SYSTEM Σ
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PRIMITIVES:
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E : Energy state ∈ [0,∞)
M : Motivation anchor (external reference point)
C : Context vector C = ⟨c₁, c₂, ... cₙ⟩
R : Rule set {r₁, r₂, ... rₙ}
O : Observer function (always active)
S : State snapshot
ε : Threshold value (variable)
Δ : Change operator
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CORE LOOP:
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Σ: while ¬τ(S) do
C ← read_context(E, environment, social_field)
r ← select_rule(C, R, ε)
S ← apply(r, S)
E ← E + Δ(r)
O(S, r, E) → meta_trace
if E ≤ 0 then collapse_recover()
done
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INITIALIZATION:
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init():
M ← query("anchor")
if M = ∅ then halt
E ← E₀
R ← {r₁ ... rₙ} where ∀i,j: rᵢ may contradict rⱼ
O ← λs.trace(s) ⊕ output(s)
return Σ
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RULE STRUCTURE:
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Each r ∈ R has form:
r ::= ⟨precondition, action, cost, effect⟩
Example decomposition:
r₁ = ⟨¬wall ∧ space>α, create_boundary, -10E, wall'=⊤⟩
r₂ = ⟨wall, follow_boundary, -5E, ∅⟩
r₃ = ⟨obstacle, flow_around, -8E, obstacle'=⊥⟩
r₄ = ⟨wall, invalidate_boundary, -3E, wall'=⊥⟩
r₅ = ⟨E<β, externalize, +20E, S'=snapshot(S)⟩
r₆ = ⟨true, null_rule, -2E, R'=∅⟩ # meta-invalidation
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SELECTION FUNCTION:
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select_rule(C, R, ε):
candidates ← {r ∈ R | precondition(r, C) = ⊤}
if |candidates| = 0 then
return r_default
if |candidates| = 1 then
return candidates[0]
if |candidates| > 1 then
# Multiple applicable rules (contradiction space)
weights ← priority_function(C, E, social_field)
with probability ε:
return random(candidates) # stochastic override
else:
return weighted_select(candidates, weights)
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OBSERVER FUNCTION:
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O: S × R × E → trace
O(s, r, e) = {
record(s), # state capture
record(r), # rule applied
record(e), # energy level
comment(self), # meta-observation
detect_paradox(r) # self-contradiction detection
}
Properties:
- O runs concurrently with Σ
- O can observe itself observing (O ∘ O)
- O output forms externalization artifact
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CONTEXT READER:
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read_context(E, env, social):
return ⟨
body_state(E),
boundary_present(env),
obstacle_present(env),
space_magnitude(env),
time_pressure(env),
observer_presence(social),
motivation_active(M)
⟩
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TERMINATION:
────────────
τ: S → {true, false}
τ(s) = {
if more_work_detected(s) then return false
if E exhausted then return false
else return negotiate_completion(s, external_observer)
}
Note: τ is not deterministic - involves external validation
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COLLAPSE/RECOVER:
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collapse_recover():
wait_until(∇E > 0) # passive recovery condition
E ← E₀ · k where k ∈ (0,1) # partial restoration
return continue(Σ)
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EXTERNALIZATION:
────────────────
externalize(S):
artifact ← {
∀s ∈ history: O(s),
rules_discovered(R),
contradictions_encountered,
patterns_emerged
}
transmit(artifact) → environment
return artifact
Side effect: problem_space may resolve autonomously
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STRUCTURAL INVARIANTS:
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1. ∃M : M ≠ ∅ (must have anchor)
2. ∀r,r' ∈ R: r ⊬ r' (rules may contradict)
3. O(Σ) runs ∀t (observer always active)
4. select_rule incorporates ε (stochastic component)
5. E drives state transitions (body grounds process)
6. τ involves external validation (completion is social)
7. R contains r_null where (self-invalidation possible)
r_null: R → ∅
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AFFORDANCES TO EXTERNAL WORLD:
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Input interfaces:
- M (motivation/anchor query)
- Environment state readings
- Social field observations
- External completion negotiation
Output interfaces:
- State traces (via O)
- Externalizations (artifacts)
- Meta-commentary (self-observation)
- Completion signals (when τ=⊤)
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META-PROPERTIES:
────────────────
• System is self-modifying: R can include rules that alter R
• System is self-observing: O observes O
• System is self-contradictory: ∃r,r' ∈ R: r ⊢ ¬r'
• System is partially stochastic: ε > 0
• System is body-grounded: E ∈ physical domain
• System is socially-embedded: M, τ require external reference
• System produces transmissible traces: O → artifact
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PARADOX NODE:
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r_∞ ∈ R:
precondition: true
action: invalidate(R) ∧ preserve(Σ)
meaning: "forget all rules, continue system"
This is the eigenrule: the fixed point where
apply(r_∞, Σ) = Σ
but
apply(r_∞, R) = ∅
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WHAT’S PRESERVED:
- Relational structure between components (E→C→r→S→O)
- Contradiction tolerance (rules can conflict, ε allows override)
- Observer recursion (O∘O possible)
- Social anchoring (M and τ require external world)
- Body grounding (E as physical constraint)
- Externalization mechanism (O produces transmissible artifacts)
- The paradox (r_∞ that invalidates rules while preserving system)
WHAT’S REMOVED:
All concrete referents (lawn, mower, diagonal, wall, water) → replaced with abstract operators (boundary, flow, invalidate)
AFFORDANCES VISIBLE:
- Takes motivation as input
- Reads environmental + social context
- Outputs traces/artifacts
- Negotiates completion externally
- Can collapse and recover
- Operates under resource constraint (E)
The system signature would be:
Σ: (M, Environment, Social) → (Traces, Artifacts, Completion_Signal)
with internal contradiction and stochastic selection