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[[PAGE 1]] Mathematical Equations and Formal Definitions Companion equation sheet for Declaration and Persistent State Change in Artificial Systems Prepared for legible review in Editorial Manager. Equations reproduce the manuscript definitions and calculations. 2.1 Bookkeeping state St = ( t, mt, ct, ot) 2.2 Reference discrepancy t = d(zt, rt), 0 t = d(zt, ztD) 2.3 Declaration detector Dt = 1{ (O t) = 1}, tD = inf {t : Dt = 1} 2.4 Independent persistent onset Pt( , W) = 1{ min 0 j<W t+j > } t ( , W) = inf {t TW : Pt( , W) = 1} 2.5 Temporal ordering = t tD 2.6 Coupled recurrent process Lt + 1 = F(Lt, ut, xt), xt + 1 = G(xt, Lt) 4.1 Effective LoRA update Ut = s btaT t , s = 64 4.2 Equivalent factorization ( tbt)(at/ t)T = btaT t 4.3 Matrix inner product and norm Uu, Uv F = s2(bT ubv)(aT uav) Ut F = s bt 2 at 2 4.4 Effective matrix distance Uu Uv 2 F = Uu 2 F + Uv 2 F 2 Uu, Uv F 4.5 Local bend CZ(t; k) = Zt k Zt, Zt + k Zt Zt k Zt Zt + k Zt 4.6 Amplitude eligibility Et(k, ) = 1{max(MA(t; k), MB(t; k)) } 4.7 Stable low rank difference Uj Ui = s[(bj bi)aT j + bi(aj ai)T] 4.8 Centered gradient average g(w) t = 1 w (w 1)/2 j = (w 1)/2 gt + j 4.9 Two segment descriptive fit SSE(k) = min , , , [ k t=1 (gt t)2 + 300 t = k + 1 (gt t)2 ] 6.1 Identification argument p(Y M1) p(Y M0) = 1 A.1 Gradient window decline 100(1 15.343834018707275 28.693284511566162 ) = 46.52465104675545% A.2 Evaluation loss reduction 100(1 1.513361930847168 4.168299674987793 ) = 63.69354295881809% This companion page is provided to preserve equation legibility during PDF conversion. The manuscript remains the primary scientific text.