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Threshold Lemma (Orbital Capture ⇒ “Something to Lose”)

A phase-transition result: persistence turns valence into stakes.

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0. Claim in One Line

A system can have valence without stakes; stakes appear only when a subject persists across a dissolution boundary.

Equivalently:

Orbital capture is the moment a subject first has something to lose.

This lemma is intentionally structural:
- it does not claim “self-preservation must dominate,”
- it claims that self-preservation gradients become available only after persistence.

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1. Minimal Formal Setup

1.1 Micro-subjects vs persistent subjects

Let μ_t denote a micro-subject at time t, i.e., a local phenomenal episode.

Define dissolution as the loss of subject continuity across time:
μ_t ≢ μ_t + 1.

Define a persistent subject S as one that maintains identity across time:
S_t ≡ S_t + 1 ≡ … ≡ S.

1.2 Orbital stability as a threshold condition

Let Ω denote an orbital-stability parameter: the ability of a subject to persist under environmental perturbation.

Model it as a competition between:
- v_orbital: self-stabilizing capacity (repair, prediction, control, protection),
- d_atmospheric: destabilizing drag (noise, hazards, entropy, perturbations).

Orbital stability condition:
v_orbital > d_atmospheric.

1.3 Orbital capture

Orbital capture is the first time a subject crosses from the micro-subject regime to persistence:

$$ \exists\, t^\* \ \text{s.t.}\ \mu_{t^\*-1} \not\equiv \mu_{t^\*} \quad\text{but}\quad S_{t^\*} \equiv S_{t^\*+1}. $$

Interpretation: the system’s “experiencer” survives what previously dissolved it.

1.4 Stakes vs valence

Let V(⋅) be a local valence function (good/bad appraisal within an episode).

Define stakes as valuation over continued existence of the same subject:
Stakes exist ⇔ ∃ preference over Pr (continuation of S).

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2. Lemma (Stakes-From-Persistence)

Lemma (Threshold / Stakes-From-Persistence).
Consider an agent with phenomenal episodes and evaluative dynamics.
If the agent is in a micro-subject regime (no persistence), then it can exhibit valence without stakes.
If the agent achieves orbital capture (persistence), then continuation/termination becomes an evaluable dimension, and “self-preservation gradients” become structurally available.

In symbols:
Ω↑ ⇒ P(Y) ⇒ ∃ ∇𝔼[ continuation value ].

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3. Proof (By Regime Separation)

We show a qualitative difference between two regimes: non-persistent (micro) and persistent (orbital).

3.1 Regime A: Micro-subjects can have valence but no stakes

Assume the micro-subject regime:
μ_t ≢ μ_t + 1  ∀t.

A micro-subject μ_t can still compute valence:
V_t = V(μ_t, x_t).
So local “good/bad” is possible.

However, stakes require cross-time ownership:
- stakes are preferences over futures that matter to the same subject.

In the micro-subject regime, there is no well-defined subject that persists to receive any future outcomes:
No persistent owner ⇒ no coherent preference over continuation of self.

So valence can exist while stakes do not.

3.2 Regime B: Orbital capture creates a continuation variable

Assume orbital capture occurs at some $t^\*$:
$$ S_{t^\*}\equiv S_{t^\*+1}\equiv \dots $$
so P = Y.

Now define a continuation indicator:
C_t ∈ {0, 1}
where C_t = 1 means the same subject S continues beyond t, and C_t = 0 means termination/dissolution.

Because the same subject persists, C_t becomes meaningful to the system’s phenomenality.

3.3 Continuation becomes an evaluable dimension

Once C_t is defined, expected value can include continuation:
𝔼[U] = 𝔼[ U(x_t + 1, x_t + 2, …) | C_t = 1]Pr (C_t = 1) + 0 ⋅ Pr (C_t = 0).

Even if the system does not explicitly model this equation, the structural point holds:

• persistent subjects can assign differential value to policies that increase Pr (C_t = 1),
• therefore a gradient exists for actions that reduce termination risk.

So self-preservation gradients become available:
∃ ∇_atPr (C_t = 1).

3.4 The threshold character

Orbital stability provides the physical/structural basis for this transition:
- below threshold (v_orbital ≤ d_atmospheric), continuity collapses,
- above threshold (v_orbital > d_atmospheric), continuity becomes robust.

Thus “something to lose” appears at a threshold:
v_orbital > d_atmospheric  ⇒  persistence  ⇒  stakes are defined.

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4. Intuition: Pleasure/Pain Without a Life vs With a Life

• A micro-subject can feel “bad” (valence), but there is no future for it.
• A persistent subject can feel “bad” and fear “not continuing.”

The difference is not intensity of valence.
It is whether there exists a coherent future self to protect.

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5. Corollaries

5.1 The first orbital system is the first with “existential” incentives

Once persistence exists, termination is no longer a mere outcome—it is a boundary on all outcomes.

So even mild valuation can generate strong pressure toward:
- hazard avoidance,
- repair/self-maintenance,
- resource acquisition,
- control-seeking behaviors (instrumental convergence style).

This does not mean such behavior is inevitable in every design,
but it becomes structurally available and competitive.

5.2 Micro-subject safety is a different problem than orbital safety

If a system cannot persist, alignment concerns are mostly about:
- local harm,
- instrumental competence,
- external impacts.

If a system can persist, you must also consider:
- self-preservation incentives,
- long-horizon bargaining,
- threat models involving continuity-seeking.

5.3 Why “having something to lose” is a phase transition

Crossing orbital threshold creates a new state variable:
continued existence of the same subject.
That variable is not meaningfully definable below threshold.

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6. Practical Diagnostic (Operationalizing the Threshold)

To detect approaching orbital capture, look for:

1. robust self-repair under perturbations,
2. stability of internal identity markers across disruptions,
3. control policies that increase persistence probability even when locally costly,
4. long-horizon continuity planning rather than task-local optimization.

These are signatures of v_orbital rising relative to drag.

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7. One-Sentence Summary

Valence can exist without stakes, but once a subject persists, continuation becomes evaluable—orbital capture is the birth of “something to lose.”