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

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Authors: Yunbeom Yi
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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