Ergodic Phonetic Manifolds: Dynamic Memory

• Ergodic phonetic manifolds are a continuous, high-dimensional framework that encodes phonetic data as persistent geometric trajectories.
• They integrate nonlinear ergodic dynamics with acoustic injection to achieve constant-time memory operations and over 3,000× compression.
• The system fuses geometric and semantic signals via a dual-process consensus, ensuring robust retrieval and suppressing semantic hallucinations.

Ergodic phonetic manifolds are a mathematical and algorithmic framework for encoding, storing, and reconstructing linguistic data as continuous, dynamical trajectories rather than discrete, static records. This paradigm underlies Phonetic Trajectory Memory (PTM), a memory system for LLMs that achieves asymptotically infinite context memory using fixed-size state, nonlinear ergodic dynamics, and resonance-based retrieval. The manifold’s topology, unitary evolution, and coupling to phonetic representations enable dramatic compression—over 3,000× relative to dense key-value (KV) caches—while maintaining high factual accuracy and suppressing semantic hallucinations. In this architecture, memory storage is recast from an accumulation of tokens to the persistence of a geometric path on a high-dimensional torus, fundamentally altering the scaling and operational characteristics of context memory (Houichime et al., 23 Dec 2025).

1. Topology and Dynamics of Ergodic Phonetic Manifolds

The central construct is the 16-dimensional torus $\mathbb{T}^{16} = \mathbb{R}^{16}/\mathbb{Z}^{16}$, forming the state space for all compressed phonetic vectors. The torus possesses finite volume ($V=1$), and distances are measured using the Lee (toroidal) metric: $$d_{\mathbb{T}}(u,v) = \sqrt{\sum_{i=1}^{16} \min\bigl(|u_i-v_i|,\;1-|u_i-v_i|\bigr)^2}.$$ Temporal evolution across the manifold is realized by iterated action of a block-diagonal orthogonal rotation operator in $SO(16)$. Each timestep’s rotation is parameterized as

$$\mathcal{R}(t) = \bigoplus_{k=1}^{8} \begin{pmatrix} \cos(\omega_k t) & -\sin(\omega_k t) \ \sin(\omega_k t) & \cos(\omega_k t) \end{pmatrix},$$

with angular frequencies $\omega_k = \pi\sqrt{p_k}$ ($p_k$: $k$-th prime), ensuring that every ratio $\omega_k/2\pi$ is irrational. By Kronecker’s and Weyl’s Equidistribution Theorems, the resulting trajectories $\{\mathcal{R}(t)x\}$ are dense and non-periodic on $\mathbb{T}^{16}$—i.e., ergodic and never repeating exactly—guaranteeing that the path preserves all injected information without self-intersection.

This evolution is strictly unitary ($\mathcal{R} \in SO(16)$, $\det\mathcal{R}=1$), preserving vector norm at every step and confining numerical drift to $$E_{\mathrm{drift}}(t) \approx \sqrt{t}\,\delta_{\mathrm{machine}}$$, with $\delta_{\mathrm{machine}}\sim10^{-7}$ (float32). For sequence lengths $t\sim10^6$, numerical noise remains well below the threshold for phonetic discrimination.

2. Language Injection, Manifold Evolution, and Resonance Retrieval

PTM decomposes interaction with the manifold at every timestep $t$ into three transformations:

1. Acoustic Injection $\Phi$: Each token $x_t$ is projected into a 16-dimensional phonetic force vector, relying on the token’s IPA features and a semi-orthogonal projection matrix $W_{\mathrm{proj}}$:

$$\Phi(x_t) = W_{\mathrm{proj}}[\operatorname{IPA}(x_t)] \in \mathbb{R}^{16}.$$

This encoding is lossless for phonetic features and prosody, allowing controlled lossiness for semantics.

2. Ergodic Evolution $\mathcal{R}$: The memory state evolves by unitary rotation and injection:

$$S_{t+1} = \mathcal{R}S_{t} \oplus \Phi(x_t)\quad (\bmod\;1),$$

strictly preserving information. This operator guarantees infinite-horizon, non-degrading storage, aligning with conservative dynamical systems.

3. Manifold Resonance $D$ (Retrieval): To retrieve a prior token $x_{t-k}$, the process inverts the rotation,

$$V_{\mathrm{rec}} = (S_t - \mathcal{R}S_{t-1}) \bmod 1,$$

extracting the phonetic component, which is then matched via cosine similarity to a vocabulary matrix $M_{\mathrm{vocab}}$. Final candidates are fused using both geometric and semantic signals (“Signal Consensus” mechanism).

Pseudocode outlining these operations is provided in the data, with each routine operating at $O(1)$ time and space per token, independent of context length (Houichime et al., 23 Dec 2025).

3. Constant-Time, Fixed-Size Memory Operations

Unlike conventional transformers, which accumulate KV memories with $O(N)$ cost, PTM maintains a fixed-size vector $S_t\in\mathbb{T}^{16}$ throughout the session. Both writing (encoding) and reading (retrieval) are reduced to constant-time primitives: a $16\times16$ matrix multiply, modular addition, and nearest-neighbor search in the manifold. As a result,

$$\mathrm{Time}(\mathrm{PTM\_Decode}) = T_{\mathrm{rotate}}^{O(1)} + T_{\mathrm{broadcast}}^{O(1)} + T_{\mathrm{LLM-inference}}$$

with no scaling in computation or memory as context depth increases. This enables practical “infinite context” operation bounded only by physical precision limits.

4. Empirical Compression, Latency, and Retrieval Fidelity

The compression capabilities of ergodic phonetic manifolds in PTM are quantified by their replacement of high-dimensional dense caches with a compact phonetic trace. Tokens are classified as “Anchors” (high-entropy, sparse KV retention) or “Bridges” (fully manifold-folded). For Bridges, a conventional dense KV uses $\sim$4096-dimension per token ($\approx$8 KB in FP16), while PTM reduces this to a single 16-d float32 vector (64 bytes), a raw compression of $4096\!:\!16=256$. With drop rates of 85–95% for Bridges, net compression exceeds 3,000×.

Quantitative results (see Table below) explicitly demonstrate PTM’s memory-latency-accuracy tradeoffs:

Test Suite	Compression	Accuracy
20,000-token narrative stream	4.4×	89.2 ± 1.4 %
Sci-Fi narrative	3.41×	92.34 % (205/222)
Historical narrative	3.64×	90.15 % (302/335)
“Blind Walk” (zero anchors)	>3000×	83.58 % (335 toks)

Retrieval latency, measured on CPU+NumPy, is ≈6.8 µs (encode) and ≈14.1 µs (decode), with worst-case generative reconstruction latency at 35.6 ms using quantized CUDA-accelerated LLM inference—well within interactive time constraints. Empirical accuracy plateaus at 89–92% on long-form, knowledge-intensive text, independent of retrieval depth. Anchor recall achieves 100% fidelity; errors manifest predominantly as phonetic mutations (e.g., “Kings”→“Zink”), not semantic drift (Houichime et al., 23 Dec 2025).

5. Signal Consensus Mechanism and Hallucination Control

PTM retrieval fuses two probability sources for each candidate token $c$:

1. Semantic Prior $P_{\theta}(c)$:

$$P_{\theta}(c) = \operatorname{softmax}_{c} \left( \log P_{\mathrm{LLM}}(c \mid C_{\mathrm{local}}) \right)$$

Evaluates the LLM’s statistical confidence.

2. Geometric Likelihood $P_{\phi}(c)$:

$$P_{\phi}(c) = \operatorname{softmax}_{c} \left( -\gamma \, \lVert (\mathcal{R}S_{t-1}\oplus\Phi(c)) - S_t \rVert_{\mathbb{T}} \right)$$

Quantifies geometric/phonetically grounded plausibility.

3. Consensus Mixture:

$$P_{\mathrm{total}}(c) = \alpha P_{\theta}(c) + (1-\alpha)P_{\phi}(c)$$

with $\alpha\approx0.4$ empirically optimal for biasing retrieval toward acoustic veracity. This mechanism suppresses semantic hallucinations—LLM outputs phonetically inconsistent with manifold state—by weighting geometric resonance, sustaining up to ≈92% factual accuracy on tasks requiring knowledge recall.

“Resonance logs” reveal that, when the LLM’s semantic evidence $P_\theta$ is weak, retrieval gracefully defers to the geometric likelihood $P_\phi$. Anchors (retained tokens) form robust “pillars” of certainty; Bridges (folded tokens) rely adaptively on the strongest available signal.

6. Limitations and Open Technical Challenges

Failure modes and sensitivities are domain-specific:

• Anchor-Selection Irrevocability: If the LLM’s attention mechanism mislabels a token as low-entropy (Bridge), the token is irretrievably absorbed into the manifold, precluding access to symbolic ground truth [Sec 8.1].
• Domain Redundancy Bias: Compression and accuracy rely upon redundancy; in low-redundancy domains (e.g., code, cryptographic material), small phonetic corruption can cause syntactic or logical errors [Sec 8.2].
• Phonetic Homomorphisms: Homophones (e.g., “raise” vs. “raze”) in ambiguous contexts are mathematically indeterminate at retrieval, forcing reliance on statistical priors [Sec 8.3].
• Precision Barrier: Domains requiring exact lexical or numeric recovery (legal, medical, computational) cannot tolerate phonetic ambiguity; adaptation requires anchoring critical tokens [Sec 8.4].
• Finite Precision and Cyclicity: Although float32 arithmetic is ultimately periodic, the cycle length ($L_{\mathrm{sys}}\sim 2^{192}$) far exceeds any practical requirement, and drift remains well below error thresholds at human timescales [Sec 2.6].

7. Broader Significance and Conceptual Reframing

The ergodic phonetic manifold, as realized in PTM, redefines textual memory for LLMs from an $O(N)$ collection of static token records to a strictly $O(1)$ dynamical trajectory in a conserved geometric space. By encoding only the acoustic trace—“address” rather than semantic “meaning”—and reconstructing through resonance, PTM demonstrates:

• Infinite-horizon fidelity without cumulative resource growth
• Constant-time access latency irrespective of context depth
• Compression ratios exceeding 3,000× relative to uncompressed dense KV retention
• Empirical recall accuracy of 89–92% across diverse narrative benchmarks
• Hallucination resistance via dual-process signal consensus

The generalization drawn is that infinite context does not necessitate infinite hardware; rather, it is achievable by exploiting the ergodicity and conservation properties of irrationally rotating high-dimensional tori. This framework recasts machine memory as a physically-conserved process—“an undying signal”—and enables new pathways for efficient, robust long-term sequence modeling (Houichime et al., 23 Dec 2025).