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294 lines
10 KiB
Markdown
294 lines
10 KiB
Markdown
# Functional Specification: SSMS-nD
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## Scalar State Manifold Segmentation — Variable Dimension
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**Document ID:** FS-SSMS-nD-2026-04-20
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**Authority:** Clean Room Implementation Protocol
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**Status:** SEALED — Source of Truth for All Implementations
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---
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## 1. Scope and Mathematical Objective
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Implement a promptable object detection system using OISC (One Instruction Set Computer) architecture operating on a **Dynamic n-Manifold** where $n \in [1, N_{\max}]$ is variable per detection instance.
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The system must lift **1D sequential data** (token streams, time series, feature tubes) into **n-dimensional submanifolds** where $n$ is determined dynamically by:
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- Intrinsic dimensionality of the detected entity
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- Prompt-driven structural constraints
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- Topological stability under $H_M(t)$ evolution
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---
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## 2. Input/Output Requirements
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### Primary Input
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$$I_{1D} \in \mathbb{R}^{L \times d}$$
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A 1D sequence of length $L$ with $d$-dimensional features per position (e.g., CLIP tokens, depth samples, audio frames).
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### Prompt Inputs
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- **Text**: UTF-8 string → embedded via frozen encoder to $\mathbb{R}^{d_{embed}}$
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- **Point**: 1D coordinate $t \in [0, L]$ (position in sequence)
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- **Structure**: Target dimensionality hint $n_{target} \in [1, N_{\max}]$
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- **Constraints**: Holonomic constraint equations $\{h_j(x) = 0\}_{j=1}^{m}$
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### Outputs
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A set of n-manifold embeddings $M = \{M_1, M_2, \dots, M_k\}$, where each:
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$$M_i = (c_i \in \mathbb{R}^n, \Sigma_i \in \mathbb{R}^{n \times n}, \theta_i \in \mathbb{R}^{p}, \sigma_i \in \{0,1\})$$
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- $c_i$: center coordinates (n-dim)
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- $\Sigma_i$: metric tensor (covariance structure)
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- $\theta_i$: orientation parameters (p-dim, $p \leq n(n-1)/2$)
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- $\sigma_i$: activation status (spawned/folded)
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---
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## 3. Core Mathematical Modules
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### Module A: Sequential Lifting Operator $\mathcal{L}_{1D \to n}$
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Lifts a 1D sequence interval $[t_0, t_1]$ into $\mathbb{R}^n$ via learned coordinate chart:
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$$\mathcal{L}_{1D \to n}: [t_0, t_1] \times \mathbb{R}^{d} \to \mathbb{R}^n$$
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$$\mathcal{L}(t, f(t)) = W_{lift} \cdot \text{Pool}(f([t_0, t_1])) + b_{lift}$$
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**Constraint**: $W_{lift} \in \mathbb{R}^{n \times d'}$ must be ternary-quantized ($\{-1, 0, 1\}$).
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**Dynamic n Selection**:
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$$n = \arg\min_{n' \in [1,N_{\max}]} \left[ \| \mathcal{L}_{n'}(I) - \text{Prompt}(I) \|^2 + \lambda \cdot \text{Complexity}(n') \right]$$
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where $\text{Complexity}(n') = n' \cdot \log(n')$ (Betti number penalty).
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---
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### Module B: Variable-n Manifold Representation
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Each manifold $M_i$ has **dynamic dimensionality** $n_i$ determined at spawn time.
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#### B.1 Scalar Node Allocation
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- Each dimension requires 1 scalar node
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- Total nodes for $M_i$: $n_i$ (centers) + $n_i(n_i+1)/2$ (upper-triangular $\Sigma$) + $p$ (orientations)
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- Stored as contiguous block in SRAM bank $b = i \mod B$
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#### B.2 Holonomic Constraints (Generalized)
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For manifold $M_i$ with dimension $n_i$, maintain $m_i$ constraints:
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$$\{h_j(x_1, \dots, x_{n_i}) = 0\}_{j=1}^{m_i}$$
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**Linear Constraints** (handled via ACI):
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$$\sum_{k=1}^{n_i} a_{jk} x_k = b_j \quad \Rightarrow \quad \text{ACI: } |\sum a_{jk} x_k - b_j| \leq \epsilon$$
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**Nonlinear Constraints** (handled via Lagrange multipliers in $V_M$):
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$$V_{constraint}(x) = \sum_{j=1}^{m_i} \lambda_j \cdot h_j(x)^2$$
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#### B.3 Yaw Generalization: SO(n) Representation
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For $n \geq 2$, orientation lives on special orthogonal group $SO(n)$.
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Storage: $n(n-1)/2$ independent parameters (Givens rotation angles or Cayley vectors).
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Holonomic constraint (orthonormality):
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$$R^T R = I_n \quad \Rightarrow \quad n(n+1)/2 \text{ constraints}$$
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ACI enforcement: $| (R^T R)_{ij} - \delta_{ij} | \leq \epsilon$ for all $i \leq j$.
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---
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### Module C: Prompt-Driven Potential Fields $V_M(x, t, n)$
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Extended potential now depends on target dimensionality $n$:
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$$V_M: \mathbb{R}^n \times \mathbb{R} \times \mathbb{N} \to \mathbb{R}$$
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#### C.1 Semantic Potential (Dimension-Agnostic)
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$$V_{semantic}^{(n)}(x) = -\langle f_{seq}(\mathcal{L}^{-1}(x)), \tilde{e}_{prompt} \rangle$$
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where $\mathcal{L}^{-1}: \mathbb{R}^n \to [0,L]$ is approximate inverse chart.
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#### C.2 Spatial Potential (1D → n)
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$$V_{spatial}^{(n)}(x; t_{prompt}) = \| x - \mathcal{L}_{1D \to n}(t_{prompt}) \|_2^2$$
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#### C.3 Structure Potential (Prompt-Driven Dimensionality)
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$$V_{structure}^{(n)}(x; n_{target}) = \begin{cases} 0 & \text{if } n = n_{target} \\ \eta \cdot |n - n_{target}| & \text{otherwise} \end{cases}$$
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#### C.4 Constraint Potential
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$$V_{constraint}(x) = \sum_{j=1}^{m} \lambda_j \cdot h_j(x)^2$$
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---
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### Module D: Betti Swoosh in Variable Dimensions
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The Betti Swoosh Hamiltonian extends to variable $n$:
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$$H_M^{(n)}(t) = -\Delta_M^{(n)} + V_M^{(n)}(x, t)$$
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where $-\Delta_M^{(n)}$ is the n-dimensional Hodge Laplacian.
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#### D.1 Dynamic ACI (Anti-Collision Identity)
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Two manifolds $M_i, M_j$ with dimensions $n_i, n_j$ collide if:
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**Case 1: $n_i = n_j = n$ (same dimension)**
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$$\| c_i - c_j \|_2 < \tau_{nms}^{(n)}$$
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**Case 2: $n_i \neq n_j$ (different dimensions)**
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Project higher to lower via $\pi: \mathbb{R}^{\max(n_i,n_j)} \to \mathbb{R}^{\min(n_i,n_j)}$:
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$$\| \pi(c_i) - \pi(c_j) \|_2 < \tau_{nms}^{(\min)}$$
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Suppression: Lower-energy manifold folded.
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#### D.2 Betti Number Tracking
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Track $\beta_k$ for all $k \in [0, n_{\max}]$ simultaneously:
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- $\beta_0$: connected components (count of active $M_i$)
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- $\beta_1$: 1D holes (loops in manifold adjacency)
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- $\beta_k$: k-dimensional cavities
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Swoosh event defined as cascade across dimensions: rank increase in $\beta_{n-1}$ followed by collapse to $\beta_n$ stability.
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---
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## 4. Implementation Constraints (Clean Room)
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### 4.1 SUBLEQ OISC Requirements
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All operations must reduce to:
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```
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M[b] ← M[b] − M[a]
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if M[b] ≤ 0: PC ← c
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```
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**Variable-n Specific Instructions**:
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- `LIFT_1D_n`: Allocate n scalar nodes, populate from 1D sequence pool
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- `CONSTRAIN_m`: Apply m holonomic constraints via ACI check
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- `PROJECT_n_m`: Project n-dim coordinates to m-dim subspace ($m < n$)
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### 4.2 Q16.16 Fixed-Point Throughout
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All calculations use 32-bit Q16.16:
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- Center coordinates: $c_i \in [-2^{15}, 2^{15}]$ metres (Q16.16)
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- Metric tensor: $\Sigma_{ij} \in [0, 2^{16}]$ (positive semi-definite enforced via ACI)
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- Orientation: Givens angles $\theta \in [-\pi, \pi]$ mapped to Q16.16
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**Dynamic Range Scaling**:
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For high-dimensional manifolds ($n > 8$), use block-floating-point:
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- Shared exponent per $M_i$ stored in scalar header
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- Mantissas: Q8.8 per dimension (16-bit packed pairs)
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### 4.3 Ternary Quantization
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All weight matrices ternary:
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$$W_{lift}, W_{orient}, W_{constraint} \in \{-1, 0, 1\}^{n \times m}$$
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MatMul-free execution via ADD/SUB accumulation:
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$$y_i = \sum_j W_{ij} x_j \Rightarrow \text{ADD if } W_{ij}=1, \text{ SUB if } W_{ij}=-1$$
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### 4.4 Butterfly Gossip Protocol
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Variable fanout based on manifold dimension:
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$$n_{contact}^{(n)} = \lceil \log_2 (k_n) \rceil$$
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where $k_n$ = count of active n-dimensional manifolds.
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Stratified gossip: separate butterfly networks per dimension $n$ to prevent crosstalk.
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---
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## 5. Verification Metrics
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### 5.1 Center-Distance AP (Per-Dimension)
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For each dimensionality $n$, compute AP based on:
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$$\text{TP}_n: \| c_{pred} - c_{gt} \|_2 < \tau_{AP}^{(n)}$$
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Thresholds scale with dimension:
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$$\tau_{AP}^{(n)} = \tau_{base} \cdot \sqrt{n}$$
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### 5.2 Holonomic Constraint Violation
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Measure ACI satisfaction rate:
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$$\text{ACI}_{score} = \frac{1}{m \cdot k} \sum_{i=1}^{k} \sum_{j=1}^{m_i} \mathbb{1}[|h_j(M_i)| \leq \epsilon]$$
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Target: $\text{ACI}_{score} > 0.99$
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### 5.3 Dimension Selection Accuracy
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When ground-truth dimension $n_{gt}$ is known:
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$$\text{DimAcc} = \frac{1}{k} \sum_{i=1}^{k} \mathbb{1}[n_i = n_{gt,i}]$$
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---
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## 6. SUBLEQ Program Layout
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### Memory Map (Per Manifold $M_i$ with dimension $n$)
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```
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M[base + 0 .. n-1]: center coordinates c[0..n-1]
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M[base + n .. n+n(n+1)/2-1]: metric tensor Σ (upper triangular)
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M[base + n(n+3)/2 .. p-1]: orientation params θ[0..p-1]
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M[base + header - 4]: dimension n
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M[base + header - 3]: constraint count m
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M[base + header - 2]: energy e_i
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M[base + header - 1]: activation σ_i
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```
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### Variable-n SUBLEQ Kernel Pseudocode
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```sUBLEQ
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; LIFT_1D_n: Populate n centers from 1D sequence
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; Input: seq_ptr, start_t, end_t, target_n, dest_base
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LIFT_LOOP:
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SUBLEQ M[seq_ptr], M[accum], CHECK_DONE ; load sequence value
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SUBLEQ M[divisor], M[accum], NEXT ; normalize
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SUBLEQ M[accum], M[dest_base + i], STORE ; store to center[i]
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SUBLEQ M[one], M[i], INC_I ; i++
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SUBLEQ M[target_n], M[i], LIFT_LOOP ; loop if i < n
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SUBLEQ M[zero], M[zero], DONE ; halt
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; CONSTRAIN_m: Apply m holonomic constraints
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CONSTRAIN_LOOP:
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SUBLEQ M[constraint_a + j], M[dot], ACCUM ; accumulate a_j · x
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SUBLEQ M[dot], M[constraint_b + j], CHECK ; compare to b_j
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SUBLEQ M[epsilon], M[residual], FAIL ; |residual| > ε?
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SUBLEQ M[one], M[j], INC_J ; j++
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SUBLEQ M[constraint_m], M[j], CONSTRAIN_LOOP
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; Betti Swoosh trigger on constraint violation
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FAIL:
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SUBLEQ M[fold_signal], M[dest_base + σ_offset], FOLD
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```
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---
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## 7. Lean 4 Formalization Requirements
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### Required Definitions
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1. **VariableDimensionManifold (n : Nat)**: Structure with dynamic $n$
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2. **LiftingOperator (d n : Nat)**: Chart $\mathcal{L}_{1D \to n}$
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3. **HolonomicConstraint (n m : Nat)**: Constraint system with $m$ equations
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4. **DynamicACI (n_i n_j : Nat)**: Cross-dimensional collision predicate
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5. **BettiSwooshND (n_max : Nat)**: Hamiltonian over all dimensions $[1, n_{max}]$
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### Required Theorems
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1. `liftingPreservesTopology`: Chart is homeomorphism onto image
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2. `holonomicConstraintACI`: $|h(x)| \leq \epsilon$ preserved under MLGRU
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3. `dynamicACICompleteness`: All collisions detected across dimensions
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4. `variableDimNmsSound`: Suppressed manifolds satisfy post-condition
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5. `bettiNumberInvariance`: $\sum_k (-1)^k \beta_k$ conserved under swoosh
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---
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## 8. Clean Room Compliance Checklist
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- [ ] No reference to SAM, SAM3, or WildDet3D source code
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- [ ] All math derived from public pinhole model + differential geometry
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- [ ] Implementation derived solely from this FS document
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- [ ] Ternary quantization from BitNet/1.58-bit paper (public)
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- [ ] SUBLEQ from Mavaddat & Parhami 1988 (public domain)
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- [ ] Betti numbers from standard algebraic topology
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- [ ] Q16.16 from DSP textbooks
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---
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**SEALED:** This specification is the sole source of truth.
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**DATE:** 2026-04-20
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**VERSION:** SSMS-nD-1.0
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