Geometric Spacetime Residuals

Narrative

Turn residuals into objects worth learning.

Rotation curves are usually treated as a contest between named physical models. This project asks a quieter question: after a chosen model has had its chance, is the remaining geometric field random noise, or does it repeat across galaxies?

Start with the inverse problem

Observed baryons predict one acceleration field. The measured rotation curve implies another. Their difference is the missing field to explain.

Give the baseline a fair fit

The experiment fits an outer-region NFW residual model first. The learned target is not the raw curve; it is what remains after that baseline is subtracted.

Learn the leftover geometry

A smooth masked dictionary compresses residual fields on a common radius grid, exposing recurring modes and coefficients that can be checked against independent diagnostics.

Mathematics

The core move is projection before learning.

The residual-of-residual construction separates baseline variation from misspecification structure. Intuitively, the algorithm learns the coherent field that survives after the fitted source family has been removed.

Residual field

The baryon-subtracted acceleration residual is the inferred field.

\[ \Delta g(R)= \frac{V_{\rm obs}^2(R)-V_{\rm bar}^2(R)}{R}. \]

After fitting a baseline source model, the learned object is:

\[ \epsilon_g(R)=\Delta g(R)-\Delta g_\theta(R). \]

Signed residuals are represented as two nonnegative channels, so dictionary coefficients remain interpretable and constrained.

\[ \epsilon_g(R)=\epsilon_g^+(R)-\epsilon_g^-(R), \qquad \epsilon_g^\pm(R)\ge 0. \]

Dictionary model

On the common normalized-radius grid, each galaxy is approximated by a small set of shared residual shapes:

\[ X_i(r)\approx \sum_{k=1}^{K} a_{ik}\,\phi_k(r), \qquad a_{ik}\ge 0. \]

Missing radial entries are handled by a mask, so the loss is evaluated only where a galaxy has observed coverage:

\[ \min_{A,\Phi\ge 0} \sum_i \left\| M_i\odot \left(X_i-\sum_{k=1}^K a_{ik}\phi_k\right) \right\|_2^2 +\lambda\,\mathcal S(\Phi). \]

Flat-tail geometry

The simplest rotation-curve fact already has geometric content. If the excess circular speed is approximately flat, then

\[ V_{\rm obs}^2(R)-V_{\rm bar}^2(R)=v_f^2, \]

so the residual acceleration is a \(1/R\) field, the weak-field potential perturbation is logarithmic, and the spherical effective-density proxy has an \(R^{-2}\) tail:

\[ \Delta g(R)=\frac{v_f^2}{R} \]

\[ \delta\Phi(R)=v_f^2\log(R/R_0)+C \]

\[ \rho_{\rm eff}(R)=\frac{v_f^2}{4\pi G R^2} \]

This is not a claim that galaxies are spherical. It is the clean diagnostic bridge from a rotation-curve residual to an inferred geometric field.

Projection theorem, in one picture

Let each profile decompose into a fitted baseline component, a latent misspecification component, and noise:

\[ x_i=b_i+s_i+\eta_i, \qquad b_i\in\mathcal B,\quad s_i\in\mathcal B^\perp. \]

Baseline projection forms the residual

\[ e_i=(I-\Pi_{\mathcal B})x_i=s_i+(I-\Pi_{\mathcal B})\eta_i. \]

Thus any cone generated by the misspecification modes is preserved after subtraction:

\[ s_i=\sum_{k=1}^{K}a_{ik}\phi_k,\quad a_{ik}\ge0 \quad\Longrightarrow\quad e_i=\sum_{k=1}^{K}a_{ik}\phi_k+\tilde\eta_i. \]

The reason for subtracting first is visible in the raw covariance. If baseline variation has scale \(c\), then

\[ C_X(c)=c^2B^\top B+S^\top S. \]

For sufficiently large \(c\), leading raw components are baseline directions. Residual learning removes \(\mathcal B\) before discovering the population structure in \(S\).

Theorem intuition

Problem

Raw profiles mix large baseline variation with smaller leftover structure.

Move

Project out the fitted baseline before learning the dictionary.

Result

Under an idealized baseline-projection model, this removes arbitrary baseline amplitude and preserves the latent misspecification modes exactly.

Meaning

The method is a model-criticism tool, not a direct declaration that one physical theory has won.

Results

The residual population is strongly compressible.

The strongest empirical claim is modest and useful: on the primary-quality SPARC sample, low-rank dictionaries reconstruct held-out radial points far better than a population mean, and the signed modes are stable enough to interpret.

131 primary-quality SPARC galaxies after coverage cuts

0.543 -> 0.098 positive residual held-out profile RMSE, mean to K=6

1.075 -> 0.168 signed NFW residual-of-residual held-out profile RMSE

0.997 median bootstrap correlation for signed mode 1

Learned positive residual and signed residual-of-residual modes — Learned population modes. The signed residual-of-residual modes describe what remains after the NFW baseline is forced to match the outer region.

Held-out reconstruction error decreases with dictionary rank — Held-out radial-point reconstruction improves through the searched rank, showing compressible population structure.

Bootstrap stability of learned modes — Bootstrap refits show that the first signed residual modes are stable enough to support interpretation.

Central overshoot diagnostic in the SPARC sample — Independent residual diagnostics supply post hoc interpretation labels, including the central-overshoot split.

Learned residual coefficient space — Learned coefficients align with the independent overshoot diagnostic, linking population modes back to physical residual behavior.

Clean learning angle

This is dictionary learning for an inverse problem with a structured nuisance baseline. The learning target is not the whole galaxy, but the projected residual field.

Physical humility

The results do not decide between dark matter and modified gravity. They show that leftover rotation-curve geometry is learnable, stable, and diagnostic.

Next extension

The immediate learning upgrade is held-out-galaxy validation: train population modes on some galaxies and predict residual structure in galaxies never used for fitting.

Artifacts

Papers and repository entry points.

The public-facing artifacts are the main conference paper and the earlier note sketches. The repository README is the entry point for code, data, and reproducibility details.

PDF

Learning the geometry left behind by galaxy rotation curves.