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Matched Queries for Curvature and Density at Branching Junctions

2026 · arxiv_cs

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Matched Queries for Curvature and Density at Branching Junctions

arXiv:2609.01319v1 [stat.ML] 1 Sep 2026

Ziqi Zhao School of Computer Science and Engineering Southeast University Nanjing, China ziqizhao@seu.edu.cn Qingjian Ni∗ School of Computer Science and Engineering Southeast University Nanjing, China nqj@seu.edu.cn

Abstract At a junction, a score field can reveal weighted tangent rays, but these rays do not determine how branches bend or how density changes away from the center. This missing information matters when local geometry is used to describe continuation beyond a single point. Small-noise expansions place curvature and density in the first correction, but do not establish whether finite observations can separate the branchwise contributions when the center is estimated. We study this inverse problem through matched queries at scales σ and λσ. For a finite union of C 2,α half-branches in RD , the normalized score satisfies Fσ = F0 + σG + O(σ 1+α ). Matched subtraction cancels the tangent term and exposes G, which is linear in branchwise curvature and log-density slope. Given tangent directions and weights, G uniquely identifies all sD branch parameters on distinct rays; a geometry-dependent condition number governs stability as rays approach one another. The sD fixed component observations are also necessary. This is a scalar-information count rather than a count of vector-valued score-network evaluations. Under coarse-localization and full-rank calibration conditions, an O(σ 2 ) center error adds D translation modes, giving (s + 1)D scalar observations except for a translation-invariant full line. We derive a perturbation bound and a conditional kernel-density-estimation (KDE) rate. Experiments recover the predicted population and N −1/5 trends and remain full rank up to D = 20 with 16 supplied branches. A known-count frontend fitted directly to finite-noise scores composes with the inverse in D = 3–5: all 135 population systems are full rank, with median relative jet error 0.132. Under strong population first-order error, matched responses reduce median parameter error by 49.4 times relative to naive tangent subtraction.

Introduction

Score fields are learned by score matching [8], drive noise-conditional generative models [19, 20], and serve as local measurements of data geometry. Near a smooth manifold, they reveal tangent spaces, intrinsic dimension, and other first-order structure [12, 21, 22]. A junction, however, has no single tangent space. Its first-order description is a set of outgoing directions and relative masses. For ∗

Corresponding author.

(a) Two finite-scale neighborhoods

(b) One weighted tangent limit

same {(vj , wj )}sj=1

same leading field F0

v2 , w 2

v1 , w 1

tangent blow-up k1′

v3 , w 3 different {(aj , kj )}sj=1 weighted tangent ray

aj and kj are absent

physical branch

Figure 1: The same tangent geometry can hide different local structure. Tangent blow-up keeps branch directions and weights but removes curvature and outward density change. local geometric recovery, that description is incomplete: it cannot tell how each branch continues away from the junction or how mass changes along it. At leading order, a junction is therefore represented only by weighted tangent rays. The two neighborhoods in Figure 1 have the same such representation, yet bend differently and carry different outward density trends. Rescaling toward the junction removes these differences and leaves the same leading score field F0 . The missing quantities survive only in the next-order term. Recovering them requires isolating a weak cross-scale signal, assigning a superposition of responses to individual branches, and controlling center error at the same asymptotic order. Variation across noise scales offers a route to this hidden structure. Forward expansions near boundaries and corners show how curvature and density derivatives enter the first small-noise correction [3]. They map known geometry to its correction, but do not establish whether a superposed correction uniquely determines every branchwise coefficient from finite observations under center error. This inverse question matters because tangent geometry cannot test whether a score representation preserves how each branch continues through a junction. The leading tangent field can be canceled rather than estimated. Consider a finite junction of one-dimensional branches in RD . Branch j has direction vj , weight wj , curvature vector kj ⊥ vj , and logarithmic density slope aj ; we call (aj , kj ) its branch jet. The normalized score obeys Fσ (z) = F0 (z) + σG(z) + O(σ 1+α ),

(1)

where F0 depends only on {vj , wj } and G is linear in the branch jets. Curvature displaces mass normally to a tangent ray, whereas density slope changes mass along it. Gaussian smoothing preserves these distinct signatures inside G. Querying the same normalized location z at scales σ and λσ exposes G directly: Dσ,λ (z) =

λσsλσ (x0 + λσz) − σsσ (x0 + σz) = G(z) + O(σ α ). (λ − 1)σ

(2)

The common field F0 cancels. With known directions and weights, scalar component evaluations become rows of a linear inverse problem. The superposed field determines every branch jet without an angular-separation assumption, and sD scalar observations are necessary and sufficient. This is an information count, not a forward-pass count: a score network usually returns all D coordinates at one location. Quantitative stability is governed by the condition number, which captures ray collisions and vanishing weights. 2

(a) Matched physical queries

(b) What survives subtraction the common tangent field F0 cancels kj

λσ

cσ + λσz matched subtraction cσ + σz

σ x0

cσ center correction curvature and density change remain

same center; same normalized z

Figure 2: Matched physical queries use the same provisional center and normalized location. Their subtraction removes the shared tangent field, leaving curvature, density change, and the center correction visible. The estimated center creates a second difficulty. Given an O(σ) coarse localization and a full-rank weak calibration system, first-order score calibration gives cσ = x0 + σ 2 b + O(σ 2+α ). This physical error is small, but normalization promotes it to the same order as G. Reusing the provisional center at both scales gives cσ Dσ,λ (z) = G(z) − λ−1 ∇F0 (z)b + O(σ α ). (3) Thus b ∈ RD contributes D known translation-shaped basis fields. Except when the tangent measure is invariant along a complete line, the branch jets and center bias remain jointly identifiable from (s + 1)D scalar observations, and estimating b refines the center to O(σ 2+α ). Contributions. 1. The superposed first correction exactly identifies every branch jet on supplied distinct rays without a fixed angular separation; quantitative stability is treated separately. 2. A locally uniform two-term expansion and matched scale difference make the correction observable by canceling the tangent field rather than estimating and subtracting it. 3. Fixed continuous schemes have scalar-information complexity sD. An augmented inverse jointly identifies the O(σ 2 ) center bias and branch jets with complexity (s + 1)D, except for the intrinsic full-line ambiguity. 4. A perturbation bound separates finite-scale, score, center, and first-stage errors and yields a conditional KDE rate. Evidence. Across 180 population fits in D ∈ {2, 3, 5}, the median remainder exponent is 0.998, matching the predicted O(σ) order. Controlled-perturbation scale-up remains full rank through D = 20 and 16 supplied branches. The phase map shows a 184-fold error contrast between nearcollision and well-separated corners, and matched responses are 49.4 times more accurate than naive subtraction under strong population first-order error. A known-count frontend fitted directly to finite-noise scores runs end to end in D = 3–5. All 135 population systems are full rank, with median relative jet error 0.132. KDE scores retain a median maximum tangent error of 0.0222 radians and provide measurable jet recovery with median

relative error 0.957 under a fixed 131,072-sample budget; at σ = 0.08, the dimensionwise medians improve to 0.688–0.810. For learned scores, median jet error reaches its minimum before score RMS. Together, these experiments test finite observations, imperfect first-order geometry, sampling, and learned scores.

Model and Second-Order Target

First-order geometry consists of tangent directions and relative masses; the second-order target is curvature and density change. Let x0 ∈ RD be the junction center. In a neighborhood of x0 , assume µloc =

s X

γj : [0, r0 ] → RD ,

(γj )# ρj (r) dr ,

(4)

j=1

plus a finite remainder µfar supported away from x0 . The curves are parameterized by arc length and satisfy γj′ (0) = vj ∈ SD−1 ,

γj (0) = x0 , ρj (0) = wj > 0,

γj′′ (0) = kj ∈ vj⊥ ,

aj = ∂r log ρj (0).

(5) (6)

Distinct branches have distinct vj . Arc-length parameterization implies kj ⊥ vj . Assumption 2.1 (Uniform jet remainder). For some 0 < α ≤ 1 and finite constants Lγ , Lρ , γj (r) − x0 − rvj − 21 r2 kj ≤ Lγ r2+α ,

(7)

|ρj (r) − wj (1 + aj r)| ≤ Lρ r1+α

(8)

for 0 ≤ r ≤ r0 , and dist(x0 , suppµfar ) ≥ ∆ > 0. The weighted tangent measure is s X

ν0 =

Z ∞

wj 0

j=1

δuvj du.

(9)

A score cannot see its total scale, so the weights may be normalized to sum to one. The known first-order geometry is {vj , wj }sj=1 ; the target is J2 (µ, x0 ) = {(aj , kj ) : j = 1, . . . , s}.

(10)

Here kj measures bending and aj measures outward log-density change. Let pσ = µ ∗ φσ , where φσ is the centered Gaussian density with covariance σ 2 ID , and define sσ (x) = ∇x log pσ (x),

Fσ (z) = σsσ (x0 + σz).

(11)

Write the repeated one-dimensional Gaussian integrals as Z ∞

Jj,m (z) = 0

∥z − uvj ∥2 u exp − 2 m

du.

(12)

The leading scalar transform and its first correction are Q0 (z) = Q1 (z) =

s X j=1 s X

wj Jj,0 (z),

(13) 1 ⟨kj , z⟩ Jj,2 (z) . 2

wj aj Jj,1 (z) +

j=1

(14)

(a) Density slope

(b) Curvature

ρj (σu)/wj = 1 + σaj u + · · ·

[γj (σu) − x0 ]/σ = uvj + σu2 kj /2 + · · · σu2 kj /2

r = σu mass changes along the ray

the branch moves normally

Figure 3: Two distinct order-σ signatures. Density slope changes mass along a tangent ray, whereas curvature moves the branch normally away from it.

Why Second-Order Geometry Appears Across Scale

The mechanism is visible before the formal expansion. At physical radius r = σu, both the density change aj r and the normal displacement r2 kj /2, after division by the observation scale σ, contribute at order σ. Gaussian smoothing therefore places both effects in the same correction G. Theorem 3.1 (Second-order junction expansion). Under Theorem 2.1, for every compact Q ⊂ RD and every fixed integer m ≥ 0, there are Cm,Q < ∞ and σ0 > 0 such that e σ − Q0 − σQ1 Q

C m (Q)

≤ Cm,Q σ 1+α ,

0 < σ ≤ σ0 ,

(15)

e σ is the locally rescaled Gaussian convolution after removal of its z-independent normalization. where Q Consequently, for m ≥ 1, ′ ∥Fσ − F0 − σG∥C m−1 (Q) ≤ Cm,Q σ 1+α , (16)

with

Q1 G=∇ . Q0

F0 = ∇ log Q0 ,

(17)

F0 records the tangent rays, whereas G superposes all branch jets. The expansion is forward; Section 4 proves that the superposition is uniquely invertible. The proof substitutes r = σu into the curve and density expansions, expands the Gaussian kernel once, and integrates under a uniform Gaussian envelope. The far-away measure is exponentially small, and positivity of Q0 permits the logarithm. Section A gives the complete derivative and remainder argument. For a fixed scale ratio λ > 1, define Dσ,λ (z) =

Fλσ (z) − Fσ (z) . (λ − 1)σ

(18)

The two scores are evaluated at x0 +λσz and x0 +σz, respectively, so they share the same normalized location z. Corollary 3.2 (Renormalized scale derivative). For fixed 1 < λ ≤ λmax and compact Q, sup ∥Dσ,λ (z) − G(z)∥ ≤ CQ,λ σ α . z∈Q

Equivalently, G(z) = ∂σ [σsσ (x0 + σz)]|σ=0+ . 5

(19)

Thus matched subtraction removes the common tangent field F0 and exposes G without differentiating a noisy score. If an exact tangent score were available, (Fσ − F0 )/σ would have the same limit. The matched difference is preferable in practice because it cancels the leading field present in the observations themselves.

Recovering Branch Jets

With the two-scale response in hand, two questions remain: does it uniquely determine the branch jets, and how many scalar observations are needed? Both become finite-dimensional once the directions and weights are known. Choose an orthonormal basis Nj = [nj,1 , . . . , nj,D−1 ] for vj⊥ and write kj = Nj κj , with κj ∈ RD−1 . Define wj Jj,1 (z) Aj (z) = ∇ , Q0 (z) wj ⟨nj,r , z⟩ Jj,2 (z) Kj,r (z) = ∇ , 2Q0 (z)

Then G(z) =

s X

aj Aj (z) +

s D−1 X X

(20) r = 1, . . . , D − 1.

(21)

κj,r Kj,r (z).

(22)

j=1 r=1

j=1

Thus each density slope and curvature coordinate multiplies a known response pattern. Theorem 4.1 (Jet identifiability). Fix distinct directions v1 , . . . , vs ∈ SD−1 and positive weights w1 , . . . , ws . The map {(aj , kj )}sj=1 7−→ G (23) is injective over aj ∈ R and kj ∈ vj⊥ . The geometric reason is that density changes act along a ray, while curvature acts normally to it; a localized observation can therefore separate both from every other branch. The proof represents the correction as a distribution on the tangent rays. Because Gaussian convolution is injective, a zero field implies a zero distribution. A test supported in a narrowing tube isolates one ray, first its normal curvature and then its tangential density change. Repeating this test proves uniqueness for distinct rays, although stability may deteriorate as two rays approach. Section B gives the full argument. Theorem 4.2 (Sharp scalar-information complexity). Under the conditions of Theorem 4.1, there exist sD scalar component evaluations of G whose values uniquely determine all branch jets. Conversely, any fixed continuous observation scheme that identifies every jet vector in a nonempty open subset of RsD requires at least sD scalar outputs. Each branch contributes one density slope and D − 1 curvature coordinates. One scalar observation is one coordinate at one location; because a network evaluation usually returns all D coordinates, sD is a scalar-information count, not a forward-pass count. Point-coordinate evaluations span the dual of the sD-dimensional space generated by Equations (20) and (21), so sD of them suffice. For necessity, invariance of domain rules out a continuous injection from an open subset of RsD into fewer coordinates. In computation, we select wellconditioned rows from a larger candidate matrix; Section C gives the formal proof. 6

Algorithm 1 Two-scale branch-jet reconstruction ŝ ), scales σ and λσ, candidate normalized grid Z. Require: First-order estimate (cσ , {v̂j , ŵj }j=1 1: Construct the branch basis fields in Equations (20) and (21); for center-robust recovery, append Equation (28). 2: Select scalar point-coordinate rows by rank-revealing QR, or retain an overdetermined design. 3: At each selected z, query the same center cσ at cσ + σz and cσ + λσz. 4: Form Equation (26) and solve least squares. 5: Return (âj , k̂j ) and, when applicable, x̂0 = cσ − σ 2 b̂.

(a) View normalized by σ

(b) View normalized by λσ

z σb/λ

σb true x0

origin cσ

true x0

origin cσ

Figure 4: The same physical center error occupies different lengths in the two normalized views: σb at scale σ and σb/λ at scale λσ. Both panels keep the query location z fixed.

Recovering Jets with an Imperfect Center

The previous section assumed the exact center x0 . A score-only first stage instead returns a nearby point. The key question is whether its error hides the order-σ branch signal or can be estimated with it. Proposition 5.1 (Center accuracy after local calibration). Fix smooth localized score tests around a bounded reference offset. If the resulting exact tangent-model least-squares matrix has full column rank, then using the finite-noise score from Theorem 3.1 returns a physical center of the form cσ = x0 + σ 2 b + O(σ 2+α )

(24)

for some bounded b ∈ RD . In normalized coordinates, the calibration error is order σ; multiplying by the noise scale makes the physical error order σ 2 . The proposition is local: a coarse procedure must first place the calibration window within O(σ) of x0 . Section D gives the full weak-system and least-squares expansion. More generally, we allow any first stage satisfying cσ = x0 + σ 2 b + rσ , ∥rσ ∥ ≤ Cσ 2+α . (25) Use this same physical center at both noise scales and define cσ Dσ,λ (z) =

λσsλσ (cσ + λσz) − σsσ (cσ + σz) . (λ − 1)σ

(26)

Theorem 5.2 (Center-bias expansion). Under Theorem 2.1 and Equation (25), uniformly on compact normalized query sets, 1 ∇F0 (z)b + O(σ α ), λ where ∇F0 is the D × D Jacobian of the tangent score. cσ Dσ,λ (z) = G(z) −

(27)

The shared physical offset becomes σb at the small scale and σb/λ at the large scale, so their mismatch has a known translation shape. The proof expands Fσ at the two normalized offsets and subtracts them. All quadratic shift terms are smaller than the retained order. The complete Taylor calculation is in Section D. Append the D known translation fields Cr (z) = −λ−1 ∇F0 (z)er ,

r = 1, . . . , D.

(28)

The inverse now has (s + 1)D coordinates. The only intrinsic exception is a tangent measure that is unchanged by translation along a nonzero direction; for a finite positive ray junction, this requires a complete line formed by opposite rays with matching density. Theorem 5.3 (Center-robust identifiability and scalar information). If ν0 is not invariant under any nonzero translation, the map ({aj , kj }sj=1 , b) 7−→ G − λ−1 ∇F0 b

(29)

is injective. Consequently, (s + 1)D fixed scalar component evaluations are sufficient, and fewer than (s + 1)D fixed continuous scalar observations cannot identify every parameter vector in an open set. Unless the junction itself is translation-invariant along a full line, a center shift cannot imitate branchwise curvature and density changes. A narrowing-tube test first removes curvature and center components normal to each ray. The endpoint term can cancel only in the full-line case. Otherwise b = 0, and Theorem 4.1 removes the density terms. Section D gives the proof and formal invariance criterion. Estimating b immediately refines the center: x̂0 = cσ − σ 2 b̂.

(30)

If b̂ − b = O(σ α ), then x̂0 − x0 = O(σ 2+α ).

Error Propagation and Finite Samples

Finite-scale approximation, score noise, and first-stage geometry error determine accuracy. Let p = sD for a known center and p = (s + 1)D for joint center recovery. Select M ≥ p scalar evaluations, stack the exact basis rows into B ∈ RM ×p , and let y be the corresponding two-scale response for parameter vector θ. Theorem 6.1 (Deterministic perturbation bound). Assume σmin (B) = γ > 0. Let the implemented b = B + E and the observed response be yb = y + e. If ∥E∥ < γ, the least-squares design be B op estimator obeys Cbias σ α + ∥e∥ + ∥E∥op ∥θ∥ θb − θ ≤ . (31) γ − ∥E∥op If the two normalized score fields have stacked errors εσ and ελσ , and the center remainder in Equation (25) is rσ , one may take ∥e∥ ≤

εσ + ελσ ∥rσ ∥ + Cctr 2 . (λ − 1)σ σ

(32)

Errors in recovered directions and weights enter through E by smooth dependence of the basis on the first-order geometry over any declared separated, positive-weight class. 8

The bound makes conditioning explicit: finite-scale bias and response noise are amplified by the inverse smallest singular value of the selected design. The bound follows from the pseudoinverse identity and Weyl’s inequality; Section E gives the proof. Its main message is the decomposition: scale differencing divides normalized-score noise by σ, while an unmodeled physical center remainder is divided by σ 2 . Corollary 6.2 (Kernel-density score rate). Assume a fixed finite query design, a positive lower bound on local tangent mass, bounded branch jets, and the conditions of Theorem 6.1. For N independent samples and fixed λ > 1, the score of the empirical Gaussian kernel density estimate (KDE) gives, with probability at least 1 − δ, 

θb − θ ≤ C σ α +



log(M/δ) log(M/δ) + + first-stage error . N σ3 N σ2

(33)

Ignoring the lower-order Bernstein term and assuming a comparably accurate first stage,

σopt ≍

log(M/δ) N

1/(2α+3)

log(M/δ) α/(2α+3) b θ − θ = OP .

(34)

The variance term reflects two losses: only N σ samples fall locally, and the scale difference contributes another factor 1/σ. The rate balances the deterministic remainder σ α against the scale-differenced sampling fluctuation. A learned score enters the same response bound through its errors at both scales. Thus cross-scale consistency, rather than accuracy at either scale alone, is the quantity that controls second-order recovery.

Experiments

Each experiment tests one proof interface: finite-scale expansion, query conditioning, sampling, first-stage composition, or learned-score consistency. All results use the retained paper configuration; Section F gives full grids, seeds, hardware, and tables. Finite-scale expansion. Across 180 population fits in D ∈ {2, 3, 5}, the median remainder exponent is 0.998 (5th–95th percentile 0.957–1.03), with median log–log R2 1. Exact- and O(σ 2 )biased-center runs follow the predicted order in Theorems 3.1 and 5.2, providing an independent numerical check of the analytic results (Figure 6). Scalar information versus stable designs. At response-noise RMS 10−3 and twice the parameter count, determinant-maximizing (D-optimal) row selection attains median relative error 0.00552, 2.89 times lower than random selection. Overdetermination and conditioning-aware rows increase the smallest singular value: Theorem 4.2 gives the information limit, whereas design controls noise amplification. How geometry controls stability and scale. The 42-cell phase map in Figure 5a has 20 seeds per cell. From (3◦ , 0.01) to (45◦ , 0.2), median error falls from 0.698 to 0.00379 (184-fold) and condition number from 5.61 × 104 to 24.6. All 200 scale-up systems remain full rank through D = 20, 16 branches, 340 unknowns, and 680 queries. These runs use perturbed supplied geometry; the frontend study below covers D = 3–5. In the largest case, strong first-order perturbation raises error from 0.0825 to 85.6, so first-stage accuracy controls error even when rank is preserved. 9

Stability improves away from collisions

30 10

−1

12 8 5

−2

median relative error

minimum angle (degrees)

3 0.01

0.02

0.04

0.08

0.12

Naive subtraction amplifies tangent error

naive / matched parameter error

population score KDE score

exact

0.2

minimum branch weight

mild

moderate

strong

first-order perturbation

Figure 5: Stability and response construction. (a) Median error over 20 seeds per angle–weight cell grows as rays collide or weights vanish. (b) Paired naive/matched error ratio; values above one favor the matched response. Bands are interquartile ranges. Why match two scales? Naive subtraction divides first-order error by the base scale. Its population error is 7.75, 26.6, and 49.4 times the matched error under mild, moderate, and strong perturbations; the smallest-scale strong ratio is 143. KDE noise reduces it to 1.96. The exact-F0 curve is an idealized reference for perfect nuisance removal; with estimated geometry, matched responses avoid the 1/σ amplification (Figure 5b). Finite-sample kernel density estimates. Across four planar junctions and twenty seeds, all seed-bootstrap 95% intervals contain the predicted N −1/5 slope, and error falls by 52.1–66%. Seed halves select bandwidth multipliers within 0.4 in 68.8% of comparisons; the dense [1.8, 3.2] sweep identifies a broad stable operating region across geometries (Figure 7). Estimated first-order frontends. On planar junctions, a Fourier-based frontend estimates branch count, directions, and weights. Blind count accuracy is 100% for population and 98.8% for KDE scores. A separate known-count frontend fits the same first-order quantities directly from finite-noise scores in D = 3–5 with up to 6 branches. All 270 second-stage systems are full rank. Population runs have median maximum tangent error 0.0197 radians and median relative jet error 0.132. With KDE scores, these medians are 0.0222 and 0.957; at σ = 0.08, the dimensionwise median jet errors range from 0.688 to 0.810 (Table 6). Changing the query rows has negligible effect (median error ratio 1). In population runs, the estimated basis absorbs part of the finite-scale remainder, giving an estimated/exact-basis error ratio of 0.611 (Figure 9). Learned scores. Across 20 base jobs, the training trajectories separate single-scale score fit from recovered-jet accuracy. In the enlarged hard-case study (width 256, 6 blocks, 50000 updates), median jet error reaches 0.511 at 40000; score RMS continues to improve to 0.0143 at the final checkpoint, where jet error is 0.676. This controlled synthetic experiment isolates cross-scale consistency as the mechanism used by the inverse (Figures 9 and 10). The support material contains the experiment code and its full configuration; each entry point writes its numerical observations as CSV files (Section F).

Related Work

Classical work reconstructs manifolds and estimates tangent spaces or curvature from samples [1, 6, 7, 14, 15, 17]; score methods recover first-order and smooth-support geometry, including weighted singular tangents [5, 9, 12, 21, 22, 25]. Our target is instead the branchwise second-order continuation of a singular junction on supplied rays. Denoising identities expose local scores [2, 23], and forward asymptotics derive their density and curvature corrections [3, 18, 24]. Those results map known geometry to a correction. We solve the converse coefficient problem: uniquely decomposing the superposed correction with finite scalar information while accounting for center translation. Unlike Prony and super-resolution, the unknowns are coefficients on supplied rays, not atom locations [4, 10, 11].

Limitations and Conclusion

Limitations. The theory covers zero-thickness C 2,α half-branches, positive C 1,α densities, Gaussian smoothing, two known scales, branch correspondence, and supplied first-order geometry. The condition number quantifies the effect of ray collisions and vanishing weights, and the KDE rate is conditioned on first-stage accuracy. Experiments separate full-pipeline recovery with known counts in D = 3–5 and blind planar counts from second-stage scaling through D = 20 and 16 branches with supplied perturbed geometry. The full-line case exhibits the intrinsic translation symmetry, while learned-score runs isolate cross-scale consistency on controlled synthetic junctions. Conclusion. The first score correction has an inverse role: on supplied distinct rays, its superposed field uniquely determines how every branch bends and how its density changes. Matched subtraction makes the correction observable with sharp scalar-information complexity, while translation modes accommodate imperfect localization. Known-count experiments carry score-estimated first-order geometry through the inverse in D = 3–5, and the supplied-geometry interface scales to larger systems. The stability map, matched-response comparison, and learned-score trajectories identify the geometric and statistical conditions that govern accuracy. Branch jets thus test whether a score field retains junction continuation and provide a target for thicker supports, higher-dimensional strata, pretrained models, and adaptive queries.

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[6] Herbert Federer. Curvature measures. Transactions of the American Mathematical Society, 93 (3):418–491, 1959. doi: 10.1090/S0002-9947-1959-0110078-1. [7] Christopher R. Genovese, Marco Perone-Pacifico, Isabella Verdinelli, and Larry Wasserman. Minimax manifold estimation. Journal of Machine Learning Research, 13(43):1263–1291, 2012. [8] Aapo Hyvärinen. Estimation of non-normalized statistical models by score matching. Journal of Machine Learning Research, 6(24):695–709, 2005. [9] Andrey Kharitenko, Zebang Shen, Riccardo De Santi, Niao He, and Florian Dörfler. Landing with the score: Riemannian optimization through denoising. In International Conference on Learning Representations, 2026. [10] Stefan Kunis, Thomas Peter, Tim Römer, and Ulrich von der Ohe. A multivariate generalization of prony’s method. Linear Algebra and its Applications, 490:31–47, 2016. [11] Stefan Kunis, H. Michael Möller, and Ulrich von der Ohe. Prony’s method on the sphere. SMAI Journal of Computational Mathematics, S5:87–97, 2019. [12] Xiang Li, Zebang Shen, Ya-Ping Hsieh, and Niao He. When scores learn geometry: Rate separations under the manifold hypothesis. In International Conference on Learning Representations, 2026. [13] James R. Munkres. Topology. Prentice Hall, 2 edition, 2000. [14] Partha Niyogi, Stephen Smale, and Shmuel Weinberger. Finding the homology of submanifolds with high confidence from random samples. Discrete & Computational Geometry, 39(1–3): 419–441, 2008. doi: 10.1007/s00454-008-9053-2. [15] Umut Ozertem and Deniz Erdogmus. Locally defined principal curves and surfaces. Journal of Machine Learning Research, 12(34):1249–1286, 2011. [16] Iosif Pinelis. Optimum bounds for the distributions of martingales in banach spaces. The Annals of Probability, 22(4):1679–1706, 1994. [17] David Preiss. Geometry of measures in Rn : Distribution, rectifiability, and densities. Annals of Mathematics, 125(3):537–643, 1987. [18] Divit Rawal. Rao-blackwellized score matching on manifolds. arXiv:2605.25567, 2026. [19] Yang Song and Stefano Ermon. Generative modeling by estimating gradients of the data distribution. In Advances in Neural Information Processing Systems, volume 32, 2019. [20] Yang Song, Jascha Sohl-Dickstein, Diederik P. Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. Score-based generative modeling through stochastic differential equations. In International Conference on Learning Representations, 2021. [21] Jan Pawel Stanczuk, Georgios Batzolis, Teo Deveney, and Carola-Bibiane Schönlieb. Diffusion models encode the intrinsic dimension of data manifolds. In International Conference on Machine Learning, 2024. [22] Enrico Ventura, Beatrice Achilli, Gianluigi Silvestri, Carlo Lucibello, and Luca Ambrogioni. Manifolds, random matrices and spectral gaps: The geometric phases of generative diffusion. In International Conference on Learning Representations, 2025. 12

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Proof of the two-term expansion

The local argument below tracks every power of σ. Constants may depend on the compact query set, the branch count, the jet bounds, r0 , ∆, and the far mass, but not on σ.

A.1

A Gaussian envelope

Lemma A.1 (Polynomial Gaussian envelope). Fix a compact Q ⊂ RD . There are c > 0 and constants Cm,Q such that the following holds. Let v ∈ SD−1 , u ≥ 0, and let qt = uv +tξ for 0 ≤ t ≤ 1, 2 with ∥ξ∥ ≤ u/2. For every fixed mixed-derivative order m of Kz (q) = e−∥z−q∥ /2 in (z, q), 2

|∂ m Kz (qt )| ≤ Cm,Q (1 + u)m e−cu ,

z ∈ Q.

(35)

The same form holds for vector and matrix derivatives. Proof. Let R = supz∈Q ∥z∥. Since u/2 ≤ ∥qt ∥ ≤ 3u/2, for u ≥ 4R we have ∥z − qt ∥ ≥ u/4. For bounded u, every polynomial factor is absorbed into the constant. Derivatives of a Gaussian equal a polynomial in z − qt times the same Gaussian, giving the claim.

A.2

Expansion of the rescaled convolution

After changing variables r = σu, remove the common factor (2π)−D/2 σ 1−D and write e σ (z) = Q

s Z r0 /σ X

ρj (σu)Kz (qj,σ (u)) du + Rfar,σ (z),

(36)

j=1 0

where qj,σ (u) = (γj (σu) − x0 )/σ. By Equation (7), qj,σ (u) = uvj + δj,σ (u),

δj,σ (u) =

σ 2 u kj + ϵj,σ (u), 2

∥ϵj,σ (u)∥ ≤ Lγ σ 1+α u2+α .

(37)

Decrease r0 if needed so that ∥δj,σ (u)∥ ≤ u/2 for 0 ≤ u ≤ r0 /σ. Taylor’s theorem and Theorem A.1 give, uniformly in z ∈ Q, Kz (qj,σ (u)) = Kz (uvj ) + ∇q Kz (uvj )T δj,σ (u) + Rj,σ (z, u), h

|Rj,σ (z, u)| + ∥∇z Rj,σ (z, u)∥ ≤ C σ 1+α P1 (u) + σ 2 P2 (u) e

−cu2

(38) −cu2

≤ Cσ 1+α P (u)e

(39)

The last inequality uses 0 < α ≤ 1 and σ ≤ 1. Since ∇q Kz (uvj ) = (z − uvj )Kz (uvj ) and kj ⊥ vj , σ σ ∇q Kz (uvj )T u2 kj = u2 ⟨kj , z⟩ Kz (uvj ). 2 2

(40)

Likewise, Equation (8) gives ρj (σu) = wj (1 + σaj u) + rj,σ (u),

|rj,σ (u)| ≤ Lρ σ 1+α u1+α .

(41)

Multiplying Equations (38) and (41), collecting the order-zero and order-σ terms, and applying the envelope to cross terms yields 1 ρj (σu)Kz (qj,σ (u)) = wj Kz (uvj ) + σwj aj u + u2 ⟨kj , z⟩ Kz (uvj ) + Rj,σ (z, u), 2

(42)

with

|Rj,σ (z, u)| + ∥∇z Rj,σ (z, u)∥ ≤ Cσ 1+α P (u)e−cu .

(43)

The right-hand side is integrable on [0, ∞). Extending the tangent integrals from r0 /σ to infinity contributes a Gaussian tail smaller than every power of σ. For the far measure, ∥x − x0 ∥ ≥ ∆. On a compact normalized query set and for small σ, ∥(x − x0 )/σ − z∥ ≥ ∆/(2σ). Hence the rescaled far denominator and its z-gradient are bounded by 2 2 a polynomial in σ −1 times e−∆ /(8σ ) , again smaller than σ 1+α . Integrating Equation (42) proves Equation (15). e σ = Q0 [1 + σQ1 /Q0 + Finally, Q0 is strictly positive and has a positive minimum on Q. Write Q Rσ /Q0 ]. For every fixed m, the preceding envelope applies to all z-derivatives through order m. The map x 7→ log x is smooth on a compact interval bounded away from zero, so composition in C m gives e σ = log Q0 + σQ1 /Q0 + OC m (σ 1+α ) log Q (44) which proves Equation (16).

Distributional proof of jet identifiability 2

Let K(z) = e−∥z∥ /2 . The tangent transform is Q0 = K ∗ ν0 . Associate to the jet parameters the tempered distribution ⟨η, ϕ⟩ =

s X

Z ∞

wj 0

j=1

1 aj uϕ(uvj ) + u2 ⟨kj , ∇ϕ(uvj )⟩ du. 2

(45)

Direct evaluation on the translated kernel y 7→ K(z − y) gives (K ∗ η)(z) =

Z ∞

wj 0

E 1 D aj uK(z − uvj ) + u2 kj , ∇y K(z − y)|y=uvj du 2

Z ∞

wj 0

1 aj u + u2 ⟨kj , z − uvj ⟩ K(z − uvj ) du = Q1 (z), 2

(46)

where ⟨kj , vj ⟩ = 0. Suppose G = 0. Since Q0 > 0 and RD is connected, ∇(Q1 /Q0 ) = 0 implies Q1 = CQ0 . Thus K ∗ (η − Cν0 ) = 0. Taking Fourier transforms in the space of tempered distributions gives

c K(ξ) ηb − C νb0 = 0.

(47)

c The Gaussian multiplier K(ξ) is everywhere positive, so η = Cν0 . Fix branch j and an interval I = (u− , u+ ) ⋐ (0, ∞) such that the compact ray segment {uvj : u ∈ I} has a tubular neighborhood disjoint from the other rays and the origin. In local coordinates y = uvj + n, n ∈ vj⊥ , choose ϕ(y) = χ(u)ψ(n) with ψ(0) = 0 and arbitrary ∇ψ(0). Then ν0 (ϕ) = 0, and all branches except j vanish by support. Equality η = Cν0 gives

wj 2

u2 χ(u) ⟨kj , ∇ψ(0)⟩ du = 0.

(48)

Arbitrary χ and ∇ψ(0) imply kj = 0. Now take ψ(0) = 1 and ∇ψ(0) = 0. Then Z

(aj u − C)χ(u) du = 0

(49)

for every χ ∈ Cc∞ (I). Hence aj u = C on I. Since I contains more than one point, aj = 0 and C = 0. Repeating for every branch proves the nullspace is trivial. Applying the result to the difference of two parameter collections proves Theorem 4.1.

Finite evaluations and the query lower bound

Lemma C.1 (Evaluation basis). Let f1 , . . . , fp : X → RD be linearly independent as functions. Then p there exist pairs (xm , rm ) ∈ X × {1, . . . , D}, m = 1, . . . , p, such that the matrix [eT rm fℓ (xm )]m,ℓ=1 is nonsingular. Proof. Let V = span{f1 , . . . , fp }. Point-coordinate evaluations Lx,r (f ) = eT r f (x) separate points of V : if every such functional vanishes on f ∈ V , then f is the zero function. Therefore their restrictions span V ∗ . Select p of them forming a basis of V ∗ ; their matrix in the basis f1 , . . . , fp is nonsingular. By Theorem 4.1, the p = sD fields in Equations (20) and (21) are linearly independent, so Theorem C.1 proves the upper bound in Theorem 4.2. For the lower bound, restrict the parameter vector to a nonempty open box in RsD . Any fixed continuous scheme with M scalar outputs induces a continuous injective map from that box to RM . If M < sD, compose with the standard coordinate embedding RM ,→ RsD . Invariance of domain [13] would make the image open in RsD , although it lies in a lower-dimensional coordinate subspace, a contradiction.

Center Calibration and Center-Robust Identifiability

D.1

Why weak calibration has a physical O(σ 2 ) error

ref Let cref σ = x0 − σb0 with bounded b0 , and use normalized coordinates y = (x − cσ )/σ. The normalized finite-noise score can be written

uσ (y) = F0 (y − b0 ) + σG(y − b0 ) + OC 1 (σ 1+α )

(50)

on every compact set containing the test supports. For a smooth localized test ψ, the corresponding weak-calibration row and response are Z

aψ (u) = rψ (u) =

ψ(y)u(y) dy, Z h

ψ(y) dy ,

(51) i

−∇ψ(y)⊤ u(y) + ψ(y){∥u(y)∥2 + y ⊤ u(y) + D} dy.

(52)

Both maps are smooth polynomial functionals of u on the compact test supports. Substitution of Equation (50) therefore gives Aσ = A0 + σA1 + O(σ 1+α ),

rσ = r0 + σr1 + O(σ 1+α ).

(53)

The exact tangent identity gives r0 = A0 β0 with β0 = (b0 , 1) because each branch is one-dimensional. If A0 has full column rank, the least-squares solution is −1 ⊤ βbσ = (A⊤ σ Aσ ) Aσ rσ .

(54)

The full-rank least-squares map is smooth in a neighborhood of (A0 , r0 ). Differentiating its normal equations, or expanding the pseudoinverse, yields −1 ⊤ 1+α βbσ = β0 + σ(A⊤ ), 0 A0 ) A0 (r1 − A1 β0 ) + O(σ

(55)

where the apparent derivatives of A⊤ σ Aσ cancel because r0 = A0 β0 . If b1 denotes the first D coordinates of the order-σ coefficient, the physical center returned by the calibration is 2 2+α b cref ). σ + σ bσ = x0 + σ b1 + O(σ

(56)

This proves Theorem 5.1. It also makes the role of the local-window assumption explicit: a separate coarse procedure must place the reference within O(σ) of x0 .

D.2

Two-scale center expansion

To prove Theorem 5.2, put hσ = (cσ − x0 )/σ = σb + O(σ 1+α ) and hλσ = (cσ − x0 )/(λσ) = (σ/λ)b + O(σ 1+α ). Uniform C 1 regularity of Equation (16) and a Taylor expansion give σsσ (cσ + σz) = Fσ (z + hσ ) = F0 (z) + σ∇F0 (z)b + σG(z) + O(σ 1+α ), (57) σ λσsλσ (cσ + λσz) = Fλσ (z + hλσ ) = F0 (z) + ∇F0 (z)b + λσG(z) + O(σ 1+α ). (58) λ Terms quadratic in h and products of h with the order-σ correction are O(σ 2 ), which is contained in O(σ 1+α ) for α ≤ 1. Subtraction proves Equation (27). Definition D.1 (Translation lineality). The tangent measure ν0 has translation lineality if there exists b = ̸ 0 such that (Ttb )# ν0 = ν0 for every t ∈ R. For a finite positive ray junction, this can occur only along a complete line represented by opposite rays with matching density. For injectivity, suppose the augmented field in Equation (29) is zero. Since ∇F0 b = ∇(bT ∇ log Q0 ), Q1 − λ−1 bT ∇Q0 ∇ Q0

= 0.

(59)

Hence Q1 − λ−1 bT ∇Q0 = CQ0 . Distributional differentiation commutes with convolution, so Gaussian injectivity yields η − λ−1 b · ∇ν0 = Cν0 . (60) Use the same tube around an interior segment of ray j. For a test function whose value is zero on the ray and whose normal gradient g(u) ∈ vj⊥ is arbitrary, Equation (60) becomes wj

Z D I

−1 1 2 2 u kj + λ Pvj⊥ b, g(u)

du = 0.

(61)

Therefore 12 u2 kj + λ−1 Pv⊥ b = 0 for all u ∈ I, which implies j

kj = 0,

Pv ⊥ b = 0

(62)

for every j. Thus b is parallel to every ray direction. With the normal part removed, b · ∇ν0 is supported at the common endpoint. Indeed, for a smooth compactly supported ϕ, ⟨b · ∇ν0 , ϕ⟩ = −

X j

Z ∞

wj 0

  X d (bT vj ) ϕ(uvj ) du =  wj bT vj  ϕ(0).

(63)

Neither Cν0 nor the remaining density-slope distribution has an atom at zero, so the coefficient must vanish. Under Equation (62), this cancellation is equivalent to b lying in a translation-lineality direction of the finite ray measure. The no-lineality assumption gives b = 0. The argument in Section B then gives aj = C = 0. Linear independence and the query count follow exactly as in Theorem C.1 and the invariance-of-domain lower bound. 17

Perturbation Composition and Finite-Sample Score Error

E.1

Deterministic least-squares composition

Let θ ∈ Rp be the branch-jet vector, with the center coefficient appended in the augmented model. After selecting M scalar point-coordinate evaluations, write ∥rfs ∥ ≤ Cbias σ α ,

y = Bθ + rfs ,

(64)

where B is the limiting design and rfs is the finite-scale remainder from Theorems 3.2 and 5.2. The b = B + E and yb = y + e. Whenever B b has full column rank, implemented design and response are B b † (yb − Bθ) b =B b † (rfs + e − Eθ), θb − θ = B b†

θb − θ ≤ B

(65)

Cbias σ + ∥e∥ + ∥E∥op ∥θ∥ .

(66)

b ≥ σmin (B) − ∥E∥ , which proves Equation (31). Weyl’s inequality gives σmin (B) op Suppose the normalized score vectors at the selected queries have stacked errors εσ and ελσ . Scale differencing contributes at most εσ + ελσ . (67) (λ − 1)σ

If the provisional center has an additional physical remainder rσ , its normalized displacements are rσ /σ and rσ /(λσ). A mean-value bound for the two normalized score fields, followed by division by (λ − 1)σ, gives ∥rσ ∥ Cctr 2 . (68) σ This proves Equation (32). Finally, for a fixed finite candidate set the map from distinct directions and positive normalized weights to every design entry is smooth: the ray-Gaussian integrals and their derivatives are dominated by polynomial Gaussian envelopes, and Q0 is uniformly positive. Hence on a compact first-order class with angular separation δ0 > 0 and minimum weight w0 > 0,

∥E∥op ≤ LB

b − w∥ max ∥vbj − vj ∥ + ∥w j

(69)

after the branch correspondence is fixed. This is the explicit “first-stage design error” term used in the end-to-end experiment.

E.2

A simultaneous empirical KDE score bound

At a physical query x, define 2

Wi = e−∥x−Xi ∥ /(2σ ) ,

Vi =

Xi − x Wi , σ

(70)

with means a = EWi and b = EVi . The normalized population and empirical KDE scores are F = b/a and Fb = V /W . The elementary envelopes 0 ≤ Wi ≤ 1, 2

2 E ∥Vi ∥2 ≤ a e

∥Vi ∥ ≤ e−1/2 , 2

follow by maximizing re−r /2 and using r2 e−r ≤ (2/e)e−r /2 . 18

(71)

Let t = log(4M/δ). Scalar Bernstein for Wi and a Hilbert-space Bernstein inequality for Vi [16], followed by a union bound over M queries, imply with probability at least 1 − δ that simultaneously r

2at t + , N 3N r 4at 4e−1/2 t V − b ≤ Q(a) := + . eN 3N

W − a ≤ D(a) :=

(72) (73)

Whenever D(a) < a, direct ratio subtraction gives Fb − F ≤

Q(a) + ∥F ∥ D(a) . a − D(a)

(74)

On a declared compact normalized query set for a one-dimensional branch junction, positivity and the p local density lower bound give a ≥ a− σ for all sufficiently small σ, while ∥F ∥ is bounded. If 2t/(N a− σ) + t/(3N a− σ) ≤ 1/2, Equation (74) simplifies to "r

max Fbσ (zm ) − Fσ (zm ) ≤ C

m≤M

t t + . Nσ Nσ

(75)

The same bound holds at λσ with constants depending on fixed λ. Substituting Equation (75) into Equation (67), absorbing the fixed finite design and λ into the constant, and adding Equations (64) and (69) yields 

θb − θ ≤ C σ α +



log(M/δ) log(M/δ) + + εfirst  . N σ3 N σ2

(76)

Here εfirst collects the center remainder and the tangent direction/weight error after multiplication by their deterministic Lipschitz constants. Under sample splitting, or under an independently validated first stage whose contribution is no larger than the displayed stochastic terms, balancing σ α and (N σ 3 )−1/2 gives Equation (34). The term N −1 σ −2 is lower order at that bandwidth for every α > 0.

E.3

Learned-score error units

If a learned physical score has pointwise error ∥sθ (x, τ ) − sτ (x)∥ ≤ ϵnet (τ ), then its normalized score error is τ ϵnet (τ ). Consequently the two-scale response error is bounded by ϵnet (σ) + λϵnet (λσ) . λ−1

(77)

The bound shows why a small average score loss does not certify the cross-scale derivative: the errors at the two scales must also be aligned, not simply small in isolation.

Experimental Design and Reproduction Details

F.1

Experiment-to-claim map

Question

Script and primary output

Does the finite-scale remainder have order σ α ?

run_population.py; parameter Numerical check of Theorems 3.1 error and refined-center error over and 5.2 alongside the analytic proof. decreasing σ run_query_design.py; σmin , Compares D-optimal, pivoted-QR, condition number, error under random, and uniform scalar rows. response noise run_kde.py; score error, two- Tests the conditional trend in Theoscale response error, jet error, ef- rem 6.2 on the retained twenty-seed fective sample size grid. run_end_to_end.py; branch Separates known-count, blind-count, count, tangent error, jet/center and exact-first-order interfaces. proxy error run_general_end_to_end.py; Tests the known-count score-to-jet center, tangent, weight, and jet pipeline without supplied directions errors in D = 3–5 or weights. run_learned_score.py; score Isolates cross-scale consistency on RMS, scale-difference RMS, jet controlled synthetic junctions. error run_stability_phase.py; singu- Separates exact identifiability from lar values, condition number, rela- uniform stability over 42 cells. tive error run_scalability.py; rank, er- Tests computation and sensitivity to ror, timing through D = 20 and first-order perturbations. 16 branches run_response_baselines.py; Compares the proposed response paired response and parameter with naive subtraction and an exacterrors F0 diagnostic. run_end_to_end_diagnostics. Separates first-order basis error, row py; crossed basis and query-row choice, finite-scale remainder, and policies KDE noise. run_learned_score.py; six hard- Width-256, six-block follow-up; case jobs through 50k updates checkpoint conclusions remain post hoc.

Can the sharp query theorem be implemented stably? Does sampling create the predicted bias–variance tradeoff? Does the first-order frontend compose with the jet inverse? Does score-estimated geometry compose beyond the plane? Does ordinary score training preserve the two-scale derivative? How do angle and minimum weight control stability? Does the inverse remain tractable in larger systems? Why match two scales instead of subtracting a tangent estimate? Which E4 interface produces the observed error? Does a larger learned score solve the inverse mismatch?

Interpretation

Table 1: Every experiment targets a distinct proof interface.

F.2

Geometries and exact population evaluation

Each synthetic branch is a constant-curvature arc γj (r) − x0 =

sin(κj r) 1 − cos(κj r) vj + nj , κj κj

(78)

with the continuous straight-ray limit at κj = 0. This curve is exactly arc-length parameterized, satisfies γj′′ (0) = κj nj , and has nonzero higher-order terms. The density is ρj (r) = wj exp(aj r + dj r2 ), 20

(79)

where the quadratic coefficient keeps the finite-radius sampling distribution well behaved without changing the target slope aj . The normalized population score is evaluated as a Gaussian-weighted first-moment/zeroth-moment ratio using fixed Gauss–Legendre quadrature. It is never obtained by numerical differentiation. The analytic design uses the posterior moment identities F0 (z) = EΠz [U ] − z,

∇F0 (z) = CovΠz (U ) − ID ,

(80)

and corresponding differentiated ray moments for the density and curvature columns. The implemented design uses this column order and sign convention, the center coefficient −1/λ, exact planar tangent reconstruction, and generic-rank constructions in D ∈ {2, 3, 5}.

F.3

Population and finite-query matrices

The population matrix crosses ambient dimension, branch count, random seed, scale ratio, true versus O(σ 2 )-biased center, and decreasing base scale. Directions are generated with a minimum angular separation, weights are positive and normalized, and curvature/density-slope coordinates are independently sampled inside declared bounds. The fitted exponent uses only the smallest configured scales. For query design, a candidate normalized grid is built independently of the unknown second-order coefficients. Every point contributes D possible scalar component rows. We compare: 1. Greedy D-optimal: sequentially maximize b⊤ A−1 b with a small ridge; 2. Pivoted QR: select a rank-revealing minimal basis, then add high-norm rows; 3. Random: uniform sampling without replacement; 4. Uniform: equally spaced row indices, retained as a deliberately geometry-agnostic baseline. The response perturbation has a configured RMS and is regenerated from a recorded seed. The code records scalar-query count, number of unique physical locations, rank, σmin , condition number, and relative parameter error.

F.4

Stability phase, scale-up, and response baselines

The stability phase experiment uses three planar rays with configured minimum angles and weights drawn from ∆min ∈ {3, 5, 8, 12, 20, 30, 45}◦ ,

wmin ∈ {0.01, 0.02, 0.04, 0.08, 0.12, 0.20}.

Each of the 42 cells has 20 seeds, response-noise RMS 10−3 , and twice as many scalar queries as unknowns. We retain the entire grid, including the high-error near-collision cells. The full grid characterizes the transition across angle and weight without imposing a universal separation threshold. The scale-up crosses (D, s) ∈ {(2, 4), (3, 8), (5, 8), (10, 16), (20, 16)}, ten seeds, and four firstorder perturbation levels. Direction/weight RMS pairs are (0, 0), (0.01, 0.005), (0.03, 0.015), and (0.06, 0.03). Every case uses σ = 0.04, λ = 2, and twice the parameter count in scalar queries. Timing separates population-response evaluation, row design, and least-squares solution. The response baseline uses four planar geometries, ten seeds, four base scales, population and KDE scores, and the same four perturbation levels. Three methods share fixed exact-geometry query rows: the matched two-scale response, naive subtraction of a tangent field, and an exact-F0 diagnostic. The exact-F0 diagnostic is an idealized reference that isolates the cost of estimating the nuisance field. Method ratios are paired before aggregation. 21

F.5

KDE and estimated frontends

A branch is selected according to its finite-radius mass and its radial coordinate is sampled by an inverse CDF. Empirical Gaussian KDE scores are evaluated exactly from the stored samples, with a per-query log-weight shift for numerical stability. The effective sample size P Wi (x))2 ESS(x) = Pi 2

(

i Wi (x)

(81)

is recorded at both scales. One sampled dataset is reused across the bandwidth sweep for a paired comparison. The planar first-stage frontend queries a tangential score on an equally spaced shell, spectrally integrates the circular log density, divides by the positive Gaussian–ray Fourier multipliers, infers Toeplitz rank, recovers directions through a matrix pencil, and fits nonnegative weights. The end-to-end experiment reports known-count and blind-count variants separately. The second-order error is evaluated only when the recovered count matches, and a separate exact-first-order row isolates the second stage. The nonplanar frontend treats the branch count as known and fits the leading tangent-ray score directly. Its variables are a normalized center offset, hyperspherical branch directions, and softmax-normalized weights. Multistart nonlinear least squares uses shell and interior score queries; the fitted center and first-order geometry then determine the second-stage query rows. Ground-truth directions and weights enter only the final permutation alignment and error calculation. The grid crosses D ∈ {3, 4, 5}, branch counts {3, 4, 6}, five seeds, population and KDE scores, and base scales {0.080, 0.055, 0.038}. KDE rows use 131,072 curve samples. Every case also includes an exact-first-order diagnostic evaluated at the same fitted center. The crossed E4 diagnostic uses the same four geometries, five seeds, four scales, known/blind counts, population/KDE scores, and 32,768 samples. For every successful first stage, it independently crosses an estimated or exact first-order basis with rows selected from estimated or exact geometry. This separates coefficient-basis error from row-selection error. A failed blind-count case produces one explicit sentinel instead of four inverse rows; count accuracy includes that sentinel, whereas error summaries contain only successful inversions. Median relative error changes by a factor of 1 when the row policy is switched. In population scores, the estimated-basis/exact-basis ratio 0.611 reflects finite-scale bias cancellation: the estimated tangent basis can absorb part of the remainder, so the comparison diagnoses bias absorption rather than geometric accuracy.

F.6

Learned-score protocol and hardware scaling

The network predicts the normalized score σsσ from a normalized noisy location and a boundedfrequency sinusoidal encoding of log σ. Clean samples are restricted to radius 1.25 around the junction, and the noise scale is drawn log-uniformly across the configured interval. Training samples Z ∼ N (0, ID ),

e = X/σ + Z, X

(82)

e σ) + Z∥2 . The residual MLP has width 128, four blocks, eight sinusoidal then minimizes 12 E∥fθ (X, frequencies up to 8, and 135042 trainable parameters. Each job starts from 32,768 physical samples, of which 7544–8110 fall within the radius cutoff. AdamW uses batch size 1,024, initial learning rate 1.5 × 10−3 , weight decay 10−6 , and gradient-norm clipping at 10. A cosine schedule decays to 5% of the initial rate over 20,000 updates. Evaluation uses an exponential moving average with decay 0.999 after a 1,000-update warmup at checkpoints {1000, 2500, 5000, 7500, 10000, 15000, 20000}.

All geometries, initializations, sample draws, minibatches, and noise draws use recorded integer seeds. The base configuration has one independent job per geometry–seed pair. The launcher assigns the 20 jobs round-robin across two NVIDIA GeForce RTX 3090 GPUs; each process sees one device as cuda:0. Median job time is 241 seconds, aggregate GPU time is 1.33 hours, and dual-GPU wall time is 2.47 × 103 seconds. The scale-up retains the two harder four-ray geometries and three seeds, increases the model to width 256 with 6 residual blocks, uses 65,536 physical samples, and trains through 50000 updates. The seven checkpoints are {1, 5, 10, 20, 30, 40, 50}k. The 6 jobs were run concurrently on one RTX 3090; median per-job wall time was 736 seconds. Independent population and empirical score evaluations diagnose the same controlled synthetic distribution. Jet-error checkpoint selection is reported post hoc; the nonmonotone parameter trajectory is the mechanism under study.

F.7

Support Material

The anonymous package contains analysis/configs/paper.yaml, ten experiment entry points, and the shared numerical routines they import. The entry points cover the eleven studies above because run_learned_score.py accepts both the base and scale-up configurations. Each script writes only CSV output to an ignored results/ directory. The package contains neither generated results nor code for manuscript figures or tables. Exact commands and the mapping from scripts to experiments appear in analysis/README.md.

F.8

Generated tables and additional figures

The following tables report aggregates from the experiment CSVs and are included here for completeness. D

center

median slope

IQR

median R2

instances

2 2 3 3 5 5

biased true biased true biased true

1.005 0.992 1.003 0.999 0.996 0.999

[0.993, 1.021] [0.978, 1.003] [0.989, 1.009] [0.992, 1.010] [0.988, 1.001] [0.992, 1.002]

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

30 30 30 30 30 30

Table 2: Population finite-scale fits from the active result profile.

design D-optimal pivoted QR random uniform

median σmin

median cond.

median relative error

95th pct.

0.468 0.321 0.138 0.109

23.2 42.7 33.6 46.6

0.00552 0.00754 0.016 0.0195

0.011 0.0151 0.0629 0.0714

Table 3: Query-design summary at approximately twice the parameter dimension.

geometry

error slope

seed-bootstrap 95% CI

theory

bandwidth slope

-0.189 -0.256 -0.231 -0.201

[-0.248, -0.134] [-0.327, -0.187] [-0.283, -0.196] [-0.266, -0.126]

-0.200 -0.200 -0.200 -0.200

-0.184 -0.163 -0.200 -0.159

clustered four-ray regular Y weak four-ray weak Y

Table 4: Best bandwidth and error slopes on the retained twenty-seed grid.

field

tangent input

count rate

median jet error

95th pct.

median angle

kde kde kde population population population

blind count known count exact first-order blind count known count exact first-order

98.8% 100.0% 100.0% 100.0% 100.0% 100.0%

1.77 1.79 1.8 0.0396 0.0396 0.0672

5.78 5.76 5.77 0.0892 0.0892 0.15

0.0216 0.0214 0 0.0177 0.0177 0

Table 5: Composition of the planar tangent frontend and the second-order inverse.

Pop. angle

Pop. jet error

KDE angle

KDE jet error

3 4 5

0.0212 0.0167 0.0209

0.142 0.124 0.125

0.0235 0.0206 0.0219

0.975 0.967 0.927

Table 6: Known-count nonplanar composition from D = 3 to D = 5. Entries are medians over 45 cases per source and dimension; angle errors are in radians, jet errors are relative, and every displayed design is full rank.

geometry

seeds

score RMS

scale-diff. RMS

jet error

KDE jet error

5 5 5 5

0.0149 0.0164 0.0169 0.0203

0.231 0.215 0.238 0.303

0.972 0.34 0.746 0.761

0.861 0.277 0.599 0.677

clustered four-ray regular Y weak four-ray weak Y

Table 7: Final-checkpoint learned-score diagnostics.

Regime near collision well separated

Angle

Min. weight

Median σmin

Median cond.

Median rel. error

3 45

0.01 0.2

0.0018 0.553

5.61e+04 24.6

0.698 0.00379

Table 8: Opposite corners of the angle–minimum-weight stability phase map.

Score source

Perturbation

Median naive/matched

IQR

population population population population KDE KDE KDE KDE

exact mild moderate strong exact mild moderate strong

0.334 7.75 26.6 49.4 0.863 0.893 1.21 1.96

[0.332, 0.337] [3.5, 15.7] [10.2, 52.4] [21.4, 115] [0.652, 1.24] [0.676, 1.32] [0.916, 1.78] [1.32, 3.09]

160 160 160 160 160 160 160 160

Table 9: Paired naive-subtraction error divided by matched-response error.

Case D2/s4 D2/s4 D2/s4 D2/s4 D3/s8 D3/s8 D3/s8 D3/s8 D5/s8 D5/s8 D5/s8 D5/s8 D10/s16 D10/s16 D10/s16 D10/s16 D20/s16 D20/s16 D20/s16 D20/s16

Perturbation

Median rel. error

Median cond.

Response time (s)

exact mild moderate strong exact mild moderate strong exact mild moderate strong exact mild moderate strong exact mild moderate strong

0.205 0.215 0.259 0.609 0.524 0.52 0.831 0.989 0.162 0.163 0.311 0.556 0.124 0.161 0.485 79.2 0.0825 0.131 0.408 85.6

40.2 40.1 38.2 38.5 169 143 174 307 44 49.7 60.3 117 99.8 104 175 8.55e+03 128 137 162 2.43e+04

0.011 0.011 0.011 0.011 0.022 0.022 0.022 0.022 0.030 0.030 0.030 0.030 0.096 0.096 0.096 0.096 0.121 0.121 0.121 0.121

Table 10: Scale-up errors and timings. Every displayed cell has ten seeds and full column rank.

Source population population population population kde kde kde kde

Basis

Rows

Projected error

Relative proxy error

Remainder

Cond.

estimated estimated exact exact estimated estimated exact exact

estimated exact estimated exact estimated exact estimated exact

0.039 0.0376 0.0785 0.0675 1.68 1.67 1.63 1.66

0.0781 0.0741 0.164 0.131 3.29 3.14 3.3 3.44

0.0219 0.0219 0.0387 0.0375 0.852 0.844 0.851 0.833

20.7 20.3 20.8 20.4 19.8 19.7 19.5 19.5

Table 11: Crossed E4 diagnostic on successful inverse rows. “Basis” and “Rows” vary independently.

Step

Score RMS

Response RMS

Parameter error

Parameter-error IQR

1k 5k 10k 20k 30k 40k 50k

0.0872 0.0345 0.0215 0.0187 0.0148 0.0145 0.0143

0.675 0.184 0.201 0.222 0.217 0.192 0.21

2.11 0.725 0.768 0.739 0.65 0.511 0.676

[1.81, 2.9] [0.705, 0.768] [0.674, 1.03] [0.686, 0.977] [0.452, 0.931] [0.473, 0.648] [0.665, 0.708]

Table 12: Width-256, six-block hard-case learned-score follow-up. Values are medians over six jobs.

joint O(σ ) center 10

−1

O(σ)

parameter error ‖θ − θ‖2

true center

D-optimal pivoted QR random uniform

relative parameter error

design condition number

noise RMS = 10−3

−1

−2

2 × 10

−2

3 × 10 4 × 10

−2

6 × 10

−2

base scale σ

−1

1.0

1.5

2.0

3.0

query budget / parameter count

(a) Predicted population order.

1.0

1.5

2.0

3.0

query budget / parameter count

(b) Conditioning controls recovery.

Figure 6: Two theorem-facing checks. Lines show medians and bands show interquartile ranges. Left: exact- and joint-center errors follow the O(σ) reference. Right: additional, well-selected scalar queries stabilize noisy recovery.

selected-bandwidth jet error

regular Y

slope −0.256 95% CI [−0.327, − 0.187]

slope −0.201 95% CI [−0.266, − 0.126]

median (IQR) 10

−1

theory N −1/5 10 2048

selected-bandwidth jet error

weak Y

8192

32768

clustered four-ray

c 10 6 × 10 4 × 10 3 × 10 2 × 10

−1

131072

2048

8192

131072

weak four-ray

slope −0.189 95% CI [−0.248, − 0.134]

32768

slope −0.231 95% CI [−0.283, − 0.196]

−1 −1 −1 −1

10 2048

8192

32768

−1

131072

2048

8192

sample size N

32768

131072

sample size N

Figure 7: KDE finite-sample rate by geometry. Faint points are individual seeds at the medianselected bandwidth; solid curves and bands show the seed median and interquartile range. Red references have slope −1/5. Insets report 5,000-resample seed-bootstrap percentile intervals.

population score field

KDE score field

jet/center proxy error

blind count: 100.0% 10

blind count: 98.8%

−1

6 × 10 4 × 10 3 × 10 2 × 10

−2

blind count known count exact first-order

−2

3 × 10

−2

4 × 10

−2

6 × 10

−2

3 × 10

base scale σ

−2

4 × 10

−2

6 × 10

−2

base scale σ

Figure 8: First-order frontend followed by second-order recovery. Bands are interquartile ranges over correct-count rows. Population error decreases with scale; at fixed sample size, KDE variance reverses the trend. Blind-count accuracy is computed on all rows.

Scaling is feasible but perturbation-sensitive

relative proxy error

exact mild

E4: data source dominates row policy

moderate strong

relative proxy error

−1

D2/

/s16

D3/

D5/

D10

D20

population score KDE score

−1

est. B est. Q

/s16

est. B exact Q

exact B est. Q

exact B exact Q

dimension / branches 10

Inverse quality peaks before 50k steps

score RMS two-scale response RMS

median (6 jobs) best median: 40k

parameter error

population diagnostic RMS

A better score need not improve the response

−1

training step

Figure 9: Supplemental diagnostics. (a) Dimension/branch scale-up under exact through strong first-order perturbations. (b) E4 with first-order basis (B) and query-row geometry (Q) crossed. (c–d) Width-256 learned-score follow-up through 50k updates; bands are interquartile ranges, and thin curves in (d) are all six jobs.

−1

optimizer updates

6 × 10 4 × 10 3 × 10 2 × 10

0 00

−1

jet parameter error

two-scale response RMS

lowest median: 5k

normalized-score RMS

−1 −1

−1

optimizer updates

learned median KDE median population median

optimizer updates

Figure 10: Learned-score diagnostics across seven checkpoints. Thin curves show all 20 jobs; thick curves and bands show the median and 5th–95th percentiles. Median jet error reaches its minimum at 5000 updates before score RMS reaches its lowest value; dotted and dashed lines are matched KDE and population baselines.

Matched Queries for Curvature and Density at Branching Junctions

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