index

The dissipation β-function describes accumulation: as systems coarse-grain, maintenance load $\eta$ rises monotonically, effective dimension falls, and the coupled flow approaches a horizon at $(\eta, d) = (1, 2)$. The geometry of self-correction establishes that non-integrable feasibility projections create an irreducible curl floor — a structural minimum on circulation that no correction field can eliminate. Together, these results characterize what happens to systems that cannot globally optimize. What they leave open is what happens when systems locally resist.

This paper treats repair as a first-class process in the constraint geometry framework. Repair consists of localized architectural operations — reconfiguration, consolidation, re-routing — that attempt to reduce excess curl generated by history and accumulated structural debt. It is neither global optimization nor inverse renormalization group flow. It is local optimization under geometric constraint, bounded by the same slack $(1 - \eta)$ available to all other system activity and further gated by overhead costs that arise because repair operations themselves must navigate the non-integrable constraint manifold. The framework identifies two empirically distinct operating modes, captures both under a single gating function, and writes the resulting repair-augmented system as a set of coupled flow equations with explicit parameter dependence.

Repair as Local Optimization

Renormalization group flow is a semigroup: coarse-graining destroys information, and the lost structure cannot be recovered without injecting both work and architectural design. This is why repair cannot be “inverse RG.” A system that has integrated out fast modes at scale $\mu$ cannot reverse that integration by any local operation — the geometric obstruction that enforces irreversibility of coarse-graining applies equally to repair operations, since they must traverse the same non-integrable feasibility map that generated the curl floor in the first place. What repair can do is introduce a counterflow that locally bends the RG vector field, partially offsetting accumulation in the region where the repair operation acts. Repair may redirect trajectories in state space and partially reopen degrees of freedom, but it cannot eliminate the fixed-point structure imposed by the constraint geometry.

The constraint is fundamental. The curl-maintenance functional $\mathcal{M}_{\mathrm{curl}}(F)$ from Theorem 4 has a floor set by the spectral gap of the Hodge Laplacian and the persistent projection defect. Repair can reduce excess curl above this floor — the curl attributable to architectural history, suboptimal routing, and accumulated structural compromises — but it cannot touch the geometric minimum itself. The floor is a property of the constraint manifold, not of the system’s history, and no local operation can change it.

This distinction between geometric curl and excess curl is what makes the framework predictive. Total observed circulation in a system is the sum of an irreducible geometric contribution and a history-dependent architectural contribution. Repair acts on the second component. Its effectiveness therefore depends on two things: how much excess curl exists (a function of history) and how much capacity the system has to perform repair operations (a function of current load).

The following lemma makes the relationship between repair and geometry precise.

Lemma (Repair–Geometry Compatibility). Let $M$ be the constraint manifold on which implemented correction fields are realized, with feasibility enforced by a state-dependent projection $\Pi_x$ that maps a nominal gradient proposal $F_0 = \nabla\phi$ to an implemented field $F = \Pi_x(F_0)$. Let $\alpha$ be the 1-form associated with $F$, and let the curl-maintenance functional $\mathcal{M}_{\mathrm{curl}}(F) = \frac{1}{2}\int_M |d\alpha|^2 \, dV$ admit a strictly positive lower bound under the non-integrability of $\Pi_x$. Define a repair operation as any localized architectural intervention that (i) reduces the history-dependent component of $\mathcal{M}_{\mathrm{curl}}$ and (ii) is itself implemented through fields subject to the same feasibility map $\Pi_x$. Then there exists a strictly positive geometric overhead floor,

$$c_{\mathrm{geom}} \geq \kappa \, \lambda_1^\perp \, |\delta\alpha_{\mathrm{rep},\perp}|^2,$$

where $\lambda_1^\perp > 0$ is the first positive eigenvalue of the Hodge Laplacian on 1-forms restricted to the orthogonal complement of harmonic forms $\mathcal{H}^1(M)$ (when $H^1(M) = 0$ this reduces to the usual $\lambda_1$; on periodic domains such as $T^3$ it is the spectral gap on the mean-zero sector), $\delta\alpha_{\mathrm{rep},\perp}$ is the non-harmonic component of the projection defect induced by the repair operation’s own implementation, and $\kappa = O(1)$ absorbs normalization and locality conventions.

Proof sketch. The argument is compositional. Non-integrability of $\Pi_x$ implies that implemented correction fields generically contain a coexact component in their Hodge decomposition that cannot be removed by gauge or gradient adjustment. Decompose the projection defect as $\delta\alpha_{\mathrm{rep}} = \delta\alpha_{\mathrm{rep,harm}} + \delta\alpha_{\mathrm{rep},\perp}$ via the Hodge decomposition; harmonic forms contribute no curl ($d\delta\alpha_{\mathrm{rep,harm}} = 0$), so all curl resides in the non-harmonic component. The coercivity of the exterior derivative on this subspace is controlled by the spectral gap $\lambda_1^\perp$, yielding a lower bound on $\mathcal{M}_{\mathrm{curl}}$ for any feasible field. A repair operation that modifies system architecture while respecting feasibility must itself be realized through $\Pi_x$, and consequently its implementation induces a non-harmonic projection defect $\delta\alpha_{\mathrm{rep},\perp}$ in the same coexact subspace. The same spectral gap bound therefore applies to the repair operation’s own implementation cost. Repair may reduce excess curl associated with architectural history, but it cannot avoid paying a strictly positive geometric overhead whenever feasibility remains non-integrable.

Corollary (Irreducible Repair Overhead). Whenever repair operations are executed through the same non-integrable feasibility map that generated excess curl, the total repair overhead admits a decomposition $c = c_{\mathrm{geom}} + c_{\mathrm{arch}}$ with $c_{\mathrm{geom}} > 0$ fixed by geometry and $c_{\mathrm{arch}} \geq 0$ encoding system-specific architectural inefficiencies. No architectural refinement can eliminate $c_{\mathrm{geom}}$.

Corollary (Thresholded Structural Repair). Because $c_{\mathrm{geom}} > 0$, there exists a finite maintenance load $\eta_{\mathrm{crit}} < 1$ at which net repair capacity vanishes. Above this threshold, repair attempts may persist locally but cannot produce durable structural improvement.

Definition (Feasibility basin criterion). A state is “good enough” if applying the feasibility map $\Pi_x$ (or its proxy) converges to a physically admissible state within a bounded number of iterations. The proposal dynamics need not satisfy feasibility identically; it suffices that proposals land within the basin of attraction of the feasibility projection¹. In the Navier–Stokes finite-residence setting, finite residence replaces solver convergence as the measurable criterion: states that would support first-order transport are visited but not retained, indicating the corresponding operator lies in the nullspace of the implemented feasibility map. Exact physical feasibility is enforced by the projection or solver, and the generative dynamics serve as initialization — the same separation used in physics-corrected diffusion for ACOPF¹.

Repair Capacity and Overhead

The capacity to perform repair is constrained by the same resource that constrains everything else: available slack. A system at maintenance load $\eta$ has residual capacity $1 - \eta$ for all non-maintenance activity, including repair. But repair is itself a structured operation that must navigate the non-integrable constraint manifold, and that navigation incurs overhead.

The repair capacity is

$$\mathcal{R}(\eta) = 1 - \eta - \frac{c}{1 - \eta},$$

where $c = c_{\mathrm{geom}} + c_{\mathrm{arch}}$ captures two distinct sources of overhead. The Repair–Geometry Compatibility lemma establishes that the geometric component satisfies $c_{\mathrm{geom}} \geq \kappa \, \lambda_1^\perp \, |\delta\alpha_{\mathrm{rep},\perp}|^2$ — it is bounded below by the same Hodge spectral gap that sets the curl floor, scaled by the non-harmonic projection defect of the repair operation itself. This is not an assumption but a consequence of the fact that repair must navigate the same non-integrable feasibility map that generated the excess curl. The architectural component $c_{\mathrm{arch}} \geq 0$ captures additional overhead from coordination, sequencing, and the system-specific logistics of implementing structural change.

The overhead term $c/(1 - \eta)$ diverges as $\eta \to 1$: when nearly all capacity is consumed by maintenance, the residual is too small to absorb the fixed costs of repair. This creates a critical threshold. Setting $\mathcal{R}(\eta) = 0$ yields the condition where repair capacity vanishes,

$$(1 - \eta)^2 = c,$$

so the critical load is $\eta_{\mathrm{crit}} = 1 - \sqrt{c}$. Below this value, repair is net-positive — it can reduce excess curl faster than accumulation generates it. Above it, repair still occurs (the system still attempts local optimization) but no longer produces net structural improvement. The system is spending more on the overhead of repair than repair returns in reduced curl.

Two Modes of Repair

The distinction between what repair attempts and what repair accomplishes produces two empirically distinct regimes, depending on whether the structural effectiveness gate responds to available slack alone or to net repair capacity.

Mode I — Persistent Local Optimization

In systems where repair activity continues for as long as any slack exists, the gate function is simply

$$\mathcal{S}(\eta) = 1 - \eta.$$

Repair attempts persist even when they are net-negative — when the overhead of reconfiguration exceeds the curl reduction achieved. Effectiveness decays monotonically with increasing load, and the system experiences gradually worsening structural integrity without an abrupt transition. This is the regime of biological aging, infrastructure decay, and organizational maintenance churn. The system continues repairing, but the repairs accomplish progressively less, and the gap between attempted and effective repair widens as $\eta$ rises.

Mode II — Net-Positive Structural Repair

In systems where repair affects structure only when it is net-positive after overhead, the gate function tracks repair capacity directly,

$$\mathcal{S}(\eta) = \mathcal{R}(\eta).$$

This produces a sharp threshold. Below $\eta_{\mathrm{crit}}$, repair can reopen degrees of freedom — it actively counteracts dimensional reduction and maintains the system’s access to higher-dimensional configuration space. Above $\eta_{\mathrm{crit}}$, repair operations may continue at the local level but produce no structural recovery. The transition is geometric, not parametric: it reflects the point where overhead costs absorb all available repair capacity. This is the regime relevant to compact astrophysical objects, where gravitational self-energy drives $\eta$ through a threshold that separates structurally active stellar evolution from the consolidated end-states of white dwarfs and neutron stars.

The Single Gate

Both regimes are instances of a single expression,

$$\mathcal{S}(\eta) = (1 - \eta)\,\sigma\!\left(\frac{\mathcal{R}(\eta)}{\Delta}\right),$$

where $\sigma$ is a smooth logistic function and $\Delta$ controls the sharpness of the transition. In the limit $\Delta \to \infty$, the gate reduces to $1 - \eta$ (Mode I: persistent optimization regardless of net capacity). In the limit $\Delta \to 0$, the gate becomes a sharp step function at $\mathcal{R} = 0$ (Mode II: structural effect only when net-positive). Physical systems occupy intermediate values of $\Delta$, and the parameter functions as a regime classifier rather than a free knob: tightly coupled systems where repair operations directly engage structural degrees of freedom (white dwarfs, brittle materials) sit at small $\Delta$, loosely coupled systems where repair acts through many intermediate layers (biological organisms, institutions) sit at large $\Delta$, and engineered systems with designed modularity occupy intermediate values where the boundary sharpness reflects the coupling architecture.

The gating structure has a direct analog in constrained diffusion for ACOPF¹, where the guidance strength $\lambda_t$ is timestep-dependent: stronger corrections are applied at early, high-noise timesteps and gradually relaxed as the sample stabilizes. The repair gate $\mathcal{S}(\eta)$ is load-dependent rather than timestep-dependent, but the control logic is identical — guidance strength is monotone in distance from feasibility, whether that distance is measured by diffusion noise or by maintenance load. Early in sampling (far from feasibility) maps to low $\eta$ (high slack): both regimes apply strong corrections because the system has capacity to absorb them. Late in sampling (near feasibility) maps to high $\eta$ (near saturation): both regimes relax corrections because overcorrection destabilizes. The correspondence is structural, not metaphorical — the same schedule principle governs both.

The Repair-Augmented System

The full dynamics couples the dissipation β-function, dimensional flow, and a repair actuation variable $r(\ell)$ that measures the density of local optimization activity at RG scale $\ell$. The system is

$$\frac{d\eta}{d\ell} = -\eta(1-\eta)\,K(d) + r\,g(X)\,\eta\,\mathcal{R}(\eta),$$ $$\frac{dd}{d\ell} = -\frac{\eta}{\rho^*}\ln\varphi + \chi\,r\,g(X)\,\mathcal{S}(\eta),$$ $$\frac{dr}{d\ell} = \alpha(1-\eta) - (\beta\eta + \gamma)\,r,$$

where $K(d) = \rho^* + \frac{d-2}{2}\ln\varphi$ is the accumulation kernel from the dissipation β-function, with $\rho^* = 4\pi\varphi^2/10 \approx 3.29$.

The first equation adds a repair counterflow to the accumulation law. The term $r\,g(X)\,\eta\,\mathcal{R}(\eta)$ is positive when repair capacity is positive, partially offsetting the negative (accumulating) flow. The factor $g(X)$ captures how repair efficiency depends on scale separation. The helicity stiffness data pin the functional form: since the stiffness ratio $\mathcal{S}(X) \approx 0.97X + 0.03$ measures resistance to topology change, each unit of repair effort accomplishes less curl reduction at higher $X$. The natural identification is $g(X) = g_0/X$, where the $1/X$ dependence mirrors the linear stiffness law and $g_0$ is an architecture-dependent normalization. The effective repair ceiling is then the product $\gamma_{\mathrm{crit}}(X) \cdot g(X)$. The operator resonance identity shows that when the hazard operator matches the dissipation operator (current-squared hazard, $p = 1$), $\gamma_{\mathrm{crit}}$ is exactly $X$-independent — the phase boundary exponent $\alpha = 0$ is an analytical identity, not a fit. The ceiling therefore scales as $X^0 \cdot X^{-1} = X^{-1}$, set entirely by stiffness. The phase boundary contributes nothing, and the repair penalty at high scale separation is the full $1/X$ that the stiffness law imposes.

The second equation allows repair to counteract dimensional reduction. The coupling $\chi$ is a bounded structural coefficient — a scalarized projection factor measuring how much effective dimension is restored per unit of structurally successful local optimization. It is bounded above by $O(1)$ because dimensional recovery cannot exceed the degrees of freedom lost, and bounded below by zero whenever repair is structurally effective, since curl reduction without dimensional recovery would contradict the identification of dimensional reduction with constraint collapse. In a more detailed treatment, $\chi$ would be replaced by a tensor quantity tracking which degrees of freedom are recoverable and at what cost; the scalar version represents an intentional coarse-graining that preserves the qualitative dynamics. The gate $\mathcal{S}(\eta)$ ensures that dimensional recovery occurs only when repair is structurally effective.

The third equation gives repair actuation its own slow dynamics. Repair is sourced by available slack $\alpha(1-\eta)$ — the system allocates repair effort in proportion to residual capacity — and decays under load-dependent damping $(\beta\eta + \gamma)r$. The damping increases with $\eta$ because higher maintenance load competes for the same resources repair requires. The constant $\gamma$ ensures repair actuation decays even at $\eta = 0$, reflecting the baseline cost of maintaining repair infrastructure.

A boundary note is worth stating explicitly. The repair capacity $\mathcal{R}(\eta)$ diverges to $-\infty$ as $\eta \to 1$, but the repair actuation $r(\ell)$ is sourced by $(1-\eta)$, so $r = O(1-\eta)$ or faster near the horizon. The product $r \, \mathcal{R}(\eta)$ therefore remains bounded, and the $\eta = 1$ fixed point of the bare β-function is preserved. Repair shuts off at the horizon by the same mechanism that shuts off all other non-maintenance activity: exhaustion of slack.

White Dwarf Cooling as Thresholded Repair

The repair framework makes a specific empirical prediction: systems in the small-$\Delta$ regime should exhibit a sharp compactness-controlled threshold rather than smooth aging-like decay. White dwarfs provide a natural test case². As a white dwarf cools, gravitational self-energy drives $\eta$ upward through the structural saturation regime identified in the triadic tension analysis. If Mode II applies, repair effectiveness should shut off sharply once $\eta$ crosses $\eta_{\mathrm{crit}} = 1 - \sqrt{c}$, producing a discrete transition in cooling behavior at a specific compactness.

To test this prediction, we examine cooling-age residuals across a compactness-stratified sample of massive white dwarfs, asking whether the data exhibit the sharp gate structure the framework requires.

The test uses 7,515 DA white dwarfs from the Gaia DR3 catalog³, pre-filtered for $R/R_S \in [500, 1500]$ and $M \geq 1.1 \, M_\odot$ (hydrogen-atmosphere, thick-envelope fits). Cooling ages are derived from Montréal evolutionary tracks⁴ (thick-hydrogen-envelope sequences, $0.20$–$1.30 \, M_\odot$), supplemented by La Plata relativistic ONe-core tracks⁵ extending coverage to $1.369 \, M_\odot$ and CO-core GR tracks⁶ as a systematic check on core-composition sensitivity. Cooling-age residuals are constructed by $k$-nearest-neighbor matching ($k = 10$) in standardized $(M, \log T_{\mathrm{eff}})$ space against a reference population at $R/R_S \in [1000, 1500]$ where no anomaly is expected. For the ultra-massive 500–700 bin, where all objects have $M > 1.32 \, M_\odot$ and no reference population exists at comparable mass, a direct Mann-Whitney comparison against the 700–900 bin replaces kNN matching. Full methodological details are given in the Appendix.

The residuals reveal a sharp threshold with a monotonic compactness gradient:

The sign convention is explicit: $\Delta t > 0$ means the anomaly-zone object appears older than its matched references — a physical cooling delay where the object remains hotter than expected for its true age. The anomaly intensifies monotonically from null at $R/R_S > 1300$ through $+108$ Myr at 700–900 to $+345$ Myr at 500–700, confined to $R/R_S \lesssim 10^3$ and vanishing over a narrow compactness interval. The 500–700 result is robust to core-composition assumptions: CO-core tracks yield $+697$ Myr ($p = 5.9 \times 10^{-10}$), and mass-matched restriction to the exact overlap range ($M \in [1.322, 1.362] \, M_\odot$) strengthens the signal to $+436$ Myr ($p < 10^{-12}$). The ONe result is reported as primary because ONe cores are physically expected for $M > 1.05 \, M_\odot$.

Globally across the 700–1000 anomaly zone, the kNN analysis yields a mean residual of $+76$ Myr ($14.5\sigma$ permutation $Z$-score, Wilcoxon $p = 4.27 \times 10^{-84}$, 95% bootstrap CI: $[+67, +85]$ Myr). The Spearman rank correlation between $M^2$ and $\Delta t$ is $\rho = +0.350$ ($p = 4.93 \times 10^{-30}$), confirming that higher-mass (more compact) objects show larger positive residuals — consistent with the compactness-controlled gradient the repair framework predicts.

Fitting a logistic gate model $\Delta t(x) = A/[1 + \exp((x - x_c)/w)]$ in $x = \log_{10}(R/R_S)$ to the Montréal-only kNN binned medians — the regime where a single matching statistic applies uniformly — yields gate center $x_c = 2.887$ ($R/R_S \approx 771$), width $w = 0.027$, and amplitude $A = 328$ Myr. The 25–75% transition width is $\Delta x_{25 \to 75} = 2w\ln 3 \approx 0.060$, corresponding to $R/R_S \approx 718 \to 828$. The amplitude $A$ is the extrapolated interior amplitude of the gate model, not the observed median in any single bin — the 700–900 bin at $x \approx 2.9$ already sits on the decaying shoulder, which is why its median ($+108$ Myr) falls below $A$. The anomaly is effectively absent by $x \approx 3.0$ ($R/R_S \approx 10^3$).

Mapping the empirical shutoff at $R/R_S \sim 10^3$ to the repair-capacity framework uses the compactness–load relationship established in the triadic tension collapse trajectory, where $R/R_S = 10^3$ corresponds to $M \approx 1.17 \, M_\odot$ with $\eta = 0.46$. This gives

$$\eta_{\mathrm{crit}} = 0.46,$$

which yields

$$c = (1 - \eta_{\mathrm{crit}})^2 = 0.54^2 = 0.292.$$

The inferred $c$ is order-unity, consistent with a geometry-bounded overhead controlled by the Hodge spectral gap as the Repair–Geometry Compatibility lemma requires. The coincidence of $\eta_{\mathrm{crit}} = 0.46$ with known percolation thresholds in three-dimensional configuration spaces⁷ is structurally suggestive: in both cases, the transition reflects loss of global connectivity rather than exhaustion of local activity.

Connection to Triadic Tension

The triadic tension framework establishes that three incompatible curvature sectors produce state-dependent constraints that break integrability and generate irreducible curl. The repair framework takes this as its starting condition and asks: given that global optimization is geometrically forbidden, what can local optimization accomplish?

The answer is structurally constrained. Repair bends the RG vector field without inverting it, reduces excess curl without touching the geometric floor, and reopens degrees of freedom only when net capacity exceeds overhead. The dissipation β-function remains the backbone of the dynamics — repair modifies its trajectory but cannot eliminate its fixed-point structure. The UV fixed point $(\eta, d) = (0, 3)$ and IR fixed point $(\eta, d) = (1, 2)$ persist; repair determines how and whether a system’s path between them includes episodes of structural recovery. A scope clarification is important here: the triadic tension framework’s one-loop exactness depends on a recursion-enforced vertex selection rule that forbids higher-order couplings at the self-similar fixed point. The repair-augmented system introduces an additional operator channel (architectural actuation) that deforms the flow, rather than correcting the original vertex expansion. This preserves the one-loop exactness of the bare β-function while treating repair as a controlled extension operating outside the symmetry-protected sector.

The self-correction framework’s curl floor provides the lower bound that makes the two-mode structure possible. If there were no geometric minimum, repair could in principle eliminate all circulation and the system would admit a global potential — contradicting Theorem 4. The existence of the floor means that even in Mode II, where repair is net-positive, there is a structural limit to recovery. The system can remove excess curl but not the irreducible geometric contribution, and this limit is what ultimately distinguishes repair from reversal.

Attack Surface

The Repair–Geometry Compatibility lemma establishes that $c_{\mathrm{geom}}$ is bounded below by $\kappa \, \lambda_1^\perp \, |\delta\alpha_{\mathrm{rep},\perp}|^2$, anchoring the repair threshold in the same spectral geometry that sets the curl floor. The white dwarf cooling analysis constrains the total overhead to $c = 0.292$, which in turn constrains the product $\kappa \, \lambda_1^\perp \, |\delta\alpha_{\mathrm{rep},\perp}|^2$ from above. Computing $\kappa$ independently from the spectral properties of the Hodge Laplacian on the white dwarf constraint manifold would close this loop: if the predicted $c_{\mathrm{geom}}$ matches the empirically inferred $c$, it would confirm that the overhead is predominantly geometric rather than architectural.

With $c = 0.292$ from the white dwarf data, $g(X) = g_0/X$ from the helicity stiffness law, and $\gamma_{\mathrm{crit}}$ pinned as $X$-independent by the operator resonance identity, the constrained quantities leave $g_0$ (repair efficiency normalization, architecture-dependent, setting the absolute scale of repair relative to accumulation) and the actuation dynamics $\alpha$, $\beta$, $\gamma$ controlling how repair effort is sourced and damped as the remaining free parameters in the coupled system.

The regime classifier $\Delta$ organizes systems by how tightly repair operations couple to structural degrees of freedom, but the framework does not currently derive $\Delta$ from microscopic properties. A mapping between system architecture (coupling density, modularity, timescale separation between repair and accumulation) and effective gate sharpness would allow the two modes to be distinguished a priori rather than identified empirically. This appears to be a system-specific question rather than a universal one — $\Delta$ encodes precisely the information that differs across physical, biological, and organizational realizations of the same geometric framework.

The coupled system $(\eta, d, r)$ admits both stable fixed points and limit cycles. The oscillatory solutions arise generically from the combination of irreducible curl (non-potential flow), delayed repair actuation, and coupling between load, dimension, and architecture — making cycling a structural outcome rather than a tuning artifact. A complete analysis of the fixed-point structure and bifurcation diagram, identifying which parameter regimes produce steady maintenance, periodic consolidation, and chaotic repair dynamics, is a natural next step. This analysis would connect to the oscillatory cycling already observed in the Navier–Stokes finite-residence setting and compressor surge dynamics, providing a single dynamical account of why frustrated systems cycle rather than converge.

Appendix: White Dwarf Cooling Analysis Methods

This appendix provides the full methodological detail supporting the white dwarf cooling results presented above. All steps are deterministic and reproducible from the inputs described.

Data

The input catalog is a pre-filtered subset of the Gaia DR3 white dwarf catalog, restricted to objects with $R/R_S \in [500, 1500]$ and spectroscopic mass $M \geq 1.1 \, M_\odot$ (hydrogen-atmosphere, thick-envelope fits). Starting count: 7,870 objects. After dropping rows with missing mass or effective temperature: 7,515. The Montréal-only grid retains 7,169 objects within the track convex hull; the extended ONe grid retains 7,482.

Cooling-Age Model Construction

Cooling ages are modeled as a deterministic function of $(M, T_{\mathrm{eff}})$ using two sets of evolutionary tracks.

The primary grid uses Montréal DA thick-hydrogen-envelope sequences ($0.20$–$1.30 \, M_\odot$ in steps of $0.05 \, M_\odot$, 23 tracks). The Montréal grid has a hard ceiling at $1.30 \, M_\odot$: any object with spectroscopic mass exceeding this value cannot be assigned a cooling age. In the target sample, 346 objects have $M > 1.30 \, M_\odot$, and all 145 objects in the $R/R_S \in [500, 700)$ bin fall in this excluded range.

To recover the 500–700 bin and extend coverage into the ultra-massive regime, the Montréal grid is supplemented with La Plata evolutionary sequences that include full general-relativistic corrections. ONe-core GR sequences⁵ ($M/M_\odot = 1.29, 1.31, 1.33, 1.35, 1.369$) are physically appropriate for $M > 1.05 \, M_\odot$, where ONe cores result from carbon burning during the super-AGB phase. CO-core GR sequences⁶ ($M/M_\odot = 1.29, 1.31, 1.33, 1.35, 1.37, 1.382$) serve as a systematic check on core-composition sensitivity. The combined ONe grid (27 tracks, ceiling $1.369 \, M_\odot$) is used as primary; the CO grid (28 tracks, ceiling $1.382 \, M_\odot$) as systematic cross-check.

For each mass track $M_j$, the tabulated cooling solution $t_{\mathrm{cool}}(T_{\mathrm{eff}}; M_j)$ is interpolated along the track in $\log T_{\mathrm{eff}}$ using piecewise cubic Hermite interpolation (PCHIP), which preserves the physical monotonicity of cooling age as a function of decreasing temperature. For intermediate masses between neighboring tracks $(M_j, M_{j+1})$, ages are interpolated linearly in mass,

$$t_{\mathrm{cool}}(M, T_{\mathrm{eff}}) = \frac{M_{j+1} - M}{M_{j+1} - M_j} \, t_{\mathrm{cool}}(T_{\mathrm{eff}}; M_j) + \frac{M - M_j}{M_{j+1} - M_j} \, t_{\mathrm{cool}}(T_{\mathrm{eff}}; M_{j+1}),$$

producing a smooth bivariate map $(M, T_{\mathrm{eff}}) \mapsto t_{\mathrm{cool}}$. No extrapolation is performed at any stage — objects outside the convex hull of the track grid are assigned NaN ages and excluded.

Method A: kNN-Matched Residuals (700–1500)

The cooling anomaly is defined relative to a matched reference population. The reference pool consists of objects with $R/R_S \in [1000, 1500]$, where no anomaly is observed. For each object $i$ in the anomaly region ($R/R_S \in [700, 1000)$), the $k = 10$ nearest neighbors in the reference pool are identified using Euclidean distance in standardized $(M/\sigma_M, \log T_{\mathrm{eff}}/\sigma_{\log T})$ space. The median reference age $\tilde{t}_{\mathrm{ref}}$ of these neighbors defines the baseline, and the cooling-age residual is $\Delta t_i = t_i - \tilde{t}_{\mathrm{ref}}$. The sign convention is explicit: $\Delta t > 0$ means the anomaly-zone object appears older than its matched references.

Method B: Mann-Whitney Comparison (500–700)

Standard kNN matching is structurally infeasible for the 500–700 bin. All objects at $R/R_S < 700$ have $M > 1.32 \, M_\odot$, but the reference zone contains no objects above $1.242 \, M_\odot$ — the mass gap exceeds $1.8\sigma$ in standardized feature space, far beyond the $0.6\sigma$ that yields reliable matches at 700–1000. An alternative approach directly compares cooling ages between ultra-massive white dwarfs ($M > 1.32 \, M_\odot$) across $R/R_S$ bins using a one-sided Mann-Whitney $U$ test, with the alternative hypothesis that 500–700 objects have systematically greater cooling ages than 700–900 objects (i.e., positive $\Delta t$, consistent with a cooling delay at higher compactness). The test is supplemented by bootstrap confidence intervals (10,000 resamples) and a mass-matched robustness check restricting both bins to their exact mass overlap range.

Gate Model and Parameter Inference

The threshold structure is characterized by fitting a logistic gate model in $x = \log_{10}(R/R_S)$,

$$\Delta t(x) = \frac{A}{1 + \exp\!\left(\frac{x - x_c}{w}\right)},$$

to the Montréal-only kNN binned medians (the regime where a single matching statistic applies uniformly). The three bin centers at $(x, \Delta t) \in \{(2.9, 118.1),\, (3.0, 5.3),\, (3.08, 0.3)\}$ Myr yield $A = 328$ Myr, $x_c = 2.887$ ($R/R_S \approx 771$), and $w = 0.027$. The amplitude $A$ is the extrapolated interior amplitude — the 700–900 bin already sits on the decaying shoulder of the gate — and the 25–75% transition width $\Delta x_{25 \to 75} = 2w\ln 3 \approx 0.060$ corresponds to $R/R_S \approx 718 \to 828$.

Statistical Tests

Significance for kNN residuals is assessed by sign-flip permutation testing ($N = 2000$), Wilcoxon signed-rank tests, and bootstrap confidence intervals ($N = 10{,}000$). The Spearman rank correlation between $M^2$ and $\Delta t$ tests for a monotonic relationship between compactness and residual magnitude. For the Mann-Whitney comparisons, the rank-biserial correlation provides a standardized effect size.

Footnotes

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3. Gentile Fusillo, N. P., Tremblay, P.-E., Cukanovaite, E., et al. (2021). “A catalogue of white dwarfs in Gaia EDR3.” Monthly Notices of the Royal Astronomical Society, 508(3), 3877–3896. https://doi.org/10.1093/mnras/stab2672 ↩

4. Bédard, A., Bergeron, P., Brassard, P., & Fontaine, G. (2020). “Synthetic colors and evolutionary sequences of hydrogen- and helium-atmosphere white dwarfs.” The Astrophysical Journal, 901, 93. https://doi.org/10.3847/1538-4357/aba151 ↩

5. Althaus, L. G., Camisassa, M. E., Torres, S., et al. (2022). “Evolutionary models for ultra-massive white dwarfs with ONe cores including GR corrections.” Astronomy & Astrophysics, 668, A58. https://doi.org/10.1051/0004-6361/202244604 ↩ ↩²

6. Althaus, L. G., Camisassa, M. E., Torres, S., Córsico, A. H., & Rebassa-Mansergas, A. (2023). “Evolutionary models for ultra-massive white dwarfs with CO cores.” Monthly Notices of the Royal Astronomical Society, 523, 4492. https://doi.org/10.1093/mnras/stad1707 ↩ ↩²

7. Stauffer, D. & Aharony, A. (1994). Introduction to Percolation Theory. 2nd ed., Taylor & Francis. ↩