LIF \(M!\) Ordering Space — Research CheckpointLIF \(M!\) Ordering Space — 目前研究 Checkpoint

§1Current system in one picture目前系統一張圖看完

The project is no longer one monolithic “SNN model.” It is best understood as four mathematically distinct modules: representation, functional evaluation, action selection, and temporal actuation. The main progress came from separating these modules so that one component cannot silently solve the job of another.

現在這個研究已經不是一個混在一起的「SNN model」。最乾淨的理解方式，是把它拆成四個數學上不同的模組：representation、functional evaluation、action selection、temporal actuation。真正的重要進展，就是把這四件事拆開，避免某個大模型偷偷把其他模組的工作全部做掉。

§2Symbols and state spaces符號與 state space

§3Training data and experimental unitsTraining data 與實驗單位

The common formal task is the frozen Moving-Shapes JEPA line. The selected functional/navigation lineage uses 100,000 train, 10,000 validation, and 10,000 test samples, with formal seeds 0/1/2. M6 is the primary mechanism-discovery size.

目前共用的 formal task 是 frozen Moving-Shapes JEPA lineage。selected functional / navigation 線使用 train 100,000、validation 10,000、test 10,000，formal seeds 為 0/1/2；M6 是主要 mechanism-discovery 尺度。

One true evaluator call yields \(L(x,\pi)\). Utilities and pairwise ranking labels can then be derived algebraically from already-evaluated candidate losses; they are not new independent task observations. This distinction prevents factorial supervision from becoming a hidden key.

一次真正 evaluator call 才產生 \(L(x,\pi)\)。utilities、pairwise ranking labels 可以由已經算過的 candidate losses 代數推導，它們不是新的 independent task observations。這個 distinction 用來防止 factorial supervision 偷偷變成 hidden key。

§4JEPA front end: prediction defines functionJEPA 前端：prediction objective 定義「有用」

Permutation states are not intrinsically good or bad. JEPA supplies the task objective. Conceptually:

permutation states 本身沒有「好或壞」。真正讓它有 functional meaning 的是 JEPA prediction objective：

The exact frozen loss \(\ell\) is authoritative. If the implementation contains normalization, cosine geometry, or auxiliary terms, it must not be silently replaced by MSE.

真正的 frozen loss \(\ell\) 必須以 implementation 為準。如果實際包含 normalization、cosine geometry 或其他項目，就不能擅自當成 MSE。

§5Continuous score geometry before spikesSpike 前的 continuous score geometry

Ordering is invariant to a common score shift, so the intrinsic score geometry has at most \(M-1\) degrees of freedom.

所有 score 一起加 common shift 不會改 ordering，所以 intrinsic score geometry 最多只有 \(M-1\) 個自由度。

The mechanism-first line prefers a fixed/constrained map rather than a generic MLP:

mechanism-first 路線偏好 fixed / constrained map，而不是 generic MLP：

For M6, \(u\in\mathbb R^5\) and \(W\in\mathbb R^{6\times5}\). Hyperplanes \(s_i=s_j\) divide the centered 5-D space into ordering chambers. All 720 M6 permutations are reachable at \(d=5\).

M6 時 \(u\in\mathbb R^5\)、\(W\in\mathbb R^{6\times5}\)。超平面 \(s_i=s_j\) 把 centered 5-D space 切成 ordering chambers；\(d=5\) 時全部 720 permutations 都可 reach。

M6 has \(\binom62=15\) pair margins, but they are redundant: \(D_{12}+D_{23}=D_{13}\). The intrinsic dimension remains five.

M6 雖然有 \(\binom62=15\) 個 pair margins，但它們是 redundant，例如 \(D_{12}+D_{23}=D_{13}\)。intrinsic dimension 仍然是 5。

§6LIF timing mapLIF timing map

In the valid regime, \(t^*\) decreases monotonically with current:

在有效區間內，\(t^*\) 對 current 單調遞減：

§7Permutation state and factorial capacityPermutation state 與 factorial capacity

Nominal capacity is not the same as reachable, used, effective, controllable, or functional capacity. Ordering cannot create information:

nominal capacity 不等於 reachable / used / effective / controllable / functional capacity。ordering 不能創造資訊：

The possible value of \(M!\) is combinatorial organization, not free information.

\(M!\) 可能帶來的是 combinatorial organization，不是免費資訊。

§8Mathematical adjacent-swap LIF actuator數學化 adjacent-swap LIF actuator

Let \(u=\pi_k\), \(v=\pi_{k+1}\) and currently \(I_u>I_v\). Take:

令 \(u=\pi_k\)、\(v=\pi_{k+1}\)，目前 \(I_u>I_v\)。取：

Choose \(\epsilon\) within neighbor and spike-feasibility margins. Then only the target pair reverses and shared monotone LIF gives:

只要 \(\epsilon\) 落在 neighbor / spike-feasibility margins 內，就只會翻轉 target pair：

The analytic actuator passed all 3600 directed edges. The minimal learned actuator selected five rank-conditioned scalar margins. After target-independent stabilization, the ordered source-target path audit passed:

analytic actuator 通過全部 3600 directed edges；minimal learned actuator 只需要 5 個 rank-conditioned scalar margins。加上 target-independent stabilization 後：

§9Task-induced functional fieldTask 誘導出的 functional field

This exact potential structure is why learned scalar potentials are attractive: \(\hat U(A\to B)=V(A)-V(B)\) guarantees antisymmetry and zero circulation.

這也是 scalar potential 很有吸引力的原因：若 \(\hat U(A\to B)=V(A)-V(B)\)，antisymmetry 與 zero circulation 自動成立。

Functional headroom remains strongfunctional headroom 並沒有隨 M 消失

At M12, about 98.5% of sampled states still have at least one beneficial adjacent move. The issue is not absence of useful actions.

M12 sampled states 裡約 98.5% 仍至少有一條 beneficial adjacent move，所以問題不是「沒有 useful action」。

§10What one-step ranking has already shownOne-step ranking 已經證明什麼

The M6 potential-ranked controller is useful as a control: compositional state information contains real one-step action signal.

M6 potential-ranked controller 現在最適合當 control：它證明 compositional state information 的確含有真實 one-step action signal。

The selected controller used 6,913 parameters with hidden width 64, without a 720-way head or factorial lookup.

selected controller 是 6,913 parameters、hidden width 64，沒有 720-way head 或 factorial lookup。

The important result is one-step learnability, not final closed-loop success. Once the policy updates \(\pi\), it changes its own input distribution; a first-step success does not guarantee recursive safety.

真正重要的是 one-step learnability，不是把這個 controller 當 final closed-loop success。一旦 policy update \(\pi\)，它就改變自己的 input distribution；第一步有效不代表後面 recursive safety 也成立。

§11Current mechanistic direction: JEPA residual action-effect factorization目前機制方向：JEPA residual action-effect factorization

Instead of directly learning scalar \(U_k\), expose the current JEPA error and the vector effect of one action:

目前改成不要直接硬學 scalar \(U_k\)，而是把 JEPA 現在錯在哪裡、以及一個 action 對 error vector 做了什麼拆開：

If the verified frozen loss is sum squared error:

如果 verified frozen loss 是 sum squared error：

For mean-MSE:

若是 mean-MSE：

This explains why the same action can flip from helpful to harmful across contexts: \(\Delta e_k\) may be similar while the current residual \(e\) changes direction. Scalar globality therefore does not automatically imply a globally complex action mechanism.

這可以解釋為什麼同一 action 在不同 context 會從 helpful 翻成 harmful：\(\Delta e_k\) 可能很類似，但 current residual \(e\) 的方向不同。所以 scalar globality 不代表 action mechanism 必然也 global。

§12Full mathematical loop: input → prediction → update → prediction → update完整數學迴圈：input → prediction → update → 再 prediction → 再 update

Step 1 · task sampletask sample

Step 2 · task representationstask representations

Step 3 · open ordering geometry打開 ordering geometry

The exact frozen \(C\) is implementation-defined; the mechanism rule is that this coordinate map cannot become an unconstrained factorial-memory network.

實際 frozen \(C\) 以 implementation 為準；mechanism rule 是這個 coordinate map 不能變成 unconstrained factorial-memory network。

Step 4 · relational staterelational state

Step 5 · real LIF timingreal LIF timing

Step 6 · prediction at current state目前 state 下 prediction

Step 7 · local candidate fieldlocal candidate field

Step 8 · decisiondecision

Autonomous operation replaces true \(U\) with a compact estimate \(\hat U\). This is the unresolved address decoder.

Autonomous operation 需要用 compact estimate \(\hat U\) 取代真實 \(U\)。這就是目前尚未解掉的 address decoder。

Step 9 · physical updatephysical update

For \(a_r=k\), \(u=\pi_{r,k}\), \(v=\pi_{r,k+1}\):

若 \(a_r=k\)，\(u=\pi_{r,k}\)、\(v=\pi_{r,k+1}\)：

after the target-independent stabilization and exact LIF timing.

在 target-independent stabilization + exact LIF timing 後成立。

Step 10 · predict again再 prediction 一次

§13What is trained, and with what loss?到底哪些東西有 training？training data / loss 是什麼？

A · JEPA / predictor training

The predictor is trained on Moving-Shapes context/target prediction and then frozen for counterfactual landscape studies.

predictor 先在 Moving-Shapes context/target prediction 上 training，之後 functional landscape 系列都 freeze。

Formal selected split: 100k train / 10k validation / 10k test. The exact \(\ell\) must follow source implementation.

selected formal split：100k train / 10k validation / 10k test。實際 \(\ell\) 必須跟 source implementation 一致。

B · Counterfactual functional labels

There is no supervised permutation class. Ranking labels are derived from candidate losses:

沒有 supervised permutation class。ranking labels 是由 candidate losses 導出：

C · Potential ranking losses

A representative pairwise objective is:

代表性的 pairwise objective：

§14Complexity and the anti-universal-key rule複雜度與 anti-universal-key rule

• forbid\(O(M!)\) permutation embeddings / logits / lookup tables
• forbidM>6 factorial training enumerationM>6 factorial training enumeration
• forbidlarge generic controller as the real mechanismlarge generic controller 成為真正 mechanism
• watchdense polynomial/global structures that are polynomial but scientifically non-compactdense polynomial / global structures，雖然 polynomial 但可能科學上仍不 compact

§15The remaining bottleneck, mathematically現在真正瓶頸，用數學講清楚

• answeredCan the state exist?state 能不能存在？
  \(M=6,d=5\Rightarrow720/720\).
• answeredCan LIF execute a requested local move?LIF 能不能執行指定 local move？
  \(3600/3600\) adjacent edges; stabilized \(518400/518400\) paths.
• openCan a compact mechanism know which move is functionally correct?小型 mechanism 能不能知道哪一條 move 對 task 是對的？

Local action does not imply local decisionLocal action 不代表 local decision

* M6 q90 used a conservative repair; M8–M12 are the stronger evidence for the globalizing trend.

* M6 q90 經過保守 repair；globalizing trend 主要以 M8–M12 為較強證據。

The unresolved source question is whether this globality is intrinsic to the task, introduced by ordering, or produced by collapsing a vector prediction error into scalar utility. A direct coordinate-linear instrument had held-out \(R^2\approx0\) and sign accuracy near 0.5, so it did not answer that question.

還沒回答的是：globality 是 task 本來就 global、ordering 造成，還是把 vector prediction error 壓成 scalar utility 後產生？direct coordinate-linear instrument 的 held-out \(R^2\approx0\)、sign accuracy 近 0.5，所以沒有回答成功。

§16Evidence that matters to the current architecture真正會影響目前架構的核心證據

§17What the architecture is now所以現在的架構到底是什麼

• resolvedState substrate: centered continuous score geometry → first-spike permutation.State substrate：centered continuous score geometry → first-spike permutation。
• resolvedState write/update: adjacent mathematical current edit → stabilized real LIF actuation.State write/update：adjacent mathematical current edit → stabilized real LIF actuation。
• resolvedTask value: frozen JEPA/task predictor defines \(L(x,\pi)\); utility is an exact loss difference.Task value：frozen JEPA/task predictor 定義 \(L(x,\pi)\)；utility 是 exact loss difference。
• partialOne-step address decoder: real signal exists, but the compact direct law is not final.One-step address decoder：確實有 signal，但 compact direct law 還不是 final。
• openRecursive address decoder: must remain safe after its own updates change the state distribution.Recursive address decoder：自己的 update 改變 state distribution 後仍要保持安全。
• currentCurrent mechanistic hypothesis: \(e+\Delta e_k+\) known JEPA loss geometry.目前 mechanistic hypothesis：\(e+\Delta e_k+\) known JEPA loss geometry。

What changed conceptually一路上真正改掉的是什麼

§18Small conclusion目前的小結論

One-sentence mental model: a compact factorial state substrate and a reliable local write mechanism already exist; the open question is whether the task admits a compact read/address law, possibly in JEPA residual geometry rather than permutation identity alone.

一句話心智模型：「compact factorial state substrate + reliable local write mechanism」已經存在；現在真正未知的是 task 有沒有 compact read/address law，而這個 law 可能要從 JEPA residual geometry 找，不一定只靠 permutation identity。