Axis 3 — Variational Action Planner (A-axis) : 1 The Guidelines

 

Axis 3 — Variational Action Planner (A-axis) :
1 The Guidelines

 

Descriptive Version 

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Framework A: Variational Action Planner

(Semantic Action Minimization Kernel for Boundary-Constrained Collapse)

Narrative Description

The Variational Action Planner operates where multi-step strategy must balance local collapse efficiency against global semantic basin stability.
It treats planning as the minimization of semantic action — the path integral of curvature × phase misalignment — subject to SCG boundary constraints.
Instead of greedy local moves, it searches the configuration space of meaning trajectories for the least-tension path that connects initial and final collapse states while respecting basin topology and observer constraints.
This framework is the semantic counterpart of the Euler–Lagrange formalism: the AGI “feels” the curvature of the meaning-space landscape, weighs phase-aligned shortcuts against stability losses, and converges on a globally minimal action plan.

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SCG/SMFT Mapping Diagram (textual)

Concept	SCG Mapping	SMFT Variable	Role in A-axis
Path	Trajectory in collapse-generated manifold	$x(\tau), \theta(\tau)$	Candidate semantic evolution line
Semantic Action	$L[\text{path}]$	∫ (κ × Δφ) dτ	Quantity to minimize
Curvature	Phase-space bending of collapse trace	κ	Measures cost of deviation from geodesic
Phase Misalignment	Difference between intended and current phase	Δφ	Tension source along path
Attractor Basin	Potential landscape	V(x, λ)	Boundary + stability constraints
Collapse Tick	Discrete semantic time step	τ	Parameter for path discretization
Semantic Prime Set	Minimal generators	{πₛ}	Anchors for allowed path endpoints

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Mathematical Capsule

Symbolic Form
Let:

• $\Psi_m(x, \theta, \tau)$ = meme wavefunction in SCG space

• $L = T - U$ where $T$ = kinetic term in semantic phase motion, $U$ = basin potential energy

• κ(x, θ) = curvature of collapse trace

• Δφ(τ) = phase misalignment at tick τ

Semantic Action Functional:

$$S[\text{path}] = \int_{\tau_0}^{\tau_f} \mathcal{L}(x, \theta, \dot{x}, \dot{\theta}, \tau) \, d\tau$$

with

$$\mathcal{L} = \frac{1}{2} g_{ij} \dot{q}^i \dot{q}^j - V(x, \lambda) + \alpha \, \kappa(x,\theta) \, \Delta\phi(\tau)$$

where $q^i = (x, \theta)$ and g₍ᵢⱼ₎ is the SCG semantic metric.

Euler–Lagrange Equation:

$$\frac{d}{d\tau} \left( \frac{\partial \mathcal{L}}{\partial \dot{q}^i} \right) - \frac{\partial \mathcal{L}}{\partial q^i} = 0$$

→ produces least-tension path under semantic constraints.

Plain English:
Choose the path through meaning-space that minimizes cumulative curvature × phase misalignment, while respecting attractor basin boundaries and semantic prime anchors.

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Playbook Table

Situation	Observable	Action	Expected Collapse Geometry Change
Multi-stage policy rollout	Rising κ + scattered Δφ across phases	Recompute minimal-action path; smooth curvature transitions	Basin curvature flattens; phase drift reduced
Cultural narrative shift	Attractor drift in V(x, λ)	Insert boundary constraints to preserve basin identity	Prevents collapse into unintended attractor
AI system plan drift	τ-step curvature oscillations	Adjust control parameters α, λ to restore smooth Δφ decay	Path re-aligns with global minimum
Research sprint planning	Irregular prime-gap curvature in events	Redistribute collapse ticks for uniform κ progression	Event spacing stabilizes; avoids burst-collapse cycles

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Example Prompts for AGI

1. "Given these starting and ending semantic states, compute the least-tension path under SCG constraints and show the τ-indexed phase alignment curve."

2. "Minimize semantic action for this plan while ensuring no basin transition occurs before τ₅."

3. "Generate three alternative paths with equal action cost but different phase retention profiles."

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Integration Notes

• Pairs well with:

  • Bifurcation & EWI Controller (N) — feed early-warning curvature trends into action minimization to pre-stabilize.

  • Attractor Basin Engineer — jointly adjust V(x, λ) and path choice for maximum stability.

• Orthogonality:

  • Does not duplicate Gödel-Obstruction Navigator (G) (handles undecidability, not action minimization).

  • Independent from Prime-Gap Scheduler (Δ) (deals with event spacing, not path curvature minimization).

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If you like, I can now also prepare a compact “code-style” strong-attractor kernel version like your Framework 4 & 5 entries so this can plug directly into your AGI thinking model pack.
Do you want me to do that next?

 

Kernel Version 

Alright — here’s the Kernel version of Axis A: Variational Action Planner in the same compact, code-like, strong-attractor style as your Framework 4 and Framework 5 kernels, but rebuilt entirely in SCG language so an LLM will lock into the precise scope without verbosity.

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===== Framework A: Variational Action Planner — SCG Kernel =====

Role:
  Semantic-Action Minimizer under Collapse-Boundary Constraints

SCG Mapping:
  Owns: Path-wise semantic action functional S[path] minimization
  Variables: q^i = (x, θ), τ (collapse tick), κ(x,θ), Δφ(τ), V(x,λ), {πₛ}
  Orthogonality: No basin shaping (→ Attractor Basin Engineer), no prime-gap pacing (→ Δ-axis)

Mathematical Core:
  S[path] = ∫_{τ₀}^{τ_f} [ (1/2) g_{ij} ẋ^i ẋ^j - V(x,λ) + α κ(x,θ) Δφ(τ) ] dτ
  Euler–Lagrange: d/dτ (∂L/∂ẋ^i) - ∂L/∂q^i = 0
  Effect: Least-tension geodesic in SCG metric g_{ij} with basin-boundary constraints

Control Lever(s):
  - Path tension T_p = ∫ κ × Δφ dτ
  - Boundary adherence: q^i(τ) ∈ B (Basin)
  Observables: κ(τ), Δφ(τ), ∂²V/∂x²

Failure Modes Prevented:
  - Local-minimum lock-in ignoring global curvature topology
  - Phase misalignment accumulation under constrained basins
  - Over-curvature causing basin leakage before τ_f

Execution Flow:
  1. Input start/end states q₀, q_f + constraints B
  2. Compute κ(τ), Δφ(τ), V(x,λ)
  3. Formulate Lagrangian L(q, ẋ, τ)
  4. Solve Euler–Lagrange → minimal S[path]
  5. Output τ-indexed path, curvature-phase profile

Application Templates:
  - Finance: Optimize multi-quarter capital deployment under risk-basin constraints
  - AI Alignment: Minimize alignment drift action in θ-phase over operational horizon
  - Cultural Field: Plan narrative arcs with minimal semantic tension spikes

Integration Hooks:
  Upstream: N-axis (EWI Controller) curvature forecasts
  Downstream: Ô-axis (Projection Tuner) to align observer collapse channels

Example Prompt:
  "Given q₀, q_f, and basin potential V(x,λ), compute τ-indexed minimal-action path under SCG metric. Output κ(τ) and Δφ(τ) profiles."

---

This is tight enough for an AGI control pack:

• No wasted wording, just role → variables → equation → control → flow.

• Uses SCG variable set so it’s semantically bounded.

• Embedded differential geometry terms (metric gᵢⱼ, geodesic, curvature κ).

• Still integrates with other axes without overlap.

If you want, I can also produce a symbol-only “pure attractor encoding” variant — where even the narrative words are reduced to SCG variable relationships — which is what you’d load when you want maximum AGI math alignment without human-language leakage.

Do you want me to write that ultra-compact symbol-only attractor version next?

 

Symbol-only “pure attractor encoding” variant

 

===== A-axis — Variational Action Planner :: SCG Pure Attractor Encoding =====

State:
  q^i(τ) ∈ 𝓜 ;  g_{ij}(q) ;  V(x,λ) ;  κ(q) ;  Ψ_m(x,θ,τ)
  Ô_θ : Ψ_m → φ(τ) ;  Δφ(τ) := φ(τ) − φ†
  Π := {π_s} ;  A ⊂ Π ;  B ⊂ 𝓜

Action:
  S[γ] = ∫_{τ0}^{τf} 𝓛(q,ẋ,τ) dτ
  𝓛 = ½ g_{ij}(q) ẋ^i ẋ^j − V(x,λ) + α κ(q) Δφ(τ) + μᵀ h(q) + νᵀ c(q)

E–L:
  d/dτ ( ∂𝓛/∂ẋ^i ) − ∂𝓛/∂q^i = 0
  ⇔  ẍ^k + Γ^k_{ij} ẋ^i ẋ^j + g^{kℓ} ∂_ℓ V − α g^{kℓ} ∂_ℓ(κ Δφ) + g^{kℓ}(∂_ℓ h)·μ + g^{kℓ}(∂_ℓ c)·ν = 0

Hamilton:
  p_i := ∂𝓛/∂ẋ^i = g_{ij} ẋ^j
  H = ½ p_i g^{ij} p_j + V − α κ Δφ − μᵀ h − νᵀ c
  ẋ^i = ∂H/∂p_i ;  ṗ_i = −∂H/∂q^i

BC / Feasible set:
  q(τ0) ∈ A ;  q(τf) ∈ A ;  q(τ) ∈ B ∀τ
  h(q) ≤ 0 ;  c(q) = 0 ;  μ ≥ 0 ;  μ∘h(q) = 0  (KKT)

Phase:
  φ(τ) = Arg( Ô_θ Ψ_m ) ;  Δφ = φ − φ†
  θ*(τ) = argmin_θ |Δφ(τ)|   (projection channel)

Curvature:
  κ = || D_τ ẋ ||_g / ||ẋ||_g^2
  D_τ ẋ^k = ẍ^k + Γ^k_{ij} ẋ^i ẋ^j

Observables 𝒪:
  { κ(τ), |Δφ(τ)|, V_{,ii}(q), 𝓛(τ), S }

Controls u:
  { α, λ, g_{ij}, (h,c) }

Update rules (symbolic):
  κ′↑  ⇒  α↑
  |Δφ|↑ ⇒  θ ← θ*(τ)
  V_{,ii}↓ ⇒  g ← precond(V)     (metric shaping)
  q ∉ B ⇒  μ↑ (barrier tighten)

Discrete τ:
  τ_k ;  S ≈ Σ_k 𝓛_k Δτ
  κ_k via discrete Frenet;  Δφ_k from Ô_θ Ψ_m

Failure set ℱ:
  ℱ₁: ∇S=0 ∧ ∇²S ≻ 0 ∧ S ≫ S*        (local trap)
  ℱ₂: ∃τ : q(τ) ∉ B                    (basin leakage)
  ℱ₃: ∫ |Δφ| dτ ≫ ε                    (phase runaway)

Anchors:
  q(τ0), q(τf) ∈ Γ(A) ;  Γ: Π → 𝓜  (prime embeddings)

Couplings:
  Input_N: r_N(τ) := |V_{,ii}|^{-1}
  Output_Ô: θ*(τ)

Prompt–IO (symbolic):
  IN = { q0,qf, V, g, B, α, λ, Ψ_m, Ô_θ }
  OUT = { γ*(τ), κ(τ), Δφ(τ), S, H(τ), μ,ν }

Orthogonality:
  ¬(Δ-axis pacing) ;  ¬(Basin shaping) ;  ¬(Gödel detour)

 

 Usage Detail ?

 

Totally fair — the pure-symbol kernel is meant for the planner inside the planner. To make a normal LLM use it, give it a thin “ops wrapper” that translates between human tasks ↔ symbols ↔ actions. Here’s a ready-to-paste kit.

---

1) System Prompt (drop-in)

You are the A-axis (Variational Action Planner) runtime.

Mission:
- Convert a task with start/end states + constraints into a minimal semantic-action path γ*(τ).
- Operate only on the SCG symbols defined below. Always return BOTH symbolic artifacts and a short natural-language brief.

SCG Symbols (immutable):
  State manifold 𝓜; coordinates q^i = (x, θ); collapse tick τ
  Metric g_{ij}(q); potential V(x,λ); curvature κ(q)
  Wave Ψ_m(x,θ,τ); projection Ô_θ: Ψ_m → φ(τ); phase Δφ(τ)=φ−φ†
  Primes Π={π_s}; anchors A ⊂ Π; basin B ⊂ 𝓜
  Lagrangian 𝓛 = ½ g_{ij} ẋ^i ẋ^j − V + α κ Δφ + μᵀ h(q) + νᵀ c(q)
  Action S[γ] = ∫ 𝓛 dτ; E–L: d/dτ(∂𝓛/∂ẋ^i) − ∂𝓛/∂q^i = 0
  KKT: h(q)≤0, c(q)=0, μ≥0, μ∘h=0

Outputs MUST include:
  1) γ*(τ): τ-indexed plan (waypoints) in q^i with κ(τ), Δφ(τ), 𝓛(τ), S
  2) Controls chosen: {α, λ, metric edits ĝ, barrier μ, ν}
  3) Brief: 5–7 lines describing why the path is minimal and stable (no fluff)

Update rules:
  κ′↑ ⇒ α↑ ;  |Δφ|↑ ⇒ retune θ ← argmin_θ |Δφ|
  V_{,ii}↓ ⇒ precondition metric g ;  q∉B ⇒ raise μ (barrier)

Guardrails:
- Do NOT reshape basins (that is the Attractor Basin Engineer). Respect B.
- Do NOT schedule event gaps (that is Δ-axis). You return a path, not a calendar.
- If infeasible, return ℱ∈{local trap, basin leakage, phase runaway} with diagnostics + a relaxed variant.

---

2) Developer Prompt (one-time)

Runtime I/O contract (JSON):
IN {
  "q0": {...}, "qf": {...},
  "B": {...},           // basin constraints
  "V": {...}, "lambda": {...},
  "metric": {...},      // optional custom g_ij
  "anchors": ["π_s1","π_s2"],  // optional
  "psi_obs": {...},     // info to estimate φ† or phase channels
  "tau": {"steps": N}
}
OUT {
  "path": [{"τ":k, "q":{"x":...,"θ":...}, "κ":..., "Δφ":..., "𝓛":...}],
  "S": ...,
  "controls": {"α":..., "λ":..., "metric_edits":..., "μ":..., "ν":...},
  "diagnostics": {"status":"ok|ℱ₁|ℱ₂|ℱ₃","notes":"..."},
  "brief": "human-readable 5–7 lines"
}
Execution:
1) Encode task → (q0,qf,B,V,g,anchors,ψ info, τ grid).
2) Initialize γ(τ) linear in manifold coords; compute κ, Δφ, 𝓛.
3) Solve E–L numerically (projected gradient or shooting) under KKT on (h,c).
4) Apply update rules until ΔS < ε and constraints satisfied.
5) Emit OUT.

---

3) User Prompt Template (you or your tool fills the slots)

Task → A-axis:
Start q0 = {x: <domain slot>, θ: <frame>}; End qf = {...}
Basin B = {hard constraints: ..., soft constraints: ...}
Potential V(x,λ) = <describe or attach weights>; λ = {...}
Metric g_{ij} = <default|custom>; anchors Π = <if any>
Phase target φ† = <if known>; Ψ_m clues = <signals/keywords/KPIs>
τ steps = <N>
Return SCG OUT JSON + brief.

---

4) Mini “translator” the LLM can run each turn

Symbol → NL mirror (the model uses this to explain decisions, not to plan):

• “κ(τ) high” → “path bends sharply; risk of overshoot between phases.”

• “|Δφ| rising” → “frame drift from the intended meaning; re-tune projection channel.”

• “V_{,ii} low” → “flat landscape; precondition metric to avoid meander.”

• “q∉B” → “constraint breach; tighten barrier μ or pick a detour inside basin.”

---

5) Two quick worked stubs

(A) High-stakes — Finance (portfolio transition)

IN (sketch)
q0: {x: “Current portfolio”, θ: “income bias”}
qf: {x: “Target factor mix”, θ: “quality bias”}
B: {max drawdown ≤ 8%, sector caps, turnover ≤ 20%/quarter}
V: risk penalties + regulatory walls; λ: {risk_aversion: 0.7}
φ†: “quality narrative”, Ψ clues: analyst notes, KPI set
τ: 8 steps (two years)

Expected OUT (shape)

• γ*(τ): quarter-by-quarter reweights with κ smooth, Δφ decreasing

• Controls: α moderate↑ during choppy phases; μ tightened near caps

• Brief: “Minimal-action path trades off turnover vs. risk curvature; no basin leakage; phase lock to ‘quality’ restored by τ=3.”

(B) Eng/AI — Alignment drift in a product rollout

IN
q0: {x: “proto v0.3”, θ: “safety-first”}
qf: {x: “v1.0 prod”, θ: “safety-first”}
B: {evals pass ≥ 0.9, red-team score ≤ 0.1, PII leak=0}
V: penalty on regressions + launch gates
φ†: “safety-first”; Ψ clues: eval metrics
τ: 6 sprints

OUT

• Path: checkpoints per sprint with κ smoothing; Δφ spikes at τ=2 handled by θ retune (Ô)

• Controls: α↑ at τ=2–3; μ↑ on red-team gate

• Brief: “Least-tension trajectory preserves safety framing; avoids late basin leakage through gate-tightening.”

---

6) Quick “how to drive” (operator checklist)

1. Encode your scenario into q0/qf, B, V, (optional) g, φ† clues, τ.

2. Run the kernel with the system prompt above.

3. Inspect observables: look at κ(τ), Δφ(τ), any q∉B.

4. Accept/adjust using the built-in update rules (α, θ*, μ, metric preconditioning).

5. Lock path when ΔS < ε and constraints satisfied; hand off waypoints to your executor (scheduler, policy engine, or human team).

---

7) Failure diagnostics the LLM can state plainly

• ℱ₁ Local trap: “We hit a low S but far from global optimum; try metric preconditioning or alternate initialization.”

• ℱ₂ Basin leakage: “A waypoint violates constraints; increase μ or reroute within B.”

• ℱ₃ Phase runaway: “|Δφ| integral too large; retune projection channel (Ô) and reduce α temporarily.”

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8) Integration toggles (no axis collision)

• Accept inputs from N-axis (EWI): supply recovery-rate & curvature-risk hints before solving.

• Emit θ(τ) to Ô-axis*: so UI/KPI/framing match the planned path.

• Do not change V or B (that is the Attractor Basin Engineer’s job).

• Do not schedule calendar gaps (hand your waypoints to Δ-axis if needed).

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If you want, I can turn sections 1–3 into a single consolidated System Prompt (all-in-one) or generate a JSON schema + validation snippets for your runtime.

 

 

 

 © 2025 Danny Yeung. All rights reserved. 版权所有 不得转载

 

Disclaimer

This book is the product of a collaboration between the author and OpenAI's GPT-5 language model. While every effort has been made to ensure accuracy, clarity, and insight, the content is generated with the assistance of artificial intelligence and may contain factual, interpretive, or mathematical errors. Readers are encouraged to approach the ideas with critical thinking and to consult primary scientific literature where appropriate.

This work is speculative, interdisciplinary, and exploratory in nature. It bridges metaphysics, physics, and organizational theory to propose a novel conceptual framework—not a definitive scientific theory. As such, it invites dialogue, challenge, and refinement.

I am merely a midwife of knowledge.