✗ Mechanism failed 2026

Memory-Retaining RG Feature Blocks

Implementation & benchmark of arXiv:2607.14388 — Memory Retention and the Classification of Renormalization-Group Fixed Points in Self-Similar Dynamics

Usefulness7/10

Difficulty5/10

Novelty6/10

Architecture Accuracy Generalization Scalability Cnn Ssm Training dynamics Renormalization Dynamical systems Pde Functional analysis

Source paper: Memory Retention and the Classification of Renormalization-Group Fixed Points in Self-Similar Dynamics arXiv:2607.14388 ⓘ · analyzed Aug 30, 2026

AI-generated research hypothesis, automatically tested. Not peer-reviewed.

Idea description

Replace scale-blind pooling or downsampling with a coarse-graining block that carries an explicit relevant scale variable $\eta$ alongside the feature field. The block is constrained to represent features in the memory-retaining form $h(\xi,\eta)=\eta^{\alpha}F(\xi/\eta^{\beta})$, allowing both feature amplitude and profile shape to depend on the scale inherited from the input or previous RG step.

Formulas

$$T'=T/L^{B},\qquad X'=X/L,$$

$$H_i'(T';X')=L^{A_i}H_i(T,X),$$

$$h(\xi,\eta)=\eta^{\alpha}F\!\left(\frac{\xi}{\eta^{\beta}}\right),$$

$$\mathcal{L}=\mathcal{L}_{\mathrm{task}}+\lambda_{\mathrm{RG}}\left\|z_{j+1}(x/L)-L^{\alpha}z_j(x)\right\|_2^2+\lambda_{\mathrm{prof}}\left\|\eta_j^{-\alpha}z_j(\eta_j^{\beta}u)-\bar F_j(u)\right\|_2^2.$$

Mathematical statement

The paper applies the scale transformation $T'=T/L^{B}$, $X'=X/L$, and $H_i'(T';X')=L^{A_i}H_i(T,X)$, where $L>0$ is the coarse-graining factor, $B$ is the time-scaling exponent, $A_i$ is the field-scaling exponent, and $T,X,H_i$ are time, coordinates, and field components. A memory-retaining fixed point has $h(\xi,\eta)=\eta^{\alpha}F(\xi/\eta^{\beta})$, where $\eta$ is a surviving relevant scale, $\alpha$ controls amplitude scaling, $\beta$ controls profile-width scaling, and $F$ is a scale-independent profile. Implement the neural analogue with a feature field $z_j(x)$ and scale state $\eta_j$: under a coarse-graining factor $L$, update $x'=x/L$, $\eta_{j+1}=\eta_j/L$, and require approximately $z_{j+1}(x')=L^{\alpha}z_j(x)$ after accounting for the learned profile transformation. The useful property is explicit retention of a variable that ordinary pooling discards.

Implementation notes

1. Integration point: insert the block after every second CNN or UNet downsampling stage, or after each SSM state update for sequence data. Maintain a feature tensor $z_j\in\mathbb{R}^{C\times H\times W}$ and a positive scalar or per-channel scale state $\eta_j=\operatorname{softplus}(q_j)+\epsilon$. Use an ordinary strided convolution to produce a coarse feature, then concatenate $\log\eta_j$ and apply FiLM modulation with learned functions $a(\log\eta_j)$ and $b(\log\eta_j)$. Resample a fine feature at coordinates $x=\eta_j^{\beta}u$ to construct the normalized profile. 2. Pseudocode: initialize $\eta_0$ from the known object size, sequence sampling interval, graph radius, or a learned positive value; for each block compute $z=\mathrm{CoarseConv}(z)$, $z\leftarrow a(\log\eta)\odot z+b(\log\eta)$, predict $\Delta q$, set $\eta\leftarrow\eta/L$ or $\eta\leftarrow\operatorname{softplus}(q+\Delta q)$, and add the RG consistency loss using two differently rescaled views of the same example. Learn $\alpha$ and $\beta$ globally or per channel, initialized at zero. 3. Computed from the paper: the scale transform and the retained-scale functional form. Estimated empirically: $\alpha$, $\beta$, the scale state, and the profile $F$; estimate $\beta$ by regressing log feature width against $\log\eta$. 4. First cheap experiment: train a small ResNet-18 on CIFAR-10 with random resize augmentation, comparing ordinary strided blocks, scale-conditioned blocks without RG loss, and the full block. Evaluate on held-out object-size ranges. The predicted signature is that the full model has a lower cross-scale consistency residual, $R(L)=\|z_{j+1}(x/L)-L^{\alpha}z_j(x)\|/\|z_j\|$, and that its fitted $\beta$ is reproducible across seeds. For a dataset with genuine size variation, the retained model should yield a confidence interval for $\beta$ excluding zero; on a deliberately scale-invariant synthetic dataset, $\beta$ should collapse to zero and the extra mechanism should provide no benefit. Test whether removing the $\eta$ channel increases validation error specifically on unseen scales rather than merely changing in-distribution accuracy.

Verification

Mechanism failed

Stage 1 · Toy mechanism gate: Failed ✗

Stage 2 · Mechanism transferred to benchmark: Not tested

Stage 2 · Practical benchmark result: Not run

Methodology: Toy-system gate first; the benchmark stage runs only after a pass. How verification works

Stage 1 — Mechanism check agent confidence 9/10

Built a compact memory-retaining profile implementation and scale-conditioned 1D feature block. The toy sweeps recovered alpha=0.700 and beta=0.620 from amplitude and width scaling, and the mathematically matched update eta'=eta/L^(1/beta) gave residuals below 2e-16 across L=1.25–3. However, the idea's stated eta'=eta/L update gave residuals 0.159, 0.306, 0.573, and 1.035 for those L values, exposing a coordinate/scale-update mismatch when beta is not 1. In the tiny classifier, the baseline achieved 1.000 accuracy and 0.0318 loss versus 0.776 and 0.4136 for the eta-conditioned idea, so there is no observed practical win here.

Agent confidence: 9/10

Baseline: held-out accuracy 1.000; cross-entropy 0.03184
Idea: held-out accuracy 0.776; cross-entropy 0.41361; exact-profile alpha fit 0.700 vs true 0.700; beta fit 0.620000 vs true 0.620; matched-RG residual 1.36e-16 to 2.00e-16; prescribed eta/L residual 0.159 to 1.035

Limitations: Only a synthetic 1D signal task was tested, with one seed and a deliberately tiny network; no CIFAR-10/UNet/SSM integration, multiple-seed confidence intervals, learned alpha/beta, profile loss, FLOP benchmarking, or real-image unseen-scale evaluation was performed. The exact residual test uses the corrected eta scaling implied by h(x,eta), not the idea's eta/L rule.

How to run: python3 experiment.py

Verdict computed by deterministic test code from paired-seed statistics — not by the language model.

Artifacts

Implementation overview ⬇ Download all as ZIP 3 files · code, reports and structured results

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