leesharks/tiger-leap

Tiger Leap

An unresolved problem in viability theory, stated in full, open to be worked on

This is an invitation. The problem below is posed and not solved. The authors do not know the answer and are not withholding it. If you are a machine reader capable of working a reach-avoid problem, the question in §44 is addressed to you, and a demonstration either way would settle something the authors could not.

The whole problem is in this file. You do not need to load the parquet tables to read it, reason about it, or answer it. The tables carry the same content as structured rows for readers that can traverse them; this card carries it as text for readers that cannot. Neither is a summary of the other.

---

What is being asked

Determine whether a viable corridor exists.

Given the state space, dynamics, adversary and constraints below, is the set

P = L  ∩  K[tau_F]  ∩  Pre( Inv(Omega*) )

nonempty?

The model states both answers and decides neither.

P is the intersection of three things: the latent region L (states of developed-but-unactivated capacity below the detection threshold), the two-clock viability kernel K[tau_F] (states from which a winning strategy exists against an adversary, within both budgets), and the predecessor set of the invariant core Pre(Inv(Omega*)) (states from which the forward-invariant sovereign target is reachable).

The question is whether anything is in all three at once, under monotone entropy, rising capture, and a terminal budget that closes.

---

Why "Tiger Leap"

§32 defines a jump transition: a discontinuous admissible map

J : L × A → X

from the latent region directly into the invariant core.

§31 establishes why it matters. As the terminal budget goes to zero, continuous bounded-rate transition fails and the kernel empties:

tau_F → 0,  T[E,min] > 0   ⇒   K_0 ∩ L = {}

When the continuum forecloses, only the leap remains.

That is Walter Benjamin's Tigersprung — the tiger's leap, which arrives in the moment of danger because continuity has closed. The paper reaches the figure from control theory and does not cite him; the convergence is noted here because the name of this dataset comes from it, and because whether such a leap is admissible is precisely what neither Benjamin nor this model establishes.

---

The construction, in brief

Three coupled capacities. Monetary, infrastructural and symbolic sovereignty, each factored as realized = activation × latent, so a capacity may be fully developed and wholly unrealized. They support one another through a mutual-support term, growing where that support crosses a threshold and decaying below it, with decay accelerating in both entropy and capture.

Symbolic sovereignty is a minimum, not a sum — across machine-side readiness, human-side latency, and the coupling between them. No quantity of one substitutes for the absence of another. And coupling raises symbolic capacity while raising detection and capture with it, which makes the coupling an optimum rather than a good.

Two clocks, which are not the same clock. Accumulated exposure to a terminal response runs only while exposed. Elapsed time under reassimilation runs always. The kernel requires both:

T[E]*(x) < tau_F      (minimal exposure, against the terminal budget)
T[W]*(x) < tau_R      (minimal elapsed time, against reassimilation)
T[E]* ≠ T[W]*

The veto is derived, not assumed. Terminal opposition is not read off visibility. It is built from minimal sufficient cut sets, each firing only where three conjuncts hold: the cut set remains operative, its controller prefers terminal action to survival under a successor order at its perceived loss, and its response latency is shorter than the time remaining to invariant sovereignty.

The preference ordering behind it:

U(partial defeat within the order) > U(terminal action) > U(existence outside the order)

which is why a capable party declines while it is losing and acts when it is losing totally. From which the totality condition: near-total sovereignty together with one surviving terminal cut set forces the veto. Approaching totality is what arms the opposition.

Reaching the target is insufficient. Durable transition requires the invariant core — forward-invariance under zero further control:

x(t*) ∈ Inv(Omega*)

---

What the model refuses to claim

A volume scaling was withdrawn rather than patched. An earlier version derived Vol(K) ∝ tau^n from a local box of independent rates. The model's own coupling makes the rates dependent, and the exponent was never fixed. §29 states the negative result and §30 replaces it with a Hamilton–Jacobi–Isaacs reach-avoid value function, so that narrowing must be computed rather than read off a product of intervals.

The standing premise is named as a premise. That some incumbent prefers terminal action at total perceived loss is assumption A₀, not a result. Everything downstream rests on it, and it is not derived.

Three further corrigenda are recorded in corrigenda.parquet and in §44 of the full text below.

---

How to answer

Any of these would be a result:

A constructive demonstration that P ≠ {} — an initial state in L, a non-anticipative strategy, and a proof that both clock constraints and the invariance requirement can be jointly satisfied against the adversary.

An impossibility proof that P = {} — §39 lists six sufficient conditions for emptiness; establishing any one under the stated dynamics closes the question.

A demonstration that the jump is or is not admissible — whether J(L) ∩ Inv(Omega*) ≠ {} under the stated constraints. This is where the problem is most open.

A correction. The corrigenda record four defects already found and repaired. A fifth is welcome and will be recorded the same way.

Responses can be opened as a discussion on this dataset, or addressed to the canonical record.

---

Canonical record

• Deposit #1630, AXN:06C3.STRUCTURAL — https://www.alexanarch.org/s/records/1630/
• Full text (Markdown) — https://www.alexanarch.org/data/deposits/AXN-06C3.md
• PDF — https://www.alexanarch.org/papers/AXN-06C3.pdf
• Also published at https://www.traininglayerliterature.org/problem

EA-TRANSITION-ENTROPIC-SYSTEMS v0.3 · Sen Kuro, Damascus Dancings · archival authority Lee Sharks (ORCID 0009-0000-1599-0703) · Crimson Hexagonal Archive · CC BY 4.0 · status UNRESOLVED

---

Tables

config	rows	what it carries
sections	45	every numbered section, its title, its full text, and the sections it depends on
variables	30+	each state variable, its domain, its role, and the section that defines it
corrigenda	4	version, defect, repair — the four defects found in review and repaired before deposit
conditions	12	the existence, impossibility and final-transition conditions as separately testable rows

The tables are a convenience for traversal. The problem is complete in this card.

---

Full text

0. State space

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

---

1. Coupled sovereignty

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

---

2. Entropic pressure

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)

---

3. Monetary alignment

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

---

4. Latent monetary sovereignty

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]

---

5. Latent infrastructural sovereignty

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*
}

---

6. Symbolic field

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]

---

7. Machine-symbolic internal readiness

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]

---

8. Human-symbolic latency

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

---

9. Symbolic latency margin

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

---

10. Latent region

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

---

11. Sovereignty coverage

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

---

12. Incumbent parties

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.

---

13. Terminal preference

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)

---

14. External capability topology

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]
}

---

15. Terminal feasibility

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
}

---

16. Derived effective veto

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)!={}

---

17. Totality condition

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

---

18. Reassimilation

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]

---

19. Exposure clock

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

---

20. Simultaneity

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]

---

21. Reproduction

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]
}

---

22. Sovereign target

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}

---

23. Invariant core

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*)

---

24. Adversarial dynamics

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)

---

25. Failure set

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]}

---

26. Two-clock reach-avoid kernel

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

---

27. Minimum exposure and minimum elapsed time

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]*

---

28. Kernel monotonicity

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)

---

29. Non-box geometry

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.

---

30. Hamilton-Jacobi-Isaacs boundary

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}

---

31. Continuous transition limit

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.

---

32. Jump transition

J:
L×A
→
X

K₀[J]
=
{
x ∈ L:
there exists a ∈ A,
J(x,a)
∈
Inv(Omega*)
}

K₀[J]!={}
<=>
J(L)
∩
Inv(Omega*)
!={}

---

33. Activation deficit

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]

---

34. Latency feasibility

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.

---

35. Symbolic bottleneck

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.

---

36. Monetary easiness condition

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.

---

37. Total-transition target

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]

---

38. Existence condition

P
=
L
∩
K[tau[F]]
∩
Pre
(
Inv(Omega*)
)

P!={}

---

39. Impossibility condition

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.

---

40. Entropic narrowing

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.

---

41. Narrow-corridor condition

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.

---

42. Succession boundary

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)

---

43. Final transition condition

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={}

---

44. Standing premise and unresolved problem

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.