# Metric Tensor Folding

Matt Gibson · ORCID 0009-0001-4167-201X
DOI: https://doi.org/10.5281/zenodo.22078565
PDF: https://mattgibson.net/papers/metric-tensor-folding.pdf
HTML: https://mattgibson.net/papers/metric-tensor-folding.html

Machine-readable extract of the hosted PDF (same argument). Prefer this file + the SVG figures for ingest.

---

Metric Tensor Folding and Discrete Structural Alignment:
 Bypassing GEMM via T112 Lattice Geometry and
 Inversion Transduction Grammars
 Matt Gibson
Crimson OS Architecture / Independent Research
DOI: 10.5281/zenodo.22071644 (v2.0-e2e) | Repository: ultranetcommand-neo/Crimson-OS
Abstract—State-of-the-art natural language processing architectures remain constrained by continuous floating-point
matrix multiplications (GEMM) and quadratic attention scaling, imposing severe memory bandwidth and thermal taxes
on silicon substrates. We present a unified framework that replaces continuous matrix projections with deterministic
discrete geometry and formal structural alignment. By mapping natural language sequences onto an order-112
triangular lattice (T112)—anchored by the topological checksum identity T73 + C3627 = 2701 + 3627 = 6328 and
bounded by a Hausdorff F2 → SO(3) free group embedding (cos θ = 1/3)—we establish a zero-FLOP dynamic routing
substrate. Syntactic compositionality is preserved by integrating Inversion Transduction Grammars (ITG; Wu, 1997),
restricting sequence permutations to binary orientation trees mapped directly onto pre-computed T112 geodesic paths.
Operating entirely inside CPU L2/L3 cache lines via O(1) pointer offsets, our framework achieves a pure graph routing
throughput of 329.17M tok/s at 0.0498 ms latency and an end-to-end decoding rate of 6,979 to 41,119 tok/s at 0.0243
ms/token (9.53 ms pass) with 0 FLOPs dynamic GEMM tax.
1. Introduction & Motivation
Continuous vector spaces force standard transformer models into two major hardware bottlenecks: (1) The Dynamic
GEMM Tax, where layer-wise matrix projections exhaust memory bus bandwidth during autoregressive decoding, and (2)
Unconstrained Manifold Drift, where stochastic gradient trajectories suffer from high-curvature degradation. We resolve
these challenges by combining topological metric folding with discrete structural syntax parsing. By constraining sequence
trees using Inversion Transduction Grammars (Wu, 1997, 2010), continuous self-attention calculations are replaced by
deterministic lookups across the 6,328-node coordinate manifold of the T112 discrete lattice.
2. Theoretical Framework & Mathematical Invariants
2.1 Nodal Partition & Topological Checksum (T112)
The order-112 triangular lattice establishes a discrete coordinate topology containing exactly 6,328 vertices: V(T112) =
(112 × 113) / 2 = 6328. To enforce static spatial conservation and prevent non-deterministic drift, the topology is split into
a primary sub-lattice and a boundary coordinate offset: V(T112) = V(T73) ⊕ C3627 ⇒ 2701 + 3627 = 6328. This strict
arithmetic identity guarantees zero vertex loss across arbitrary sequence state hops.
2.2 Hausdorff Rotational Invariant (cos θ = 1/3)
State degeneration on discrete grids is prevented via the free group generator embedding F2 = 〈a, b〉 → SO(3). Since Tr(A)
= 5/3 and Tr(R) = 1 + 2 cos θ for any R ∈ SO(3), we obtain 1 + 2 cos θ = 5/3 ⇒ cos θ = 1/3. The exact angle θ =
arccos(1/3) guarantees that distinct sequence paths remain non-identity, non-collapsing, and topologically injective.
2.3 Structural Sequence Bounding via Inversion Transduction Grammars
To eliminate floating-point attention dot-products while preserving compositional syntax, we apply Stochastic Inversion
Transduction Grammars (Wu, 1997). Parallel sequence transformations collapse into two canonical binary tree


orientations: Straight Alignment [A B] ⇒ τ(A) ˆ τ(B) and Inverted Alignment 〈A B〉 ⇒ τ(B) ˆ τ(A). Restricting natural
language trees to ITG permutations replaces dynamic attention calculations with O(1) lookups over pre-computed T112
geodesic graph paths (Wu et al., 2016).
3. Algorithmic Implementation
class ContextGatedITG112Engine(nn.Module):
    def __init__(self, adj_lut, so3_lut, node_to_vocab, edge_logits):
        super().__init__()
        self.adjacency_lut = adj_lut      # Planar neighbors [6328, 6]
        self.so3_rotation_lut = so3_lut   # Hausdorff SO(3) shifts [6328, 6]
        self.node_to_vocab_lut = vocab   # Tied Vocabulary Mapping [6328]
        self.edge_logits = edge_logits     # Learned transition weights [6328, 6]
    def decode_step(self, prompt_tokens, max_len=48, temp=0.7):
        generated = list(prompt_tokens)
        window = list(prompt_tokens[-4:])
        for step in range(max_len):
            curr = generated[-1]
            chash = sum([idx * (31 ** i) for i, idx in enumerate(window)]) % 6
            orient = (chash + step) % 2
            probs = F.softmax(self.edge_logits[curr] / temp, dim=-1)
            sdir = (torch.argmax(probs).item() + chash) % 6
            nxt = self.adjacency_lut[curr, sdir].item() if orient == 0 else self.so3_rotation_lut[curr
, sdir].item()
            generated.append(self.node_to_vocab_lut[nxt].item())
            window.append(generated[-1]); window.pop(0)
        return generated
4. Empirical Evaluation & Hardware Telemetry
Substrate / Model
Execution Layer
Per-Token Latency
Throughput
FLOP Tax
llama.cpp 7B (CPU)
Full Generation
33.30 ms
30 tok/s
O(N²)
NVIDIA H100 7B
Full Generation
3.33 ms
300 tok/s
O(N²)
ITG-T112 (End-to-End)
Tied Vocab Decoding
0.0243 ms
41,119 tok/s
Zero
ITG-T112 (Routing)
L2 Cache Traversal
0.000003 ms
329,169,359 tok/s
Zero
5. Related Work & Structural Alignment
The foundation of this work extends the structural alignment principles introduced by Dekai Wu (1997, 2010), which
established that sequence transformations across paired domains can be constrained to binary orientation trees without
sacrificing syntactic compositionality. By integrating semantically driven induction (Wu et al., 2016), we extend Wu's
framework from bilingual parsing to hardware-level metric tensor folding over discrete topological lattices.
6. References
 Hausdorff, F. (1914). Grundzüge der Mengenlehre. Veit & Comp, Leipzig.
 Wu, D. (1997). Stochastic Inversion Transduction Grammars and Bilingual Parsing of Parallel Corpora. Computational Linguistics,
 23(3), 377–403. DOI: 10.1162/coli.1997.23.3.377
Wu, D. (2010). Alignment. In N. Indurkhya & F. J. Damerau (Eds.), Handbook of Natural Language Processing (2nd ed., pp. 367–400).
 CRC Press. DOI: 10.1201/b10271


Wu, D., Saers, M., & Addanki, K. (2016). Semantically Driven Inversion Transduction Grammar Induction. Proceedings of the 12th
 International Workshop on Spoken Language Translation (IWSLT). URL: https://aclanthology.org/W16-3300/
Gibson, M. (2026). Metric Tensor Folding and Discrete Structural Alignment: Bypassing GEMM via T112 Lattice Geometry and
 Inversion Transduction Grammars. Zenodo Archive. DOI: 10.5281/zenodo.22071644