Gauge-Invariant, Parameter-Insensitive Regularization for Potential Recovery from Flow on Directed Graphs

# Gauge-Invariant, Parameter-Insensitive Regularization for Potential Recovery from Flow on Directed Graphs
Source: [https://arxiv.org/html/2607.13609](https://arxiv.org/html/2607.13609)
###### Abstract

Recovering a latent potential from observed flow on a directed graph \(a discrete Poisson problem with Dirichlet boundaries\) is ill\-posed, and the standard fix backfires: ridge regularization shrinks toward a gauge\-meaningless origin, collapsing and reversing the recovered ordering \(\+0\.81→−0\.42\+0\.81\\to\-0\.42rank correlation against a planted ground truth\)\. The gauge\-invariant graph Dirichlet energy removes the hazard and delivers*parameter\-insensitivity*: the estimate is stable across four orders of magnitude inλ\\lambda, whereas ridge inverts the ordering for everyλ\>0\\lambda\>0\. We prove the reduced solve is SPD and preserves dynamic range exactly where ridge collapses it, and localize absorbing boundaries from flow alone via a Poisson residual\. TheH1H^\{1\}seminorm is classical; what is new is the gauge diagnosis, the parameter\-insensitivity it buys, and an ablation showing the result is robust to the extraction method\. On three public clickstream corpora the gauge\-invariant estimate retains2828–41%41\\%of the interior dynamic range while ridge collapses to as little as0\.2%0\.2\\%\. The same gauge invariance carries into graph neural networks —neutralizing the constant mode per layer prevents the oversmoothing that collapses a deep directed GCN—linking this classical inverse problem to a central question in graph learning\.

## 1Introduction

Regularizing an ill\-posed inverse problem on a graph requires two decisions: which penalty to add, and how strongly\. They are usually treated separately, on the assumption that a reasonable penalty degrades gracefully asλ\\lambdavaries\. This paper concerns a setting where that fails: the conventional penalty does not degrade gracefully but*inverts*the solution, soλ\\lambdabecomes a choice between a usable answer and a confidently wrong one\. The setting is recovery of a latent scalar*potential*from observed directed flow—a traffic network through flow counts, a supply chain through inventory movement, a navigation log through transition frequencies: the discrete analogue of solving a Poisson equation backwards from divergence to potential\. We identify when this happens, explain why, and show a different penalty removes it, leaving the answer insensitive toλ\\lambdaacross orders of magnitude\.

#### Setup and degeneracy\.

LetG=\(V,E,W\)G=\(V,E,W\)be a weighted directed acyclic graph with absorbing boundaryB⊂VB\\subset V\. A potentialϕ\\phiinduces the flux

qi​j=Wi​j​\(ϕj−ϕi\),q\_\{ij\}=W\_\{ij\}\(\\phi\_\{j\}\-\\phi\_\{i\}\),\(1\)the discrete analogue of Ohm’s and Fick’s laws, with net divergencebi=∑jqj​i−∑jqi​jb\_\{i\}=\\sum\_\{j\}q\_\{ji\}\-\\sum\_\{j\}q\_\{ij\}\. Substituting gives the directed discrete Poisson equationL​ϕ=bL\\phi=bwithLLthe weighted Laplacian\. It recoversϕ\\phifrom a noisy empiricalbbsubject to Dirichlet values onBB, and is ill\-posed \(L​𝟏=0L\\mathbf\{1\}=0, low\-divergence chains, finite data\), so regularization is unavoidable; the standard choice is general\-form Tikhonov,arg​minϕ⁡12​‖L​ϕ−b‖22\+λ​‖R​ϕ‖22\\operatorname\*\{arg\\,min\}\_\{\\phi\}\\tfrac\{1\}\{2\}\\left\\lVert L\\phi\-b\\right\\rVert\_\{2\}^\{2\}\+\\lambda\\left\\lVert R\\phi\\right\\rVert\_\{2\}^\{2\}, with the default ridgeR=IR=I\.

#### The gauge mismatch\.

The potential is fixed only up to an additive constant, set when boundary values are assigned\. Ridge shrinks toward0, but0is not distinguished in a Dirichlet problem—it is wherever the boundary placed the origin\. On a positive\-boundary graph the pull is asymmetric, dragging interior states toward the abandon boundary regardless of their true position, collapsing the range and, past a smallλ\\lambda, reversing the order\. On a controlled instrument with planted ground truth \([Section˜7](https://arxiv.org/html/2607.13609#S7)\), the ridge rank correlation falls from\+0\.81\+0\.81atλ=0\\lambda=0to≈−0\.42\{\\approx\}\-0\.42, inverting the ordering, with linear correlation crossing zero \([Figure˜1](https://arxiv.org/html/2607.13609#S5.F1)\); the only safe ridge setting isλ→0\\lambda\\to 0—no regularization\.

#### Parameter\-insensitivity\.

The fix is to make the penalty blind to the gauge by takingRRto be the incidence operator, so the penalty is the graph Dirichlet energyϕ⊤​LG​ϕ=∑\(u,v\)∈Ewu​v​\(ϕv−ϕu\)2\\phi^\{\\top\}L\_\{G\}\\phi=\\sum\_\{\(u,v\)\\in E\}w\_\{uv\}\(\\phi\_\{v\}\-\\phi\_\{u\}\)^\{2\}\(the graphH1H^\{1\}seminorm\); we call the resulting estimator*graph\-Sobolev regularization*\. BecauseLG​𝟏=0L\_\{G\}\\mathbf\{1\}=0, this penalizes differences, not amplitude, and is flat along the gauge mode\. The consequence is practical: the estimate is origin\-invariant and stable inλ\\lambda, holding rank correlation at\+0\.81\+0\.81and linear correlation in\[\+0\.76,\+0\.85\]\[\+0\.76,\+0\.85\]acrossλ∈\[10−3,10\]\\lambda\\in\[10^\{\-3\},10\], so the strength cannot be misset\. Where ridge forces a knife\-edge choice, the Dirichlet energy removes it\.

#### What is, and is not, new\.

The penalty itself is old: theH1H^\{1\}seminorm is the classical alternative to the identity seminorm \(general\-form vs\. standard\-form Tikhonov\(Hansen,[1998](https://arxiv.org/html/2607.13609#bib.bib5); Tikhonov and Arsenin,[1977](https://arxiv.org/html/2607.13609#bib.bib16)\)\), andϕ⊤​L​ϕ\\phi^\{\\top\}L\\phiis the standard smoothness functional in graph signal processing and semi\-supervised learning\(Shuman et al\.,[2013](https://arxiv.org/html/2607.13609#bib.bib15); Zhou et al\.,[2004](https://arxiv.org/html/2607.13609#bib.bib20); Belkin and Niyogi,[2003](https://arxiv.org/html/2607.13609#bib.bib1); Zhu et al\.,[2003](https://arxiv.org/html/2607.13609#bib.bib22)\)\. What is new is: \(i\) those priors interpolate a*partially observed*node signal, whereas ours is an*inhomogeneous*inverse problem deconvolving a potential from its divergence with Dirichlet boundaries, where the decisive property is gauge invariance, not smoothness; \(ii\) ridge is not merely suboptimal but*actively harmful*, inverting the ordering, with the gauge diagnosis explaining why; \(iii\) the parameter\-insensitivity this yields; \(iv\) a Poisson\-residual boundary diagnostic; and \(v\) a behavioral\-flow application with a planted\-ground\-truth instrument\. Edge\-preserving gauge\-invariant penalties \(total variation, thepp\-Laplacian\(Rudin et al\.,[1992](https://arxiv.org/html/2607.13609#bib.bib12)\)\) are discussed in[Section˜2](https://arxiv.org/html/2607.13609#S2)\.

#### Contributions\.

\(1\)Parameter\-insensitivity: the gauge\-invariant penalty makes the recovered potential stable across four orders of magnitude inλ\\lambda\([Section˜5](https://arxiv.org/html/2607.13609#S5)\);\(2\)the ridge inversion and its gauge diagnosis \(\+0\.81→−0\.42\+0\.81\\\!\\to\\\!\-0\.42rank correlation\);\(3\)discrete guarantees—gauge invariance \([Proposition˜5](https://arxiv.org/html/2607.13609#Thmproposition5)\), an SPD reduced system that scales to10410^\{4\}nodes by conjugate gradients \([Theorem˜1](https://arxiv.org/html/2607.13609#Thmtheorem1)\), and exact range preservation on chains \([Theorem˜2](https://arxiv.org/html/2607.13609#Thmtheorem2)\); and\(4\)a reproducible pipeline111Code:[https://github\.com/MohammadForouhesh/gauge\-flow\-recovery](https://github.com/MohammadForouhesh/gauge-flow-recovery)with a Poisson\-residual boundary diagnostic \([Section˜6](https://arxiv.org/html/2607.13609#S6)\), an extraction ablation, real\-corpus validation, and a downstream task where the gauge\-invariant potential is a usable node feature and ridge’s is not \([Sections˜7](https://arxiv.org/html/2607.13609#S7)and[8](https://arxiv.org/html/2607.13609#S8)\)\. We are precise about scope: the penalty guarantees preservation, not signal—whatever ordering the data supports is preserved acrossλ\\lambda, not destroyed\.

## 2Related Work

#### Inverse problems and flows on graphs\.

Recovering a potential from observed flow is the discrete counterpart of an elliptic inverse problem\. The electrical\-network analogy is classical: a resistor network with prescribed boundary potentials obeys a discrete Laplace equation whose harmonic functions encode absorption probabilities of the associated walk\(Doyle and Snell,[1984](https://arxiv.org/html/2607.13609#bib.bib3); Grady,[2006](https://arxiv.org/html/2607.13609#bib.bib4)\)\. Closest to our setting,Jia et al\. \([2019](https://arxiv.org/html/2607.13609#bib.bib7)\)recover edge flows by Hodge\-based semi\-supervised learning, andSchaub and Segarra \([2018](https://arxiv.org/html/2607.13609#bib.bib13)\)denoise edge flows by projecting onto Hodge gradient/curl subspaces\. We differ in recovering the*node potential*through the directed forward operator with Dirichlet boundaries, and in diagnosing the gauge\-induced*inversion*that the identity seminorm causes, a failure mode about the regularizer, not the representation\.

#### Regularization and graph signal processing\.

Tikhonov regularization is the default stabilizer for ill\-posed systems\(Tikhonov and Arsenin,[1977](https://arxiv.org/html/2607.13609#bib.bib16); Hansen,[1998](https://arxiv.org/html/2607.13609#bib.bib5)\), the seminorm matrix encoding prior structure; total\-variation and graph trend\-filtering penalties trade quadratic smoothness for sharp transitions\(Rudin et al\.,[1992](https://arxiv.org/html/2607.13609#bib.bib12); Wang et al\.,[2016](https://arxiv.org/html/2607.13609#bib.bib17)\)\. Whether the penalty seminorm annihilates the forward operator’s null space governs the estimator’s bias\. Known in principle, we make it concrete and consequential for the directed Dirichlet problem: the identity seminorm does not annihilate the gauge mode, and the resulting bias is not lost resolution but a reversal of order\. The same quadratic formϕ⊤​LG​ϕ\\phi^\{\\top\}L\_\{G\}\\phiis the standard smoothness functional in graph signal processing and Laplacian semi\-supervised learning\(Shuman et al\.,[2013](https://arxiv.org/html/2607.13609#bib.bib15); Belkin and Niyogi,[2003](https://arxiv.org/html/2607.13609#bib.bib1); Zhu et al\.,[2003](https://arxiv.org/html/2607.13609#bib.bib22); Zhou et al\.,[2004](https://arxiv.org/html/2607.13609#bib.bib20)\), the “trivial” structural choice that sheaf and connection Laplacians generalize\(Bodnar et al\.,[2022](https://arxiv.org/html/2607.13609#bib.bib2)\)—but there for a signal*partially observed on the nodes*, not, as here, an inhomogeneous inverse problem where gauge invariance is decisive\.

#### Directed graphs and the Hodge decomposition\.

Directed graph neural networks encode direction in a complex Hermitian \(magnetic\) Laplacian\(Zhang et al\.,[2021](https://arxiv.org/html/2607.13609#bib.bib18); He et al\.,[2022](https://arxiv.org/html/2607.13609#bib.bib6)\)or directionality\-aware message passing\(Rossi et al\.,[2023](https://arxiv.org/html/2607.13609#bib.bib11)\), and directed graph signal processing studies edge directionality\(Marques et al\.,[2020](https://arxiv.org/html/2607.13609#bib.bib10)\); these target node/edge representation learning, whereas we keep a*real*directed Laplacian as a forward operator and regularize its inverse\. The Helmholtz–Hodge decomposition\(Jiang et al\.,[2011](https://arxiv.org/html/2607.13609#bib.bib8); Lim,[2020](https://arxiv.org/html/2607.13609#bib.bib9); Schaub et al\.,[2020](https://arxiv.org/html/2607.13609#bib.bib14)\)splits an edge flow into gradient, curl, and harmonic parts; we use the gradient only to orient flow into an acyclic support, which[Sections˜7](https://arxiv.org/html/2607.13609#S7.SS0.SSS0.Px2)and[7\.1](https://arxiv.org/html/2607.13609#S7.SS1)show is not load\-bearing\. Graph total variation, trend filtering, and thepp\-Laplacian\(Rudin et al\.,[1992](https://arxiv.org/html/2607.13609#bib.bib12); Wang et al\.,[2016](https://arxiv.org/html/2607.13609#bib.bib17)\)are gauge\-invariant alternatives; the dividing line for the ridge pathology is gauge invariance, which they share and the identity seminorm lacks, not the exponent\.

#### Connection to graph representation learning\.

Operationally the gauge\-invariant potential is a directed\-graph*node feature*, useful downstream where the ridge potential is not \([Section˜8\.4](https://arxiv.org/html/2607.13609#S8.SS4)\)\. Its collapse is an instance of*oversmoothing*—repeated propagation driving node representations toward the constant mode\(Bodnar et al\.,[2022](https://arxiv.org/html/2607.13609#bib.bib2)\), just as the magnitude penalty collapses the potential’s range; we verify it \([Table˜6](https://arxiv.org/html/2607.13609#A3.T6)\), a vanilla directed GCN’s conversion AUC falling from0\.860\.86\(22layers\) to0\.810\.81\(3232\) as its node energy collapses\. The same gauge invariance fixes it: neutralizing the constant mode at each layer—the per\-layer analogue ofLG​𝟏=0L\_\{G\}\\mathbf\{1\}=0—holds AUC flat \(0\.850\.85at3232layers\), recovering PairNorm and Dirichlet\-energy\-constrained networks\(Zhao and Akoglu,[2020](https://arxiv.org/html/2607.13609#bib.bib19); Zhou et al\.,[2021](https://arxiv.org/html/2607.13609#bib.bib21)\)from one principle: flatness along the gauge mode both stabilizes the inverse problem and prevents oversmoothing\. Where magnetic\- and sheaf\-Laplacian networks\(Zhang et al\.,[2021](https://arxiv.org/html/2607.13609#bib.bib18); Bodnar et al\.,[2022](https://arxiv.org/html/2607.13609#bib.bib2)\)learn embeddings, we recover one interpretable scalar per node through an explicit forward operator\.

## 3From Raw Flows to an Acyclic Support

The inverse problem is posed on a directed acyclic graph, but raw flow is neither acyclic nor purely gradient: it carries reciprocal traffic, transient loops, and a solenoidal component no potential can explain\. Extraction is upstream machinery;[Section˜7\.1](https://arxiv.org/html/2607.13609#S7.SS1)shows the paper’s claims are*invariant*to how it is done, so we keep the description brief and defer mechanics to[appendix˜A](https://arxiv.org/html/2607.13609#A1)\.

#### Flow and dominance\.

Sessions are state sequences; collapsing consecutive repeats and counting transitions gives the empirical flowFu​vF\_\{uv\}, used as the conductanceWu​v=Fu​vW\_\{uv\}=F\_\{uv\}\. Reciprocal traffic is filtered by a*dominance*test: forρ≥1\\rho\\geq 1, the pair\(u,v\)\(u,v\)is dominant ifFu​v≥ρ​Fv​uF\_\{uv\}\\geq\\rho F\_\{vu\}\.

#### Orientation\.

An acyclic support needs an orientation of the dominant edges\. When session order is available, the default is a cheap*topological sort*: order states by mean visit position \(or net inflow\), keeping dominant edges that respect the order\. This needs no linear solve, and[Section˜7\.1](https://arxiv.org/html/2607.13609#S7.SS1)shows it recovers the potential at least as well as the alternatives\.

When only an aggregate flow matrix is available \(no session order to sort by\), we instead orient by the discrete Helmholtz–Hodge projection: the gradient potentialϕ0\\phi\_\{0\}solvingG⊤​G​ϕ0=G⊤​ωG^\{\\top\}G\\,\\phi\_\{0\}=G^\{\\top\}\\omegafor the skew flowωu​v=Fu​v−Fv​u\\omega\_\{uv\}=F\_\{uv\}\-F\_\{vu\}, with edges oriented by increasingϕ0\\phi\_\{0\}\(Jia et al\.,[2019](https://arxiv.org/html/2607.13609#bib.bib7); Schaub and Segarra,[2018](https://arxiv.org/html/2607.13609#bib.bib13)\)\. It needs only the flow, but the ablation finds it no more accurate than the topological sort and slower; we keep it solely for the order\-free setting\. Retention rules for both are in[appendix˜A](https://arxiv.org/html/2607.13609#A1)\.

#### Acyclicity and the Poisson right\-hand side\.

###### Proposition 1\(Acyclicity\)\.

Either orientation induces a topological order of the retained supportGδ=\(V,Eδ\)G\_\{\\delta\}=\(V,E\_\{\\delta\}\); henceGδG\_\{\\delta\}is acyclic\.

LetAAbe the substochastic transition operator onGδG\_\{\\delta\}\(outgoing weights normalized, absorbing rows at sinks\)\. SinceGδG\_\{\\delta\}is a DAG,AAis nilpotent\.

###### Proposition 2\(Nilpotency\)\.

If the longest directed path inGδG\_\{\\delta\}has lengthLL, thenAm=0A^\{m\}=0for somem≤L\+1m\\leq L\+1\.

Nilpotency is the algebraic certificate that extraction succeeded: a single residual cycle would makeAm≠0A^\{m\}\\neq 0for allmm; we verify‖Am‖F=0\\left\\lVert A^\{m\}\\right\\rVert\_\{F\}=0to machine precision on the instrument \([Section˜7](https://arxiv.org/html/2607.13609#S7)\)\. The empirical divergencebi=∑jFj​i−∑jFi​jb\_\{i\}=\\sum\_\{j\}F\_\{ji\}\-\\sum\_\{j\}F\_\{ij\}is the Poisson right\-hand side \([Section˜4](https://arxiv.org/html/2607.13609#S4)\)\.[Propositions˜1](https://arxiv.org/html/2607.13609#Thmproposition1)and[2](https://arxiv.org/html/2607.13609#Thmproposition2)are proved in[appendix˜B](https://arxiv.org/html/2607.13609#A2)\.

## 4The Poisson Inverse Problem

#### Constitutive law\.

The flux law[Eq\.˜1](https://arxiv.org/html/2607.13609#S1.E1)is the optimality condition of a minimum\-dissipation principle, pinning down the operator the regularizer must respect\.

###### Proposition 3\(Minimum dissipation\)\.

Among all edge flowsqqwith prescribed divergencediv⁡\(q\)=b\\operatorname\{div\}\(q\)=b, the one minimizing12​∑\(i,j\)∈EWi​j−1​qi​j2\\tfrac\{1\}\{2\}\\sum\_\{\(i,j\)\\in E\}W\_\{ij\}^\{\-1\}q\_\{ij\}^\{2\}is a gradient flow: there existsϕ\\phiwithqi​j=Wi​j​\(ϕj−ϕi\)q\_\{ij\}=W\_\{ij\}\(\\phi\_\{j\}\-\\phi\_\{i\}\), andϕ\\phisolvesL​ϕ=bL\\phi=b\.

#### Least\-squares objective and its degeneracies\.

Given a noisy empiricalbb, the natural estimator minimizes the flow residual

E​\(ϕ\)=12​‖L​ϕ−b‖22,L⊤​L​ϕ=L⊤​b\.E\(\\phi\)=\\tfrac\{1\}\{2\}\\left\\lVert L\\phi\-b\\right\\rVert\_\{2\}^\{2\},\\qquad L^\{\\top\}L\\,\\phi=L^\{\\top\}b\.\(2\)Two degeneracies make[Eq\.˜2](https://arxiv.org/html/2607.13609#S4.E2)ill\-posed even on exactbb:L​𝟏=0L\\mathbf\{1\}=0leavesϕ\\phidetermined only up to an additive gauge, and interior chains carry no divergence, which flattens the solution\.

###### Lemma 1\(Chain saturation\)\.

On an interior chainv1→⋯→vk→sv\_\{1\}\\to\\cdots\\to v\_\{k\}\\to swithbvi=0b\_\{v\_\{i\}\}=0for every interiorviv\_\{i\}andϕ​\(s\)\\phi\(s\)pinned, any solution ofL​ϕ=bL\\phi=bsatisfiesϕ​\(v1\)=⋯=ϕ​\(vk\)=ϕ​\(s\)\\phi\(v\_\{1\}\)=\\cdots=\\phi\(v\_\{k\}\)=\\phi\(s\)\.

[Lemma˜1](https://arxiv.org/html/2607.13609#Thmlemma1)is why the regularizer choice matters: a chain encodes the*ordering*of its states, but the unregularized solve flattens it\. A penalty that further compresses the chain \(as ridge does\) erases the ordering; one biasing toward a smooth ramp recovers it—the distinction[Section˜5](https://arxiv.org/html/2607.13609#S5)makes precise\.

#### Dirichlet boundaries fix the gauge\.

We resolve the constant\-mode degeneracy by pinning the potential on the absorbing boundaryBB:*conversion*sinks at11and an*abandon*sink at0\. PartitioningV=I∪BV=I\\cup Band writingL,bL,bin block form, the reduced unknownϕI\\phi\_\{I\}has effective right\-hand sidebI−LI​B​ϕBb\_\{I\}\-L\_\{IB\}\\phi\_\{B\}and operatorLI​I⊤​LI​IL\_\{II\}^\{\\top\}L\_\{II\}\. This removes the gauge freedom but, as[Section˜5](https://arxiv.org/html/2607.13609#S5)shows, does not prevent a magnitude penalty from biasing the interior toward the boundary\-induced origin\.[Propositions˜3](https://arxiv.org/html/2607.13609#Thmproposition3)and[1](https://arxiv.org/html/2607.13609#Thmlemma1)are proved in[appendix˜B](https://arxiv.org/html/2607.13609#A2)\.

## 5Graph\-Sobolev Regularization

We show that ridge \(standard\-form Tikhonov\) regularization inverts the recovered ordering \([Section˜5\.1](https://arxiv.org/html/2607.13609#S5.SS1)\), introduce the gauge\-invariant Dirichlet\-energy penalty \([Section˜5\.2](https://arxiv.org/html/2607.13609#S5.SS2)\), and establish three properties: gauge invariance \([Section˜5\.3](https://arxiv.org/html/2607.13609#S5.SS3)\), an SPD reduced system \([Section˜5\.4](https://arxiv.org/html/2607.13609#S5.SS4)\), and exact range preservation on chains \([Section˜5\.5](https://arxiv.org/html/2607.13609#S5.SS5)\)\.[Section˜5\.7](https://arxiv.org/html/2607.13609#S5.SS7)reports the headline*parameter\-insensitivity*: the estimate is stable across four orders of magnitude inλ1\\lambda\_\{1\}, ridge only atλ1→0\\lambda\_\{1\}\\\!\\to\\\!0\. Proofs are in[appendix˜B](https://arxiv.org/html/2607.13609#A2)\.

### 5\.1The ridge inversion and the gauge mismatch

The standard estimator augments[Eq\.˜2](https://arxiv.org/html/2607.13609#S4.E2)with a magnitude penalty,

ϕ^Tik=arg​minϕ⁡12​‖L​ϕ−b‖22\+λ1​‖ϕ‖22,\\hat\{\\phi\}\_\{\\mathrm\{Tik\}\}=\\operatorname\*\{arg\\,min\}\_\{\\phi\}\\ \\tfrac\{1\}\{2\}\\left\\lVert L\\phi\-b\\right\\rVert\_\{2\}^\{2\}\+\\lambda\_\{1\}\\left\\lVert\\phi\\right\\rVert\_\{2\}^\{2\},\(3\)subject to the Dirichlet boundary\. The penalty pulls every interior value toward0\. But on a Dirichlet problem0is not distinguished: it is merely where the boundary placed the abandon sink\. The estimator therefore has a bias toward the abandon boundary set by the boundary labelling, not the data, and[Proposition˜4](https://arxiv.org/html/2607.13609#Thmproposition4)makes its effect exact: the interior range shrinks like1/λ11/\\lambda\_\{1\}, so the ordering flattens and then reverses\.

###### Proposition 4\(Magnitude shrinkage collapses the range\)\.

Letϕ^Tik​\(λ1\)\\hat\{\\phi\}\_\{\\mathrm\{Tik\}\}\(\\lambda\_\{1\}\)solve[Eq\.˜3](https://arxiv.org/html/2607.13609#S5.E3)on an interior chain of lengthLLwith the sink pinned at11and unit divergence injected at the free source\. Then the interior dynamic rangeΔ​ϕ^Tik=maxI⁡ϕ^−minI⁡ϕ^\\Delta\\hat\{\\phi\}\_\{\\mathrm\{Tik\}\}=\\max\_\{I\}\\hat\{\\phi\}\-\\min\_\{I\}\\hat\{\\phi\}satisfiesΔ​ϕ^Tik​\(λ1\)≤C/λ1→0\\Delta\\hat\{\\phi\}\_\{\\mathrm\{Tik\}\}\(\\lambda\_\{1\}\)\\leq C/\\lambda\_\{1\}\\to 0asλ1→∞\\lambda\_\{1\}\\to\\infty, with a constantCC*independent of the chain lengthLL*: the magnitude penalty extracts no benefit from a longer chain\.

[Proposition˜4](https://arxiv.org/html/2607.13609#Thmproposition4)does not claim the rank correlation tends to−1\-1; empirically it settles to a negative plateau near−0\.42\-0\.42\([Figure˜1](https://arxiv.org/html/2607.13609#S5.F1)\), because the collapsed order is dominated by residual divergence structure anticorrelated with the planted field\. The damage is decisive: ridge*inverts*the ordering for everyλ1\>0\\lambda\_\{1\}\>0, linear correlation crossing to zero \([Table˜1](https://arxiv.org/html/2607.13609#S5.T1)\)\.

### 5\.2The graph\-Sobolev penalty

###### Definition 1\(Graph\-Sobolev objective\)\.

LetLGL\_\{G\}be the symmetric, conductance\-weighted graph Laplacian of the undirected support ofGδG\_\{\\delta\}, so that

ϕ⊤​LG​ϕ=∑\(u,v\)∈Ewu​v​\(ϕv−ϕu\)2\.\\phi^\{\\top\}L\_\{G\}\\phi=\\sum\_\{\(u,v\)\\in E\}w\_\{uv\}\(\\phi\_\{v\}\-\\phi\_\{u\}\)^\{2\}\.\(4\)The graph\-Sobolev estimator is

ϕ^Sob=arg​minϕ⁡12​‖L​ϕ−b‖22\+λ1​ϕ⊤​LG​ϕ,\\hat\{\\phi\}\_\{\\mathrm\{Sob\}\}=\\operatorname\*\{arg\\,min\}\_\{\\phi\}\\ \\tfrac\{1\}\{2\}\\left\\lVert L\\phi\-b\\right\\rVert\_\{2\}^\{2\}\+\\lambda\_\{1\}\\,\\phi^\{\\top\}L\_\{G\}\\phi,\(5\)subject to the Dirichlet boundary\.

We use the*weighted*energy: each edge contributeswu​v​\(ϕv−ϕu\)2w\_\{uv\}\(\\phi\_\{v\}\-\\phi\_\{u\}\)^\{2\}with the flux conductance of[Eq\.˜1](https://arxiv.org/html/2607.13609#S1.E1)\(the penalty in the released code; every number uses it\)\. The unweighted incidence formLG=G⊤​GL\_\{G\}=G^\{\\top\}Gis the casewu​v≡1w\_\{uv\}\\equiv 1\.

### 5\.3Gauge invariance

A regularizerRRis*gauge\-invariant*ifR​\(ϕ\+c​𝟏\)=R​\(ϕ\)R\(\\phi\+c\\mathbf\{1\}\)=R\(\\phi\)for allc∈ℝc\\in\\mathbb\{R\}\.

###### Proposition 5\(Gauge invariance\)\.

The Dirichlet energyϕ⊤​LG​ϕ\\phi^\{\\top\}L\_\{G\}\\phiis gauge\-invariant, whereas the magnitude penalty‖ϕ‖22\\left\\lVert\\phi\\right\\rVert\_\{2\}^\{2\}is not\.

The conceptual heart: sinceLG​𝟏=0L\_\{G\}\\mathbf\{1\}=0the Dirichlet energy penalizes only*differences*and is indifferent to the origin, whereas‖ϕ‖22\\left\\lVert\\phi\\right\\rVert\_\{2\}^\{2\}imposes a preferred mean\-zero gauge that has no meaning in the Dirichlet problem and fights the boundary conditions\.

### 5\.4Well\-posedness

###### Theorem 1\(SPD reduced system\)\.

Suppose every interior node has a directed path to the boundary inGδG\_\{\\delta\}\. Then for everyλ1\>0\\lambda\_\{1\}\>0the reduced graph\-Sobolev operator

M=\(L⊤​L\)I​I\+λ1​\(LG\)I​IM=\\bigl\(L^\{\\top\}L\\bigr\)\_\{II\}\+\\lambda\_\{1\}\\,\(L\_\{G\}\)\_\{II\}\(6\)is symmetric positive definite, so[Eq\.˜5](https://arxiv.org/html/2607.13609#S5.E5)has a unique interior solution\. The added PSD term also improves the conditioning of the unregularized operator without disturbing the gauge\.

Under this hypothesisMMis SPD and a Cholesky factorization solves[Eq\.˜5](https://arxiv.org/html/2607.13609#S5.E5)directly\. In practice, however, many interior nodes have no path to a sink, so the hypothesis fails andMMis only positive*semi*definite; we therefore solve by a symmetric eigensolve that inverts only the nonzero spectrum, giving the minimum\-norm solution\. This keeps the gauge mode \(in the range ofMM\) exact and leaves undetermined nodes at the smooth default, excluded from scoring \([Section˜5\.7](https://arxiv.org/html/2607.13609#S5.SS7)\); a Cholesky solve with a diagonal floor would instead amplify roundoff along the near\-null directions\.

### 5\.5Range preservation on chains

###### Theorem 2\(Range preservation, exact\)\.

On the length\-LLconstant\-conductance chain of[Proposition˜4](https://arxiv.org/html/2607.13609#Thmproposition4), with the trace\-normalized Dirichlet\-energy penalty \([Section˜5\.6](https://arxiv.org/html/2607.13609#S5.SS6)\), the graph\-Sobolev estimator has interior dynamic range exactly

Δ​ϕ^Sob​\(λ1\)=2​L\+12​L\+1\+2​λ1=11\+λ1​κ​\(L\),κ​\(L\)=22​L\+1=O​\(1/L\)\.\\Delta\\hat\{\\phi\}\_\{\\mathrm\{Sob\}\}\(\\lambda\_\{1\}\)=\\frac\{2L\+1\}\{\\,2L\+1\+2\\lambda\_\{1\}\\,\}=\\frac\{1\}\{1\+\\lambda\_\{1\}\\,\\kappa\(L\)\},\\qquad\\kappa\(L\)=\\frac\{2\}\{2L\+1\}=O\(1/L\)\.\(7\)Hence for any fixedλ1\\lambda\_\{1\}the retained range tends to the full harmonic rangeΔ​ϕtrue=1\\Delta\\phi\_\{\\mathrm\{true\}\}=1asL→∞L\\to\\infty\. Contrasted with[Proposition˜4](https://arxiv.org/html/2607.13609#Thmproposition4), the Sobolev range depends onλ1\\lambda\_\{1\}only throughλ1/\(2​L\+1\)\\lambda\_\{1\}/\(2L\+1\), whereas ridge collapses like1/λ11/\\lambda\_\{1\}independently ofLL: a longer chain helps the Dirichlet\-energy estimate and never helps ridge\.

On a length\-3030chain,[Eq\.˜7](https://arxiv.org/html/2607.13609#S5.E7)givesΔ​ϕ^Sob=61/63=0\.968\\Delta\\hat\{\\phi\}\_\{\\mathrm\{Sob\}\}=61/63=0\.968atλ1=1\\lambda\_\{1\}=1and61/81=0\.75361/81=0\.753atλ1=10\\lambda\_\{1\}=10, versus0\.5360\.536and0\.0910\.091for ridge \([Figure˜4](https://arxiv.org/html/2607.13609#A3.F4),[appendix˜B](https://arxiv.org/html/2607.13609#A2)\)\.

### 5\.6Trace normalization and cost

To makeλ1\\lambda\_\{1\}comparable across graphs, we normalize the penalty block by its trace,λ1​\(LG\)I​I↦λ1​\(LG\)I​I/tr⁡\[\(LG\)I​I\]\\lambda\_\{1\}\(L\_\{G\}\)\_\{II\}\\mapsto\\lambda\_\{1\}\(L\_\{G\}\)\_\{II\}/\\operatorname\{tr\}\[\(L\_\{G\}\)\_\{II\}\]; all sweeps use this\. The dense solve isO​\(\|I\|3\)O\(\\lvert I\\rvert^\{3\}\): Cholesky would suffice under[Theorem˜1](https://arxiv.org/html/2607.13609#Thmtheorem1)\(MMSPD\), but real extractions violate that hypothesis, so we use the symmetric eigensolve of[Section˜5\.4](https://arxiv.org/html/2607.13609#S5.SS4)throughout to return the minimum\-norm solution\. It is negligible for the tens\-to\-hundreds of interior states here; a conjugate\-gradient solve atO​\(κ​\(M\)​nnz\)O\(\\kappa\(M\)\\,\\mathrm\{nnz\}\)is available when\|I\|\\lvert I\\rvertis large\.

### 5\.7Parameter\-insensitivity: preservation versus collapse

We compare the two penalties on the synthetic instrument of[Section˜7](https://arxiv.org/html/2607.13609#S7)\(planted harmonicϕtrue\\phi\_\{\\mathrm\{true\}\},\|V\|=277\\lvert V\\rvert=277, five sinks\), scoring on the*determined*interior: then=95n=95well\-visited states \(≥100\\geq 100visits\) with a directed path to the boundary \(nodes without one are undetermined by the data,[Theorem˜1](https://arxiv.org/html/2607.13609#Thmtheorem1)\)\. Rank correlation is computed on estimates rounded to numerical tolerance so the saturated near\-zero cluster \([Lemma˜1](https://arxiv.org/html/2607.13609#Thmlemma1)\) ties deterministically rather than by roundoff\.[Figures˜1](https://arxiv.org/html/2607.13609#S5.F1)and[1](https://arxiv.org/html/2607.13609#S5.T1)report the headline—robustness toλ1\\lambda\_\{1\}, not a single best setting\. Atλ1=0\\lambda\_\{1\}=0both estimators recover the planted ordering \(ρs=\+0\.81\\rho\_\{\\mathrm\{s\}\}=\+0\.81,r=\+0\.76r=\+0\.76\); asλ1\\lambda\_\{1\}grows, graph\-Sobolev holds this unchanged \(rank correlation constant at\+0\.807\+0\.807acrossλ1∈\[10−3,10\]\\lambda\_\{1\}\\in\[10^\{\-3\},10\],r∈\[\+0\.76,\+0\.85\]r\\in\[\+0\.76,\+0\.85\]\), while ridge*inverts*the instant regularization is applied, dropping to a negative plateau near−0\.42\-0\.42\(linear correlation crossing zero\)\. The≈\+1\.23\{\\approx\}\+1\.23gap is itself*flat inλ1\\lambda\_\{1\}*—no ridge setting recovers the lost structure \(the pointwise collapse is[Figure˜3](https://arxiv.org/html/2607.13609#A3.F3),[appendix˜C](https://arxiv.org/html/2607.13609#A3)\)\. The same holds for*top\-kkpower*: NDCG@55stays in\[0\.97,1\.0\]\[0\.97,1\.0\]across the wholeλ1\\lambda\_\{1\}range under Sobolev but drops to0\.760\.76for ridge \([Table˜1](https://arxiv.org/html/2607.13609#S5.T1)\), so parameter\-insensitivity shows in top\-of\-ranking quality, not only correlation\.

Table 1:Recovery and top\-55ranking power versusλ1\\lambda\_\{1\}\(determined interior,n=95n=95; HHD extraction\)\. The gauge\-invariant penalty is insensitive toλ1\\lambda\_\{1\}\(rank correlation constant at\+0\.81\+0\.81and NDCG@55in\[0\.97,1\.0\]\[0\.97,1\.0\]across four orders of magnitude\), while ridge*inverts*the ordering for everyλ1\>0\\lambda\_\{1\}\>0\(Spearman to≈−0\.42\{\\approx\}\-0\.42, NDCG@55from0\.970\.97to0\.760\.76\)\. Metrics vs\. the plantedϕtrue\\phi\_\{\\mathrm\{true\}\}\(graded relevance for NDCG\); Spearman uses estimates rounded to tolerance so the saturated near\-zero cluster does not break ties by roundoff\.![Refer to caption](https://arxiv.org/html/2607.13609v1/x1.png)Figure 1:Recovery of the planted potential versusλ1\\lambda\_\{1\}\. Graph\-Sobolev \(blue\) preserves both rank and linear correlation across four orders of magnitude; Tikhonov \(red\) inverts the rank correlation to≈−0\.42\{\\approx\}\-0\.42and collapses the linear correlation to zero\. Determined interior,n=95n=95\.

## 6Boundary Identification from Flow Geometry

When the absorbing boundary is unknown, ranking states by the recovered potential fails on multi\-sink graphs: with several sinks pinned at the same value, the interior potential is rank\-deficient and sinks do not separate from high\-potential interior states\. The*Poisson residual*ru=\|\[L​ϕ\]u−bu\|r\_\{u\}=\\lvert\[L\\phi\]\_\{u\}\-b\_\{u\}\\rvertsucceeds instead\.

###### Theorem 3\(Residual concentration at absorbing nodes\)\.

Letuube an absorbing sink with no outgoing DAG edges\. Then\[L​ϕ\]u=0\[L\\phi\]\_\{u\}=0exactly, soru=\|bu\|r\_\{u\}=\\lvert b\_\{u\}\\rvertequals the empirical inflow atuu\. If every interior node satisfies the Poisson equation to toleranceη\\etaandminu∈B⁡\|bu\|\>η\\min\_\{u\\in B\}\\lvert b\_\{u\}\\rvert\>\\eta, then every sink ranks strictly above every interior node underrr\.

A sink is where mass accumulates without a modeled outflow: the forward operator predicts zero net flux but the data shows large inflow, and that mismatch is the residual; interior nodes, where the solve balances flux, have small residual\. On the synthetic instrument \(\|V\|=277\\lvert V\\rvert=277, five planted sinks\) the residual ranks all five true sinks in the top seven states \(worst rank77\), while potential ranking scatters them to worst rank108108; the separation is invariant across abandonment rates\{0\.05,0\.20,0\.50\}\\\{0\.05,0\.20,0\.50\\\}, making the residual a regime\-invariant boundary detector\.[Theorem˜3](https://arxiv.org/html/2607.13609#Thmtheorem3)is proved in[appendix˜B](https://arxiv.org/html/2607.13609#A2)\.

## 7Synthetic Validation

We validate on a controlled instrument: recovery can only be scored against a known potential, which no observational corpus provides\. The instrument plants a ground\-truth potential, samples flow from it, and hands the algorithm only the flow; every number comes from the released generator at one fixed configuration\.

#### The instrument\.

The generator builds a multi\-branch conversion funnel: a source feeds shared upper states; each ofK=5K=5branches is a chain terminating in a conversion sink; one abandon sink absorbs drop\-off at a depth\-decaying rate\. The ground\-truth potential is the harmonic extension under\{conversion=1,abandon=0\}\\\{\\text\{conversion\}=1,\\text\{abandon\}=0\\\}, i\.e\.ϕtrue​\(u\)=Pr⁡\[absorbed at a conversion sink before abandon∣start at​u\]\\phi\_\{\\mathrm\{true\}\}\(u\)=\\Pr\[\\text\{absorbed at a conversion sink before abandon\}\\mid\\text\{start at \}u\], exactly the object the Poisson solve targets\. The default point uses2020shared states,5050interior states per branch, abandonment0\.050\.05, and50,00050\{,\}000sessions, giving\|V\|=277\\lvert V\\rvert=277and terminal\-sink entropyH​\(π\)≈2\.27H\(\\pi\)\\approx 2\.27bits; sessions are reproducible bit\-for\-bit at a given seed\.

#### Structural certificates and recovery\.

The extracted operator is nilpotent \(‖Am‖F=0\\left\\lVert A^\{m\}\\right\\rVert\_\{F\}=0at the predictedm=7m=7\), confirming[Proposition˜2](https://arxiv.org/html/2607.13609#Thmproposition2)\(no residual cycle\)\. The Hodge gradient componentϕ0\\phi\_\{0\}alone does*not*recover the planted direction \(conductance\-weighted cosine−0\.06\-0\.06\): an honest negative showing the gradient projection is a structural device, not an estimator\. Recovery is the regularized solve’s job \([Section˜5\.7](https://arxiv.org/html/2607.13609#S5.SS7)\): graph\-Sobolev holds rank correlation\+0\.81\+0\.81while ridge*inverts*to≈−0\.42\{\\approx\}\-0\.42, and preserves97%97\\%of the chain range against ridge’s54%54\\%\([Figure˜4](https://arxiv.org/html/2607.13609#A3.F4)\), independent of the Hodge step \([Section˜7\.1](https://arxiv.org/html/2607.13609#S7.SS1)\)\.

### 7\.1Does the Hodge projection matter? An extraction ablation

We ask whether any acyclic support—in particular a cheap topological sort— serves as well as the Hodge projection, replacing it with two dominance\-thresholded topological sorts, the rest of the pipeline fixed\. The orders come from data alone:*visit\-position*\(mean fractional session position\) and*net\-inflow*\(∑v\(Fv​u−Fu​v\)\\sum\_\{v\}\(F\_\{vu\}\-F\_\{uv\}\)\); an edge is kept when net\-forward, dominant, and order\-consistent \(acyclic by construction\)\.

[Table˜2](https://arxiv.org/html/2607.13609#S7.T2)reports recovery under each extraction\. Two conclusions follow\. First, the regularizer contrast, the paper’s central claim, is*invariant to the extraction*: graph\-Sobolev recovers a strong positive ordering under all three supports while ridge*inverts*under all three \(ρs=−0\.42,−0\.32,−0\.56\\rho\_\{\\mathrm\{s\}\}=\-0\.42,\-0\.32,\-0\.56; margin\+1\.23\+1\.23to\+1\.52\+1\.52\)\. Second, Hodge is*not*the best extraction for recovery: both topological sorts give higher Sobolev rank correlation \(\+0\.96,\+0\.99\+0\.96,\+0\.99\) than Hodge \(\+0\.81\+0\.81\)\. We treat the topological sort as the default, keeping Hodge for its generality \(only the flow matrix\) and the nilpotency certificate, not accuracy\. Either way, the contribution is isolated in the regularizer, not a Hodge artifact\.

Table 2:Extraction ablation \(λ1=1\\lambda\_\{1\}=1, determined interior:n=95n=95for Helmholtz–Hodge,n=150n=150for the topological sorts, which reach the boundary from every well\-visited state\)\. The regularizer contrast holds under every acyclic support: graph\-Sobolev recovers a strong positive ordering while ridge inverts\. Hodge is not required for recovery; all supports are exactly acyclic\.
### 7\.2Regime dependence and scope

The*magnitude*of the recoverable correlation is a regime property, not the regularizer’s: at low abandonment the planted field is compressed and the unregularized solve recoversρs≈\+0\.81\\rho\_\{\\mathrm\{s\}\}\\approx\+0\.81, while as abandonment rises the well\-visited interior shrinks and recovery degrades \(toρs≈\+0\.23\\rho\_\{\\mathrm\{s\}\}\\approx\+0\.23at abandonment0\.50\.5\)\. The regularizer’s role is invariant throughout: graph\-Sobolev*preserves*whatever ordering the unregularized solve attains and ridge*degrades*it; what is regime\-dependent is whether there is signal to preserve\.

We confirm this is not an artifact of the single configuration of[Table˜1](https://arxiv.org/html/2607.13609#S5.T1): across twelve instrument variants—five seeds plus single\-axis perturbations of branch count, chain depth, abandonment, and sink entropy—the Sobolev\-minus\-Tikhonov rank margin atλ1=1\\lambda\_\{1\}=1is positive in*all twelve*under both extractions \([Table˜7](https://arxiv.org/html/2607.13609#A3.T7),[appendix˜C](https://arxiv.org/html/2607.13609#A3)\)\. On the topological\-sort support the ordering is configuration\-invariant \(ρs=\+0\.99±0\.00\\rho\_\{\\mathrm\{s\}\}=\+0\.99\\pm 0\.00vs\. ridge−0\.34±0\.06\-0\.34\\pm 0\.06\); on the Hodge support the Sobolev correlation tracks the regime \(down to\+0\.23\+0\.23at abandonment0\.50\.5\) yet stays positive, with margin never below\+0\.45\+0\.45\.

### 7\.3Scaling to10410^\{4\}nodes

Because the reduced system is symmetric PSD \([Theorem˜1](https://arxiv.org/html/2607.13609#Thmtheorem1)\), the solve admits sparse conjugate gradients \(CG\), which converge to the same minimum\-norm estimate as the dense path\. On a planted funnel with\|V\|=10,202\\lvert V\\rvert=10\{,\}202\(roughly10410^\{4\}nodes\), graph\-Sobolev recoversρs=\+0\.94\\rho\_\{\\mathrm\{s\}\}=\+0\.94while ridge inverts to−0\.18\-0\.18, and CG returns in0\.080\.08s where the dense solve is killed for memory\.[Table˜4](https://arxiv.org/html/2607.13609#A3.T4)\([appendix˜C](https://arxiv.org/html/2607.13609#A3)\) times CG against the dense eigensolve: CG matches it to10−910^\{\-9\}, scales near\-linearly, and handles4×1044\\times 10^\{4\}nodes in0\.390\.39s \(dense exhausts memory beyond∼4,000\{\\sim\}4\{,\}000\), using the topological\-sort extraction of[Section˜7\.1](https://arxiv.org/html/2607.13609#S7.SS1)\.

## 8Real\-Data Validation

No clickstream corpus carries a ground\-truth potential, so we validate claims that need no planted field: the recovered potential is interpretable \([Section˜8\.1](https://arxiv.org/html/2607.13609#S8.SS1)\); the regularizer contrast holds*internally*\([Section˜8\.2](https://arxiv.org/html/2607.13609#S8.SS2)\); and the residual concentrates at the known boundary whileϕ\\phiis stable under resampling \([Section˜8\.3](https://arxiv.org/html/2607.13609#S8.SS3)\)\. We use three public corpora, each reduced to an event\-type state space \([Section˜3](https://arxiv.org/html/2607.13609#S3.SS0.SSS0.Px1)\) and summarized in[Table˜3](https://arxiv.org/html/2607.13609#S8.T3):RetailRocket\(a tiny funnel\),Trivago\(a medium action graph\), andOTTO\(a large multi\-sink graph\)\.

Table 3:Real corpora, reduced to event\-type state spaces\. Conversion is the fraction of sessions terminating at a conversion sink\.### 8\.1Recovered potentials are interpretable

The graph\-Sobolev potential \(λ1=1\\lambda\_\{1\}=1\) reproduces the known funnel ordering \([Figure˜5](https://arxiv.org/html/2607.13609#A3.F5),[appendix˜C](https://arxiv.org/html/2607.13609#A3)\)\. On RetailRocket,view\(−0\.60\-0\.60\) sits below the abandon origin andaddtocart\(\+0\.32\+0\.32\) between the boundaries; on Trivago, clickout is highest and searches lowest, with one flagged anomaly \(interaction item image,−1\.52\-1\.52\)\. On OTTO \(the only genuinely multi\-sink graph\) the solve separatescarts\(mean\+0\.26\+0\.26, straddling the cart sink\) fromclicks\(mean−0\.01\-0\.01\), recoveringclicks<carts<orders\\text\{clicks\}<\\text\{carts\}<\\text\{orders\}from flow alone\.

### 8\.2The regularizer contrast holds without ground truth

With no truth, we measure how each penalty atλ1=1\\lambda\_\{1\}=1perturbs the unregularized solve along two axes: interior*ordering*\(Spearman vs\. the base\) and*dynamic range*\(fraction retained\)\.[Tables˜5](https://arxiv.org/html/2607.13609#A3.T5)and[2](https://arxiv.org/html/2607.13609#A3.F2)reproduce the synthetic finding cleanly: graph\-Sobolev retains2828–41%41\\%of the range across all three corpora, while ridge retains21%21\\%,5%5\\%, and, on the105105\-state OTTO graph, just0\.2%0\.2\\%\. The contrast sharpens with graph size; on OTTO ridge retains essentially none \(rank agreement0\.420\.42vs\. graph\-Sobolev’s0\.660\.66\), as[Proposition˜4](https://arxiv.org/html/2607.13609#Thmproposition4)predicts\.

### 8\.3Residual concentration and stability

The Poisson residual ranks the known conversion sink in the top one to three of up to105105states \(RetailRocket3/43/4, Trivago1/111/11, OTTO2/1052/105\), confirming[Theorem˜3](https://arxiv.org/html/2607.13609#Thmtheorem3); being single\-sink, these corpora do not exercise its multi\-sink advantage over potential ranking\. Bootstrap resampling \(2020resamples\) gives small coefficients of variation on well\-visited states \(median0\.0100\.010,0\.0210\.021,0\.1270\.127\);ϕ\\phiis stable wherever the divergence is well\-sampled\.

### 8\.4The recovered potential is a usable node feature

Does the potential carry*downstream*signal? We predict OTTO session conversion with a logistic regression on four per\-session order statistics —mean, max, min, last—of the interior states’ potentials; the task is leakage\-guarded, as the label is the terminal sink, the features exclude every sink, and the potential is fit on the training sessions only \([appendix˜C](https://arxiv.org/html/2607.13609#A3.SS0.SSS0.Px1)\)\. The graph\-Sobolev potential adds\+3\.1\+3\.1ROC\-AUC points over raw session statistics \(0\.828→0\.8590\.828\\\!\\to\\\!0\.859;[Table˜6](https://arxiv.org/html/2607.13609#A3.T6)\) while ridge adds nothing \(0\.8290\.829\); standalone, the44\-D Sobolev feature scores0\.8360\.836against ridge’s0\.7170\.717\. The gauge\-invariant penalty yields a usable \(if compressed\) node feature where the collapsed ridge potential does not \([Table˜6](https://arxiv.org/html/2607.13609#A3.T6)\)\.

## 9Conclusion

We showed that the standard regularizer for directed Poisson inverse problems fails specifically: the ridge \(identity\-seminorm\) penalty imposes a meaningless origin and, as its strength grows, collapses the dynamic range and reverses the ordering, so the only safe setting is no regularization\. The gauge\-invariant Dirichlet energy removes the hazard, with one defining consequence,*parameter\-insensitivity*: the recovered potential is stable across four orders of magnitude inλ1\\lambda\_\{1\}, whereas ridge inverts the ordering for everyλ1\>0\\lambda\_\{1\}\>0\. We proved gauge invariance \([Proposition˜5](https://arxiv.org/html/2607.13609#Thmproposition5)\), an SPD reduced system \([Theorem˜1](https://arxiv.org/html/2607.13609#Thmtheorem1)\), and exact range preservation on chains \([Theorem˜2](https://arxiv.org/html/2607.13609#Thmtheorem2)\); gave a Poisson\-residual boundary diagnostic; and positioned the contribution against general\-form Tikhonov and graph smoothness priors\. The penalty is classical, but its necessity and parameter\-insensitivity here are not\. The guarantee is preservation, not signal: whatever ordering the data supports, the gauge\-invariant penalty keeps it acrossλ1\\lambda\_\{1\}and ridge does not\.

#### Future work\.

Open directions: characterizing when the divergence carries enough of the harmonic field for recovery \(a theorem for the regime dependence of[Section˜7\.2](https://arxiv.org/html/2607.13609#S7.SS2)\); nonlinear constitutive laws; edge\-preserving gauge\-invariant penalties \(total variation, thepp\-Laplacian\); and which acyclic supports best aid recovery \([Section˜7\.1](https://arxiv.org/html/2607.13609#S7.SS1)\)\. All experiments, figures, and tables are reproduced by the released code from a single fixed\-seed command\.

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## Appendix AExtraction Mechanics

This appendix gives the edge\-retention rule deferred from[Section˜3](https://arxiv.org/html/2607.13609#S3), for both orientations\. Write the oriented coordinate asψ\\psi: for the topological sortψ\\psiis the mean visit position \(or net inflow\); for the Helmholtz–Hodge projectionψ=ϕ0\\psi=\\phi\_\{0\}, the gradient potential solvingG⊤​G​ϕ0=G⊤​ωG^\{\\top\}G\\,\\phi\_\{0\}=G^\{\\top\}\\omega\. With the gradient\-dominance ratio

δ​\(u,v\)=\|ψ​\(v\)−ψ​\(u\)\|\|ωu​v\|\+ϵ,ωu​v=Fu​v−Fv​u,\\delta\(u,v\)=\\frac\{\\lvert\\psi\(v\)\-\\psi\(u\)\\rvert\}\{\\lvert\\omega\_\{uv\}\\rvert\+\\epsilon\},\\qquad\\omega\_\{uv\}=F\_\{uv\}\-F\_\{vu\},\(8\)the retained edge set is

Eδ=\{\(u,v\):ψ​\(v\)\>ψ​\(u\)​and​δ​\(u,v\)≥τ\},E\_\{\\delta\}=\\bigl\\\{\(u,v\):\\psi\(v\)\>\\psi\(u\)\\ \\text\{and\}\\ \\delta\(u,v\)\\geq\\tau\\bigr\\\},\(9\)followed by top\-kkpruning per source \(retain each node’skkheaviest outgoing edges\)\. The thresholdτ\\tautrades support density against strictness: a largerτ\\taukeeps only edges whose oriented drop dominates the raw flow, and on small dense graphs \(for example the eleven\-state event graph of[Section˜8](https://arxiv.org/html/2607.13609#S8)\) a lowerτ\\tauis needed to avoid disconnecting interior states\. All reported runs useρ=2\\rho=2,k=10k=10, andτ\\tauas stated per experiment\.

## Appendix BProofs

#### The chain instrument\.

Both proofs use the chain implemented in the released code\. The nodes are0,1,…,L\+10,1,\\dots,L\+1with unit\-conductance edgesi→i\+1i\\to i\+1fori=0,…,Li=0,\\dots,L\. The sinkL\+1L\+1is the only Dirichlet node, pinned atϕL\+1=1\\phi\_\{L\+1\}=1; unit divergence is injected at the source and absorbed at the sink,

b0=−1,bL\+1=\+1,bi=0​\(1≤i≤L\)\.b\_\{0\}=\-1,\\qquad b\_\{L\+1\}=\+1,\\qquad b\_\{i\}=0\\ \(1\\leq i\\leq L\)\.The interior unknown isx=\(ϕ0,…,ϕL\)x=\(\\phi\_\{0\},\\dots,\\phi\_\{L\}\)\. The directed Laplacian acts as\(L​ϕ\)i=ϕi−ϕi\+1\(L\\phi\)\_\{i\}=\\phi\_\{i\}\-\\phi\_\{i\+1\}fori=0,…,Li=0,\\dots,L\(and\(L​ϕ\)L\+1=0\(L\\phi\)\_\{L\+1\}=0since the sink has no out\-edge\)\. Writing the interior blockLI​IL\_\{II\}and eliminating the pinned sink gives the effective right\-hand sidebeff=bI−LI​B​ϕBb^\{\\mathrm\{eff\}\}=b\_\{I\}\-L\_\{IB\}\\phi\_\{B\}, which has a−1\-1in coordinate0and a\+1\+1in coordinateLLand is zero elsewhere\. It is convenient to use the*edge gaps*

di=ϕi−ϕi\+1,i=0,…,L,so​ϕi=1\+∑j=iLdj,d\_\{i\}\\;=\\;\\phi\_\{i\}\-\\phi\_\{i\+1\},\\qquad i=0,\\dots,L,\\qquad\\text\{so \}\\phi\_\{i\}=1\+\\textstyle\\sum\_\{j=i\}^\{L\}d\_\{j\},sinceϕL\+1=1\\phi\_\{L\+1\}=1\. A short computation gives\(LI​I​x−beff\)0=d0\+1\(L\_\{II\}x\-b^\{\\mathrm\{eff\}\}\)\_\{0\}=d\_\{0\}\+1,\(⋅\)i=di\(\\,\\cdot\\,\)\_\{i\}=d\_\{i\}for1≤i≤L1\\leq i\\leq L, so

‖LI​I​x−beff‖22=\(d0\+1\)2\+∑i=1Ldi2,ϕ⊤​LG​ϕ=∑i=0Ldi2\.\\left\\lVert L\_\{II\}x\-b^\{\\mathrm\{eff\}\}\\right\\rVert\_\{2\}^\{2\}=\(d\_\{0\}\+1\)^\{2\}\+\\sum\_\{i=1\}^\{L\}d\_\{i\}^\{2\},\\qquad\\phi^\{\\top\}L\_\{G\}\\phi=\\sum\_\{i=0\}^\{L\}d\_\{i\}^\{2\}\.\(10\)The interior dynamic range isΔ​ϕ^=max0≤i≤L⁡ϕi−min0≤i≤L⁡ϕi\\Delta\\hat\{\\phi\}=\\max\_\{0\\leq i\\leq L\}\\phi\_\{i\}\-\\min\_\{0\\leq i\\leq L\}\\phi\_\{i\}\.

### B\.1Proof of[Proposition˜4](https://arxiv.org/html/2607.13609#Thmproposition4)

Ridge minimizes12​‖LI​I​x−beff‖22\+λ1​‖x‖22\\tfrac\{1\}\{2\}\\left\\lVert L\_\{II\}x\-b^\{\\mathrm\{eff\}\}\\right\\rVert\_\{2\}^\{2\}\+\\lambda\_\{1\}\\left\\lVert x\\right\\rVert\_\{2\}^\{2\}\(up to the positive data\-normalization constant, which only rescalesλ1\\lambda\_\{1\}\), with normal equations\(A\+λ1​I\)​x=c\(A\+\\lambda\_\{1\}I\)x=c, whereA=LI​I⊤​LI​I⪰0A=L\_\{II\}^\{\\top\}L\_\{II\}\\succeq 0andc=LI​I⊤​beffc=L\_\{II\}^\{\\top\}b^\{\\mathrm\{eff\}\}\. BecauseA⪰0A\\succeq 0, the matrix\(A\+λ1​I\)−1\(A\+\\lambda\_\{1\}I\)^\{\-1\}has spectral norm at most1/λ11/\\lambda\_\{1\}, so

‖x‖2=‖\(A\+λ1​I\)−1​c‖2≤‖c‖2λ1,henceΔ​ϕ^Tik≤2​‖x‖∞≤2​‖x‖2≤2​‖c‖2λ1\.\\left\\lVert x\\right\\rVert\_\{2\}=\\left\\lVert\(A\+\\lambda\_\{1\}I\)^\{\-1\}c\\right\\rVert\_\{2\}\\leq\\frac\{\\left\\lVert c\\right\\rVert\_\{2\}\}\{\\lambda\_\{1\}\},\\qquad\\text\{hence\}\\qquad\\Delta\\hat\{\\phi\}\_\{\\mathrm\{Tik\}\}\\leq 2\\left\\lVert x\\right\\rVert\_\{\\infty\}\\leq 2\\left\\lVert x\\right\\rVert\_\{2\}\\leq\\frac\{2\\left\\lVert c\\right\\rVert\_\{2\}\}\{\\lambda\_\{1\}\}\.It remains to bound‖c‖2\\left\\lVert c\\right\\rVert\_\{2\}independently ofLL\. The matrixLI​IL\_\{II\}is upper bidiagonal with11on the diagonal and−1\-1on the superdiagonal \(and a lone11in the last row\), so columnjjofLI​IL\_\{II\}has entries\(LI​I\)j​j=1\(L\_\{II\}\)\_\{jj\}=1and\(LI​I\)j−1,j=−1\(L\_\{II\}\)\_\{j\-1,j\}=\-1\. Thuscj=\(LI​I⊤​beff\)j=bjeff−bj−1effc\_\{j\}=\(L\_\{II\}^\{\\top\}b^\{\\mathrm\{eff\}\}\)\_\{j\}=b^\{\\mathrm\{eff\}\}\_\{j\}\-b^\{\\mathrm\{eff\}\}\_\{j\-1\}, and withbeff=\(−1,0,…,0,1\)b^\{\\mathrm\{eff\}\}=\(\-1,0,\\dots,0,1\)this givesc=\(−1,1,0,…,0,1\)⊤c=\(\-1,\\,1,\\,0,\\dots,0,\\,1\)^\{\\top\}, so‖c‖2=3\\left\\lVert c\\right\\rVert\_\{2\}=\\sqrt\{3\}for everyLL\. ThereforeΔ​ϕ^Tik​\(λ1\)≤2​3/λ1→0\\Delta\\hat\{\\phi\}\_\{\\mathrm\{Tik\}\}\(\\lambda\_\{1\}\)\\leq 2\\sqrt\{3\}/\\lambda\_\{1\}\\to 0, with a constant independent ofLL\. ∎

### B\.2Proof of[Theorem˜2](https://arxiv.org/html/2607.13609#Thmtheorem2)

With the trace normalization of[Section˜5\.6](https://arxiv.org/html/2607.13609#S5.SS6), the Sobolev objective is

f​\(x\)=12​bsc​\[\(d0\+1\)2\+∑i=1Ldi2\]\+λ12​τL​∑i=0Ldi2,f\(x\)=\\frac\{1\}\{2\\,b\_\{\\mathrm\{sc\}\}\}\\Big\[\(d\_\{0\}\+1\)^\{2\}\+\\sum\_\{i=1\}^\{L\}d\_\{i\}^\{2\}\\Big\]\+\\frac\{\\lambda\_\{1\}\}\{2\\,\\tau\_\{L\}\}\\sum\_\{i=0\}^\{L\}d\_\{i\}^\{2\},using[Eq\.˜10](https://arxiv.org/html/2607.13609#A2.E10), wherebsc=‖beff‖22=2b\_\{\\mathrm\{sc\}\}=\\left\\lVert b^\{\\mathrm\{eff\}\}\\right\\rVert\_\{2\}^\{2\}=2is the data\-normalization constant andτL=tr⁡\[\(LG\)I​I\]=1\+2​\(L−1\)\+2=2​L\+1\\tau\_\{L\}=\\operatorname\{tr\}\[\(L\_\{G\}\)\_\{II\}\]=1\+2\(L\-1\)\+2=2L\+1is the penalty trace \(node0has degree11, nodes1,…,L1,\\dots,Lhave degree22\)\. The crucial point is thatffis*separable*in the gapsd0,…,dLd\_\{0\},\\dots,d\_\{L\}: eachdid\_\{i\}appears in exactly one data term and one penalty term\. Minimizing term by term,

di⋆=0\(1≤i≤L\),d0⋆=arg⁡mind⁡\(d\+1\)22​bsc\+λ12​τL​d2=−11\+λ1​bsc/τL=−\(2​L\+1\)2​L\+1\+2​λ1,d\_\{i\}^\{\\star\}=0\\ \\ \(1\\leq i\\leq L\),\\qquad d\_\{0\}^\{\\star\}=\\arg\\min\_\{d\}\\ \\frac\{\(d\+1\)^\{2\}\}\{2b\_\{\\mathrm\{sc\}\}\}\+\\frac\{\\lambda\_\{1\}\}\{2\\tau\_\{L\}\}d^\{2\}=\\frac\{\-1\}\{\\,1\+\\lambda\_\{1\}b\_\{\\mathrm\{sc\}\}/\\tau\_\{L\}\\,\}=\\frac\{\-\(2L\+1\)\}\{\\,2L\+1\+2\\lambda\_\{1\}\\,\},where we usedbsc=2b\_\{\\mathrm\{sc\}\}=2\. Sinced1⋆=⋯=dL⋆=0d\_\{1\}^\{\\star\}=\\dots=d\_\{L\}^\{\\star\}=0, the recovered potential is constant on\{1,…,L\+1\}\\\{1,\\dots,L\+1\\\}and drops by\|d0⋆\|\|d\_\{0\}^\{\\star\}\|at the source, so

Δ​ϕ^Sob​\(λ1\)=\|d0⋆\|=2​L\+12​L\+1\+2​λ1=11\+λ1​κ​\(L\),κ​\(L\)=22​L\+1\.\\Delta\\hat\{\\phi\}\_\{\\mathrm\{Sob\}\}\(\\lambda\_\{1\}\)=\|d\_\{0\}^\{\\star\}\|=\\frac\{2L\+1\}\{\\,2L\+1\+2\\lambda\_\{1\}\\,\}=\\frac\{1\}\{1\+\\lambda\_\{1\}\\kappa\(L\)\},\\qquad\\kappa\(L\)=\\frac\{2\}\{2L\+1\}\.The bound11\+λ1​κ​\(L\)≥1−λ1​κ​\(L\)\\tfrac\{1\}\{1\+\\lambda\_\{1\}\\kappa\(L\)\}\\geq 1\-\\lambda\_\{1\}\\kappa\(L\)follows from11\+t≥1−t\\tfrac\{1\}\{1\+t\}\\geq 1\-tfort≥0t\\geq 0, andΔ​ϕtrue=1\\Delta\\phi\_\{\\mathrm\{true\}\}=1is theλ1=0\\lambda\_\{1\}=0value\. AsL→∞L\\to\\infty,κ​\(L\)→0\\kappa\(L\)\\to 0andΔ​ϕ^Sob→1\\Delta\\hat\{\\phi\}\_\{\\mathrm\{Sob\}\}\\to 1\. ∎

#### Numerical confirmation\.

The released solver reproducesΔ​ϕ^Sob=\(2​L\+1\)/\(2​L\+1\+2​λ1\)\\Delta\\hat\{\\phi\}\_\{\\mathrm\{Sob\}\}=\(2L\+1\)/\(2L\+1\+2\\lambda\_\{1\}\)to four decimals at every tested\(L,λ1\)\(L,\\lambda\_\{1\}\)\(for example61/63=0\.96861/63=0\.968at\(L,λ1\)=\(30,1\)\(L,\\lambda\_\{1\}\)=\(30,1\)and61/81=0\.75361/81=0\.753at\(30,10\)\(30,10\)\), and confirms thatΔ​ϕ^Tik\\Delta\\hat\{\\phi\}\_\{\\mathrm\{Tik\}\}is independent ofLL\(identical to four decimals forL=30L=30andL=100L=100at eachλ1\\lambda\_\{1\}\), as[Proposition˜4](https://arxiv.org/html/2607.13609#Thmproposition4)predicts\.

### B\.3Remark on the empirical Tikhonov plateau

[Proposition˜4](https://arxiv.org/html/2607.13609#Thmproposition4)givesΔ​ϕ^Tik→0\\Delta\\hat\{\\phi\}\_\{\\mathrm\{Tik\}\}\\to 0but makes no claim about the limiting rank correlation\. On a graph with nonzero interior divergence the collapsed estimate is not identically constant: theO​\(1/λ1\)O\(1/\\lambda\_\{1\}\)leading term is∝c=L⊤​beff\\propto c=L^\{\\top\}b^\{\\mathrm\{eff\}\}, whose sign pattern reflects local divergence rather than global position between boundaries\. The rank correlation withϕtrue\\phi\_\{\\mathrm\{true\}\}therefore tends to a regime\-dependent negative plateau \(empirically≈−0\.42\{\\approx\}\-0\.42in[Table˜1](https://arxiv.org/html/2607.13609#S5.T1)\), not to−1\-1: the recovered order is anticorrelated with the planted field but not its exact reverse\. The linear correlation likewise crosses to zero as the range collapses \([Table˜1](https://arxiv.org/html/2607.13609#S5.T1)\)\.

### B\.4Proofs of the structural results

###### Proof of[Proposition˜1](https://arxiv.org/html/2607.13609#Thmproposition1)\(acyclicity\)\.

By[Eq\.˜9](https://arxiv.org/html/2607.13609#A1.E9), every retained edge\(u,v\)\(u,v\)hasψ​\(v\)\>ψ​\(u\)\\psi\(v\)\>\\psi\(u\)for the oriented coordinateψ\\psi\. A directed cyclev1→⋯→vk→v1v\_\{1\}\\to\\cdots\\to v\_\{k\}\\to v\_\{1\}would forceψ​\(v1\)<⋯<ψ​\(vk\)<ψ​\(v1\)\\psi\(v\_\{1\}\)<\\cdots<\\psi\(v\_\{k\}\)<\\psi\(v\_\{1\}\), a contradiction; henceψ\\psistrictly increases along every directed path and is a topological order\. ∎

###### Proof of[Proposition˜2](https://arxiv.org/html/2607.13609#Thmproposition2)\(nilpotency\)\.

\(Am\)u​v\(A^\{m\}\)\_\{uv\}sums over directed walks of lengthmmfromuutovv\. On a DAG every walk is a path, of length at mostLL; hence form\>Lm\>Lno such walk exists andAm=0A^\{m\}=0\. ∎

###### Proof of[Proposition˜3](https://arxiv.org/html/2607.13609#Thmproposition3)\(minimum dissipation\)\.

Introduce a multiplierϕ\\phifor the divergence constraint and formℒ=12​∑Wi​j−1​qi​j2\+ϕ⊤​\(div⁡\(q\)−b\)\\mathcal\{L\}=\\tfrac\{1\}\{2\}\\sum W\_\{ij\}^\{\-1\}q\_\{ij\}^\{2\}\+\\phi^\{\\top\}\(\\operatorname\{div\}\(q\)\-b\)\. Stationarity inqi​jq\_\{ij\}givesWi​j−1​qi​j=ϕi−ϕjW\_\{ij\}^\{\-1\}q\_\{ij\}=\\phi\_\{i\}\-\\phi\_\{j\}up to orientation, i\.e\.qi​j=Wi​j​\(ϕj−ϕi\)q\_\{ij\}=W\_\{ij\}\(\\phi\_\{j\}\-\\phi\_\{i\}\); substituting intodiv⁡\(q\)=b\\operatorname\{div\}\(q\)=byieldsL​ϕ=bL\\phi=b\. ∎

###### Proof of[Lemma˜1](https://arxiv.org/html/2607.13609#Thmlemma1)\(chain saturation\)\.

Zero divergence atviv\_\{i\}means inflow equals outflow,Wvi−1​vi​\(ϕvi−1−ϕvi\)=Wvi​vi\+1​\(ϕvi−ϕvi\+1\)W\_\{v\_\{i\-1\}v\_\{i\}\}\(\\phi\_\{v\_\{i\-1\}\}\-\\phi\_\{v\_\{i\}\}\)=W\_\{v\_\{i\}v\_\{i\+1\}\}\(\\phi\_\{v\_\{i\}\}\-\\phi\_\{v\_\{i\+1\}\}\), which telescopes; with a single equal\-weight in\- and out\-edge this forcesϕvi=ϕvi\+1\\phi\_\{v\_\{i\}\}=\\phi\_\{v\_\{i\+1\}\}for allii, and pinning the terminal value propagatesϕvi=ϕ​\(s\)\\phi\_\{v\_\{i\}\}=\\phi\(s\)backward\. ∎

###### Proof of[Proposition˜5](https://arxiv.org/html/2607.13609#Thmproposition5)\(gauge invariance\)\.

SinceLG​𝟏=0L\_\{G\}\\mathbf\{1\}=0,\(ϕ\+c​𝟏\)⊤​LG​\(ϕ\+c​𝟏\)=ϕ⊤​LG​ϕ\+2​c​1⊤​LG​ϕ\+c2​𝟏⊤​LG​𝟏=ϕ⊤​LG​ϕ\(\\phi\+c\\mathbf\{1\}\)^\{\\top\}L\_\{G\}\(\\phi\+c\\mathbf\{1\}\)=\\phi^\{\\top\}L\_\{G\}\\phi\+2c\\,\\mathbf\{1\}^\{\\top\}L\_\{G\}\\phi\+c^\{2\}\\mathbf\{1\}^\{\\top\}L\_\{G\}\\mathbf\{1\}=\\phi^\{\\top\}L\_\{G\}\\phi\. By contrast‖ϕ\+c​𝟏‖22=‖ϕ‖22\+2​c​1⊤​ϕ\+c2​\|V\|\\left\\lVert\\phi\+c\\mathbf\{1\}\\right\\rVert\_\{2\}^\{2\}=\\left\\lVert\\phi\\right\\rVert\_\{2\}^\{2\}\+2c\\,\\mathbf\{1\}^\{\\top\}\\phi\+c^\{2\}\\lvert V\\rvertdepends oncc, with a unique minimizing gaugec⋆=−ϕ¯c^\{\\star\}=\-\\bar\{\\phi\}\. ∎

###### Proof of[Theorem˜1](https://arxiv.org/html/2607.13609#Thmtheorem1)\(SPD reduced system\)\.

MMis symmetric\. Forx≠0x\\neq 0on the interior,LI​IL\_\{II\}has trivial null space:LI​I​x=0L\_\{II\}x=0makesxxconstant on each interior component, and the path\-to\-boundary hypothesis forces that constant to match a pinned boundary value, sox=0x=0; hencex⊤​\(L⊤​L\)I​I​x=‖LI​I​x‖22\>0x^\{\\top\}\(L^\{\\top\}L\)\_\{II\}x=\\left\\lVert L\_\{II\}x\\right\\rVert\_\{2\}^\{2\}\>0\. The Dirichlet block\(LG\)I​I\(L\_\{G\}\)\_\{II\}is PSD, and a positive\-definite plus a PSD matrix is positive definite, sox⊤​M​x\>0x^\{\\top\}Mx\>0\. ∎

###### Proof of[Theorem˜3](https://arxiv.org/html/2607.13609#Thmtheorem3)\(residual concentration\)\.

For an absorbing sinkuuwith no out\-edges,\[L​ϕ\]u=∑j:\(u,j\)∈EWu​j​\(ϕu−ϕj\)\[L\\phi\]\_\{u\}=\\sum\_\{j:\(u,j\)\\in E\}W\_\{uj\}\(\\phi\_\{u\}\-\\phi\_\{j\}\)has no terms, so it is0andru=\|bu\|r\_\{u\}=\\lvert b\_\{u\}\\rvert\. For an interior noderu=\|\[L​ϕ\]u−bu\|≤ηr\_\{u\}=\\lvert\[L\\phi\]\_\{u\}\-b\_\{u\}\\rvert\\leq\\etaby hypothesis, andminB⁡\|bu\|\>η\\min\_\{B\}\\lvert b\_\{u\}\\rvert\>\\etathen orders all sinks above all interior nodes\. ∎

## Appendix CAdditional Figures and Tables

Table 4:Solver scaling: sparse conjugate gradients \(CG\) versus the dense eigensolve for the graph\-Sobolev solve \(λ1=1\\lambda\_\{1\}=1\) on layered DAGs of growing interior size\. CG matches the dense solve to10−910^\{\-9\}where both run, scales near\-linearly in edges, and handles4×1044\\times 10^\{4\}nodes in under half a second; the dense solve exhausts memory beyond∼4,000\{\\sim\}4\{,\}000interior nodes\. This backs the CG path of[Section˜7\.3](https://arxiv.org/html/2607.13609#S7.SS3)\([Theorem˜1](https://arxiv.org/html/2607.13609#Thmtheorem1)\)\. Times are single\-core wall\-clock and reproduced by the releasedrun\_scalingscript\.Table 5:Regularizer contrast on real data atλ1=1\\lambda\_\{1\}=1, versus the unregularized solve \(exact values for[Figure˜2](https://arxiv.org/html/2607.13609#A3.F2)\)\. “Range ret\.” is interior dynamic range as a fraction of the base; “Rank agree\.” is the Spearman correlation of the interior ordering with the base\.Table 6:Downstream utility on OTTO: session\-level conversion prediction \([Section˜8\.4](https://arxiv.org/html/2607.13609#S8.SS4)\), held\-out ROC\-AUC \(mean±\\pmsd over three splits;27%27\\%conversion base rate,186,271186\{,\}271sessions\)\. Features use only the interior \(non\-sink\) states a session visits; the label is the terminal sink, so the task is leakage\-free\. The graph\-Sobolev potential is a usable node feature—adding\+3\.1\+3\.1points over raw statistics and beating the ridge potential by\+11\.8\+11\.8points standalone—whereas the collapsed ridge potential adds nothing\. A trained directed GNN \(run\_gnn, mean/in/out message passing, node embeddings pooled over the same visited states\) is stronger \(0\.8610\.861\) but*oversmooths with depth*, its AUC falling to0\.8070\.807at3232layers as node\-embedding Dirichlet energy collapses; neutralizing the constant/gauge mode each layer holds it flat \(0\.8540\.854\)—the gauge principle of[Section˜2](https://arxiv.org/html/2607.13609#S2)applied to the network, recovering PairNorm\(Zhao and Akoglu,[2020](https://arxiv.org/html/2607.13609#bib.bib19)\)\. Bag\-of\-states \(full105105\-dim visit counts\) is the state\-identity ceiling\. Architecture and gauge\-centering in[appendix˜C](https://arxiv.org/html/2607.13609#A3.SS0.SSS0.Px2); reproduced by the releasedconversion\_predictionandrun\_gnnexperiments\.#### Downstream protocol \([Table˜6](https://arxiv.org/html/2607.13609#A3.T6)\)\.

Each session is labelled by its terminal sink \(converted iff it ends atorders\) and featurised only from the interior \(non\-sink\) states it visits, so the pinned label cannot leak into the features\. The potential features are the four order statistics—mean, max, min, and last\-visitedϕ^\\hat\{\\phi\}—over those states; the raw baseline is\[\#​visits,\#​distinct states\]\[\\\#\\text\{visits\},\\ \\\#\\text\{distinct states\}\], and “raw\+\+” rows concatenate the two\. The classifier is anL2L\_\{2\}\-regularised logistic regression \(scikit\-learn defaults,C=1C\{=\}1\); we hold out30%30\\%of sessions and average over three random splits, and—to avoid leakage across the split—fit the potential on the training sessions only, then apply it to both\. Becauseϕ^\\hat\{\\phi\}is a function of the state, the potential features are a11\-D\-per\-state*compression*of node identity, and therefore sit below the bag\-of\-states ceiling \(0\.8990\.899\) by construction: they cannot exceed the full105105\-dimensional identity\. The finding is not that they do, but that the gauge\-invariant compression recovers about a third of the raw\-to\-ceiling AUC gap in four features while the ridge compression recovers essentially none; a learned encoder \(e\.g\. GraphSAGE\) onϕ^\\hat\{\\phi\}could close the remainder\. Reproduced by the releasedconversion\_predictionexperiment\.

#### Directed\-GNN baseline and gauge\-centering \([Table˜6](https://arxiv.org/html/2607.13609#A3.T6)\)\.

The GNN rows use a directed graph convolutional network on the extracted DAGGδG\_\{\\delta\}\(topological\-sort support,N=105N=105nodes\)\. With learnable node embeddingsH0∈ℝN×dH^\{0\}\\in\\mathbb\{R\}^\{N\\times d\}\(d=32d=32\), each layer aggregates from in\- and out\-neighbours separately,

Hl\+1=ReLU​\(A^out​Hl​Wol\+A^in​Hl​Wil\),H^\{l\+1\}=\\mathrm\{ReLU\}\\\!\\left\(\\hat\{A\}\_\{\\mathrm\{out\}\}H^\{l\}W^\{l\}\_\{\\mathrm\{o\}\}\+\\hat\{A\}\_\{\\mathrm\{in\}\}H^\{l\}W^\{l\}\_\{\\mathrm\{i\}\}\\right\),\(11\)whereA^out\\hat\{A\}\_\{\\mathrm\{out\}\}is the row\-normalized adjacency ofGδG\_\{\\delta\}andA^in\\hat\{A\}\_\{\\mathrm\{in\}\}that of its transpose \(self\-loops enter through the pure propagation of[Section˜2](https://arxiv.org/html/2607.13609#S2)\)\. A session is scored by mean\-pooling the final embeddings over the interior states it visits—the same pooling and leakage guard as the potential features \(above\)—followed by a logistic layer; we train end\-to\-end with Adam \(learning rate10−210^\{\-2\}, weight decay10−410^\{\-4\},8080epochs, binary cross\-entropy\) on the same70/3070/30session split\.

Oversmoothing drivesHlH^\{l\}toward the constant \(gauge\) mode, the smallest eigenvector of the symmetric normalized LaplacianLsymL\_\{\\mathrm\{sym\}\}, which isu0∝d1/2u\_\{0\}\\propto d^\{1/2\}\(node degreesdd\)\.*Gauge\-centering*removes this component after every layer,

Hl←Hl−u0​\(u0⊤​Hl\),‖u0‖2=1,H^\{l\}\\leftarrow H^\{l\}\-u\_\{0\}\\,\\bigl\(u\_\{0\}^\{\\top\}H^\{l\}\\bigr\),\\qquad\\\|u\_\{0\}\\\|\_\{2\}=1,\(12\)i\.e\. it subtracts the*degree\-weighted*mean embedding—the per\-layer analogue ofLG​𝟏=0L\_\{G\}\\mathbf\{1\}=0\(for a regular graphu0∝𝟏u\_\{0\}\\propto\\mathbf\{1\}and this is the ordinary mean\)\. The “gauge\+\+scale” variant additionally rescalesHlH^\{l\}to fixed Frobenius norm, recovering PairNorm\(Zhao and Akoglu,[2020](https://arxiv.org/html/2607.13609#bib.bib19)\)\. Oversmoothing is measured by the node Dirichlet energyℰ​\(H\)=tr⁡\(H⊤​Lsym​H\)/‖H‖F2\\mathcal\{E\}\(H\)=\\operatorname\{tr\}\(H^\{\\top\}L\_\{\\mathrm\{sym\}\}H\)/\\\|H\\\|\_\{F\}^\{2\}, which vanishes as representations collapse\. Across depths\{2,4,8,16,32\}\\\{2,4,8,16,32\\\}the vanilla GCN’s test AUC falls0\.86→0\.810\.86\\\!\\to\\\!0\.81asℰ\\mathcal\{E\}collapses \(to0\.520\.52at1616layers\), while gauge\-centering holds AUC flat \(0\.850\.85at3232layers\)\. Reproduced by the releasedrun\_gnnexperiment\.

![Refer to caption](https://arxiv.org/html/2607.13609v1/x2.png)Figure 2:Regularizer contrast on real data \(λ1=1\\lambda\_\{1\}=1\) versus the unregularized solve\. Left: interior dynamic range retained\. Right: rank agreement with the base solve\. Ridge’s collapse deepens as the graph grows, reaching near\-total loss on OTTO; graph\-Sobolev preserves both throughout\. Exact values in[Table˜5](https://arxiv.org/html/2607.13609#A3.T5)\.Table 7:Robustness of the regularizer contrast across the instrument’s configuration space, deferred from[Section˜7\.2](https://arxiv.org/html/2607.13609#S7.SS2)\. Twelve variants: the default funnel at five seeds\{42,1,7,123,2024\}\\\{42,1,7,123,2024\\\}and seven single\-axis perturbations \(branch count∈\{3,8\}\\in\\\{3,8\\\}, chain depth∈\{30,80\}\\in\\\{30,80\\\}, abandonment∈\{0\.2,0\.5\}\\in\\\{0\.2,0\.5\\\}, sink entropy0\.30\.3\)\. Entries are mean±\\pmsd of the interior Spearman across the twelve variants atλ1=1\\lambda\_\{1\}=1; “margin” is graph\-Sobolev−\-Tikhonov, “min” its smallest value over the twelve, and “\>0\>0” the count with positive margin\. The contrast holds under every variant; on the topological\-sort support the graph\-Sobolev ordering is itself configuration\-invariant\. Reproduced by the releasedregularizer\_robustnessexperiment\.![Refer to caption](https://arxiv.org/html/2607.13609v1/x3.png)Figure 3:Recoveredϕ^\\hat\{\\phi\}versus plantedϕtrue\\phi\_\{\\mathrm\{true\}\}atλ1=1\\lambda\_\{1\}=1\(synthetic instrument\)\. Graph\-Sobolev \(left\) retains a positive relationship with the planted field; Tikhonov \(right\) collapses every interior state into a flat band near the abandon origin, the range collapse of[Proposition˜4](https://arxiv.org/html/2607.13609#Thmproposition4)\.![Refer to caption](https://arxiv.org/html/2607.13609v1/x4.png)Figure 4:Interior dynamic rangeΔ​ϕ^\\Delta\\hat\{\\phi\}on a length\-3030chain versusλ1\\lambda\_\{1\}\. Graph\-Sobolev stays near the unit harmonic target; Tikhonov overshoots at smallλ1\\lambda\_\{1\}then collapses toward zero\.Table 8:Blind sink discovery on the synthetic instrument\. The five planted conversion sinks, ranked among277277states by Poisson residual versus by recovered potential \(lower is better\); the residual places all sinks in the top seven, invariantly across abandonment regimes\.![Refer to caption](https://arxiv.org/html/2607.13609v1/x5.png)Figure 5:Recovered graph\-Sobolev potential \(λ1=1\\lambda\_\{1\}=1\) on RetailRocket and Trivago\. Grey bars are pinned sinks; blue bars are recovered interior states\. The ordering reproduces the known funnel; the negativeinteraction item imagestate on Trivago is discussed in[Section˜8\.1](https://arxiv.org/html/2607.13609#S8.SS1)\.![Refer to caption](https://arxiv.org/html/2607.13609v1/x6.png)Figure 6:OTTO recovered potential by event type \(105105\-state multi\-sink graph\)\. Each point is an interior state; black bars are event\-type means; dashed lines are the three Dirichlet sink levels\. The solve separatescartsstates \(mean\+0\.26\+0\.26, straddling the cart sink\) fromclicksstates \(mean−0\.01\-0\.01, near abandon\)\.

## Appendix DUncertainty Quantification

The divergencebbis an empirical estimate, and its sampling covariance propagates toϕ\\phiin closed form\. On the reduced interior systemϕI=LI​I−1​bIeff\\phi\_\{I\}=L\_\{II\}^\{\-1\}b\_\{I\}^\{\\mathrm\{eff\}\}\(atλ1=0\\lambda\_\{1\}=0for clarity\), the delta method gives

Cov⁡\(ϕI\)=LI​I−1​Cov^​\(bI\)​LI​I−⊤,\\operatorname\{Cov\}\(\\phi\_\{I\}\)\\;=\\;L\_\{II\}^\{\-1\}\\,\\widehat\{\\operatorname\{Cov\}\}\(b\_\{I\}\)\\,L\_\{II\}^\{\-\\top\},\(13\)withCov^​\(b\)\\widehat\{\\operatorname\{Cov\}\}\(b\)estimated from the multinomial transition counts\. Forλ1\>0\\lambda\_\{1\}\>0replaceLI​I−1L\_\{II\}^\{\-1\}byM−1​LI​I⊤M^\{\-1\}L\_\{II\}^\{\\top\}withMMas in[Eq\.˜6](https://arxiv.org/html/2607.13609#S5.E6)\.

## Appendix EReproduction

The released repository \([https://github\.com/MohammadForouhesh/gauge\-flow\-recovery](https://github.com/MohammadForouhesh/gauge-flow-recovery), with a self\-containedREADME\) contains the core solver \(poisson\_inverse/core\.py\), the synthetic instrument \(poisson\_inverse/synthetic\.py\), the experiments \(poisson\_inverse/experiments\.py\), and two scripts:scripts/run\_experiments\.pywrites all numerical results to JSON, andscripts/make\_figures\.pyregenerates[Figures˜1](https://arxiv.org/html/2607.13609#S5.F1),[3](https://arxiv.org/html/2607.13609#A3.F3)and[4](https://arxiv.org/html/2607.13609#A3.F4)\. The default seed reproduces every value reported in[Tables˜1](https://arxiv.org/html/2607.13609#S5.T1)and[8](https://arxiv.org/html/2607.13609#A3.T8)and in[Sections˜7](https://arxiv.org/html/2607.13609#S7.SS0.SSS0.Px2)and[5\.7](https://arxiv.org/html/2607.13609#S5.SS7)\.

The real\-data results of[Section˜8](https://arxiv.org/html/2607.13609#S8)are reproduced bypoisson\_inverse/loaders\.py\(the RetailRocket, Trivago, and OTTO loaders, which reduce each corpus to its event\-type state space\) together withscripts/run\_real\_data\.py\(which runs the interpretability, regularizer\-contrast, residual, and bootstrap checks\) andscripts/make\_real\_figures\.py\(which regenerates[Figures˜2](https://arxiv.org/html/2607.13609#A3.F2),[5](https://arxiv.org/html/2607.13609#A3.F5)and[6](https://arxiv.org/html/2607.13609#A3.F6)from the per\-dataset result files\)\. Only the single raw event file per corpus is required; the loaders construct the state space and Dirichlet boundary automatically\.