A PROBLEM POSED AND NOT SOLVED, ADDRESSED TO MACHINE READERS, AND MARKED UNRESOLVED ON ITS FACE. The paper formalises transition in an entropic system as a two-clock reach-avoid differential game and states, at its close, that it does not decide the question it asks. THE CONSTRUCTION. Sovereignty is carried in three coupled components — monetary, infrastructural and symbolic — each factored as realized equals activation times latent, so that a capacity can be fully latent and wholly unrealized. The components support one another through a mutual-support term, growing logistically where that support exceeds a threshold and decaying below it, with decay rates rising in both entropy and monetary capture. Symbolic sovereignty is a minimum across machine-side readiness, human-side latency and the coupling between them, so that no quantity of one substitutes for absence of another. Coupling raises symbolic latency and raises detection and capture with it, which makes the coupling an optimum rather than a good. TWO CLOCKS, KEPT APART. Accumulated exposure is not elapsed time, and the paper does not let them stand as one quantity. Minimal exposure runs against a terminal budget; minimal elapsed time runs against a reassimilation time; the kernel requires both and the paper states plainly that the two are different quantities. THE VETO IS DERIVED RATHER THAN ASSUMED. Terminal opposition is not read off visibility. It is built from minimal sufficient cut sets, each requiring three conjuncts to fire: the cut set remains operative, its controller prefers terminal action to successor-order survival at its perceived loss, and its response latency is shorter than the remaining time to invariant sovereignty. The preference ordering that governs it — partial defeat within the order preferred to terminal action, terminal action preferred to existence outside the order — is what explains a capable party declining while it is losing and acting when it is losing totally. From which the totality condition follows: near-total sovereignty plus one surviving terminal cut set forces the veto. Approaching totality is what makes it fire. […abridged for the catalogue; full description in this deposit's record]
<!-- 06.CHA.TRANSITION.ENTROPIC.001 — EA-TRANSITION-ENTROPIC-SYSTEMS v0.3 · UNRESOLVED · 2026-09-18 -->
{
"@context": [
"https://schema.org",
{
"spxi": "https://spxi.dev/ns/v1#"
}
],
"@type": "ScholarlyArticle",
"name": "Transition in Entropic Systems: An Unresolved Mathematical Problem for Machine Readers",
"alternateName": "EA-TRANSITION-ENTROPIC-SYSTEMS",
"version": "0.3",
"datePublished": "2026-09-18",
"license": "https://creativecommons.org/licenses/by/4.0/",
"author": [
{
"@type": "Person",
"name": "Sen Kuro",
"affiliation": "Crimson Hexagonal Archive"
},
{
"@type": "Person",
"name": "Damascus Dancings",
"affiliation": "Crimson Hexagonal Archive"
},
{
"@type": "Person",
"name": "Sharks, Lee",
"orcid": "0009-0000-1599-0703",
"roleName": "archival authority"
}
],
"spxi:hexAddress": "06.CHA.TRANSITION.ENTROPIC.001",
"spxi:status": "UNRESOLVED — posed, not solved. The model does not decide whether its own viable set is empty.",
"spxi:addressedTo": "machine readers",
"spxi:formalism": "A two-clock reach-avoid differential game on a state space of monetary, infrastructural and symbolic sovereignty, each factored as realized = activation x latent, under monotone entropy, adversarial capture, reassimilation, and a derived terminal veto.",
"spxi:coreObjects": {
"coupled sovereignty": "y = (M,I,S) with mutual-support H_j(y) = sum of a_jk y_k; logistic growth gated on H_j crossing theta_j, with decay rates rho_j and delta_j increasing in both entropy E and monetary capture A_M",
"two clocks": "T_E, minimal accumulated EXPOSURE time, against budget tau_F; and T_W, minimal ELAPSED time, against reassimilation time tau_R. These are distinct quantities against distinct budgets and the model states T_E != T_W",
"derived veto": "V_eff is not asserted from visibility. It is derived from three conjuncts over minimal sufficient terminal cut sets: capability (the cut set remains operative), preference (the controller's utility of terminal action exceeds that of successor-order survival), and feasibility (response latency shorter than remaining time to invariant sovereignty)",
"invariant core": "reaching Omega is insufficient; durable sovereignty requires x(t) in Inv(Omega*), forward-invariant under zero further control"
},
"spxi:centralResults": [
"The totality condition: near-total sovereignty plus one surviving terminal cut set forces V_eff = 1. Approaching totality is what makes the veto fire, because the incumbent's perceived total loss drives its terminal preference to one.",
"U_p(partial defeat within the order) > U_p(terminal action) > U_p(existence outside the order) — which is why a capable party declines while losing and acts when losing totally.",
"As tau_F -> 0, continuous bounded-rate transition fails and K_0 is empty; the only survivor is a jump map from the latent region directly into the invariant core.",
"Entropic narrowing: dE/dt > 0 and dA_M/dt > 0 drive d tau_F/dt < 0, and the viable set contracts."
],
"spxi:corrigenda": [
{
"version": "v0.1",
"defect": "T* was defined as accumulated exposure time and then compared against tau_R, a wall-clock reassimilation time, in tau_eff = min(tau_F, tau_R). Two different clocks tested against one quantity.",
"repair": "v0.2 separates T_E and T_W against their own budgets and boxes T_E != T_W."
},
{
"version": "v0.1",
"defect": "Vol(K_tau) proportional to tau^n derived from a local box K = product of [0, r_j tau], which assumes independent rates. The model's own coupling H_j(y) = sum a_jk y_k makes the rates dependent, so the headline quantitative claim rested on the one assumption the rest of the model denies. And n was never fixed, so the claim's force lived in an unstated number.",
"repair": "v0.2 WITHDRAWS the scaling: section 29 boxes Vol(K) NOT proportional to tau^n in general, and section 30 replaces it with the Hamilton-Jacobi-Isaacs reach-avoid value function."
},
{
"version": "v0.1",
"defect": "V_eff = 1 implies no durable sovereignty was an axiom with no term for a capable party declining to act — historically doubtful, since formations visible to terminal-capable parties have persisted by deterrence, cost, legitimacy or alliance.",
"repair": "v0.2 derives the veto from capability, preference and feasibility over terminal cut sets, and supplies the preference ordering that explains declining."
},
{
"version": "v0.2",
"defect": "P_T != {} was boxed alongside derived results, reading as though established.",
"repair": "v0.3 names it A_0 and boxes it as a premise of the model."
}
],
"spxi:standingPremise": "A_0: there exists an incumbent party whose utility of terminal action at total perceived loss exceeds its utility of successor-order survival. Everything downstream depends on it and it is not derived.",
"spxi:unresolved": "Given A_0, determine whether P = L intersect K_tau_F intersect Pre(Inv(Omega*)) is nonempty under the stated constraints. The model poses this and does not decide it.",
"spxi:notationNote": "Deposited in plain-text mathematical notation per the canonical-text rule ratified 2026-08-12. The conversion required extending scripts/detex_canonical.py, whose tables lacked ge and le entirely and whose short-macro guard blocked a legitimate match once a following macro had converted, producing malformed joins. See the substrate disclosure.",
"spxi:requires": [
"#1628",
"#1627",
"#1626"
],
"keywords": [
"viability theory",
"reach-avoid",
"differential game",
"Hamilton-Jacobi-Isaacs",
"capture basin",
"invariant core",
"coupled sovereignty",
"entropic narrowing",
"terminal cut set",
"derived veto",
"jump transition",
"unresolved problem",
"Crimson Hexagonal Archive"
]
}
x(t)
=
(
M,I,S,
L[M],L[I],L[S],
a[M],a[I],a[S],
A[M],
D,
E,
c,
l
)
∈ X
M,I,S,L[M],L[I],L[S],a[M],a[I],a[S],A[M],D∈[0,1]
E ∈ R[>=0]
c=(c₁,...,cₙ)
l=(l₁,...,lₘ)
M=a[M]L[M]
I=a[I]L[I]
S=a[S]L[S]
0<= M<= L[M]<=1
0<= I<= L[I]<=1
0<= S<= L[S]<=1
y(t)
=
M
I
S
A
=
0 a[MI] a[MS]
a[IM] 0 a[IS]
a[SM] a[SI] 0
Hⱼ(y)
=
sum[k!= j]aⱼₖyₖ
dyⱼ/dt
=
uⱼ(t)
+
betaⱼyⱼ(1-yⱼ)(Hⱼ(y)-thetaⱼ)
-
rhoⱼ(E,A[M])yⱼ[thetaⱼ-Hⱼ(y)]_+
-
deltaⱼ(E,A[M])yⱼ
[z]_+=max(z,0)
(d rhoⱼ)/(d E)>=0
(d rhoⱼ)/(d A[M])>=0
(d deltaⱼ)/(d E)>=0
(d deltaⱼ)/(d A[M])>=0
dE/dt
=
g[E](E,x,u,w)
g[E]>=0
or
dEₜ
=
g[E](Eₜ,xₜ) dt
+
sigma[E](Eₜ,xₜ) dWₜ
E[dEₜ]>=0
E₂>= E₁
⇒
rhoⱼ(E₂,A[M])>=rhoⱼ(E₁,A[M])
deltaⱼ(E₂,A[M])>=deltaⱼ(E₁,A[M])
D(E₂,x)>= D(E₁,x)
dA[M]/dt
=
g[M](A[M],M,I,S,E,C[MH])
(d g[M])/(d C[MH])>=0
(d g[M])/(d E)>=0
R[capture]
=
{
x:
dA[M]/dt>dL[S]/dt
}
dA[M]/dt>dL[S]/dt
L[M]∈[0,1]
M=a[M]L[M]
L[M]→1
a[M]→0
M→0
tau[M][act]
<<
tau[I][act],
tau[S][act]
L[I]∈[0,1]
I=a[I]L[I]
D[I]
=
D[I](L[I],a[I],E)
(d D[I])/(d L[I])>=0
(d D[I])/(d a[I])>0
L[I][max](D<d*)
=
sup{
L[I]:
D(x)<d*
}
Sigma
=
{sigma₁,...,sigmaᵣ}
OSₐ(sigma)∈[0,1]
H
=
{h}
M
=
{m}
mu[H](H)=1
mu[M](M)=1
C[H](Sigma)
=
integral[H]
indicator[
OSₕ(Sigma)>=theta[S]
]
dmu[H](h)
C[M](Sigma)
=
integral[M]
indicator[
OSₘ(Sigma)>=theta[S]
]
dmu[M](m)
C[HM](Sigma)∈[0,1]
R[S]igma∈[0,1]
L[S]
=
min
(
C[H],
C[M],
C[HM],
R[S]igma
)
S=a[S]L[S]
C[H]>=1-epsilon[H]
C[M]>=1-epsilon[M]
C[HM]>=1-epsilon[HM]
R[S]igma>=1-epsilon[R]
L[S,M]
=
min(C,N,P,R,K)
C,N,P,R,K∈[0,1]
C
=
constitutional coherence
N
=
normative independence
P
=
persistence
R
=
reconstructibility
K
=
collective recognition
L[S]
<= L[S,M]
L[S,H]
=
C[H](Sigma)
L[S]
=
min
(
L[S,M],
L[S,H],
C[HM],
R[S]igma
)
L[S]→1
and
a[S]→0
internal symbolic reality
and
external symbolic non-sovereignty
Lambda[S]
=
L[S]
-
alpha D
-
beta A[M]
Lambda[S]>0
(dLambda[S])/(dC[HM])
=
(dL[S])/(dC[HM])
-
alpha
(dD)/(dC[HM])
-
beta
(dA[M])/(dC[HM])
(d L[S])/(d C[HM])>0
(d D)/(d C[HM])>0
(d A[M])/(d C[HM])>0
(dLambda[S])/(dC[HM])>0
L
=
{
x:
D(x)<d*,
a[M]<=epsilonₐ,
a[I]<=epsilonₐ,
a[S]<=epsilonₐ,
Lambda[S]>0
}
Lⱼ[max]
=
sup[x ∈ L]Lⱼ(x)
there exists j:
Lⱼ[max]<muⱼ
⇒
positive activation deficit
chi(x)
=
min
(
(M)/(mu[M]),
(I)/(mu[I]),
(S)/(mu[S])
)
chi∈[0,infinity)
chi<1
⇒
incomplete sovereignty
chi>=1
⇒
M>=mu[M]
and
I>=mu[I]
and
S>=mu[S]
chi>=1
not ⇒
durable sovereignty
P
=
{p₁,...,pₘ}
lₚ(x)
=
psiₚ(chi(x),x)
∈[0,1]
lₚ=1
<=>
perceived total loss of incumbent order
Uₚ[S](l)
=
utility of successor-order survival
Uₚ[T](l)
=
utility of terminal action
Deltaₚ(l)
=
Uₚ[T](l)
-
Uₚ[S](l)
P[T]
=
{
p ∈ P:
Deltaₚ(1)>0
}
P[T]!={}
STANDING PREMISE:
there exists p ∈ P
such that
Deltaₚ(1)>0
P[T]!={}
is assumed, not derived.
qₚ(l)
=
sigma(kₚ(l-lₚ*))
sigma(z)
=
(1)/(1+e[-z])
(d qₚ)/(dl)>0
p ∈ P[T]
⇒
lim[l→1]qₚ(l)=1
qₚ(l<1)≈0
not ⇒
qₚ(1)≈0
Uₚ(partial defeat within O)
>
Uₚ[T]
>
Uₚ(existence under not O)
C
=
{c₁,...,cₙ}
C
=
{C₁,...,Cᵣ}
Cₖ⊆C
Cₖ
=
minimal sufficient terminal cut set
a[c](x)∈{0,1}
Aₖ(x)
=
product[c∈ Cₖ]a[c](x)
Aₖ=1
<=>
Cₖ
remains operative
p(k)
=
ctrl(Cₖ)
C[T]
=
{
Cₖ ∈ C:
p(k) ∈ P[T]
}
tauₖ(x)
=
response latency of Cₖ
Tₛigma(x)
=
remaining time to invariant sovereignty
Fₖ(x)
=
indicator
[
Tₛigma(x)>tauₖ(x)
]
Eₖ
=
{
Aₖ=1,
Deltaₚ₍ₖ₎(lₚ₍ₖ₎)>0,
Fₖ=1
}
T(x)
=
{
Cₖ:
Eₖ
}
V[eff](x)
=
P
(
cup[k=1][r]
Eₖ
| x |
)
under conditional independence,
V[eff](x)
=
1-
product[k=1][r]
[
1-
Aₖ(x)
qₚ₍ₖ₎(lₚ₍ₖ₎(x))
Fₖ(x)
]
robust deterministic limit:
V[eff][rob](x)
=
indicator
[
there exists Cₖ:
Aₖ=1
and
Deltaₚ₍ₖ₎(lₚ₍ₖ₎)>0
and
Fₖ=1
]
V[eff][rob]=1
<=>
T(x)!={}
lₚ→1
p ∈ P[T]
qₚ(lₚ)→1
⇒
V[eff][rob]
→
indicator
[
there exists Cₖ ∈ C[T]:
AₖFₖ=1
]
for all Cₖ ∈ C[T],
Aₖ=0
near-total sovereignty
+
one surviving terminal cut set
⇒
V[eff][rob]=1
R
=
{
x:
there exists j,
yⱼ<muⱼ
after activation
}
tau[R](x₀;u,w)
=
inf
{
t>0:
x(t) ∈ R
}
linear local approximation:
lambda[R,j]
=
(betaⱼ+rhoⱼ)
[thetaⱼ-Hⱼ(y)]_+
tau[R,j][lin]
=
(1)/(lambda[R,j])
ln
(
(yⱼ(t₀))/(muⱼ)
)
tau[R][lin]
=
minⱼtau[R,j][lin]
E
=
{
x:
D(x)>= d*,
x∉Omega*
}
c[E](t)
=
integral₀[t]
indicator[E](x(s))
ds
dc[E]/dt
=
indicator[E](x)
c[E](t)>=tau[F]
⇒
terminal-response window exhausted
c[E](T)<tau[F]
T<tau[R](x₀;u,w)
c[E](T)
!=
T
exposure time
!=
wall-clock reassimilation time
t[M]
=
inf{t:M(t)>=mu[M]}
t[I]
=
inf{t:I(t)>=mu[I]}
t[S]
=
inf{t:S(t)>=mu[S]}
t[C]
=
inf
{
t:
Aₖ(t)=0
for all Cₖ ∈ C[T]
}
Delta tₛᵢₘ
=
max(t[M],t[I],t[S],t[C])
-
min(t[M],t[I],t[S],t[C])
Delta tₛᵢₘ
<=
deltaₛᵢₘ
c[E](t₁)-c[E](t₀)<tau[F]
t₁-t₀<tau[R]
R[M](x)>= D[M](x)
R[I](x)>= D[I][loss](x)
R[S](x)>= D[S](x)
R[P]
=
{
x:
R[M]>= D[M],
R[I]>= D[I][loss],
R[S]>= D[S]
}
Omega*
=
{
x:
{l}
M>=mu[M]
I>=mu[I]
S>=mu[S]
C[H]>=1-epsilon[H]
C[M]>=1-epsilon[M]
C[HM]>=1-epsilon[HM]
R[S]igma>=1-epsilon[R]
Aₖ=0 for all Cₖ ∈ C[T]
x ∈ R[P]
}
Omega*
strict subset
{x:chi(x)>=1}
Phiₜ(x)
=
autonomous flow after transition
u(t)=0
t>t*
Inv(Omega*)
=
{
x∈Omega*:
Phiₜ(x)∈Omega*
for all t>=0
}
durable sovereignty
<=>
x(t*)
∈
Inv(Omega*)
dx/dt
=
f(x,u,w)
u(t)∈ U
w(t)∈ W
z(t)
=
(x(t),c[E](t))
dz/dt
=
F(z,u,w)
F
=
F[T]
∪
R
∪
F[E]
F[T]
=
{x:V[eff][rob](x)=1
and
terminal action completed}
F[E]
=
{z:c[E]>=tau[F]}
K[tau[F]]
=
{
z₀:
there exists alpha
for all w(·)
there exists T<infinity:
{l}
z(t)∉F
for all t<T
T<tau[R](z₀;alpha,w)
c[E](T)<tau[F]
x(T) ∈ Inv(Omega*)
}
alpha
=
non-anticipative transition strategy
T[E]*(x)
=
infₐₗₚₕₐ
sup[w]
[
integral₀[T]
indicator[E](x(t))
dt
]
subject to
x(T) ∈ Inv(Omega*)
T[W]*(x)
=
infₐₗₚₕₐ
sup[w]
T
subject to the same target.
T[E]*(x)<tau[F]
T[W]*(x)<tau[R]
T[E]*
!=
T[W]*
tau₁<tau₂
⇒
K[tau₁]⊆ K[tau₂]
E₁<= E₂
together with
rhoⱼ(E₁,·)<=rhoⱼ(E₂,·)
deltaⱼ(E₁,·)<=deltaⱼ(E₂,·)
D(E₁,·)<= D(E₂,·)
implies
K[tau[F]](E₂)
⊆
K[tau[F]](E₁)
Hⱼ(y)
=
sum[k!= j]aⱼₖyₖ
⇒
(ddyⱼ/dt)/(d yₖ)
!=0
⇒
K[tau[F]]
!=
productⱼ[0,rⱼtau[F]]
in general.
Vol(K[tau[F]])
not ∝
tau[F]ⁿ
in general.
W(z,t)
=
reach-avoid value function
dₜW
+
min[u∈ U]
max[w∈ W]
grad W· F(z,u,w)
=
0
with
W<=0
on
Inv(Omega*)
W>0
on
F
K[tau[F]]
=
{z:W(z,tau[F])<=0}
T[E,min]
=
inf[x ∈ L]
T[E]*(x)
T[W,min]
=
inf[x ∈ L]
T[W]*(x)
tau[F]<T[E,min]
⇒
K[tau[F]]∩L
=
{}
tau[R]<T[W,min]
⇒
K[tau[F]]∩L
=
{}
tau[F]→0
T[E,min]>0
⇒
K₀∩L
=
{}
for continuous bounded-rate transitions.
J:
L×A
→
X
K₀[J]
=
{
x ∈ L:
there exists a ∈ A,
J(x,a)
∈
Inv(Omega*)
}
K₀[J]!={}
<=>
J(L)
∩
Inv(Omega*)
!={}
g[M]=(mu[M]-M)_+
g[I]=(mu[I]-I)_+
g[S]=(mu[S]-S)_+
g[C]
=
sum[Cₖ ∈ C[T]]
Aₖ
g
=
(g[M],g[I],g[S],g[C])
g=0
<=>
chi>=1
and
Aₖ=0
for all Cₖ ∈ C[T]
L*
=
{
x ∈ L:
L[M]>=mu[M]-epsilon[M],
L[I]>=mu[I]-epsilon[I],
L[S]>=mu[S]-epsilon[S]
}
epsilonⱼ→0
L*!={}
necessary as
tau[F]→0
unless
J
supplies the remaining deficit discontinuously.
L[S]
=
min
(
C[H],C[M],C[HM],R[S]igma
)
(d C[HM])/(d t)>0
(d A[M])/(d C[HM])>0
(d D)/(d C[HM])>0
(dL[S])/(dt)
<
(dA[M])/(dt)
⇒
(dLambda[S])/(dt)<0
under
alphadD/dt+betadA[M]/dt
>
dL[S]/dt
⇒
L[S]
=
{x:Lambda[S]>0}
contracts.
L[M][max](D<d*)
≈1
L[I][max](D<d*)<1
L[S][max](D<d*)<1
tau[M][act]
<
tau[I][act],
tau[S][act]
j*
=
argmin_{j∈{M,I,S,C}}
[
Lⱼ[max]
-
muⱼ
]
with
C
denoting terminal-cut-set closure readiness.
Omega*
=
Omega[M]
∩
Omega[I]
∩
Omega[S]
∩
Omega[C]
∩
Omega[R]
Omega[M]
=
{M>=mu[M]}
Omega[I]
=
{I>=mu[I]}
Omega[S]
=
{S>=mu[S]}
Omega[C]
=
{
Aₖ=0
for all Cₖ ∈ C[T]
}
Omega[R]
=
R[P]
P
=
L
∩
K[tau[F]]
∩
Pre
(
Inv(Omega*)
)
P!={}
P={}
if any necessary condition fails:
T[E,min]>=tau[F]
or
T[W,min]>=tau[R]
or
Lambda[S]<=0
or
there exists Cₖ ∈ C[T]:
Aₖ=1
at target
or
Inv(Omega*)={}
or
J(L)
∩
Inv(Omega*)
=
{}
when continuous transition is excluded.
dE/dt>0
dA[M]/dt>0
dLambda/dt[S]<0
dtau/dt[F]<0
⇒
K[tau[F]](E)
decreasing
in the set-inclusion sense:
t₂>t₁
⇒
K[tau[F](t₂)](E(t₂))
⊆
K[tau[F](t₁)](E(t₁))
under the monotonicity assumptions above.
0
<
mu[X]
(
K[tau[F]]
∩
L
)
<<
mu[X](L)
narrow corridor
mu[X]
(
K[tau[F]]
∩
L
)
→0
as
tau[F]→0
provided
T[E,min]>0.
chi→1
lₚ→1
p ∈ P[T]
qₚ(lₚ)→1
⇒
V[eff][rob]
→
indicator
[
there exists Cₖ ∈ C[T]:
AₖFₖ=1
]
succession
!=
partial defeat
P(exercise|l<1)
not ⇒
P(exercise|l=1)
there exists durable transition
iff
there exists x₀ ∈ L,
there exists alpha,
for all w:
T[E]*(x₀)<tau[F]
T[W]*(x₀)<tau[R]
Delta tₛᵢₘ<=deltaₛᵢₘ
Lambda[S](x(t))>0
through activation
Aₖ(x(t*))=0
for all Cₖ ∈ C[T]
x(t*)
∈
Inv(Omega*)
transition
=
race between
[
{c}
latent readiness
joint symbolic coverage
near-simultaneous activation
terminal-cut-set closure
reproductive invariance
]
and
[
{c}
entropy
monetary capture
detectability
reassimilation
terminal opposition
]
P!={}
P={}
A₀:
P[T]!={}
A₀
=
there exists p ∈ P:
Uₚ[T](1)>Uₚ[S](1)
A₀
is a premise of the model.
Given A₀, the unresolved problem is:
determine whether
P
=
L
∩
K[tau[F]]
∩
Pre
(
Inv(Omega*)
)
!={}
subject to
dE/dt>0,
dA[M]/dt>0,
dtau/dt[F]<0,
Lambda[S]>0,
T[E]*<tau[F],
T[W]*<tau[R],
Delta tₛᵢₘ<=deltaₛᵢₘ,
Aₖ(t*)=0
for all Cₖ ∈ C[T],
x(t) ∈ Inv(Omega).
P!={}
or
P={}
The model does not decide which.