The full n-dimensional curvature decomposes as R^N = R^M + Sigma_{k=4}^{n-1} [ (K^{(k)})^2 - K^{(k)}_{mu nu} K^{(k) mu nu} ] + cross terms from the nested normal frames. (E28) Lemma 4 (Projected Consistency). The projection of (E9) onto the tangent and normal directions of each intermediate submanifold yields: n^A n^B E_{AB} = 0 (Hamiltonian constraint), (E29) n^A e^B_mu E_{AB} = 0 (momentum constraint), (E30) e^A_mu e^B_nu E_{AB} = 0 (dynamical equations). (E31) ================================================================================ SECTION 3: DIMENSIONAL REDUCTION AND THE FOUR STRUCTURES ================================================================================ Definition 8 (Fibration). The submanifold M is the base of a fibration pi : N -> M with compact fiber F = pi^{-1}(x), where dim(F) = d = n - 4. The exact sequence of tangent bundles is 0 -> VF -> TN -> pi^* TM -> 0, (E32) where VF = ker(d pi) is the vertical subbundle. An Ehresmann connection HN complementary to VF gives TN = HN + VF. In adapted coordinates (x^mu, y^a) with mu in {0,1,2,3} and a in {1,...,d}, the metric decomposes as gamma_{AB} = [ g_{mu nu}(x) + h_{ab}(x,y) A^a_mu A^b_nu h_{bc} A^b_mu ] [ h_{ac} A^c_nu h_{ab}(x,y) ], (E33) where g_{mu nu} is the metric on M, h_{ab} the metric on F, and A^a_mu are connection 1-forms. Equivalently, with theta^a = dy^a + A^a_mu dx^mu, gamma = g_{mu nu} dx^mu tensor dx^nu + h_{ab} theta^a tensor theta^b. (E34) Definition 9 (Fiber Laplacian). Let Delta_F = d_F d_F^* + d_F^* d_F denote the Laplace-de Rham operator on F, with {Upsilon_alpha^{(p)}} a complete orthonormal basis of eigen-p-forms: Delta_F Upsilon_alpha^{(p)} = lambda_alpha^{(p)} Upsilon_alpha^{(p)}, lambda_alpha^{(p)} >= 0. (E35) The orthonormality condition is integral_F Upsilon_alpha^{(p)} wedge star_F Upsilon_beta^{(p)} = delta_{alpha beta}. (E36) Axiom A7 (Fiber Spectral Structure). The compact fiber F is geometrically distinguished such that the zero eigenspaces of Delta_F in degrees p = 0, 1, 2 satisfy dim ker Delta_F^{(0)} = 1, dim ker Delta_F^{(1)} = 1, dim ker Delta_F^{(2)} = 2. (E37) This yields exactly four lowest modes: Psi^{(1)} = Y_0 (constant scalar, degree 0), Psi^{(2)} = omega (harmonic 1-form, degree 1), Psi^{(3)} = eta_1 (harmonic 2-form, degree 2, first), Psi^{(4)} = eta_2 (harmonic 2-form, degree 2, second). (E38) Lemma 5 (Universality of Four Modes). For any n >= 6 (equivalently d >= 2), the fiber dimension d grows but the lowest harmonic modes of Delta_F remain exactly four, by Axiom A7. The higher modes correspond to strictly positive eigenvalues and decouple at the level of the zero-eigenspace sector. Definition 10 (Harmonic Decomposition). The fundamental field Phi decomposes as Phi(x,y) = Sigma_{k=1}^4 Phi^{(k)}(x) wedge Psi^{(k)}(y) + (higher modes). (E39) The expansion coefficients Phi^{(k)}(x) are differential forms on M. Via exterior differentiation and Hodge duality on M, all four structures induce 2-tensors. Definition 11 (The Four Emergent Tensors). The four tensor structures on M are: F^{(1)}_{mu nu} := D_mu phi_nu + D_nu phi_mu (symmetric 2-tensor), (E40) F^{(2)}_{mu nu} := partial_mu A_nu - partial_nu A_mu (antisymmetric 2-tensor), (E41) F^{(k)}_{mu nu} := star_M (d B_k)_{mu nu rho} dx^rho = epsilon_{mu nu rho sigma} g^{sigma lambda} D_lambda B_k, k = 3, 4, (E42) where D_mu is the Levi-Civita covariant derivative on (M, g), phi_mu is the 1-form coefficient from the k=1 mode, A_mu from the k=2 mode, B_3 and B_4 are scalar coefficients from the k=3,4 modes, and epsilon_{mu nu rho sigma} is the volume form on M. Alternatively, for k = 3, 4, F^{(k)}_{mu nu} = partial_mu C^{(k)}_nu - partial_nu C^{(k)}_mu, (E43) where C^{(k)} are 1-forms on M emerging from the degree-2 harmonics on F via the holonomy reduction. Lemma 6 (Field Equations on N). The fundamental field Phi satisfies d Phi = 0, d star Phi = 0. (E44) Definition 12 (Emergent Currents). Projecting (E44) onto the k-th harmonic mode by fiber integration against Psi^{(k)} yields the currents J^{(k)}_mu := integral_F Psi^{(k)} wedge star_F (d Phi)_mu, (E45) for k in {1, 2, 3, 4}, where the subscript mu denotes the horizontal component. Explicitly: J^{(1)}_mu = D^nu F^{(1)}_{mu nu} - (1/2) D_mu F^{(1)}, (E46) J^{(2)}_mu = D^nu F^{(2)}_{mu nu}, (E47) J^{(k)}_mu = D^nu F^{(k)}_{mu nu} + O_k(A, B), k = 3, 4, (E48) where F^{(1)} = g^{mu nu} F^{(1)}_{mu nu} is the trace and O_k denotes covariant coupling terms arising from the holonomy structure of F. Theorem 4 (Decoupled Field Equations on M). Integrating the master equations (E44) over the fiber and using orthonormality (E36) yields, for each k in {1, 2, 3, 4}: D^nu F^{(k)}_{mu nu} = J^{(k)}_mu. (E49) Additionally, for k = 3, 4, the holonomy constraint gives epsilon^{mu nu rho sigma} D_nu F^{(k)}_{rho sigma} = H^{(k)}(F^{(2)}, F^{(3)}, F^{(4)}), (E50) where H^{(k)} encodes the topological coupling from the structure constants of the fiber holonomy algebra. Definition 13 (Coupling Constants). The kinetic term on N for Phi is S_kin = integral_N Phi wedge star Phi. (E51) Substituting the harmonic expansion (E39) and using orthonormality (E36), the cross terms vanish and the fiber integrals give normalization factors N_k := integral_F Psi^{(k)} wedge star_F Psi^{(k)}. (E52) The coupling constants are defined by g_k^{-2} := N_k = Vol(F) * lambda_k^{(d-2)/2}, d = n - 4, (E53) where lambda_k is the eigenvalue corresponding to mode k, and for k = 1 the convention lambda_1 = 1 applies in the exponent (the constant mode has eigenvalue 0 and the normalization is purely volumetric). Lemma 7 (Fiber Integration of Quadratic Form). The master quadratic form decomposes as integral_F Phi wedge star Phi = Sigma_{k=1}^4 (integral_F Psi^{(k)} wedge star_F Psi^{(k)}) Phi^{(k)} wedge star_M Phi^{(k)}. (E54) Identifying the emergent field strengths F^{(k)}_{mu nu} with the components of Phi^{(k)}, integral_{N/M} F wedge star F = Sigma_{k=1}^4 g_k^{-2} F^{(k)}_{mu nu} F^{(k) mu nu}, (E55) with indices raised by g^{mu nu} on M. ================================================================================ SECTION 4: THE LOOKUP TABLE VERIFICATION SYSTEM ================================================================================ Definition 14 (State Space). The configuration manifold is Sigma = R^{36}, coordinatized by the state vector q = (r_1, ..., r_6, p_1, ..., p_6) in Sigma, (E56) where for each i in {1, ..., 6}: r_i = (r_i^1, r_i^2, r_i^3) in R^3, p_i = (p_i^1, p_i^2, p_i^3) in R^3. (E57) The symplectic form on Sigma is omega = Sigma_{i=1}^6 dp_i^a wedge dr_i^a, (E58) with standard symplectic matrix J in R^{36 x 36} satisfying J^2 = -I. Definition 15 (Separation Vector). For any pair (i, j) with i < j, r_{ij} := r_i - r_j in R^3, (E59) with Euclidean norm |r_{ij}| := sqrt{delta_{ab} r_{ij}^a r_{ij}^b}. Definition 16 (Hamiltonian). The Hamiltonian H : Sigma -> R is H(q) = T(p) + U^{(2)}(r) + U^{(3)}(r) + U^{(>=4)}(r, p), (E60) where T(p) = Sigma_{i=1}^6 (p_i . p_i) / (2 m_i), (E61) U^{(2)}(r) = - Sigma_{1 <= i < j <= 6} G m_i m_j / |r_{ij}|, (E62) U^{(3)}(r) = Sigma_{1 <= i < j < k <= 6} Q_{ijk} / (|r_{ij}|^2 |r_{jk}|^2), (E63) U^{(>=4)}(r, p) = Sigma_{l=4}^6 U_l(r, p). (E64) Here m_i > 0 are scalar parameters, G > 0 is a coupling constant, and Q_{ijk} are three-body coupling coefficients. Each U_l collects all l-point interactions and p-dependent contributions from the compact fiber. Definition 17 (Hamilton's Equations). The dynamics on Sigma are governed by dr_i/dt = partial H / partial p_i, dp_i/dt = -partial H / partial r_i, for each i. (E65) Definition 18 (Flow Map). The time-t flow map of the Hamiltonian vector field X_H = (dr/dt, dp/dt) is Phi_H^t : Sigma -> Sigma, Phi_H^t(q_0) = q(t), (E66) where q(t) is the unique solution to (E65) with initial condition q(0) = q_0. By construction, Phi_H^t preserves the symplectic form: (Phi_H^t)^* omega = omega. Definition 19 (Reference Trajectory). Let q_ref : [0, T] -> Sigma denote reference trajectory data, where for each t in [0, T], q_ref(t) = (r_1^{ref}(t), ..., r_6^{ref}(t), p_1^{ref}(t), ..., p_6^{ref}(t)). (E67) Definition 20 (Error Functional). The error functional E[Phi_H] is E[Phi_H] = || Phi_H^t(q_0) - q_ref(t) ||_{L^2[0,T]} (E68) = [ integral_0^T || Phi_H^t(q_0) - q_ref(t) ||^2_Sigma dt ]^{1/2}, (E69) where the norm on Sigma is ||q||^2_Sigma = Sigma_{i=1}^6 ( m_i delta_{ab} r_i^a r_i^b + delta_{ab} p_i^a p_i^b / m_i ). (E70) Definition 21 (Stationarity Constraints). The flow map Phi_H minimizes the error functional if and only if the following stationarity conditions hold: delta E / delta H = 0. (E71) Explicitly, let q(t) = Phi_H^t(q_0) and define the residual eta(t) := q(t) - q_ref(t) in Sigma. (E72) Then (E71) is equivalent to integral_0^T < eta(t), delta X_H(q(t)) >_Sigma dt = 0 (E73) for all admissible variations delta X_H, where <.,.>_Sigma is the inner product inducing (E70). The coupling parameters (m_i, G, Q_{ijk}, and higher-order coefficients) are constrained by partial E[Phi_H] / partial G = 0, (E74) partial E[Phi_H] / partial Q_{ijk} = 0, for all (i,j,k), (E75) partial E[Phi_H] / partial m_i = 0, for all i. (E76) Together, (E74)-(E76) yield a closed nonlinear system for the parameters. Definition 22 (Verification). Let epsilon > 0 be a specified bound. The emergent geometric field verifies the lookup table Phi_H to accuracy epsilon if and only if E[Phi_H] < epsilon. (E77) Equivalently, sup_{t in [0,T]} ||Phi_H^t(q_0) - q_ref(t)||_Sigma < epsilon / sqrt{T}. (E78) A sequence of Hamiltonians {H_n} converges to the reference lookup table if lim_{n -> infinity} E[Phi_{H_n}] = 0. (E79) ================================================================================ SECTION 5: SELF-CONSISTENCY AND CLOSURE ================================================================================ Lemma 8 (Contributions from the Four Structures). The four emergent structures indexed by k in {1, 2, 3, 4} contribute to the Hamiltonian as follows. H_{k=1}: modifies the pairwise potential via a geometric correction factor: U_{k=1}^{(2)}(r) = - Sigma_{i=4)}(r) = Sigma_{i=4)}(r) + U_{fiber}(r, p), (E85) where U_{fiber}(r, p) collects residual fiber-curvature effects. Definition 23 (Source Field). The scalar source field on R^3 induced by the state configurations is rho(r, t) = Sigma_{i=1}^6 m_i delta^3(r - r_i(t)). (E86) Definition 24 (Effective Geometric Potential). The effective geometric potential Phi_eff : R^3 x [0,T] -> R satisfies nabla^2 Phi_eff = 4 pi G rho + (1/s^2) partial_t^2 Phi_eff + Lambda_eff, (E87) where nabla^2 = delta^{ab} partial_a partial_b is the spatial Laplacian on R^3, partial_t^2 = partial^2 / partial t^2, and Lambda_eff is the effective curvature term from the fiber. Lemma 9 (Decomposition of Lambda_eff). The effective curvature term decomposes over the four emergent structures as Lambda_eff = Lambda_1 + Lambda_2 + Lambda_3 + Lambda_4, (E88) where Lambda_1(r, t) = -(1/2) Sigma_{i=4} (-1)^n lambda_n R^{(n)}(r; {r_i}_{i=1}^6), (E93) where R^{(n)} denotes the n-th order Riemann curvature invariant of the emergent submanifold M evaluated at the configuration points, and lambda_n are normalization constants from dimensional reduction. Theorem 5 (Self-Consistency). Let Phi_eff be the solution to (E87) with the decomposition (E88)-(E93). Let H_{full} be the Hamiltonian (E85). Then E[Phi_{H_{full}}] < epsilon (E94) if and only if the following coupled system has a solution: (i) Hamilton's equations for H_{full} yield q(t) = Phi_{H_{full}}^t(q_0), (ii) Phi_eff satisfies (E87) with source rho from (E86), (iii) The coupling parameters satisfy (E74)-(E76), (iv) The verification bound (E77) holds. Furthermore, the emergent field Phi_eff is the unique solution to the constrained variational problem on the lookup table: among all fields satisfying the stationarity constraints (E71) and the coupling equations (E74)-(E76), the field that achieves E[Phi_H] < epsilon is unique by the strict convexity of the error functional in the neighborhood of the minimum. ================================================================================ SECTION 6: CORE EQUATION SYSTEM ================================================================================ The complete mathematical system comprises six core equations: +--------------------------------------------------------------------------------+ | (C1) E_{AB} = Z(Phi) G_{AB} + T_{AB} = 0 on N | | | | (C2) O_n Phi = 0 on N | | | | (C3) D^nu F^{(k)}_{mu nu} = J^{(k)}_mu on M, k=1,2,3,4 | | | | (C4) g_k^{-2} = Vol(F) * lambda_k^{(d-2)/2} d = n - 4 | | | | (C5) nabla^2 Phi_eff = 4 pi G rho + s^{-2} partial_t^2 Phi_eff + Lambda_eff | | | | (C6) E[Phi_{H_{full}}] < epsilon | +--------------------------------------------------------------------------------+ Equation (C1) is the geometric tensor equation from the n-dimensional action variation. Equation (C2) is the fundamental operator eigenvalue equation on N. Equation (C3) gives the four decoupled field equations on M, one for each of the emergent tensor structures arising from the harmonic analysis on the fiber. Equation (C4) defines the four coupling constants from the fiber geometry. Equation (C5) is the effective geometric potential equation with fiber curvature source. Equation (C6) is the verification bound closing the system. The dimension n enters the system only through d = n - 4 in (C4). For any n >= 6, the same four structures emerge because the lowest harmonic modes of the fiber Laplacian remain four by Axiom A7: one scalar mode, one 1-form mode, and two 2-form modes. The fiber volume Vol(F) and the eigenvalues lambda_k depend on n, but the number of emergent structures is invariant. ================================================================================ END OF DERIVATION ================================================================================