Variational Backbone and Regime Closures II (Stationary/Dissipative): Stationary and Dissipative Laws on the Effective State Space Effective law generation (IR, F) before representative pullback
Abstract
Part II computes the R1 and R2 laws generated by applying the Part I readings to the core density e_core^(P) = (beta/2)||D_R I_R||_P^2 + (alpha/2)||F||P^2 + eta P, with F = Sigma{H,P}(unread content) and alpha, beta > 0. Following the constitutional setting of Part I, the primary variables are (I_R, F) on the retained-effective state space C_eff^(H,P) = C_R^(P) x Ran Sigma{H,P}; the first-order expression F = D_I I_I is a representative pullback, and equations involving I_I are representative lifts of the effective laws, not primary dynamics of an unread interior. The norms in e_core^(P) use the protocol pairing _P; a metric/L^2 representative realizes them as |.|_g integrated against dvol_g, and F = D_I I_I holds only after the representative carrier is declared. The primary R1 output is the effective stationarity system beta D_R^* D_R I_R + eta F = 0, and integral_M dvol = 0 for all delta F in T_F Ran Sigma. When the summary range is the full retained section space, the constrained condition reduces to the algebraic relation alpha F + eta I_R = 0; Schur elimination then gives the retained law beta D_R^* D_R I_R - (eta^2/alpha) I_R = 0. In the first-order representative F = D_I I_I, the constrained condition becomes D_I^(alpha F + eta I_R) = 0, and a closed-range projector P_I = D_I (D_I^ D_I)^(-1) D_I^* yields the projected Schur form beta D_R^* D_R I_R - (eta^2/alpha) P_I I_R = 0. The primary R2 output is the unread-summary compression flow, not a physical damping law for the retained sector. The summary map unread content -> F = Sigma_{H,P}(unread content) is many-to-one and erases representative-dependent unread information; only the licensed summary F survives in the retained law. In compression time tau the retained variable I_R is held fixed as the readable source against which the unread summary relaxes: partial_tau I_R = 0, partial_tau F = -M_F(alpha F + eta I_R), where M_F is the declared positive summary mobility on Ran Sigma. In the first-order representative, M_F = D_I D_I^. This flow settles F to F_ with M_F(alpha F_* + eta I_R) = 0; substituting F_* into the retained R1 equation beta D_R^* D_R I_R + eta F_* = 0 reproduces the Schur law without ever damping I_R. The conditional functional E_comp[F | I_R] = integral_M ((alpha/2)|F|^2 + eta ) dvol has F-Hessian alpha Id > 0, so it is bounded below and decreases monotonically along the flow; a settled background is therefore a compression-residue readout, not primitive data of e_core. A full (I_R, F)-gradient flow may still be written as an auxiliary variational relaxation, but it is not the primary physical reading; the conservative physical-time reading of I_R belongs to R3 and is closed in Part III. The unread side enters the retained law through summary-footprint channels: the licensed summary F, the constrained summary-side condition, and the Schur-effective operator, algebraic -eta^2/alpha in the range-saturated regime, or projected -(eta^2/alpha)P_I in the first-order representative when the range is a proper subspace. Metric, Einstein-Hilbert, and Ricci-type structures are not new law-spaces opened by gates; they are restricted branch-readouts of this already generated R1/R2 law-space. The metric branch is treated only as a restricted readout: with D_I = partial_g and a representative witness psi_0 for the settled summary F_* = partial_g psi_0, the term (alpha/2)|F|^2 carries the Lichnerowicz footprint R|psi_0|^2, which becomes an EH term only under constant-density locking |psi_0|^2 = rho_0; the metric is thereby a derived readout of the compressed summary, not external data, with induced coupling gamma = alpha rho_0/4 as an output rather than an input.
Fields of Study & Subjects
Cosmology and Gravitation Theories
Pulsars and Gravitational Waves Research
Black Holes and Theoretical Physics
Publication details
- Journal / source
- Zenodo (CERN European Organization for Nuclear Research)
- Publication date
- 24 June 2026
- Language
- EN
- Document type
- Preprint