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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.