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Hidden Gauge Controls Feature Specialization in ReLU Networks
Tongxi Wang
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 89%
Last extracted: 8/10/2026, 3:16:25 AM
Summary
This paper investigates how hidden gauge parameters (positive-homogeneous scaling) in overparameterized ReLU networks control feature specialization and ownership. Using a Gaussian teacher-student model, the authors demonstrate that assigning opposite gauges to initially duplicate neurons creates a sharp $\Theta(D^2)$ separation in specialization time. The favorable gauge deterministically selects one neuron as the 'owner' of the teacher feature, while redundant neurons undergo functional pruning. The mechanism is attributed to different mobilities for changing a feature's coefficient versus its direction, proving that hidden parameter choices can dictate representation dynamics invisible to the initial predictor.
Entities (8)
Relation Signals (5)
Hidden Gauge → controls → Feature Ownership
confidence 95% · assigning the favorable gauge to one neuron deterministically selects it as the owner
Hidden Gauge → causes → Specialization Time Separation
confidence 93% · Opposite gauges produce distinct feature trajectories and a sharp $\Theta(D^2)$ separation in specialization time
ReLU Network → exhibits → Functional Pruning
confidence 90% · drives the remaining functional contribution to zero... prove global selection and functional pruning
Reaction-Transport Decomposition → attributes → Hidden Gauge Effect
confidence 88% · An exact reaction–transport decomposition attributes the effect to different mobilities for changing a feature's coefficient and direction.
Gaussian Teacher-Student Model → enables → Proof of Global Selection
confidence 85% · In a tractable Gaussian teacher–student model... We prove global selection and functional pruning
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Abstract
Abstract:Training changes a network's predictions while allocating task-relevant structure across its internal units. In an overparameterized ReLU network, several neurons can begin with exactly the same functional role, yet one may acquire a teacher feature while the others become redundant. We call the identity of that neuron feature ownership and ask whether it can be controlled by a parameter choice invisible to the initial predictor. In a tractable Gaussian teacher--student model, we fix the complete initial function and vary only a positive-homogeneous scaling gauge. Opposite gauges produce distinct feature trajectories and a sharp $\Theta(D^2)$ separation in specialization time that no global change of clock can explain. Among any fixed number of initially duplicate students, assigning the favorable gauge to one neuron deterministically selects it as the owner and drives the remaining functional contribution to zero. An exact reaction--transport decomposition attributes the effect to different mobilities for changing a feature's coefficient and direction. We prove global selection and functional pruning, extend finite-time selection to visible perturbations and small-step full-batch gradient descent, and verify the predicted loss, alignment, pruning, and dissipation trajectories in population and finite-sample training. The initial predictor therefore determines neither when the feature is learned nor which neuron learns it.
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- Source: https://arxiv.org/abs/2608.06766v1
- Canonical: https://arxiv.org/abs/2608.06766v1
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Hidden Gauge Controls Feature Specialization in ReLU Networks Tongxi Wang School of Future Technology Southeast University Nanjing, China tongxi_wang@seu.edu.cn Abstract Training changes a network’s predictions while allocating task-relevant structure across its internal units. In an overparameterized ReLU network, several neurons can begin with exactly the same functional role, yet one may acquire a teacher feature while the others become redundant. We call the identity of that neuron feature ownership and ask whether it can be controlled by a parameter choice invisible to the initial predictor. In a tractable Gaussian teacher–student model, we fix the complete initial function and vary only a positive-homogeneous scaling gauge. Opposite gauges produce distinct feature trajectories and a sharp Θ(D2) (D^2) separation in specialization time that no global change of clock can explain. Among any fixed number of initially duplicate students, assigning the favorable gauge to one neuron deterministically selects it as the owner and drives the remaining functional contribution to zero. An exact reaction–transport decomposition attributes the effect to different mobilities for changing a feature’s coefficient and direction. We prove global selection and functional pruning, extend finite-time selection to visible perturbations and small-step full-batch gradient descent, and verify the predicted loss, alignment, pruning, and dissipation trajectories in population and finite-sample training. The initial predictor therefore determines neither when the feature is learned nor which neuron learns it. 1 Introduction During training, neural networks fit input–output mappings and construct task-adaptive internal representations. The neural tangent kernel and related lazy-training theories explain optimization when features remain close to initialization (Jacot et al., 2018; Chizat et al., 2019). Rich-regime analyses instead track representations and data-dependent kernels that evolve substantially during training (Yang and Hu, 2021; Atanasov et al., 2022; Kunin et al., 2024). These theories clarify when feature learning occurs and how far representations move. We study a more local assignment problem: when several neurons are initially redundant, what determines which one becomes the owner of a task-relevant feature? Here feature ownership identifies the initially duplicate neuron that ultimately carries the teacher ridge. Teacher–student models make this question precise. Existing analyses show that ReLU students can align with teacher features, that over-realized networks can contain specialized copies of teacher units, and that such specialization can emerge during optimization (Tian, 2017, 2020; Akiyama and Suzuki, 2021; Zhou et al., 2021). These results establish that feature ownership can emerge. They do not isolate how ownership is assigned when candidate neurons start with the same feature, the same functional contribution, and the same population signal. In that symmetric situation, the initial predictor offers no reason for one neuron to win over another. Positive homogeneity supplies a hidden source of asymmetry. A ReLU neuron can be rescaled by multiplying its output weight and inversely rescaling its input weight without changing the represented function. Prior work shows that such parameter lifts are dynamically meaningful: relative layer scales determine kernel and adaptive regimes, homogeneous gradient flow carries conserved imbalance quantities, and function-preserving rescaling can change or deliberately condition training trajectories (Williams et al., 2019; Du et al., 2018; Marcotte et al., 2023; Kunin et al., 2024; Lebeurrier et al., 2026). Building on this optimization geometry, we ask for a sharper feature-learning consequence: How strongly can a function-invisible scale separate specialization times, and can it deterministically assign a feature to one of several functionally identical neurons? We answer both questions in a two-layer Gaussian ReLU teacher–student model. The comparison is counterfactual: the networks have the same initial predictor, loss, feature directions, and functional coefficients, and differ only in a hidden scaling gauge. For a single student, opposite gauges create non-collinear feature-learning paths whose specialization times differ by a sharp factor Θ(D2) (D^2). Their visible-state vector fields point in different directions, which rules out every scalar learning-rate or global clock explanation. Figure 1 summarizes this separation at the level of mechanism, visible trajectory, and specialization time. The effect becomes more consequential in an overparameterized student. Start with any fixed number of duplicate neurons, so they are interchangeable from the predictor’s viewpoint. Give one neuron the favorable gauge and the others the opposite gauge. Gradient descent selects the marked neuron as the unique owner and drives the redundant functional mass to zero. Permuting the hidden gauge assignment permutes the winner without changing the initial function. In this controlled setting, a parameterization choice invisible at initialization determines feature ownership. The mechanism is simple enough to state without the full formalism. A ReLU neuron can learn by changing its functional coefficient or by rotating its switching boundary. The hidden gauge reallocates mobility between these two motions. Opposite gauges have the same instantaneous reaction mobility at a shared visible state, yet radically different transport mobility for rotating the feature. This node-level consequence of ReLU homogeneity persists in arbitrary-depth feedforward networks. The Gaussian teacher–student geometry then allows the local law to be integrated into sharp global specialization and ownership results. Section 3 formalizes the mechanism through an exact reduced dynamics and loss-dissipation identity. Contributions. • Sharp non-clock specialization separation. We prove matching upper and lower bounds showing a Θ(D2) (D^2) gap between functionally identical opposite-gauge initializations, and prove that their feature-space vector fields are not related by a global time reparameterization. • Gauge-programmed feature ownership and functional pruning. For any fixed number of initially duplicate students, one favorable gauge deterministically selects the owner. The selected unit captures the teacher, the redundant functional coefficients vanish, and the post-capture motion of redundant features is quadratically smaller in the gauge magnitude. • Mechanism, robustness, and trajectory-level evidence. We derive an exact decomposition that separates changes in functional coefficient from changes in feature direction. We establish finite-time robustness to visible perturbations, a conditional long-time extension for nonsymmetric redundant dictionaries, and a small-step full-batch gradient-descent theorem. Reduced, original-parameter, and finite-sample dynamics agree across complete loss, alignment, functional-pruning, and dissipation trajectories. The strongest unconditional global statement concerns exactly duplicate students, which is the setting that cleanly isolates feature assignment. Around that state, finite-time winner selection persists on an open set; long-time coefficient-wise pruning for a fully nonsymmetric redundant dictionary requires explicit nondegeneracy conditions. We state these boundaries rather than treating them as generic properties of deep networks. The result is a tractable but complete example in which predictor-level information is insufficient to determine representation-level dynamics: the hidden lift controls both the time scale of specialization and the identity of the neuron that specializes. Figure 1: Same predictor, different specialization paths. (a) Functionally identical opposite-gauge initializations share the same visible state and reaction mobility, but their direction mobilities differ by Θ(D2) (D^2). (b) At D=16D=16, their paths in visible (c,q)(c,q) space are non-collinear, so no global time reparameterization can match them. (c) The measured specialization-time ratio follows D2D^2 (fitted exponent 1.971.97), with T+D∼D−1T_+D D^-1 and T−D∼DT_-D D. 2 Related work 2.1 Lazy and feature-learning regimes The neural tangent kernel gives a tractable description of wide-network training when the kernel remains effectively fixed (Jacot et al., 2018). This behavior belongs to a broader lazy regime in which the model stays close to its linearization (Chizat et al., 2019). Feature-learning theories study evolving representations, including infinite-width parameterizations with nontrivial feature motion (Yang and Hu, 2021), data-adaptive kernel evolution (Atanasov et al., 2022), and exact models connecting initialization imbalance to rich learning (Kunin et al., 2024). These works characterize whether and how far features move. We resolve the timing and ownership of one feature among functionally redundant neurons. 2.2 Homogeneous gauges and function-preserving rescaling Gradient flow in homogeneous networks conserves differences of adjacent squared layer norms (Du et al., 2018), and Marcotte et al. (2023) provide a general framework for identifying such conservation laws. Using path-lifting coordinates, Marcotte et al. (2025) show that arbitrary-depth ReLU gradient flow admits a lower-dimensional intrinsic dynamics that depends on initialization. In a shallow univariate ReLU model, Williams et al. (2019) derive a nonredundant function parameterization whose dynamics depend on the initialization lift and distinguish kernel from adaptive knot-motion regimes. Kunin et al. (2024) further show how unbalanced initialization and learning-rate asymmetry modify rich learning through conserved quantities. Recent optimization methods explicitly exploit function-preserving rescaling: Path-conditioned Training chooses a better-conditioned representative on a rescaling orbit (Lebeurrier et al., 2026), while soft gauge fixing adds a balancing force that reduces scale redundancy (Terin, 2026). Path-SGD builds an optimization geometry designed to be invariant to function-preserving node rescalings (Neyshabur et al., 2015). This provides a useful control: ordinary Euclidean geometry generates our effect, while an exactly gauge-equivariant update preserves gauge-related function trajectories. A complementary identifiability literature asks when a deep ReLU function determines its parameters modulo permutation and positive rescaling (Bona-Pellissier et al., 2022, 2023). Our question starts after that functional quotient is recognized: even when scale is unidentifiable from the function, the chosen representative can still determine Euclidean training mobility. Together, these papers establish that the lift of a ReLU function can affect training. We make the incoming/outgoing mobility reallocation explicit at an arbitrary-depth feedforward ReLU node, then derive two global consequences in a solvable model: a sharp non-clock Θ(D2) (D^2) specialization gap at a fixed predictor and deterministic ownership with functional pruning among duplicate neurons. 2.3 Teacher–student specialization and overparameterization Analytic Gaussian ReLU gradients and symmetry breaking in teacher–student models were developed by Tian (2017). Tian (2020) proves specialization of over-realized students under finite width and input dimension. Local convergence near over-realized teacher representations is known (Zhou et al., 2021), and global measure-space recovery can be obtained under regularization (Akiyama and Suzuki, 2021). Overparameterization also changes optimization speed and landscape structure (Safran et al., 2021; Xu and Du, 2023). These works study whether teacher features are recovered or specialized. Our controlled initialization asks which of several initially indistinguishable students acquires the feature, and proves that the hidden gauge alone can assign the winner. 2.4 Reduced training-dynamics theories Mean-field limits describe the distributional evolution of two-layer networks (Mei et al., 2019), while dynamical mean-field theories can predict complete train/test trajectories across data, width, depth, and parameterization in tractable models (Bordelon and Pehlevan, 2025). Our reduced system addresses a narrower ownership problem and follows the same trajectory-level standard: one closed description predicts loss, feature alignment, redundant mass, and both components of dissipation throughout training. 3 Theory We first derive coordinates that separate function-visible state from the hidden gauge, then use them to connect a local mobility law to specialization time and feature ownership. Throughout, d is the input dimension and D=|δ|D=|δ| is the gauge magnitude in large-gauge statements. All population results assume that the data distribution assigns zero mass to the relevant ReLU switching hyperplanes. 3.1 Exact marked dynamics Consider the two-layer ReLU predictor and population square loss fΘ(x)=∑i=1mai[wi⊤x]+,ℒ(Θ)=12[(fΘ(x)−y(x))2].f_ (x)= _i=1^ma_i[w_i x]_+, ( )= 12E\! [(f_ (x)-y(x))^2 ]. (1) For each regular neuron wi≠0w_i≠ 0, introduce the marked coordinates ri=‖wi‖,si=wiri,qi=airi,δi=ai2−ri2.r_i=\|w_i\|, s_i= w_ir_i, q_i=a_ir_i, _i=a_i^2-r_i^2. (2) The predictor depends on (qi,si)(q_i,s_i) and is blind to δi _i. Writing RΘ=fΘ−yR_ =f_ -y, define the scalar and vector residual moments together with the two mobilities BΘ(s) B_ (s) =[RΘ(x)[s⊤x]+], =E\! [R_ (x)[s x]_+ ], Θ(s) _ (s) =[RΘ(x)s⊤x>0x], =E\! [R_ (x)1_\s x>0\x ], (3) μ(q,δ) μ(q,δ) =δ2+4q2, = δ^2+4q^2, χ(q,δ) χ(q,δ) =2qμ(q,δ)−δ. = 2qμ(q,δ)-δ. Ordinary Euclidean gradient flow is then exactly equivalent, on the regular chart, to q˙i=−μiBΘ(si),s˙i=−χi(I−sisi⊤)Θ(si),δ˙i=0, q_i=- _iB_ (s_i), s_i=- _i(I-s_is_i )T_ (s_i), δ_i=0, (4) where μi=μ(qi,δi) _i=μ(q_i, _i) and χi=χ(qi,δi) _i=χ(q_i, _i). Moreover, −ℒ˙=∑i[μiBΘ(si)2⏟reaction+μi+δi2‖(I−sisi⊤)Θ(si)‖2⏟transport].- L= _i [ _iB_ (s_i)^2_reaction+ _i+ _i2 \|(I-s_is_i )T_ (s_i) \|^2_transport ]. (5) Thus qiq_i is the functional coefficient, sis_i is the feature direction, and the conserved δi _i reallocates mobility between coefficient reaction and directional transport. Appendix A proves the bidirectional reduction and dissipation identity in full generality. The hidden dependence can already be read off exactly at a shared visible state. For q>0q>0 and opposite marks ±D± D, the residual moments and reaction mobility coincide, whereas χ(q,+D)χ(q,−D)=(D2+4q2+D2q)2=Θ(D2/q2). χ(q,+D)χ(q,-D)= ( D^2+4q^2+D2q )^2= (D^2/q^2). The same functional residual signal therefore produces different directional motion. Since both q and s are function-visible, the induced visible-state vector field depends on the omitted mark δ. Definition 3.1 (Dynamical outcomes). In the one-teacher setting, a unit specializes when its nonvanishing feature aligns with the teacher. Selection or capture is the finite-time event in which a designated unit uniquely enters a winner-dominant set. Locking means that the trajectory remains in that set and converges to the selected representation. For a redundant index set ℛR, functional pruning means ∥∑j∈ℛqj[sj⊤⋅]+∥L2→0\| _j q_j[s_j ·]_+\|_L^2→ 0, while coefficient-wise pruning is the stronger conclusion qj→0q_j→ 0 for every j∈ℛj . Feature ownership combines specialization of one selected unit with functional pruning of the others. 3.2 From local mobility to specialization time The mobility reallocation in (4) is a node-level consequence of ReLU homogeneity. Consider a hidden node in any feedforward ReLU network without normalization or parameter sharing. Let u¯=(u,b) u=(u,b) collect its incoming affine parameters and let v collect its outgoing weights. The rescaling gρ:(u¯,v)↦(u¯/ρ,ρv)g_ρ:( u,v) ( u/ρ,ρ v), ρ>0ρ>0, satisfies fgρΘ f_g_ρ =fΘ, =f_ , (6) ∇u¯(ρ)ℒ _ u^(ρ)L =ρ∇u¯ℒ, =ρ _ uL, ∇v(ρ)ℒ _v^(ρ)L =ρ−1∇vℒ, =ρ^-1 _vL, dtu¯(ρ) \,d\,dt u^(ρ) =ρ2dtu¯^, =ρ^2 \,d\,dt u, dtv(ρ) \,d\,dt v^(ρ) =ρ−2dtv^. =ρ^-2 \,d\,dt v. For a two-layer neuron with fixed q>0q>0, reciprocal lifts ρ and ρ−1ρ^-1 have marks δ=±q(ρ2−ρ−2)δ=± q(ρ^2-ρ^-2) and an input-direction mobility ratio ρ4=Θ(D2)ρ^4= (D^2). Proposition A.2 proves (6), including jointly rescaled node biases. The optimizer geometry determines whether this hidden state matters. An exactly gauge-equivariant update preserves gauge-related function trajectories. Under symmetric ℓ2 _2 weight decay with coefficient λwd _ wd, the mark obeys δ˙=−2λwdδ δ=-2 _ wdδ and decays exponentially. These two controls locate the mechanism within ordinary Euclidean optimization; see sections˜A.2 and A.1. We next integrate the local law in a solvable model. Let x∼(0,Id)x (0,I_d), f⋆(x)=q⋆[s⋆⊤x]+f_ (x)=q_ [s_ x]_+, and c=s⊤s⋆c=s s_ . With the arc-cosine kernel κ(c)=1−c2+(π−arccosc)c2π,κ(c)= 1-c^2+(π- c)c2π, the marked flow closes exactly: q˙=−μ(q,δ)(q2−q⋆κ(c)),c˙=χ(q,δ)q⋆κ′(c)(1−c2),δ˙=0. q=-μ(q,δ) ( q2-q_ κ(c) ), c=χ(q,δ)q_ κ (c)(1-c^2), δ=0. (7) Theorem 3.2 (Same-predictor specialization gap; informal). Fix q0>0q_0>0 and −1<c0<1-1<c_0<1. Every solution of (7) converges to (q⋆,1)(q_ ,1). Realize the same visible initial state (q0,c0)(q_0,c_0) with opposite marks δ=±Dδ=± D, and let Tδ(ε)=inft:cδ(t)≥1−εT_δ( )= \t:c_δ(t)≥ 1- \. For all sufficiently large D, T+D(ε)=Θ(D−1log1ε),T−D(ε)=Θ(Dlog1ε),T−DT+D=Θ(D2).T_+D( )= \! (D^-1 1 ), T_-D( )= \! (D 1 ), T_-DT_+D= (D^2). (8) At every common nonstationary visible state, the two vector fields are non-collinear; hence no scalar learning-rate or global time reparameterization matches their feature paths. The formal convergence, two-sided hitting-time bounds, and non-clock statement are theorems˜A.9, A.10 and A.5, respectively. The two scales in (8) make the ratio sharp: the favorable lift specializes on the accelerated scale D−1D^-1, while the unfavorable lift requires the retarded scale D. Non-collinearity is a separate geometric conclusion, because a monotone change of time can alter speed but not tangent direction. The implied constants hold for fixed (q0,c0,q⋆,ε)(q_0,c_0,q_ , ) and need not remain uniform near degenerate boundary values. 3.3 Gauge-selected feature ownership Let the teacher be f⋆(x)=[u⊤x]+f_ (x)=[u x]_+ and initialize m≥2m≥ 2 students at the same visible state qi(0)=1m,si(0)=s0,ϕ=∠(s0,u)∈Iϕ:=[0.77,0.90].q_i(0)= 1m, s_i(0)=s_0, φ= (s_0,u)∈ I_φ:=[0.77,0.90]. Assign one designated index k the mark +D+D and every other index the mark −D-D. Theorem 3.3 (Hidden-gauge ownership and functional pruning; informal). For each fixed m and ϕ∈Iϕφ∈ I_φ, every sufficiently large D selects unit k as the unique owner. The trajectory enters a winner tube by tcap≤CmD−1t_ cap≤ C_mD^-1 and then satisfies ℒ(t)≤ℒ(tcap)e−cmD(t−tcap),qk(t)sk(t)→u,‖fred(t)‖L2→0.L(t) (t_ cap)e^-c_mD(t-t_ cap), q_k(t)s_k(t)→ u, \|f_ red(t)\|_L^2→ 0. For exact duplicate initialization, every redundant coefficient tends to zero and ∫tcap∞‖s˙j(t)‖dt≤CmD−2(j≠k). _t_ cap^∞\| s_j(t)\|\,\,dt≤ C_mD^-2 (j≠ k). Permuting the unique positive mark permutes the owner while leaving the complete initial predictor unchanged. The proof has two named stages. Fast gauge-selected capture uses exchangeability to reduce the system to one designated unit and one aggregate redundant block. On fast time τ=Dtτ=Dt, positive-gauge transport is order one while negative-gauge transport is order D−2D^-2; a certified multiplicity-weighted arc-cosine flow reaches the winner basin, and the finite-D trajectory shadows it to order D−2D^-2. Post-capture locking uses local feature-Gram coercivity and the dissipation identity to obtain exponential convergence and summable redundant transport. The formal result is theorem˜B.7; theorems˜C.1 and C.2 extend its capture basin and quantify sufficient multiplicity dependence. This construction also identifies the assignment mechanism cleanly. At initialization, every unit has the same (qi,si)(q_i,s_i) and the complete predictor is invariant under permutations of unit labels; the location of the unique positive mark is the only index-dependent input. Permutation equivariance of gradient flow and the theorem together show that moving this mark moves the owner. The conclusion is therefore an intervention at a fixed function, rather than a correlation between initial feature quality and eventual specialization. It does not assert that large gauge disparities arise typically under standard random initialization; it shows that when such hidden disparities are present, they can be causally decisive. 3.4 Robustness and discrete gradient descent For one positive and m−1m-1 negative marks, measure departure from duplicate initialization by Δ0=maxi(|qi(0)−1m|+‖si(0)−s0‖)+maxi||δi(0)|D−1|. _0= _i ( |q_i(0)- 1m |+\|s_i(0)-s_0\| )+ _i | | _i(0)|D-1 |. (9) Theorem 3.4 (Open-set finite-time selection; informal). For every fixed m, there are εm>0 _m>0 and D0(m)<∞D_0(m)<∞ such that, if ϕ∈Iϕφ∈ I_φ, Δ0≤εm _0≤ _m, and D≥2D0(m)D≥ 2D_0(m), the positive-gauge unit is the unique unit to enter a fixed winner-dominant set by time (T⋆+1)/D(T_ +1)/D. The proof supplies the conservative sufficient scales εm≥ae−bm3,D0(m)≤AeBm2 _m≥ ae^-bm^3, D_0(m)≤ Ae^Bm^2 for universal positive constants a,b,A,Ba,b,A,B. This finite-time statement is unconditional under the displayed assumptions; its formal version is theorem˜C.6. Proposition 3.5 (Conditional locking after perturbed capture; informal). Suppose a selected post-capture state has positive winner–redundant quotient transversality, an active redundant coefficient frame, and the explicit continuation margins of theorem˜C.11. Then the winner tube is forward invariant, the loss decays as e−cDte^-cDt, the selected feature converges to the teacher, every redundant coefficient vanishes, and total redundant-direction motion is O(D−2)O(D^-2). The active-frame hypothesis controls coefficient cancellation modes that functional pruning alone cannot identify. The complete conditions and proof appear in theorem˜C.11. Proposition 3.6 (Discrete full-batch gradient descent; informal). For exact duplicates, let η=h/Dη=h/D and write the pre-capture state as Y=(q,P,θ,ψ)Y=(q,P,θ,ψ). For every fixed m, sufficiently small h and sufficiently large D select the same positive-gauge owner, with maxn≤ncap‖Yn−Y∞(nh)‖≤Cm(h+D−2),ncap≤⌈T⋆+1h⌉. _n≤ n_ cap\|Y_n-Y_∞(nh)\|≤ C_m(h+D^-2), n_ cap≤ T_ +1h . After capture, ℒn+1≤(1−cmh)ℒnL_n+1≤(1-c_mh)L_n; the total gauge drift is Om(h/D)O_m(h/D) and the redundant-direction path is Om(D−2)O_m(D^-2). The formal discrete theorem, including chart preservation and the loss-weighted Euler remainder, is theorem˜C.19. 4 Experiments All experiments use the Gaussian teacher–student model analyzed above. We compare the multiplicity-weighted fast system, the exact finite-gauge marked ODE, the original (ai,wi)(a_i,w_i) population flow, and the original-parameter empirical flow. The capture criterion is fixed before every sweep. The four levels serve distinct checks. Agreement between marked and original-parameter population flows audits the exact coordinate reduction; convergence of the finite-gauge flow to the fast system tests the singular limit; and empirical trajectories probe sampling error without entering any proof. We compare full functionally identifiable trajectories rather than selected endpoints. For each coupled seed, finite-sample datasets are nested across N, and uncertainty is resampled at the seed level. No theoretical coefficient is fit to empirical trajectories, and the displayed reference slopes are anchored guides. Appendix D gives the fixed capture tube, trajectory metric, solver audit, and complete sweep tables. 4.1 Controlled same-predictor counterfactual Figure 1 isolates the central intervention. The two students represent the same initial function and receive the same initial functional signal; only the hidden gauge differs. Their trajectories leave the shared visible state along distinct curves. The favorable gauge rapidly rotates toward the teacher, whereas the opposite gauge first changes its functional coefficient with negligible alignment. The specialization-time ratio follows the predicted quadratic law. Both singular-limit residuals converge at order D−2D^-2; figure˜D.1 reports these numerical checks. 4.2 Feature ownership and full-trajectory correspondence Figure 2 shows a representative (m,D,N)=(8,32,8192)(m,D,N)=(8,32,8192) run. One reduced theory simultaneously predicts population loss, growth of the selected coefficient, redundant-mass decay, feature alignment, and the reaction–transport dissipation split. Across the complete population grid, the original-parameter and marked flows differ by less than 1.3×10−71.3× 10^-7, and population gauge drift stays below 2.2×10−112.2× 10^-11. Figure 2: Feature ownership and full-trajectory correspondence. (a) The programmed owner captures the teacher coefficient while redundant functional mass vanishes; the shared vertical line marks τcap _ cap. (b)–(d) The same four dynamical levels agree on population loss, functionally weighted feature residuals, and reaction–transport dissipation. Solid and dashed curves denote the exact marked and fast systems; open circles and diamonds denote the original-parameter population and N=8192N=8192 finite-sample flows. 4.3 Predicted scaling laws We sweep m∈2,4,8,16,32m∈\2,4,8,16,32\ and D∈8,16,32,64,128D∈\8,16,32,64,128\. Across multiplicities, fitted capture-time slopes range from −1.00-1.00 to −0.995-0.995, post-capture redundant-drift slopes from −2.00-2.00 to −1.99-1.99, and finite-gauge approximation slopes from −2.00-2.00 to −1.99-1.99; every corresponding R2R^2 exceeds 0.99990.9999. Panels (a)–(b) of figure˜3 display the ownership-level laws, while figure˜D.1 reports the finite-gauge approximation. For finite samples we use m∈2,4,8,16m∈\2,4,8,16\, N∈512,1024,2048,4096,8192N∈\512,1024,2048,4096,8192\, and 20 coupled seeds per configuration (400 trajectories). Every run captures the programmed owner. The fitted functional-state error slopes range from −0.45-0.45 to −0.49-0.49, compatible with fixed-system Monte Carlo scaling N−1/2N^-1/2. Panel (c) reports paired-bootstrap 95% confidence intervals over coupled seeds. The guide is empirical and does not assert a width or propagation-of-chaos theorem. Figure 3: Scaling laws and discrete shadowing. (a) Winner capture follows D−1D^-1. (b) Post-capture redundant-direction drift follows D−2D^-2. (c) Finite-sample full-trajectory error is compatible with N−1/2N^-1/2; light points are individual seeds and bars are paired-bootstrap 95% confidence intervals. (d) Discrete full-batch GD shadows gradient flow to first order in h=ηDh=η D. Reference lines are anchored at D=32D=32, N=2048N=2048, and h=0.1h=0.1 and indicate predicted slopes rather than additional fits. 4.4 Robustness and discrete training The angle certificate covers ϕ∈[0.77,0.90]φ∈[0.77,0.90], and all 24 angle–multiplicity tests capture the programmed owner. At (m,D)=(8,32)(m,D)=(8,32), all 50 random near-duplicate runs succeed for perturbations up to 5%5\%, well beyond the conservative theorem radius. Across m=2m=2 to 256256, the observed rescaled capture time DtcapDt_ cap decreases from 25.225.2 to 18.518.5. Discrete full-batch GD with h=ηD∈0.025,0.05,0.1,0.2,0.4h=η D∈\0.025,0.05,0.1,0.2,0.4\ always selects the same owner; its trajectory error has slope 1.121.12 in h, consistent with the first-order guide in panel (d) of figure˜3. Appendix D reports the remaining robustness audits. 5 Conclusion Feature learning allocates task-relevant structure to particular internal units. We identified a setting in which that allocation depends on information hidden from the initial function. At a fixed ReLU predictor, opposite gauges create a sharp non-clock Θ(D2) (D^2) specialization-time gap. Among duplicate students, the gauge assignment deterministically selects the feature owner and functionally prunes the remaining neurons. The reaction–transport decomposition explains this behavior and predicts complete population and finite-sample trajectories. The strongest global theorem concerns exact duplicates. Finite-time selection persists under visible perturbations, while long-time coefficient-wise pruning for a fully nonsymmetric redundant dictionary requires explicit transversality and active-frame conditions. The discrete theorem treats small-step full-batch gradient descent, and the finite-sample scaling is empirical. Within this scope, predictor-level descriptions discard mobility information that decisively changes representation-level competition. The marked coordinates show what a dynamics-level description must retain when internal allocation matters. Residual moments specify the functional signal, while the hidden mark specifies the metric converting it into coefficient reaction and feature transport. Gauge-equivariant optimization removes this dependence, whereas symmetric weight decay gradually erases the mark. These controls offer concrete ways to distinguish or suppress gauge-driven ownership. 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Jin (2021) A local convergence theory for mildly over-parameterized two-layer neural network. In Conference on Learning Theory, p. 4577–4632. Cited by: §1, §2.3. Appendix roadmap The appendices follow the proof dependencies of the main text. • Appendix A derives the exact marked dynamics, proves the node-mobility law, and establishes one-neuron specialization and the non-clock quadratic time gap. • Appendix B proves global feature ownership and pruning for exactly duplicate students through fast capture and post-capture locking. • Appendix C extends the capture analysis to an angle interval, quantifies multiplicity dependence, proves open-set finite-time selection and conditional locking, and treats discrete full-batch gradient descent. • Appendix D specifies the numerical protocol and reports the trajectory, scaling, robustness, and discretization audits corresponding to Figures 1–3. Appendix B builds on Appendix A, and Appendix C builds on both. Appendix D provide numerical validation. Appendix A Exact marked dynamics and one-neuron specialization A.1 Population gradient flow and the regular marked chart Let π be a probability measure on ℝdR^d with finite second moment and assume that it assigns zero mass to every hyperplane used below. Let y∈L2(π)y∈ L^2(π) and σ(z)=z+σ(z)=z_+. For a finite positive particle measure ϱ on :=ℝ×(ℝd∖0),P:=R×(R^d \0\), define fϱ(x)=∫aσ(w⊤x)ϱ(da,dw),ℒ(ϱ)=12∫(fϱ(x)−y(x))2π(dx).f_ (x)= a\,σ(w x)\, (\,da,\,dw), ( )= 12 (f_ (x)-y(x) )^2\,π(\,dx). (A.1) Write Rϱ=fϱ−yR_ =f_ -y. Along a characteristic of the Wasserstein/population gradient flow, a˙=−∫Rϱ(x)σ(w⊤x)π(dx),w˙=−a∫Rϱ(x)w⊤x>0xπ(dx). a=- R_ (x)σ(w x)\,π(\,dx), w=-a R_ (x)1_\w x>0\x\,π(\,dx). (A.2) This is the standard two-layer mean-field characteristic system (Du et al., 2018; Mei et al., 2019). We derive the exact marked reduction needed below. For w≠0w≠ 0, set r=‖w‖,s=wr∈d−1,q=ar,δ=a2−r2.r=\|w\|, s= wr ^d-1, q=ar, δ=a^2-r^2. (A.3) Define μ(q,δ):=δ2+4q2,r2(q,δ)=μ(q,δ)−δ2,χ(q,δ):=2qμ(q,δ)−δ.μ(q,δ):= δ^2+4q^2, r^2(q,δ)= μ(q,δ)-δ2, χ(q,δ):= 2qμ(q,δ)-δ. (A.4) The marked state space is :=(q,s,δ):s∈d−1,μ(q,δ)>δ.X:= \(q,s,δ):s ^d-1,\ μ(q,δ)>δ \. (A.5) The condition is exactly r>0r>0. The formula for χ is regular at q=0q=0 whenever δ<0δ<0 and equals a/ra/r. The inverse chart is r=μ−δ2,a=qr,w=rs.r= μ-δ2, a= qr, w=rs. (A.6) Hence the map (a,w)↦(q,s,δ)(a,w) (q,s,δ) is a smooth bijection from P onto X. Let ρ be the pushforward of ϱ . Positive homogeneity gives fρ(x)=∫qσ(s⊤x)ρ(dq,ds,dδ).f_ρ(x)= q\,σ(s x)\,ρ(\,dq,\,ds,\,dδ). (A.7) Define the residual moments Bρ(s):=∫Rρ(x)σ(s⊤x)π(dx),ρ(s):=∫Rρ(x)s⊤x>0xπ(dx).B_ρ(s):= R_ρ(x)σ(s x)\,π(\,dx), _ρ(s):= R_ρ(x)1_\s x>0\x\,π(\,dx). (A.8) By homogeneity, Bρ(s)=s⊤ρ(s).B_ρ(s)=s T_ρ(s). (A.9) Let s⟂=I−ss⊤ P_s =I-s . Theorem A.1 (Exact gauge-marked reduction). Let (ϱt)t∈[0,T]( _t)_t∈[0,T] be a characteristic solution of (A.2) whose support remains in P. Its marked pushforward ρt _t satisfies ∂tρ+∂q(Vqρ)+divd−1(Vsρ)=0,Vδ=0, _tρ+ _q(V_qρ)+div_S^d-1(V_sρ)=0, V_δ=0, (A.10) with the exact velocities Vq=−μ(q,δ)Bρ(s),Vs=−χ(q,δ)s⟂ρ(s),Vδ=0.V_q=-μ(q,δ)B_ρ(s), V_s=-χ(q,δ) P_s T_ρ(s), V_δ=0. (A.11) Conversely, every characteristic solution of (A.10) that remains in X pulls back through (A.6) to a solution of (A.2). Thus the two systems are bidirectionally equivalent up to the first exit from the regular chart. Moreover, the loss satisfies the exact reaction–transport dissipation identity dtℒ(ρt)=−∫[μ(q,δ)Bρt(s)2⏟coefficient reaction+μ(q,δ)+δ2‖s⟂ρt(s)‖2⏟directional transport]ρt(dq,ds,dδ). \,d\,dtL( _t)=- [ μ(q,δ)B_ _t(s)^2_coefficient reaction+ μ(q,δ)+δ2 \| P_s T_ _t(s) \|^2_directional transport ] _t(\,dq,\,ds,\,dδ). (A.12) In particular, the entire δ-marginal is conserved. Proof. Since σ(w⊤x)=rσ(s⊤x)σ(w x)=rσ(s x), (A.2) becomes a˙=−rBρ(s),w˙=−aρ(s). a=-rB_ρ(s), w=-aT_ρ(s). (A.13) Taking radial and tangential components gives r˙=s⊤w˙=−aBρ(s),s˙=1rs⟂w˙=−ars⟂ρ(s). r=s w=-aB_ρ(s), s= 1r P_s w=- ar P_s T_ρ(s). (A.14) Therefore q˙ q =ra˙+ar˙=−(r2+a2)Bρ(s)=−μ(q,δ)Bρ(s), =r a+a r=-(r^2+a^2)B_ρ(s)=-μ(q,δ)B_ρ(s), (A.15) δ˙ δ =2aa˙−2rr˙=−2arBρ(s)+2arBρ(s)=0. =2a a-2r r=-2arB_ρ(s)+2arB_ρ(s)=0. (A.16) The identity a/r=χ(q,δ)a/r=χ(q,δ) yields the stated s-velocity. Applying the chain rule to test functions proves the pushforward continuity equation. The inverse calculation gives the converse. For dissipation, the original Euclidean gradient flow gives dtℒ \,d\,dtL =−∫[r2Bρ(s)2+a2‖ρ(s)‖2]dρ =- [r^2B_ρ(s)^2+a^2\|T_ρ(s)\|^2 ]\,dρ =−∫[(r2+a2)Bρ(s)2+a2‖s⟂ρ(s)‖2]dρ, =- [(r^2+a^2)B_ρ(s)^2+a^2\| P_s T_ρ(s)\|^2 ]\,dρ, (A.17) where (A.9) was used to decompose =Bs+s⟂T=Bs+ P_s T. Finally, r2+a2=μ(q,δ),a2=μ(q,δ)+δ2,r^2+a^2=μ(q,δ), a^2= μ(q,δ)+δ2, which proves (A.12). ∎ For fixed q≠0q≠ 0, the transport share a2/(a2+r2)=12(1+δ/δ2+4q2)a^2/(a^2+r^2)= 12(1+δ/ δ^2+4q^2) increases from 0 to 11 as δ ranges from −∞-∞ to +∞+∞. The reaction mobility depends on |δ||δ|, whereas the transport mobility distinguishes its sign. Remark A.2 (Symmetric weight decay). Consider the regularized objective ℒλwd(a,w)=ℒ(a,w)+λwd2(a2+‖w‖2),λwd≥0.L_ _ wd(a,w)=L(a,w)+ _ wd2 (a^2+\|w\|^2 ), _ wd≥ 0. The raw flow becomes a˙=−rB−λwda,w˙=−a−λwdw. a=-rB- _ wda, w=-aT- _ wdw. On every regular neuron, the marked coordinates satisfy the exact identities q˙=−μ(q,δ)B−2λwdq,s˙=−χ(q,δ)s⟂,δ˙=−2λwdδ. q=-μ(q,δ)B-2 _ wdq, s=-χ(q,δ) P_s T, δ=-2 _ wdδ. (A.18) Hence δ(t)=δ(0)e−2λwdtδ(t)=δ(0)e^-2 _ wdt: symmetric weight decay preserves the sign while eroding the gauge magnitude. On the O(D−1)O(D^-1) positive-gauge capture scale, fixed λwd _ wd changes the mark by a relative O(λwd/D)O( _ wd/D) amount. On the O(D)O(D) negative-gauge specialization scale, the decay is perturbative only when λwdD _ wdD is small. This calculation specifies the weight-decay boundary of the unregularized mechanism. A.2 Node-gauge mobility beyond two layers The following proposition isolates the architecture-level calculation behind the marked two-layer mobility. It is intentionally local in time: Euclidean gradient flow is not gauge equivariant, so two gauge-related initial states need not remain gauge related after training begins. Proposition A.3 (Node-gauge mobility in arbitrary-depth ReLU networks). Consider a finite feedforward ReLU computation graph with independently parameterized edges and no normalization layers. Fix a hidden node with preactivation z=u⊤h+b,z=u h+b, outgoing parameter vector v, and all remaining parameters collected in ξ. Write the incoming affine parameter as u¯:=(u,b) u:=(u,b); the bias-free case is b=0b=0. Assume u¯≠0 u≠ 0 and define, for ρ>0ρ>0, u¯(ρ):=u¯/ρ,v(ρ):=ρv,Θ(ρ):=gρΘ:=(u¯(ρ),v(ρ),ξ). u^(ρ):= u/ρ, v^(ρ):=ρ v, ^(ρ):=g_ρ :=( u^(ρ),v^(ρ),ξ). Then the realized network function is invariant: fgρΘ=fΘ.f_g_ρ =f_ . (A.19) Let ℒL be any differentiable loss that depends on the parameters only through the realized network outputs. At every differentiability point, ∇u¯(ρ)ℒ(Θ(ρ))=ρ∇u¯ℒ(Θ),∇v(ρ)ℒ(Θ(ρ))=ρ−1∇vℒ(Θ),∇ξℒ(Θ(ρ))=∇ξℒ(Θ). _ u^(ρ)L( ^(ρ))=ρ\, _ uL( ), _v^(ρ)L( ^(ρ))=ρ^-1 _vL( ), _ξL( ^(ρ))= _ξL( ). (A.20) Under Euclidean gradient flow, write u¯^=u¯/‖u¯‖ u= u/\| u\| and, when v≠0v≠ 0, v^=v/‖v‖ v=v/\|v\|. The instantaneous normalized-direction velocities at the two gauge-related states satisfy dtu¯(ρ)^|Θ(ρ)=ρ2dtu¯^|Θ,dtv(ρ)^|Θ(ρ)=ρ−2dtv^|Θ. . \,d\,dt u^(ρ) |_ ^(ρ)=ρ^2 . \,d\,dt u |_ , . \,d\,dt v^(ρ) |_ ^(ρ)=ρ^-2 . \,d\,dt v |_ . (A.21) Thus node rescaling reallocates Euclidean mobility between incoming-affine feature motion and outgoing-combination motion. If both normalized direction velocities are nonzero and ρ≠1ρ≠ 1, their joint two-block velocity is not a scalar multiple of the original one. Proof. Positive homogeneity gives σ((u/ρ)⊤h+b/ρ)=ρ−1σ(u⊤h+b).σ\! ((u/ρ) h+b/ρ )=ρ^-1σ(u h+b). Multiplying every outgoing edge by ρ therefore leaves each downstream preactivation, and hence the complete network output, unchanged. This proves (A.19). The identity ℒ(u¯/ρ,ρv,ξ)=ℒ(u¯,v,ξ)L( u/ρ,ρ v,ξ)=L( u,v,ξ) holds for all (u¯,v,ξ)( u,v,ξ) in every differentiability region. Differentiating it with respect to the three parameter blocks yields (A.20). Under Euclidean gradient flow, u¯˙(ρ)=−∇u¯(ρ)ℒ(Θ(ρ))=ρu¯˙,v˙(ρ)=−∇v(ρ)ℒ(Θ(ρ))=ρ−1v˙. u^(ρ)=- _ u^(ρ)L( ^(ρ))=ρ u, v^(ρ)=- _v^(ρ)L( ^(ρ))=ρ^-1 v. For any nonzero vector z, dtz‖z‖=1‖z‖(I−z^z^⊤)z˙. \,d\,dt z\|z\|= 1\|z\| (I- z z ) z. Since ‖u¯(ρ)‖=‖u¯‖/ρ\| u^(ρ)\|=\| u\|/ρ, ‖v(ρ)‖=ρ‖v‖\|v^(ρ)\|=ρ\|v\|, and normalized directions are unchanged by positive scaling, (A.21) follows. The final claim follows because the two nonzero blocks are multiplied by distinct factors ρ2ρ^2 and ρ−2ρ^-2. ∎ Corollary A.4 (Relation to the marked two-layer gauge). For a two-layer scalar-output ReLU neuron with fixed q=ar>0q=ar>0, start from the balanced lift a=r=qa=r= q and apply reciprocal node gauges ρ and ρ−1ρ^-1. Their imbalance marks are δ+=q(ρ2−ρ−2),δ−=−q(ρ2−ρ−2), _+=q(ρ^2-ρ^-2), _-=-q(ρ^2-ρ^-2), and their input-direction mobilities are ρ2ρ^2 and ρ−2ρ^-2. Hence the mobility ratio is ρ4ρ^4. If D=|δ±|D=| _±| and q is fixed, then ρ2=Θ(D)ρ^2= (D) and ρ4=Θ(D2)ρ^4= (D^2) as D→∞D→∞. Remark A.5 (Local law versus global consequence). Section˜A.2 is independent of data distribution, loss choice, and network depth, but it is an instantaneous statement at gauge-related states. The global specialization times in equation˜8 require uniform control of the residual signal along the trajectory, and the winner/pruning theorem additionally requires the global Gaussian competition basin. Those global consequences are not asserted for arbitrary deep networks. Allowing a jointly scaled node bias in section˜A.2 does not change this boundary: the closed Gaussian theorems in the main text remain bias-free. Remark A.6 (Gauge-equivariant update rules). Let U be a deterministic one-step update map and suppose it is exactly equivariant under the node gauge: (gρΘ)=gρ(Θ)for every admissible (ρ,Θ).U(g_ρ )=g_ρU( ) every admissible (ρ, ). If Θ~0=gρΘ0 _0=g_ρ _0, induction gives Θ~n=gρΘn _n=g_ρ _n for every n, so fΘ~n=fΘnf_ _n=f_ _n throughout training. The continuous-time analogue is a vector field satisfying V(gρΘ)=Dgρ(Θ)V(Θ)V(g_ρ )=Dg_ρ( )V( ). Thus an exactly gauge-equivariant optimizer removes function-level divergence caused solely by the choice of lift. Path-SGD is motivated by a rescaling-invariant path geometry (Neyshabur et al., 2015); the conclusion applies whenever the implemented update satisfies exact equivariance. At function level, a finite network f(x)=∑i=1mqiσ(si⊤x),‖si‖=1,f(x)= _i=1^mq_iσ(s_i x), \|s_i\|=1, has signed distributional curvature Dx2f=∑iqisi⊗siℋd−1↾si⊤x=0.D_x^2f= _iq_i\,s_i s_i\,H^d-1\! _\s_i x=0\. Both the function and its curvature depend on (qi,si)(q_i,s_i) and are blind to δi _i. The mark therefore changes future directional transport without changing the current predictor. A.3 Gaussian ReLU kernel and closed ODE Let x∼(0,Id)x (0,I_d). Fix a teacher f⋆(x)=q⋆σ(s⋆⊤x),q⋆>0,‖s⋆‖=1,f_ (x)=q_ σ(s_ x), q_ >0, \|s_ \|=1, (A.22) and a student f(x)=qσ(s⊤x),q>0,‖s‖=1.f(x)=qσ(s x), q>0, \|s\|=1. Set c=s⊤s⋆c=s s_ . Define the arc-cosine kernel κ(c):=[σ(s⊤x)σ(s⋆⊤x)]=1−c2+(π−arccosc)c2π.κ(c):=E[σ(s x)σ(s_ x)]= 1-c^2+(π- c)c2π. (A.23) It satisfies κ(−1)=0,κ(1)=12,κ′(c)=π−arccosc2π>0(−1<c≤1).κ(-1)=0, κ(1)= 12, κ (c)= π- c2π>0 (-1<c≤ 1). (A.24) Analytical population-gradient formulae for Gaussian teacher–student ReLU models are classical; here they are combined with the conserved gauge mark. (Tian, 2017) Lemma A.7 (Gaussian residual moments). For the one-teacher model, B(s)=q2−q⋆κ(c),B(s)= q2-q_ κ(c), (A.25) and s⟂(s)=−q⋆κ′(c)s⟂s⋆. P_s T(s)=-q_ κ (c) P_s s_ . (A.26) The population loss is ℒ(q,c)=q24+q⋆24−qq⋆κ(c).L(q,c)= q^24+ q_ ^24-q_ κ(c). (A.27) Proof. The identities [σ(s⊤x)2]=1/2E[σ(s x)^2]=1/2 and (A.23) give (A.25) and (A.27). For a tangent perturbation v⟂sv s, v⊤[xs⊤x>0σ(s⋆⊤x)]=dϵ|ϵ=0κ(s+ϵv‖s+ϵv‖⋅s⋆)=κ′(c)v⊤s⋆.v E [x1_\s x>0\σ(s_ x) ]= \,d\,dε |_ε=0κ\! ( s+ε v\|s+ε v\|· s_ )=κ (c)v s_ . The self term is parallel to s, so tangential projection yields (A.26). ∎ Theorem A.8 (Exact gauge-marked specialization ODE). Fix a gauge mark δ∈ℝδ . On the regular region q>0q>0 and −1<c<1-1<c<1, population gradient flow is exactly q˙=−μ(q,δ)(q2−q⋆κ(c)),c˙=χ(q,δ)q⋆κ′(c)(1−c2),δ˙=0. q=-μ(q,δ) ( q2-q_ κ(c) ), c=χ(q,δ)q_ κ (c)(1-c^2), δ=0. (A.28) The loss dissipates according to ℒ˙=−μ(q,δ)(q2−q⋆κ(c))2−μ(q,δ)+δ2q⋆2κ′(c)2(1−c2). L=-μ(q,δ) ( q2-q_ κ(c) )^2- μ(q,δ)+δ2q_ ^2κ (c)^2(1-c^2). (A.29) Proof. Insert (A.25)–(A.26) into theorem˜A.1; use ‖s⟂s⋆‖2=1−c2.\| P_s s_ \|^2=1-c^2. ∎ A.4 Invariant region and global convergence Define h(c):=2q⋆κ(c).h(c):=2q_ κ(c). (A.30) Then the coefficient equation is q˙=−μ(q,δ)2(q−h(c)). q=- μ(q,δ)2 (q-h(c) ). (A.31) Theorem A.9 (Global specialization). Assume q(0)=q0>0,−1<c(0)=c0<1.q(0)=q_0>0, -1<c(0)=c_0<1. Let q¯:=minq0,h(c0)>0,q¯:=maxq0,q⋆. q:= \q_0,h(c_0)\>0, q:= \q_0,q_ \. (A.32) Then the solution of (A.28) exists for all t≥0t≥ 0 and satisfies q¯≤q(t)≤q¯,c0≤c(t)<1. q≤ q(t)≤ q, c_0≤ c(t)<1. (A.33) Moreover, c(t)↗1,q(t)⟶q⋆,ℒ(q(t),c(t))⟶0.c(t) 1, q(t) q_ , (q(t),c(t)) 0. (A.34) Proof. Since h is increasing and 0<h(c)≤q⋆0<h(c)≤ q_ on c∈[c0,1]c∈[c_0,1], the vector field at q=q¯q= q points inward: q=q¯⟹q−h(c)≤q¯−h(c0)≤0⟹q˙≥0.q= q q-h(c)≤ q-h(c_0)≤ 0 q≥ 0. At q=q¯q= q, q=q¯⟹q−h(c)≥q¯−q⋆≥0⟹q˙≤0.q= q q-h(c)≥ q-q_ ≥ 0 q≤ 0. Thus q∈[q¯,q¯]q∈[ q, q]. On this compact interval χ(q,δ)>0χ(q,δ)>0. Since κ′(c)>0κ (c)>0 on (−1,1)(-1,1), c is strictly increasing until it reaches 11, and c=1c=1 is invariant. Let c∞=limtc(t)c_∞= _tc(t). If c∞<1c_∞<1, then on the compact rectangle [q¯,q¯]×[c0,c∞][ q, q]×[c_0,c_∞] the coefficient χ(q,δ)q⋆κ′(c)(1−c2)χ(q,δ)q_ κ (c)(1-c^2) has a strictly positive lower bound, contradicting convergence of c(t)c(t). Hence c∞=1c_∞=1. Finally h(c(t))→q⋆h(c(t))→ q_ . If q≥q⋆+2ηq≥ q_ +2η for large time, then q−h(c)≥ηq-h(c)≥η and (A.31) forces a uniform negative drift; similarly q≤q⋆−2ηq≤ q_ -2η forces a uniform positive drift. Therefore q(t)→q⋆q(t)→ q_ . Equation (A.27) gives ℒ→0L→ 0. ∎ A.5 Same function, opposite gauge, quadratic time-scale separation Fix D>0D>0 and the same function state (q,s)(q,s). The two gauges δ=±Dδ=± D are realized by a±2 a_±^2 =D2+4q2±D2, = D^2+4q^2± D2, r±2 r_±^2 =D2+4q2∓D2, = D^2+4q^2∓ D2, a±r± a_±r_± =q. =q. (A.35) They have identical predictor, loss, switching interface, and signed function curvature. At the same state (q,c)(q,c), their reaction mobility is identical, μ(q,+D)=μ(q,−D)=D2+4q2,μ(q,+D)=μ(q,-D)= D^2+4q^2, (A.36) while the transport mobilities are χ+(q,D)=D2+4q2+D2q,χ−(q,D)=D2+4q2−D2q,χ+χ−=1. _+(q,D)= D^2+4q^2+D2q, _-(q,D)= D^2+4q^2-D2q, _+ _-=1. (A.37) For 0<ε<1−c00< <1-c_0, define the specialization hitting time Tδ(ε):=inft≥0:cδ(t)≥1−ε.T_δ( ):= \t≥ 0:c_δ(t)≥ 1- \. (A.38) Let Ic0(ε):=∫c01−εdc1−c2=12log((2−ε)(1−c0)ε(1+c0)),I_c_0( ):= _c_0^1- \,dc1-c^2= 12 \! ( (2- )(1-c_0) (1+c_0) ), (A.39) and k0:=κ′(c0)>0k_0:=κ (c_0)>0. Theorem A.10 (Quadratic gauge gap in specialization time). Under the assumptions of theorem˜A.9, let D≥q¯D≥ q. For the two trajectories with the same initial (q0,c0)(q_0,c_0) and gauges δ=±Dδ=± D, q¯Dq⋆Ic0(ε) qDq_ I_c_0( ) ≤T+D(ε)≤q¯Dq⋆k0Ic0(ε), ≤ T_+D( )≤ qDq_ k_0I_c_0( ), (A.40) 2Dq¯q⋆Ic0(ε) 2D qq_ I_c_0( ) ≤T−D(ε)≤2Dq¯q⋆k0Ic0(ε). ≤ T_-D( )≤ 2D qq_ k_0I_c_0( ). (A.41) Consequently, 2k0q¯,2D2≤T−D(ε)T+D(ε)≤2q¯,2k0D2. 2k_0 q^,2D^2≤ T_-D( )T_+D( )≤ 2 q^,2k_0D^2. (A.42) In particular, uniformly for ε↓0 0, T+D(ε)=Θ(D−1log1ε),T−D(ε)=Θ(Dlog1ε).T_+D( )= \! (D^-1 1 ), T_-D( )= \! (D 1 ). (A.43) Proof. Because c is strictly increasing, Tδ(ε)=∫c01−εdcχ(qδ(c),δ)q⋆κ′(c)(1−c2).T_δ( )= _c_0^1- \,dcχ(q_δ(c),δ)q_ κ (c)(1-c^2). (A.44) By theorem˜A.9, q¯≤qδ(c)≤q¯ q≤ q_δ(c)≤ q. For D≥q¯D≥ q, Dq¯≤χ+(q,D)≤2Dq¯, D q≤ _+(q,D)≤ 2D q, (A.45) where the upper bound uses D2+4q2≤D+2q2D. D^2+4q^2≤ D+ 2q^2D. Similarly, q¯2D≤χ−(q,D)≤q¯D, q2D≤ _-(q,D)≤ qD, (A.46) using χ−=2q/(D2+4q2+D) _-=2q/( D^2+4q^2+D). Finally, k0≤κ′(c)≤12(c0≤c≤1).k_0≤κ (c)≤ 12 (c_0≤ c≤ 1). Insert these bounds into (A.44) to obtain (A.40)–(A.41). Taking ratios gives (A.42); (A.39) is asymptotic to 12log(1/ε) 12 (1/ ). ∎ Corollary A.11 (The two gauges do not differ by a global clock). At every state with q2−q⋆κ(c)≠0,−1<c<1, q2-q_ κ(c)≠ 0, -1<c<1, the vector fields for δ=+Dδ=+D and δ=−Dδ=-D are not collinear. Therefore their phase-plane trajectories cannot be matched by a scalar learning-rate or global time rescaling. Proof. At a common state, the q-components of the two vector fields are equal by (A.36) and nonzero by assumption. Any collinearity factor must therefore equal one. But the c-components differ because χ+χ−=1 _+ _-=1 and, for D>0D>0, χ+≠χ− _+≠ _-. Hence the vector fields are not collinear. ∎ A.6 Fast and slow limiting systems The hitting-time theorem is nonasymptotic. The following limits explain the mechanism more finely. Proposition A.12 (Transport-enabled fast limit for δ=+Dδ=+D). Let QD+(τ)=q+D(τ/D),CD+(τ)=c+D(τ/D).Q_D^+(τ)=q_+D(τ/D), C_D^+(τ)=c_+D(τ/D). For every finite τmax _max, sup0≤τ≤τmax‖(QD+(τ),CD+(τ))−(Q+(τ),C+(τ))‖≤CτmaxD2, _0≤τ≤ _max \|(Q_D^+(τ),C_D^+(τ))-(Q_+(τ),C_+(τ)) \|≤ C_ _maxD^2, (A.47) where dQ+dτ=−(Q+2−q⋆κ(C+)),dC+dτ=q⋆Q+κ′(C+)(1−C+2). \,dQ_+\,dτ=- ( Q_+2-q_ κ(C_+) ), \,dC_+\,dτ= q_ Q_+κ (C_+)(1-C_+^2). (A.48) Proof. On the invariant rectangle [q¯,q¯]×[c0,1][ q, q]×[c_0,1], μ(q,+D)D=1+O(D−2),χ(q,+D)D=1q+O(D−2) μ(q,+D)D=1+O(D^-2), χ(q,+D)D= 1q+O(D^-2) uniformly with uniformly Lipschitz vector fields. Standard continuous dependence and Gronwall’s inequality give (A.47). ∎ Proposition A.13 (Fast reaction layer and slow transport for δ=−Dδ=-D). Let h(c)=2q⋆κ(c)h(c)=2q_ κ(c) and eD(t)=q−D(t)−h(c−D(t))e_D(t)=q_-D(t)-h(c_-D(t)). There is a constant C independent of D such that |eD(t)|≤|eD(0)|e−Dt/2+CD2(t≥0).|e_D(t)|≤|e_D(0)|e^-Dt/2+ CD^2 (t≥ 0). (A.49) On the fast scale τ=Dtτ=Dt, (q−D(τ/D),c−D(τ/D))(q_-D(τ/D),c_-D(τ/D)) converges on compact intervals to dQfdτ=−(Qf2−q⋆κ(c0)),dCfdτ=0, \,dQ_f\,dτ=- ( Q_f2-q_ κ(c_0) ), \,dC_f\,dτ=0, (A.50) so Qf(τ)=h(c0)+(q0−h(c0))e−τ/2,Cf(τ)=c0.Q_f(τ)=h(c_0)+ (q_0-h(c_0) )e^-τ/2, C_f(τ)=c_0. (A.51) On the slow scale σ=t/Dσ=t/D, CD−(σ):=c−D(Dσ)C_D^-(σ):=c_-D(Dσ) converges uniformly on compact intervals to the solution of dC−dσ=2q⋆2κ(C−)κ′(C−)(1−C−2),C−(0)=c0. \,dC_-\,dσ=2q_ ^2κ(C_-)κ (C_-)(1-C_-^2), C_-(0)=c_0. (A.52) For every σ0>0 _0>0, supσ0≤σ≤σmax|q−D(Dσ)−2q⋆κ(C−(σ))|=O(D−2). _ _0≤σ≤ _max |q_-D(Dσ)-2q_ κ(C_-(σ)) |=O(D^-2). (A.53) Proof. Set A(c)=q⋆κ′(c)(1−c2)A(c)=q_ κ (c)(1-c^2). Since q∈[q¯,q¯]q∈[ q, q], |h′(c)|≤q⋆,0≤A(c)≤q⋆2,0≤χ(q,−D)≤q¯D.|h (c)|≤ q_ , 0≤ A(c)≤ q_ 2, 0≤χ(q,-D)≤ qD. Differentiating eD=q−h(c)e_D=q-h(c) gives e˙D=−μ(q,−D)2eD−h′(c)χ(q,−D)A(c). e_D=- μ(q,-D)2e_D-h (c)χ(q,-D)A(c). Since μ≥Dμ≥ D, variation of constants yields (A.49). The fast-scale limit follows from μ(q,−D)D=1+O(D−2),χ(q,−D)D=O(D−2). μ(q,-D)D=1+O(D^-2), χ(q,-D)D=O(D^-2). For the slow scale, the exact identity |Dχ(q,−D)−q|=4|q|3(D2+4q2+D)2≤q¯,3D2 |Dχ(q,-D)-q |= 4|q|^3 ( D^2+4q^2+D )^2≤ q^,3D^2 (A.54) shows dσCD−(σ)=q−D(Dσ)A(CD−(σ))+O(D−2). \,d\,dσC_D^-(σ)=q_-D(Dσ)A(C_D^-(σ))+O(D^-2). The transient term in (A.49) has slow-time integral O(D−2)O(D^-2), while the remaining tracking error is O(D−2)O(D^-2). Gronwall’s inequality therefore gives uniform convergence of CD−C_D^- to (A.52); the coefficient statement follows from (A.49). ∎ Appendix B Global ownership under exact duplication Let X∼(0,Id)X (0,I_d) and let the teacher be f⋆(x)=σ(u⊤x),‖u‖=1.f_ (x)=σ(u x), \|u\|=1. The student network is f(x)=∑i=1mqi(t)σ(si(t)⊤x),‖si(t)‖=1.f(x)= _i=1^mq_i(t)σ(s_i(t) x), \|s_i(t)\|=1. We use population square loss ℒ=12[(f(X)−f⋆(X))2].L= 12E [(f(X)-f_ (X))^2 ]. All students begin from the same visible state, qi(0)=1m,si(0)=s0,∠(s0,u)=ϕ:=π4.q_i(0)= 1m, s_i(0)=s_0, (s_0,u)=φ:= π4. (B.1) Choose one index k and assign δk=+D,δj=−D(j≠k), _k=+D, _j=-D (j≠ k), where δi=ai2−‖wi‖2 _i=a_i^2-\|w_i\|^2 is the conserved gauge mark and qi=ai‖wi‖q_i=a_i\|w_i\|. The underlying positive factors are uniquely reconstructed from ai2=D2+4qi2+δi2,‖wi‖2=D2+4qi2−δi2.a_i^2= D^2+4q_i^2+ _i2, \|w_i\|^2= D^2+4q_i^2- _i2. Thus every choice of the selected index gives the same initial function and loss. Set M:=m−1.M:=m-1. By permutation symmetry, the M negative-gauge students remain identical. Write q=qk,p=qj(j≠k),P:=Mp,q=q_k, p=q_j\ (j≠ k), P:=Mp, and let θ and ψ be the signed angles of the winner and the common loser direction relative to u in the invariant plane spanu,s0span\u,s_0\. The population function is exactly f(x)=qσ(sθ⊤x)+Pσ(sψ⊤x).f(x)=qσ(s_θ x)+Pσ(s_ψ x). Define the angular ReLU kernel g(α)=sinα+(π−α)cosα2π,ω(α):=−g′(α)=(π−α)sinα2π.g(α)= α+(π-α) α2π, ω(α):=-g (α)= (π-α) α2π. This is the angular form of the kernel in Appendix A: g(α)=κ(cosα)g(α)=κ( α) for α∈[0,π]α∈[0,π]. The reduced loss is ℰ(q,P,θ,ψ)=14(q2+P2+1)+qPg(|ψ−θ|)−qg(|θ|)−Pg(|ψ|). E(q,P,θ,ψ)= 14(q^2+P^2+1)+qPg(|ψ-θ|)-qg(|θ|)-Pg(|ψ|). (B.2) This identity already absorbs all (M2) M2 loser–loser interactions: because their directions coincide, their self and pair terms sum to P2/4P^2/4. Let μ+(q)=D2+4q2,μ−(p)=D2+4p2, _+(q)= D^2+4q^2, _-(p)= D^2+4p^2, χ+(q)=μ+(q)+D2q,χ−(p)=2pμ−(p)+D. _+(q)= _+(q)+D2q, _-(p)= 2p _-(p)+D. Proposition B.1 (Exact multiplicity-reduced marked flow). On the duplicate-loser manifold, ordinary population gradient flow is exactly q˙ q =−μ+(q)∂qℰ, =- _+(q)\, _q E, P˙ P =−Mμ−(P/M)∂Pℰ, =-M _-(P/M)\, _P E, (B.3) θ˙ θ =−χ+(q)q∂θℰ, =- _+(q)q\, _θ E, ψ˙ ψ =−χ−(P/M)P∂ψℰ, =- _-(P/M)P\, _ψ E, (B.4) where the last expression has its continuous extension at P=0P=0. Moreover, dtℰ= \,d\,dt E= −μ+(q)(∂qℰ)2−Mμ−(P/M)(∂Pℰ)2 - _+(q)( _q E)^2-M _-(P/M)( _P E)^2 −χ+(q)q(∂θℰ)2−χ−(P/M)P(∂ψℰ)2. - _+(q)q( _θ E)^2- _-(P/M)P( _ψ E)^2. (B.5) Proof. For one loser, the coefficient gradient is ∂pjℒ=P2+qg(|ψ−θ|)−g(|ψ|)=∂Pℰ. _p_jL= P2+qg(|ψ-θ|)-g(|ψ|)= _P E. Therefore P˙=Mp˙ P=M p gives the second equation in (B.3). The common angular derivative satisfies ∂ψℰ=∑j≠k∂ψjℒ, _ψ E= _j≠ k _ _jL, and each summand is 1/M1/M of the total. Dividing the individual angular gradient by the individual functional coefficient p=P/Mp=P/M gives the final equation in (B.4). The selected-unit equations follow from the exact marked reduction in theorem˜A.1. Pairing the four state velocities with the four derivatives of (B.2) yields (B.5). ∎ B.1 The multiplicity-weighted fast capture flow Set τ=Dt.τ=Dt. For every fixed M and on compact sets with q>0q>0, μ+(q)D=1+O(D−2),μ−(P/M)D=1+O(D−2), _+(q)D=1+O(D^-2), _-(P/M)D=1+O(D^-2), χ+(q)Dq=1q2+O(D−2),χ−(P/M)DP=O(D−2). _+(q)Dq= 1q^2+O(D^-2), _-(P/M)DP=O(D^-2). Hence the common loser direction freezes at leading order: ψ(τ)≡ϕ.ψ(τ)≡φ. Define EM(q,P,θ):=14(q2+P2+1)+qPg(ϕ−θ)−qg(|θ|)−Pg(ϕ),E_M(q,P,θ):= 14(q^2+P^2+1)+qPg(φ-θ)-qg(|θ|)-Pg(φ), (B.6) for −ϕ<θ<ϕ-φ<θ<φ. The fast system is q′=−∂qEM,P′=−M∂PEM,θ′=−q−2∂θEM.q =- _qE_M, P =-M _PE_M, θ =-q^-2 _θE_M. (B.7) with initial condition q(0)=q0:=1M+1,P(0)=P0:=M+1,θ(0)=ϕ.q(0)=q_0:= 1M+1, P(0)=P_0:= MM+1, θ(0)=φ. (B.8) The subscript M records the metric weight, not a change in the energy landscape. Indeed, dτEM=−(∂qEM)2−M(∂PEM)2−q−2(∂θEM)2. \,d\,dτE_M=-( _qE_M)^2-M( _PE_M)^2-q^-2( _θE_M)^2. (B.9) For later use define d:=ϕ−θ,Z:=qg(d)−g(ϕ),W:=Pg(d)−g(|θ|).d:=φ-θ, Z:=qg(d)-g(φ), W:=Pg(d)-g(|θ|). The coefficient equations become q′=−(q2+W),P′=−M(P2+Z).q =- ( q2+W ), P =-M ( P2+Z ). (B.10) The scalar arc-cosine inequalities The proof uses the same four strict arc-cosine inequalities as the two-student basin theorem. For 0≤x≤ϕ0≤ x≤φ, g(x)g(ϕ−x)−ω(x)ω(ϕ−x) g(x)g(φ-x)-ω(x)ω(φ-x) ≤12g(ϕ), ≤ 12g(φ), (B.11) 2g(ϕ)g(ϕ−x) 2g(φ)g(φ-x) ≤g(x), ≤ g(x), (B.12) g(x)g(ϕ+x)+ω(x)ω(ϕ+x) g(x)g(φ+x)+ω(x)ω(φ+x) ≤12g(ϕ), ≤ 12g(φ), (B.13) 2g(ϕ)g(ϕ+x) 2g(φ)g(φ+x) <g(x)(x>0). <g(x) (x>0). (B.14) The first three follow from explicit derivative factorizations and endpoint values; the strict negative-angle critical-point exclusion uses the one-dimensional interval certificate reproduced in Appendix B.5. Importantly, none of these inequalities depends on M. B.2 Finite entrance for arbitrary multiplicity For M≤3M≤ 3, the initial point already satisfies W(0)≤0W(0)≤ 0. For larger multiplicities, W(0)=M2(M+1)−g(ϕ)>0,W(0)= M2(M+1)-g(φ)>0, so the two-student invariant sector cannot be invoked directly. The next lemma supplies the finite entrance argument needed for arbitrary fixed multiplicity. Lemma B.2 (Finite entrance into the canonical basin). Let (q,P,θ)(q,P,θ) solve (B.7)–(B.8). Put A:=12−g(ϕ),ω0:=ω(ϕ/2),γent:=A22ω02.A:= 12-g(φ), _0:=ω(φ/2), _ ent:= A^22 _0^2. For ϕ=π/4φ=π/4, 0<γent<0.267.0< _ ent<0.267. There is a finite time τ∗≥0 _ ≥ 0 such that W(τ∗)=0,Z(τ∗)≤0,ϕ/2≤θ(τ∗)<ϕ,W( _ )=0, Z( _ )≤ 0, φ/2≤θ( _ )<φ, and τ∗≤q0[W(0)]+ω02,q(τ∗)≥(1−γent)q0. _ ≤ q_0[W(0)]_+ _0^2, q( _ )≥(1- _ ent)q_0. (B.15) Immediately after τ∗ _ , one has W<0W<0. Proof. If W(0)≤0W(0)≤ 0, set τ∗=0 _ =0. Assume W(0)>0W(0)>0 and work until the first zero of W. First, P cannot reach zero while W>0W>0, and at P=1P=1 the vector field points inward because P2+Z≥12−g(ϕ)>0. P2+Z≥ 12-g(φ)>0. Thus 0<P≤P0<10<P≤ P_0<1. Similarly q<1q<1. At a hypothetical first point with Z=0Z=0 and 0≤θ≤ϕ0≤θ≤φ, (B.11) gives Z′ Z =−g(ϕ)2−Pg(d)2+ω(d)2+g(θ)g(d)−ω(θ)ω(d) =- g(φ)2-P\g(d)^2+ω(d)^2\+g(θ)g(d)-ω(θ)ω(d) ≤0. ≤ 0. Hence Z≤0Z≤ 0 throughout this entrance phase. Because W>0W>0 and P≤1P≤ 1, g(d)>g(θ).g(d)>g(θ). The function g is strictly decreasing, so d<θd<θ, equivalently θ>ϕ/2.θ>φ/2. Furthermore, (B.12) implies P2>g(θ)2g(d)≥g(ϕ). P2> g(θ)2g(d)≥ g(φ). Consequently ∂PEM=P2+Z>qg(d)≥0. _PE_M= P2+Z>qg(d)≥ 0. For 0≤θ≤ϕ0≤θ≤φ, set S:=Pω(d)+ω(θ).S:=Pω(d)+ω(θ). Direct differentiation gives W′=−M(P2+Z)g(d)−S2q≤−ω02q.W =-M ( P2+Z )g(d)- S^2q≤- _0^2q. (B.16) Thus W is strictly decreasing. Since q′=−(q/2+W)<0q =-(q/2+W)<0, one has q≤q0q≤ q_0, and (B.16) yields τ∗≤q0W(0)ω02. _ ≤ q_0W(0) _0^2. Using −W′≥ω02/q-W ≥ _0^2/q as a change-of-variables estimate, q0−q(τ∗) q_0-q( _ ) =∫0τ∗(q2+W)dτ = _0 _ ( q2+W )\,dτ ≤1ω02∫0W(0)(q2+W)qdW ≤ 1 _0^2 _0^W(0) ( q2+W )q\,\,dW ≤q02ω02(q0W(0)+W(0)2). ≤ q_02 _0^2 (q_0W(0)+W(0)^2 ). Now W(0)=A−q02,W(0)=A- q_02, so q0W(0)+W(0)2=A2−q024≤A2.q_0W(0)+W(0)^2=A^2- q_0^24≤ A^2. This proves the lower bound in (B.15). Finally, (B.16) is strict at W=0W=0, so the trajectory crosses into W<0W<0. ∎ B.3 A multiplicity-independent global basin Define the canonical sector ℬ:=0<q≤1,0≤P≤1,−ϕ<θ<ϕ,Z=qg(ϕ−θ)−g(ϕ)≤0,W=Pg(ϕ−θ)−g(|θ|)≤0.B:= \ array[]c0<q≤ 1, 0≤ P≤ 1, -φ<θ<φ,\\ Z=qg(φ-θ)-g(φ)≤ 0,\\ W=Pg(φ-θ)-g(|θ|)≤ 0 array \. (B.17) Lemma B.3 (Weighted arc-cosine basin). For every M≥1M≥ 1, the sector ℬB is forward invariant under (B.7). Every trajectory in ℬB converges to (q,P,θ)=(1,0,0).(q,P,θ)=(1,0,0). Proof. Only the P-equation differs from the two-student flow, and it is multiplied by the positive factor M. We nevertheless record the sign argument. At q=1q=1, ∂qEM=12+W≥12−g(|θ|)≥0, _qE_M= 12+W≥ 12-g(|θ|)≥ 0, so the vector field points inward. Since W≤0W≤ 0, q′=−q2−W≥−q2,q =- q2-W≥- q2, and therefore q remains positive for every finite time. At P=0P=0, P′=−MZ≥0P =-MZ≥ 0; at P=1P=1, ∂PEM=12+Z≥12−g(ϕ)>0. _PE_M= 12+Z≥ 12-g(φ)>0. The angular faces are inward-pointing: at θ=ϕθ=φ, θ′=−ω(ϕ)/q<0θ =-ω(φ)/q<0, while at θ=−ϕθ=-φ, θ′≥ω(ϕ)−ω(2ϕ)q>0.θ ≥ ω(φ)-ω(2φ)q>0. At Z=0Z=0, (B.11) for θ≥0θ≥ 0 and (B.13) for θ<0θ<0 imply Z′≤0Z ≤ 0. At W=0W=0, (B.12) or (B.14) implies ∂PEM=P2+Z≥0, _PE_M= P2+Z≥ 0, and the exact identity W′=−M(∂PEM)g(ϕ−θ)−(θ,P)2qW =-M( _PE_M)g(φ-θ)- S(θ,P)^2q shows W′≤0W ≤ 0, where (θ,P)=Pω(ϕ−θ)+ω(θ),θ≥0,Pω(ϕ−θ)−ω(−θ),θ<0. S(θ,P)= casesPω(φ-θ)+ω(θ),&θ≥ 0,\\ Pω(φ-θ)-ω(-θ),&θ<0. cases Thus ℬB is invariant. The energy identity is EM′=−(∂qEM)2−M(∂PEM)2−(θ,P)2.E_M =-( _qE_M)^2-M( _PE_M)^2- S(θ,P)^2. (B.18) The stationary equations do not depend on M. There is no stationary point with 0<θ<ϕ0<θ<φ because >0 S>0. At θ=0θ=0, stationarity forces P=0P=0 and q=1q=1. For θ=−x∈(−ϕ,0)θ=-x∈(-φ,0), coefficient stationarity yields P^(x)=12g(ϕ)−g(ϕ+x)g(x)14−g(ϕ+x)2. P(x)= 12g(φ)-g(φ+x)g(x) 14-g(φ+x)^2. The scalar certificate in Appendix B.5 proves P^(x)ω(ϕ+x)<ω(x),0<x≤ϕ, P(x)ω(φ+x)<ω(x), 0<x≤φ, so ≠0 S≠ 0 and no negative-angle stationary point exists. It remains to exclude an omega-limit with q=0q=0. Along any omega-limit sequence, (B.18) forces ∂qEM→0,∂PEM→0,→0. _qE_M→ 0, _PE_M→ 0, S→ 0. If q=0q=0, then ∂PEM=0 _PE_M=0 gives P=2g(ϕ)P=2g(φ). For θ=−x<0θ=-x<0, the equation ∂qEM=0 _qE_M=0 would require 2g(ϕ)g(ϕ+x)=g(x),2g(φ)g(φ+x)=g(x), contradicting (B.14). For θ≥0θ≥ 0, one has =Pω(ϕ−θ)+ω(θ)>0. S=Pω(φ-θ)+ω(θ)>0. Hence no zero-q omega-limit exists. The unique invariant subset of the zero-dissipation set is therefore (1,0,0)(1,0,0), and LaSalle’s principle gives the claim. ∎ Theorem B.4 (Global capture in the multiplicity-weighted fast flow). For every fixed M≥1M≥ 1, the solution of (B.7)–(B.8) satisfies q(τ)→1,P(τ)→0,θ(τ)→0.q(τ)→ 1, P(τ)→ 0, θ(τ)→ 0. Moreover, it enters the sector ℬB after the finite entrance time of Lemma B.2. Proof. Lemma B.2 gives a point in ℬB with q bounded away from zero. Lemma B.3 then gives global convergence. ∎ B.4 Return to the finite-gauge population flow Proposition B.5 (Finite-D approximation). Fix m and T<∞T<∞. Let (qD,PD,θD,ψD)(q_D,P_D, _D, _D) solve the exact reduced flow (B.3)–(B.4) in fast time, and let (q,P,θ)(q,P,θ) solve (B.7). Then sup0≤τ≤T(|qD−q|+|PD−P|+|θD−θ|+|ψD−ϕ|)≤Cm,TD2. _0≤τ≤ T (|q_D-q|+|P_D-P|+| _D-θ|+| _D-φ| )≤ C_m,TD^2. Proof. By Theorem B.4, the fast trajectory stays in a compact set with q bounded away from zero on [0,T][0,T]. On that set, μ+(q)D=1+O(D−2),μ−(P/M)D=1+O(D−2), _+(q)D=1+O(D^-2), _-(P/M)D=1+O(D^-2), χ+(q)Dq=1q2+O(D−2),χ−(P/M)DP=O(D−2), _+(q)Dq= 1q^2+O(D^-2), _-(P/M)DP=O(D^-2), uniformly with one derivative. Hence the exact fast-time vector field equals the limiting vector field plus a C1C^1 perturbation of size CmD−2C_mD^-2. Gronwall’s inequality proves the estimate. ∎ Local winner coercivity At the zero-loss winner state q=1,θ=0,P=0,ψ=ϕ,q=1, θ=0, P=0, ψ=φ, the common loser direction is functionally invisible. The functional transverse derivative is spanned by [u⊤x]+,u⊤x>0v⊤x,[sϕ⊤x]+,[u x]_+, 1_\u x>0\v x, [s_φ x]_+, where v⟂uv u lies in the invariant plane. These functions are linearly independent in Gaussian L2L^2. Lemma B.6 (Symmetric winner-tube coercivity). For every fixed m, there is a neighborhood mV_m of the winner manifold and constants cm,Cm>0c_m,C_m>0, independent of D, such that cm((q−1)2+θ2+P2) c_m ((q-1)^2+θ^2+P^2 ) ≤ℰ≤Cm((q−1)2+θ2+P2), ≤ E≤ C_m ((q-1)^2+θ^2+P^2 ), (B.19) (∂qℰ)2+(∂Pℰ)2+q−2(∂θℰ)2 ( _q E)^2+( _P E)^2+q^-2( _θ E)^2 ≥cmℰ. ≥ c_m E. (B.20) Furthermore, |∂ψℰ|≤Cm|P|ℰon m.| _ψ E|≤ C_m|P| E V_m. (B.21) Proof. Positive definiteness of the Gram matrix of the three displayed tangent functions gives (B.19) by Taylor expansion of the finite-dimensional Gaussian kernel loss. The same Hessian gives the local gradient inequality (B.20). Since ∂ψℰ _ψ E vanishes identically at P=0P=0 and its remaining factor vanishes at the winner state, another Taylor expansion gives (B.21). ∎ Theorem B.7 (Global gauge-selected capture and pruning). Fix m≥2m≥ 2 and the duplicate initialization (B.1). Select any index k, assign δk=+D _k=+D, and assign δj=−D _j=-D for every j≠kj≠ k. There exist constants D0(m),Cm,cm>0D_0(m), C_m, c_m>0 and a winner tube mV_m, all independent of D, such that for every D≥D0(m)D≥ D_0(m): 1. the ordinary ReLU population-gradient trajectory enters mV_m at a time tcap≤CmD−1;t_cap≤ C_mD^-1; 2. after entry, ℒ(t)≤ℒ(tcap)e−cmD(t−tcap);L(t) (t_cap)e^-c_mD(t-t_cap); 3. the selected student globally captures the teacher and all redundant coefficients are pruned: qk(t)→1,sk(t)→u,qj(t)→0(j≠k);q_k(t)→ 1, s_k(t)→ u, q_j(t)→ 0 (j≠ k); 4. every loser has post-capture directional displacement ∫tcap∞‖s˙j(t)‖dt≤CmD−2,j≠k. _t_cap^∞\| s_j(t)\|\,dt≤ C_mD^-2, j≠ k. Permuting the gauge marks permutes the global winner while leaving the complete initial predictor unchanged. Proof. Let M=m−1M=m-1. Theorem B.4 implies that the limiting fast trajectory enters a strictly interior subset of mV_m after a finite fast time TmT_m. Proposition B.4 then places the exact finite-D trajectory in mV_m by physical time (Tm+1)/D(T_m+1)/D for all large D. Inside mV_m, exact dissipation (B.5), the bounds μ+(q)≥D,Mμ−(P/M)≥D,χ+(q)q≥Dq2, _+(q)≥ D, M _-(P/M)≥ D, _+(q)q≥ Dq^2, and Lemma B give −ℰ˙≥cmDℰ.- E≥ c_mD E. This proves exponential contraction. The local loss equivalence then yields q→1,θ→0,P→0.q→ 1, θ→ 0, P→ 0. Since every loser coefficient equals p=P/Mp=P/M, each qj→0q_j→ 0. For the common loser direction, |χ−(P/M)|≤|P|MD. | _-(P/M) |≤ |P|MD. Combining this with (B.21) and (B.19) gives |ψ˙|≤CmDℰ.| ψ|≤ C_mD E. Integrating the exponential loss bound produces ∫tcap∞|ψ˙|dt≤CmD−2. _t_cap^∞| ψ|\,dt≤ C_mD^-2. All losers share this direction by exact exchange symmetry. Finally, the equations are equivariant under permutation of student labels, whereas the initial function depends only on the common visible state; permuting the unique positive gauge therefore permutes the selected winner without changing the initial predictor. ∎ Remark B.8 (Dependence on multiplicity). The theorem is valid for every fixed m, but its constants may depend on m. In particular, the initial selected coefficient is 1/m1/m, the fast-flow metric weight is m−1m-1, and the time needed to enter a fixed winner tube need not remain bounded as m→∞m→∞. The statement is finite-width and does not take an m→∞m→∞ limit. B.5 Scalar arc-cosine certificate The only non-elementary scalar sign used in Lemma B.3 is the exclusion of a negative-angle stationary state. Put P^(x)=12g(ϕ)−g(ϕ+x)g(x)14−g(ϕ+x)2, P(x)= 12g(φ)-g(φ+x)g(x) 14-g(φ+x)^2, and define F(x)=(14−g(ϕ+x)2)ω(x)−(12g(ϕ)−g(ϕ+x)g(x))ω(ϕ+x).F(x)= ( 14-g(φ+x)^2 )ω(x)- ( 12g(φ)-g(φ+x)g(x) )ω(φ+x). The included script scripts/certify_kernel.py evaluates the exact expression for F′F with outward-rounded 80-decimal interval arithmetic on 2048 equal subintervals of [0,π/4][0,π/4]. The certified lower bound stored in data/kernel_certificate.json is strictly positive. Therefore F(x)>0F(x)>0 for x>0x>0, which is equivalent to P^(x)ω(ϕ+x)<ω(x). P(x)ω(φ+x)<ω(x). The multiplicity M never enters this scalar certificate. B.6 A convenient closed-form entrance constant For reference, A=12−g(π/4),ω0=ω(π/8),A= 12-g(π/4), _0=ω(π/8), and γent=A22ω02=0.26678…<1. _ ent= A^22 _0^2=0.26678…<1. Thus the entrance argument retains at least 73.3%73.3\% of the initial winner coefficient before the canonical invariant sector is reached, uniformly over all multiplicities. This uniform fraction enables the arbitrary-fixed-m extension even though the aggregate redundant coefficient approaches one as m grows. Appendix C Robustness and discrete gradient descent Let X∼(0,Id)X (0,I_d) and let the teacher be f⋆(x)=[u⊤x]+,‖u‖=1.f_ (x)=[u x]_+, \|u\|=1. The student has m≥2m≥ 2 raw ReLU units, f(x)=∑i=1mqi[si⊤x]+,‖si‖=1,f(x)= _i=1^mq_i[s_i x]_+, \|s_i\|=1, trained by population square-loss gradient flow. Each functional coefficient and direction arise from ordinary parameters (ai,wi)(a_i,w_i) through qi=ai‖wi‖,si=wi‖wi‖,δi=ai2−‖wi‖2.q_i=a_i\|w_i\|, s_i= w_i\|w_i\|, _i=a_i^2-\|w_i\|^2. The marks δi _i are conserved. We designate unit 11 as the programmed winner and set δ1=+D,δj=−D(j≥2). _1=+D, _j=-D (j≥ 2). At the exactly duplicate visible initialization, qi(0)=1m,si(0)=s0,∠(s0,u)=ϕ,q_i(0)= 1m, s_i(0)=s_0, (s_0,u)=φ, the initial predictor is independent of the choice of winner. The exact duplicate model provides the cleanest same-predictor comparison and the unconditional global selection theorem. This appendix develops four complementary extensions of that result: a certified interval of initial angles, explicit sufficient dependence on the fixed multiplicity m, stability of finite-time selection under visible perturbations, and a small-step full-batch gradient-descent analogue. For fully nonsymmetric redundant dictionaries, long-time coefficient-wise pruning is stated under explicit transversality and active-frame conditions that rule out functional cancellation modes. For α∈[0,π]α∈[0,π], write g(α)=sinα+(π−α)cosα2π,ω(α)=(π−α)sinα2π=−g′(α).g(α)= α+(π-α) α2π, ω(α)= (π-α) α2π=-g (α). On the duplicate-loser manifold, put M=m−1M=m-1, let q be the winner coefficient, P the aggregate loser coefficient, and θ,ψθ,ψ their signed angles from u. In fast time τ=Dtτ=Dt, the limiting capture flow is q′=−∂qEϕ,P′=−M∂PEϕ,θ′=−q−2∂θEϕ,ψ≡ϕ,q =- _qE_φ, P =-M _PE_φ, θ =-q^-2 _θE_φ, ψ≡φ, (C.1) where Eϕ(q,P,θ)=14(q2+P2+1)+qPg(ϕ−θ)−qg(|θ|)−Pg(ϕ).E_φ(q,P,θ)= 14(q^2+P^2+1)+qPg(φ-θ)-qg(|θ|)-Pg(φ). (C.2) The exact finite-D marked flow differs from (C.1) by O(D−2)O(D^-2) on compact regular sets. C.1 Certified initial-angle basin The global basin argument uses four arc-cosine inequalities. The first result shows that the mechanism is not tied to the reference angle ϕ=π/4φ=π/4. Theorem C.1 (Certified angle basin). For every ϕ∈Iϕ:=[0.77,0.90],φ∈ I_φ:=[0.77,0.90], all scalar inequalities used in the multiplicity-weighted basin proof hold uniformly. Consequently, for every fixed m≥2m≥ 2 the fast flow (C.1), initialized at q(0)=1m,P(0)=m−1m,θ(0)=ϕ,q(0)= 1m, P(0)= m-1m, θ(0)=φ, converges to (q,P,θ)=(1,0,0)(q,P,θ)=(1,0,0). The entrance lemma retains at least 1−γent⋆>0.65881- _ ent >0.6588 of the initial winner coefficient, uniformly over m and ϕ∈Iϕφ∈ I_φ. Proof. For 0≤x≤ϕ0≤ x≤φ, the three non-strict inequalities admit elementary derivative factorizations. If R1,R2,R3R_1,R_2,R_3 denote their nonnegative residuals, then R1′(x)=(ϕ−2x)sinxsin(ϕ−x)2π2(0≤x≤ϕ/2),R_1 (x)= (φ-2x) x (φ-x)2π^2 (0≤ x≤φ/2), and R1(x)=R1(ϕ−x)R_1(x)=R_1(φ-x). Further, R2′(x)=−ω(x)−2g(ϕ)ω(ϕ−x)<0,R2(ϕ)=0,R_2 (x)=-ω(x)-2g(φ)ω(φ-x)<0, R_2(φ)=0, while R3′(x)=(2π−ϕ−2x)sinxsin(ϕ+x)2π2≥0,R3(0)=0.R_3 (x)= (2π-φ-2x) x (φ+x)2π^2≥ 0, R_3(0)=0. The remaining strict negative-angle inequality, the negative-angle stationary-point exclusion, the inward angular-face margin, and the entrance constants were certified by outward-rounded interval arithmetic on IϕI_φ; see data/angle_interval_certificate.json. The certified margins include infK4>0.20727,infF′>0.01659,infω(ϕ)−ω(2ϕ)>0.00780. K_4>0.20727, F >0.01659, \ω(φ)-ω(2φ)\>0.00780. Finally, γent(ϕ)=(1/2−g(ϕ))22ω(ϕ/2)2≤0.34116. _ ent(φ)= (1/2-g(φ))^22ω(φ/2)^2≤ 0.34116. The entrance and invariant-sector arguments in the proof of theorem˜B.7 therefore apply uniformly throughout IϕI_φ. ∎ Remark C.2. The certified interval is not claimed maximal. Its purpose is to replace a single hand-picked angle by a quantitatively nontrivial open basin containing π/4π/4. C.2 Dependence on multiplicity We next separate the part of the capture dynamics that is uniform over fixed multiplicities from the conservative m-dependence introduced by finite-D shadowing. For ϕ∈Iϕφ∈ I_φ, let ℬϕB_φ be the invariant arc-cosine sector used in the proof of theorem˜B.7. Choose a fixed winner tube ⋆(ϕ)=(q,P,θ):(q−1)2+P2+θ2<r⋆2⋐ℬϕV_ (φ)=\(q,P,θ):(q-1)^2+P^2+θ^2<r_ ^2\ _φ with r⋆>0r_ >0 uniform over IϕI_φ. Put Γϕ(q,P,θ)=(∂qEϕ)2+(∂PEϕ)2+q−2(∂θEϕ)2 _φ(q,P,θ)=( _qE_φ)^2+( _PE_φ)^2+q^-2( _θE_φ)^2 when q>0q>0, and use the continuous nonsingular form Γ~ϕ=(∂qEϕ)2+(∂PEϕ)2+ϕ2 _φ=( _qE_φ)^2+( _PE_φ)^2+ S_φ^2 on the closure. The critical-point classification in theorem˜C.1 implies γ⋆:=infϕ∈Iϕinfℬϕ¯∖⋆(ϕ)Γ~ϕ>0. _ := _φ∈ I_φ _ B_φ _ (φ) _φ>0. Let E⋆E_ be a uniform upper bound on the initial energy and let Tent⋆T_ ent be the uniform entrance-time bound supplied by theorem˜C.1. Define T⋆:=Tent⋆+E⋆γ⋆.T_ :=T_ ent + E_ _ . (C.3) Proposition C.3 (Uniform fast capture and explicit finite-gauge threshold). There exist positive constants A0,L0,r⋆,c⋆,C⋆A_0,L_0,r_ ,c_ ,C_ , depending only on IϕI_φ, such that the following holds. For every m≥2m≥ 2 and ϕ∈Iϕφ∈ I_φ, the fast flow enters ⋆(ϕ)V_ (φ) by fast time T⋆T_ , independently of m. On the exact finite-D duplicate-loser flow, the pre-capture vector field has a Lipschitz bound Lm≤L0m2L_m≤ L_0m^2 and differs from the fast vector field by at most A0D−2A_0D^-2. Hence the sufficient threshold D0(m):=⌈(4A0T⋆r⋆)1/2exp(L0T⋆2m2)⌉D_0(m):= ( 4A_0T_ r_ )^1/2 \! ( L_0T_ 2m^2 ) (C.4) ensures entry into the winner tube by physical time tcap≤(T⋆+1)D−1.t_ cap≤(T_ +1)D^-1. Inside the tube, the loss contraction constant can be chosen uniformly in m: ℒ(t)≤ℒ(tcap)e−c⋆D(t−tcap).L(t) (t_ cap)e^-c_ D(t-t_ cap). For the symmetric loser block, the common loser direction satisfies the sharper bound ∫tcap∞‖s˙lose(t)‖dt≤C⋆(m−1)D2. _t_ cap^∞\| s_ lose(t)\|\,dt≤ C_ (m-1)D^2. Proof. The entrance lemma is uniform in m because its time is at most C/mC/m and it retains a fixed fraction of q(0)=1/mq(0)=1/m. Once inside the canonical sector, dτEϕ=−(∂qEϕ)2−M(∂PEϕ)2−ϕ2≤−Γ~ϕ. \,d\,dτE_φ=-( _qE_φ)^2-M( _PE_φ)^2- S_φ^2≤- _φ. Outside ⋆V_ , the right-hand side is at most −γ⋆- _ , proving the uniform fast-time bound (C.3). Before capture, q remains bounded below by c/mc/m; differentiating the polar-coordinate fast field therefore gives Lm≤L0m2L_m≤ L_0m^2. The exact mobility identities give, uniformly on the same compact set, |μ+D−1|≤2D2,M|μ−D−1|≤2D2, | _+D-1 |≤ 2D^2, M | _-D-1 |≤ 2D^2, |χ+Dq−1q2|≤1D2,χ−D|P|≤1(m−1)D2. | _+Dq- 1q^2 |≤ 1D^2, _-D|P|≤ 1(m-1)D^2. Thus the finite-D vector-field defect is at most A0D−2A_0D^-2. Gronwall gives supτ≤T⋆‖zD(τ)−z∞(τ)‖≤A0T⋆D−2eL0m2T⋆, _τ≤ T_ \|z_D(τ)-z_∞(τ)\|≤ A_0T_ D^-2e^L_0m^2T_ , which is below r⋆/4r_ /4 under (C.4). The local Gaussian feature Gram on the winner tube is uniformly positive over IϕI_φ. Exact reaction–transport dissipation and Mμ−≥DM _-≥ D therefore yield the uniform PL inequality −ℒ˙≥c⋆Dℒ.- L≥ c_ DL. Finally, χ−(P/M)≤|P|(m−1)D,|∂ψE|≤C|P|ℒ, _-(P/M)≤ |P|(m-1)D, | _ψE|≤ C|P| L, and local coercivity gives |P|≤Cℒ|P|≤ C L. Integrating the exponential loss bound proves the stated loser-direction estimate. ∎ Remark C.4 (Conservative versus observed m-dependence). The exponential threshold in (C.4) is a direct polar-coordinate Gronwall bound, not a claim of sharpness. In the numerical audit, DtcapDt_ cap decreases from 25.225.2 at m=2m=2 to 18.518.5 at m=256m=256. The theorem makes the sufficient dependence explicit; sharpening D0(m)D_0(m) is left open. C.3 Open-set selection and conditional locking Exact duplication is a clean same-function counterfactual, while finite-time selection should persist under perturbations of the visible state. Long-time pruning is more delicate because zero-loss over-realized ReLU representations contain functionally neutral directions. We therefore distinguish: 1. an unconditional finite-time selection theorem on an open set around duplicate initialization; 2. an asymptotic locking and pruning theorem under explicit quotient-transversality and active redundant-frame conditions. The second statement requires two distinct nondegeneracy controls: quotient transversality separates the winner tangent from the redundant function space, while the active redundant frame controls the actual coefficient vector. For the full m-student system, define the visible/gauge perturbation size Δ0:=maxi(|qi(0)−m−1|+∥si(0)−s0∥)+maxi||δi(0)|D−1|, _0:= _i (|q_i(0)-m^-1|+\|s_i(0)-s_0\| )+ _i | | _i(0)|D-1 |, (C.5) where δ1>0 _1>0 and δj<0 _j<0 for j≥2j≥ 2. Put fred(x,t)=∑j=2mqj(t)[sj(t)⊤x]+.f_ red(x,t)= _j=2^mq_j(t)[s_j(t) x]_+. Let ZD∘(τ)Z_D (τ) denote the exact duplicate reference trajectory in fast time τ=Dtτ=Dt, and let preU_ pre be a compact pre-capture neighborhood containing ZD∘([0,T⋆+1])Z_D ([0,T_ +1]) for every D≥2D0(m)D≥ 2D_0(m). Lemma C.5 (Finite-horizon full-system stability). For every fixed m, the full marked vector field on preU_ pre is locally Lipschitz with Lip(Fm,D)≤L¯m,L¯m≤L¯m3,Lip(F_m,D)≤ L_m, L_m≤ Lm^3, and its dependence on the relative gauge magnitudes is bounded by A¯m2Δ0 Am^2 _0. Consequently, sup0≤τ≤T⋆+1‖ZDpert(τ)−ZD∘(τ)‖≤eL¯m3(T⋆+1)(1+A¯m2(T⋆+1))Δ0. _0≤τ≤ T_ +1\|Z_D pert(τ)-Z_D (τ)\|≤ e Lm^3(T_ +1)(1+ Am^2(T_ +1)) _0. (C.6) Proof. On preU_ pre all qiq_i stay in a compact interval bounded away from zero and all pairwise angles stay away from the nonsmooth antipodal endpoint. The Gaussian ReLU kernel κ(si⊤sj)κ(s_i s_j) and its first two angular derivatives are therefore uniformly bounded. Differentiating the m coefficient equations and the m spherical transport equations gives at most O(m2)O(m^2) pair interactions per row and O(m)O(m) rows in the Euclidean product norm, hence the conservative bound L¯m3 Lm^3. The mobility maps (q,|δ|/D)↦μ/D(q,|δ|/D) μ/D and (q,|δ|/D)↦χ/D(q,|δ|/D) χ/D have bounded first derivatives on the same compact set. The gauge-magnitude perturbation therefore contributes at most A¯m2Δ0 Am^2 _0. The estimate follows from the standard continuous-dependence inequality for ODEs. ∎ Theorem C.6 (Open-set finite-time winner selection). There are universal constants a,b>0a,b>0 such that, for every fixed m≥2m≥ 2, every ϕ∈Iϕφ∈ I_φ, and Δ0≤εm:=ae−bm3,D≥2D0(m), _0≤ _m:=ae^-bm^3, D≥ 2D_0(m), (C.7) the designated positive-gauge student is the unique student to enter the fixed winner-dominant set sel:=q1≥1−r⋆,∠(s1,u)≤r⋆,‖fred‖L2(π)≤r⋆W_ sel:= \q_1≥ 1-r_ ,\ (s_1,u)≤ r_ ,\ \|f_ red\|_L^2(π)≤ r_ \ by time tsel≤(T⋆+1)/Dt_ sel≤(T_ +1)/D. Every negative-gauge student remains outside the corresponding teacher-alignment threshold at tselt_ sel. Proof. The exact duplicate reference orbit enters the interior of selW_ sel at fast time at most T⋆T_ , with a strictly positive winner/loser margin. Choose a,ba,b so that the right-hand side of (C.6), together with the O(D−2)O(D^-2) finite-gauge error, is smaller than one quarter of this margin. The visible state, the redundant function, and every teacher alignment are continuous functions of the marked state on preU_ pre. The perturbed trajectory therefore enters selW_ sel before fast time T⋆+1T_ +1, while every negative-gauge student remains below the winner threshold. This proves a genuine open-set selection statement without invoking any post-capture nondegeneracy. ∎ The active quotient frame. Let ℋ:=L2(γd)H:=L^2( _d) and write φs(x):=[s⊤x]+,τu,v(x):=u⊤x>0v⊤x,v⟂u. _s(x):=[s x]_+, _u,v(x):=1_\u x>0\v x, v u. For a redundant direction tuple =(s2,…,sm)s=(s_2,…,s_m) define the redundant synthesis map Sβ:=∑j=2mβjφsj,K:=kerS.S_sβ:= _j=2^m _j _s_j, K_s:= S_s. Only the redundant coefficient block is quotiented: the active coordinate space is ℰ:=ℝ×u⟂×(ℝm−1/K),E_s:=R× u × (R^m-1/K_s ), with norm ‖(α,v,[β])‖q2:=|α|2+‖v‖2+infk∈K‖β+k‖2.\|(α,v,[β])\|_ q^2:=|α|^2+\|v\|^2+ _k∈ K_s\|β+k\|^2. The associated linearized realization frame is F(α,v,[β]):=αφu+τu,v+Sβ.F_s(α,v,[β]):=α _u+ _u,v+S_sβ. (C.8) We say that the winner and redundant dictionary are quotient-transverse with constant λq>0 _ q>0 if ‖Fξ‖ℋ2≥λq‖ξ‖q2(ξ∈ℰ).\|F_sξ\|_H^2≥ _ q\|ξ\|_ q^2 (ξ _s). (C.9) This condition rules out a cancellation between the winner coefficient/directional tangent and the function generated by the redundant ridges. It still permits coefficient zero modes among the redundant ridges themselves. For continuation of the actual redundant coefficients we use the stronger active coefficient-frame condition ‖Sβ‖ℋ2≥λred‖β‖2(β∈ℝm−1).\|S_sβ\|_H^2≥ _ red\|β\|^2 (β ^m-1). (C.10) Unlike (C.9), this excludes redundant coefficient kernels. The exact duplicate theorem of Appendix B does not require (C.10): there the exchange symmetry reduces all redundant coefficients to the single aggregate variable P. The condition is needed only for a fully nonsymmetric open-basin locking statement. Lemma C.7 (Gaussian ReLU ridge expansion). There are constants rridge,Cridge>0r_ ridge,C_ ridge>0, depending only on the input dimension, with the following property. For v⟂uv u, ‖v‖≤rridge\|v\|≤ r_ ridge, let s(v)=Expu(v)s(v)=Exp_u(v). Then ‖φs(v)−φu−τu,v‖ℋ \| _s(v)- _u- _u,v\|_H ≤Cridge‖v‖3/2, ≤ C_ ridge\|v\|^3/2, (C.11) ‖Dvφs(v)−D0φs(0)‖op \|D_v _s(v)-D_0 _s(0)\|_ op ≤Cridge‖v‖1/2. ≤ C_ ridge\|v\|^1/2. (C.12) Here tangent spaces are identified by parallel transport along the minimizing spherical geodesic. Proof. Let AvA_v be the sign-disagreement wedge Av=x:sign(u⊤x)≠sign(s(v)⊤x).A_v=\x:sign(u x) (s(v) x)\. Rotational invariance of the Gaussian law gives γd(Av)≤C‖v‖ _d(A_v)≤ C\|v\|. On AvcA_v^c the ReLU is linear on the same half-space, and the sphere-chart remainder is O(‖v‖2‖x‖)O(\|v\|^2\|x\|). On AvA_v the first-order residual is bounded by C‖v‖‖x‖C\|v\|\|x\|. Cauchy–Schwarz and the Gaussian fourth moment therefore give ‖φs(v)−φu−τu,v‖L2(Avc)≤C‖v‖2,‖φs(v)−φu−τu,v‖L2(Av)≤C‖v‖3/2,\| _s(v)- _u- _u,v\|_L^2(A_v^c)≤ C\|v\|^2, \| _s(v)- _u- _u,v\|_L^2(A_v)≤ C\|v\|^3/2, which proves (C.11). The derivatives differ only by the smooth change of tangent frame and by the indicator difference s(v)⊤x>0−u⊤x>01_\s(v) x>0\-1_\u x>0\. Its squared L2L^2 operator contribution is bounded by a Gaussian second moment on AvA_v, hence by C‖v‖C\|v\|, proving (C.12). ∎ For a state in a winner chart write q1=1+α,s1=s(v)=Expu(v),β=(q2,…,qm),q_1=1+α, s_1=s(v)=Exp_u(v), β=(q_2,…,q_m), and keep the redundant directions =(s2,…,sm)s=(s_2,…,s_m) as marked base variables. The realization residual is ℛ(α,v,β):=(1+α)φs(v)+Sβ−φu.R_s(α,v,β):=(1+α) _s(v)+S_sβ- _u. (C.13) Let JZJ_Z denote its derivative with respect to the active coordinates (α,v,β)(α,v,β), with the redundant coefficient block interpreted on the quotient. Lemma C.8 (Nonlinear quotient normal form). Assume (C.9) with lower bound λq _ q. There is r0=r0(λq,m,d)>0r_0=r_0( _ q,m,d)>0 such that, whenever ξ=(α,v,[β])ξ=(α,v,[β]) satisfies ‖ξ‖q≤r0\|ξ\|_ q≤ r_0, ℛ(α,v,β) _s(α,v,β) =Fξ+N(ξ), =F_sξ+N(ξ), ‖N(ξ)‖ℋ \|N(ξ)\|_H ≤C‖ξ‖q3/2, ≤ C\|ξ\|_ q^3/2, (C.14) ‖JZ−F‖op \|J_Z-F_s\|_ op ≤C‖ξ‖q1/2. ≤ C\|ξ\|_ q^1/2. (C.15) Consequently, λq2‖ξ‖q≤‖ℛ(α,v,β)‖ℋ≤C‖ξ‖q, _ q2\|ξ\|_ q≤\|R_s(α,v,β)\|_H≤ C\|ξ\|_ q, (C.16) and ‖JZ∗ℛ(α,v,β)‖2≥cλq‖ℛ(α,v,β)‖ℋ2.\|J_Z^*R_s(α,v,β)\|^2≥ c _ q\|R_s(α,v,β)\|_H^2. (C.17) Proof. Expanding the winner ridge and using section˜C.3 gives (1+α)φs(v)−φu=αφu+τu,v+N(α,v),(1+α) _s(v)- _u=α _u+ _u,v+N(α,v), with ‖N(α,v)‖≤C(|α|‖v‖+‖v‖3/2)≤C‖ξ‖q3/2.\|N(α,v)\|≤ C (|α|\|v\|+\|v\|^3/2 )≤ C\|ξ\|_ q^3/2. The redundant coefficient contribution is exactly linear, proving (C.14). Equation (C.12) gives (C.15). Choose the minimum-norm representative of [β][β]. By (C.9), ‖Fξ‖≥λq‖ξ‖q\|F_sξ\|≥ _ q\|ξ\|_ q. Shrinking r0r_0 so that Cr01/2≤λq/2Cr_0^1/2≤ _ q/2 proves the lower bound in (C.16); the upper bound is immediate. For the PL estimate, write r=Fξ+Nr=F_sξ+N and JZ=F+EJ_Z=F_s+E. On the quotient, the nonzero spectrum of F∗F_s^*F_s is bounded below by λq _ q, hence ‖F∗Fξ‖≥λq‖ξ‖q.\|F_s^*F_sξ\|≥ _ q\|ξ\|_ q. The remaining terms satisfy ‖F∗N‖+‖E∗r‖≤C‖ξ‖q3/2.\|F_s^*N\|+\|E^*r\|≤ C\|ξ\|_ q^3/2. Reducing r0r_0 once more gives ‖JZ∗r‖≥(λq/2)‖ξ‖q\|J_Z^*r\|≥( _ q/2)\|ξ\|_ q. Combining this with the upper bound in (C.16) proves (C.17). ∎ Lemma C.9 (Robust local reaction–transport PL). Suppose q1∈[1/2,3/2]q_1∈[1/2,3/2], the gauge signs are fixed, the winner has gauge +D+D, and all redundant neurons have gauge −D-D. Under the hypotheses of section˜C.3, for all sufficiently large D, −ℒ˙≥cDℒ.- L≥ cDL. (C.18) Proof. The coefficient components of JZ∗ℛJ_Z^*R are precisely the reaction gradients B(si)B(s_i). Its winner-direction component is the geodesic derivative q1(I−s1s1⊤)(s1)q_1(I-s_1s_1 )T(s_1), up to a uniformly conditioned chart identification. The exact dissipation weights every coefficient gradient by μi≥D _i≥ D. For the positive-gauge winner, a12q12=μ(q1,D)+D2q12≥cD, a_1^2q_1^2= μ(q_1,D)+D2q_1^2≥ cD, so the winner directional-gradient coordinate is also weighted by at least cDcD. Dropping the nonnegative redundant-direction terms and applying (C.17) yields −ℒ˙≥cD‖JZ∗ℛ‖2≥cD‖ℛ‖2=2cDℒ.- L≥ cD\|J_Z^*R\|^2≥ cD\|R\|^2=2cDL. ∎ Lemma C.10 (Coefficient control and loser transport under an active redundant frame). Assume, in addition, the redundant frame condition (C.10) with lower bound λred>0 _ red>0. In a sufficiently small winner tube, ∑j=2m|qj|2≤Cℒ. _j=2^m|q_j|^2≤ CL. (C.19) If the trajectory remains in that tube, then ∑j=2m∫t0∞‖s˙j(t)‖dt≤CD2ℒ(t0). _j=2^m _t_0^∞\| s_j(t)\|\,\,dt≤ CD^2L(t_0). (C.20) In particular, every redundant coefficient tends to zero and every redundant direction has finite post-entry motion. Proof. The full redundant frame gives ‖β‖2≤λred−1‖Sβ‖2\|β\|^2≤ _ red^-1\|S_sβ\|^2. The quotient normal form and transversality separate the winner error from the redundant function, so ‖Sβ‖2≤C‖ℛ‖2=2Cℒ\|S_sβ\|^2≤ C\|R\|^2=2CL, proving (C.19). For a negative gauge, |χ(qj,−D)|=2|qj|D2+4qj2+D≤|qj|D.|χ(q_j,-D)|= 2|q_j| D^2+4q_j^2+D≤ |q_j|D. Moreover ‖(I−sjsj⊤)(sj)‖≤Cℒ\|(I-s_js_j )T(s_j)\|≤ C L by Cauchy–Schwarz. Thus ∑j=2m‖s˙j‖≤CDℒ∑j=2m|qj|≤CDℒ. _j=2^m\| s_j\|≤ CD L _j=2^m|q_j|≤ CDL. Integrating (C.18) gives ∫t0∞ℒ(t)dt≤Cℒ(t0)/D _t_0^∞L(t)\,\,dt≤ CL(t_0)/D, proving (C.20). The local decomposition and exponential loss decay imply qj→0q_j→ 0. ∎ Theorem C.11 (Conditional asymptotic locking under an active frame). Assume the open-set finite-time selection theorem and let t0t_0 be a selected-state entry time. Suppose that at t0t_0: 1. the active quotient and redundant coefficient frames satisfy λq≥2λ0 _ q≥ 2 _0 and λred≥2λred,0 _ red≥ 2 _ red,0; 2. every redundant direction is separated from the teacher by at least 2β02 _0; 3. the state has a positive margin 0 m_0 to the remaining chart, coefficient, and gauge-sign faces of the winner tube. There is a constant CmovC_ mov, depending on the fixed tube but not on D, such that if CmovD2ℒ(t0)≤120, C_ movD^2L(t_0)≤ 12 m_0, (C.21) then the tube is forward invariant and ℒ(t)≤ℒ(t0)e−cD(t−t0),q1(t)s1(t)→u,qj(t)→0(j≥2).L(t) (t_0)e^-cD(t-t_0), q_1(t)s_1(t)→ u, q_j(t)→ 0 (j≥ 2). The redundant directions have the total-motion bound (C.20). Proof. Let texitt_ exit be the first exit time from the tube. On [t0,texit)[t_0,t_ exit), section˜C.3 gives exponential loss decay. The quotient normal form and the redundant frame imply that the winner coordinates and all redundant coefficients shrink, so the corresponding coordinate and coefficient faces cannot be reached. The angle and frame functions are Lipschitz on the compact tube. By section˜C.3, the total change of every redundant direction is at most Cmovℒ(t0)/D2C_ movL(t_0)/D^2. Hence the teacher-angle margins, the quotient-frame eigenvalue, and the redundant-frame eigenvalue each lose less than one half of their entry margin under (C.21). Gauge signs are exactly conserved in continuous time. Thus no boundary face can be reached, contradicting a finite texitt_ exit. Forward invariance and the limiting statements follow from the exponential decay and the local decomposition. ∎ Remark C.12 (Scope of the open-basin result). The exact duplicate global-capture theorem remains unconditional. A genuine open neighborhood enjoys unconditional finite-time winner selection. Long-time locking for a fully nonsymmetric redundant dictionary requires both the quotient-transversality condition and the active redundant-frame condition, because only the latter controls the actual coefficient vector rather than its quotient class. C.4 Discrete full-batch gradient descent Let Θ=(a1,w1,…,am,wm) =(a_1,w_1,…,a_m,w_m) denote the original parameters and consider explicit full-batch gradient descent Θn+1=Θn−η∇ℒ(Θn),η=hD. _n+1= _n-η ( _n), η= hD. (C.22) We treat the exact duplicate initialization. Duplicate negative-gauge students remain exactly exchangeable under every raw Euler step, so the original m-student iterate reduces without approximation to one winner and one aggregate loser. Before capture we retain the regular state Y=(q,P,θ,ψ).Y=(q,P,θ,ψ). (C.23) Let Y∞Y_∞ be the multiplicity-weighted fast flow of Appendix B. For Tc:=T⋆+1T_c:=T_ +1, positivity of the continuous fast solution gives p¯m:=12min0≤τ≤Tcq∞(τ),P∞(τ)>0. p_m:= 12 _0≤τ≤ T_c\q_∞(τ),P_∞(τ)\>0. (C.24) After winner-tube entry we use the nonsingular functionally weighted state X=(q,P,qsθ,Psψ).X=(q,P,qs_θ,Ps_ψ). (C.25) Lemma C.13 (Raw Euler expansion for one neuron). Write (s)=B(s)s+G(s),G(s)⟂s,ζ:=ηa/r.T(s)=B(s)s+G(s), G(s) s, ζ:=η a/r. One raw Euler step is a+=a−ηrB,w+=r((1−ζB)s−ζG).a^+=a-η rB, w^+=r ((1-ζ B)s-ζ G ). Whenever |ζ|(|B|+‖G‖)≤1/4|ζ|(|B|+\|G\|)≤ 1/4, r+ r^+ =r−ηaB+η2a22r‖G‖2+Rr, =r-η aB+ η^2a^22r\|G\|^2+R_r, (C.26) s+ s^+ =s−ζG−ζ2(BG+12‖G‖2s)+Rs, =s-ζ G-ζ^2 (BG+ 12\|G\|^2s )+R_s, (C.27) q+ q^+ =q−η(a2+r2)B+η2qB2+η2a32r‖G‖2+Rq, =q-η(a^2+r^2)B+η^2qB^2+ η^2a^32r\|G\|^2+R_q, (C.28) with |Rr|≤Cr|ζ|3(|B|+‖G‖)3,‖Rs‖≤C|ζ|3(|B|+‖G‖)3,|R_r|≤ Cr|ζ|^3(|B|+\|G\|)^3, \|R_s\|≤ C|ζ|^3(|B|+\|G\|)^3, |Rq|≤C(|a|+r)r|ζ|3(|B|+‖G‖)3.|R_q|≤ C(|a|+r)r|ζ|^3(|B|+\|G\|)^3. The gauge drift identity is exact: δ+−δ=η2[(rB)2−a2‖(s)‖2].δ^+-δ=η^2 [(rB)^2-a^2\|T(s)\|^2 ]. (C.29) Proof. Since G⟂sG s, r+r=(1−ζB)2+ζ2‖G‖2. r^+r= (1-ζ B)^2+ζ^2\|G\|^2. The scalar Taylor expansion of the square root, with an integral remainder on the interval determined by the smallness hypothesis, gives (C.26). Multiplying ((1−ζB)s−ζG)((1-ζ B)s-ζ G) by the reciprocal expansion of r+/r^+/r gives (C.27). Multiplying a+a^+ and r+r^+ gives (C.28). Finally, expanding (a+)2−‖w+‖2(a^+)^2-\|w^+\|^2 cancels the complete first-order term and yields (C.29). ∎ Lemma C.14 (Pre-capture one-step consistency and chart preservation). Fix m and the compact fast-flow segment Y∞([0,Tc])Y_∞([0,T_c]). There exist h0,Cm>0h_0,C_m>0 such that, for h≤h0h≤ h_0, D sufficiently large, and every discrete state satisfying q,P≥p¯m,dist(Y,Y∞([0,Tc]))≤1,q,P≥ p_m, (Y,Y_∞([0,T_c]))≤ 1, one raw original-parameter step remains in the regular chart and Yn+1=Yn+hF∞(Yn)+Rn,‖Rn‖≤Cm(h2+hD−2).Y_n+1=Y_n+hF_∞(Y_n)+R_n, \|R_n\|≤ C_m(h^2+hD^-2). (C.30) The one-step gauge drift is bounded by Cmh2/DC_mh^2/D. Proof. On the stated compact set the residual moments B,GB,G and their first derivatives are uniformly bounded. For the positive-gauge winner, a=Θ(D)a= ( D), r=Θ(D−1/2)r= (D^-1/2), and hence ζ=h/q+O(hD−2)ζ=h/q+O(hD^-2). For a negative-gauge loser, a=Θ(D−1/2)a= (D^-1/2), r=Θ(D)r= ( D), and ζ=O(hD−2)ζ=O(hD^-2). Applying section˜C.4 to the winner and to one common loser, and multiplying the loser coefficient increment by m−1m-1, gives the fast marked vector field. The second-order terms are Om(h2)O_m(h^2) and the finite-gauge mobility defects are Om(hD−2)O_m(hD^-2), which proves (C.30). The same estimates imply relative factor changes Om(h)O_m(h), preserving positivity and w≠0w≠ 0 for h0h_0 small. Equation (C.29) and the raw factor sizes give the stated gauge-drift bound. ∎ Lemma C.15 (Discrete pre-capture shadowing). For nh≤Tcnh≤ T_c, max0≤k≤n‖Yk−Y∞(kh)‖≤Cm(h+D−2). _0≤ k≤ n\|Y_k-Y_∞(kh)\|≤ C_m(h+D^-2). (C.31) Consequently, for sufficiently small h and sufficiently large D, the iterates satisfy qk,Pk≥p¯mq_k,P_k≥ p_m before capture and enter the winner tube in at most ncap≤⌈Tch⌉n_ cap≤ T_ch steps. Proof. The exact fast flow has the local expansion Y∞((n+1)h)=Y∞(nh)+hF∞(Y∞(nh))+Om(h2)Y_∞((n+1)h)=Y_∞(nh)+hF_∞(Y_∞(nh))+O_m(h^2). Subtract this from (C.30) and use the Lipschitz constant of F∞F_∞ on the compact segment. A bootstrap discrete Gronwall argument proves (C.31); choosing the right-hand side smaller than p¯m p_m closes the bootstrap and preserves the regular chart. The strict continuous entry margin then gives discrete winner-tube entry. ∎ Lemma C.16 (Raw Hessian bound in the symmetric winner tube). On the fixed symmetric winner tube there is Cm>0C_m>0, independent of D, such that ‖∇2ℒ(Θ)‖op≤CmD.\|∇^2L( )\|_ op≤ C_mD. (C.32) Moreover, the exact population gradient satisfies ‖∇ℒ(Θ)‖2≥cmDℒ(Θ).\| ( )\|^2≥ c_mDL( ). (C.33) Proof. Write the population loss using the homogeneous Gaussian ReLU kernel K(w,v)=‖w‖‖v‖κ(w^⊤v^).K(w,v)=\|w\|\|v\|κ( w v). On the nonzero winner tube, direct differentiation of the arc-cosine formula gives ‖∇wK(w,v)‖≤C‖v‖,‖∇wv2K(w,v)‖≤C,‖∇ww2K(w,v)‖≤C‖v‖w‖,\| _wK(w,v)\|≤ C\|v\|, \|∇^2_wvK(w,v)\|≤ C, \|∇^2_wK(w,v)\|≤ C \|v\|\|w\|, with the self-interaction K(w,w)=‖w‖2/2K(w,w)=\|w\|^2/2 treated explicitly. The aa Hessian blocks are kernel values and are Om(D)O_m(D). The awaw blocks contain one raw factor and one first kernel derivative and are Om(D)O_m(D). The ww blocks contain aiaja_ia_j times a second kernel derivative; the positive/negative gauge reconstructions |ai|2+‖wi‖2=D2+4qi2|a_i|^2+\|w_i\|^2= D^2+4q_i^2 show that all such blocks are Om(D)O_m(D). This proves (C.32). The squared raw gradient norm equals the negative loss derivative under gradient flow. On the symmetric winner tube, the selected/aggregate-redundant feature Gram of appendix˜B gives the local gradient inequality, while all coefficient mobilities and the selected directional mobility are bounded below by cDcD. Hence (C.33). ∎ Lemma C.17 (Loss-weighted post-capture Euler transport). Let an exact-duplicate discrete iterate lie in the symmetric winner tube and satisfy the negative-gauge bootstrap bound δ−,n≤−D/2 _-,n≤-D/2. Write pn=Pn/(m−1)p_n=P_n/(m-1) for the common loser coefficient and decompose its residual moment as n(sn)=Bnsn+Gn,Gn⟂sn.T_n(s_n)=B_ns_n+G_n, G_n s_n. There is a constant Cm>0C_m>0, independent of D and h, such that |pn|+|Bn|+‖Gn‖≤Cmℒn.|p_n|+|B_n|+\|G_n\|≤ C_m L_n. (C.34) Consequently, for η=h/Dη=h/D and h sufficiently small, ‖sn+1−sn‖≤Cm(hD2ℒn+h2D4ℒn2+h3D6ℒn3).\|s_n+1-s_n\|≤ C_m ( hD^2L_n+ h^2D^4L_n^2+ h^3D^6L_n^3 ). (C.35) In particular, every term in the one-step directional remainder vanishes with the task residual. Proof. The symmetric local loss equivalence (B.19) gives Pn2≤CmℒnP_n^2≤ C_mL_n and hence |pn|≤Cmℒn|p_n|≤ C_m L_n for fixed m. Let Rn=fΘn−f⋆R_n=f_ _n-f_ . Since Bn=[Rn(X)[sn⊤X]+]B_n=E[R_n(X)[s_n X]_+] and n(sn)=[Rn(X)sn⊤X>0X]T_n(s_n)=E[R_n(X)1_\s_n X>0\X], Gaussian Cauchy–Schwarz yields |Bn|≤ℒn,‖n(sn)‖≤2ℒn,|B_n|≤ L_n, \|T_n(s_n)\|≤ 2L_n, up to the fixed normalization of ℒ=12R2L= 12ER^2. Because Gn=(I−snsn⊤)n(sn)G_n=(I-s_ns_n )T_n(s_n), this proves (C.34). Under the bootstrap bound δ−,n≤−D/2 _-,n≤-D/2, rn2=δn2+4pn2−δn2≥D2.r_n^2= _n^2+4p_n^2- _n2≥ D2. Using an/rn=pn/rn2a_n/r_n=p_n/r_n^2 and η=h/Dη=h/D, we obtain |ζn|=η|anrn|≤Ch|pn|D2.| _n|=η | a_nr_n |≤ C h|p_n|D^2. The fixed winner tube and (C.34) imply |ζn|(|Bn|+‖Gn‖)≤1/4| _n|(|B_n|+\|G_n\|)≤ 1/4 after reducing h1(m)h_1(m) if necessary, so section˜C.4 applies. The first-order angular term is bounded by |ζn|‖Gn‖≤Cmhℒn/D2| _n|\|G_n\|≤ C_mhL_n/D^2. The explicit quadratic term is at most |ζn|2(|Bn|‖Gn‖+12‖Gn‖2)≤Cmh2D4ℒn2.| _n|^2 (|B_n|\|G_n\|+ 12\|G_n\|^2 )≤ C_m h^2D^4L_n^2. Finally, the cubic remainder satisfies ‖Rs,n‖≤C|ζn|3(|Bn|+‖Gn‖)3≤Cmh3D6ℒn3.\|R_s,n\|≤ C| _n|^3(|B_n|+\|G_n\|)^3≤ C_m h^3D^6L_n^3. Summing these three estimates proves (C.35). ∎ Lemma C.18 (Discrete descent, first-exit invariance, and gauge drift). There is h1(m)>0h_1(m)>0 such that, if 0<h≤h1(m)0<h≤ h_1(m) and a discrete iterate enters the symmetric winner tube with the fixed continuous-theory entry margin, then for every later iterate ℒn+1≤(1−cmh)ℒn.L_n+1≤(1-c_mh)L_n. (C.36) The tube is forward invariant, and the common loser direction obeys the explicit path-length bound ∑n≥ncap‖sn+1−sn‖≤CmD2(ℒncap+hD2ℒncap2+h2D4ℒncap3). _n≥ n_ cap\|s_n+1-s_n\|≤ C_mD^2 (L_n_ cap+ hD^2L_n_ cap^2+ h^2D^4L_n_ cap^3 ). (C.37) Moreover, supn|δi,n−δi,0|≤CmhD. _n| _i,n- _i,0|≤ C_m hD. (C.38) Proof. Let nexit>ncapn_ exit>n_ cap be the first putative exit index. For every ncap≤n<nexitn_ cap≤ n<n_ exit, the descent lemma, (C.32), (C.33), and η=h/Dη=h/D give (C.36). The symmetric local loss equivalence therefore keeps the winner coefficient, aggregate loser coefficient, and winner angle strictly inside their tube faces. Apply section˜C.4 up to index nexit−1n_ exit-1. Writing ϖ=1−cmh =1-c_mh, geometric contraction gives, for ℓ=1,2,3 =1,2,3, ∑k=0nexit−1−ncapℒncap+kℓ≤ℒncapℓ1−ϖℓ≤Cm,ℓhℒncapℓ. _k=0^n_ exit-1-n_ capL_n_ cap+k^\, ≤ L_n_ cap^\, 1- ≤ C_m, hL_n_ cap^\, . Substitution into (C.35) yields the finite-horizon version of (C.37), uniformly in nexitn_ exit. For D sufficiently large and h sufficiently small, this quantity is less than one half of the fixed loser-angle entry margin. Hence the loser-angle face cannot be the first exit. It remains to exclude loss of the gauge-magnitude bootstrap δ+,n≥D/2 _+,n≥ D/2 and δ−,n≤−D/2 _-,n≤-D/2. The pre-capture drift is Om(h/D)O_m(h/D) by section˜C.4 over O(h−1)O(h^-1) steps. Up to the putative exit, the exact identity (C.29) and the descent lemma imply ∑n=ncapnexit−1|δi,n+1−δi,n|≤η2∑n=ncapnexit−1‖∇iℒ(Θn)‖2≤2ηℒncap≤CmhD. _n=n_ cap^n_ exit-1| _i,n+1- _i,n|≤η^2 _n=n_ cap^n_ exit-1\| _iL( _n)\|^2≤ 2 _n_ cap≤ C_m hD. This is negligible compared with the entry gauge magnitude D, so no gauge-magnitude, gauge-sign, or regular-chart face can be reached. All possible first exit faces have now been excluded, which proves forward invariance. Letting nexit→∞n_ exit→∞ gives (C.37); combining the pre- and post-capture estimates gives (C.38). ∎ Theorem C.19 (Discrete-GD capture and functional pruning). For every fixed m≥2m≥ 2 and ϕ∈Iϕφ∈ I_φ, there exist h0(m)>0h_0(m)>0 and D~0(m)<∞ D_0(m)<∞ such that the exact duplicate full-batch gradient-descent iterates with η=h/Dη=h/D, 0<h≤h0(m)0<h≤ h_0(m), and D≥D~0(m)D≥ D_0(m) select the programmed positive-gauge winner. They obey the pre-capture shadowing estimate (C.31), enter the winner tube within ⌈Tc/h⌉ T_c/h iterations, satisfy the geometric contraction (C.36), have gauge drift (C.38), and satisfy the explicit post-capture redundant-direction bound (C.37). Proof. Combine sections˜C.4, C.4, C.4 and C.4. ∎ Appendix D Experimental and numerical validation The validation compares the following objects without refitting any dynamics to the finite systems. Fast theory. The three-dimensional multiplicity-weighted limit z∞=(q,P,θ)z_∞=(q,P,θ) from the global-capture theorem, where q is the selected functional coefficient, P is the aggregate redundant coefficient, and θ is the selected angle. Exact marked population flow. The finite-D symmetric system zD=(q,P,θ,ψ)z_D=(q,P,θ,ψ), including the slowly moving redundant direction ψ. Original-parameter population flow. The full m-student gradient flow in (ai,wi)(a_i,w_i) coordinates. No symmetry reduction is imposed by the implementation. Original-parameter empirical flow. Full-batch gradient flow on N i.i.d. samples xn∼(0,I2)x_n (0,I_2) with labels yn=[u⊤xn]+y_n=[u x_n]_+, again in the full (ai,wi)(a_i,w_i) coordinates. The global-capture theorem predicts, for every fixed multiplicity, tcap t_ cap =Θm(D−1), = _m(D^-1), (D.1) ∫tcap∞‖s˙lose(t)‖dt _t_ cap^∞\!\| s_ lose(t)\|\,\,dt =Om(D−2), =O_m(D^-2), supτ≤T‖zD(τ)−z∞(τ)‖ _τ≤ T\|z_D(τ)-z_∞(τ)\| =Om,T(D−2). =O_m,T(D^-2). The population sweep uses m∈2,4,8,16,32,D∈8,16,32,64,128.m∈\2,4,8,16,32\, D∈\8,16,32,64,128\. The empirical sweep fixes D=32D=32 and uses m∈2,4,8,16,N∈512,1024,2048,4096,8192,20 coupled seeds.m∈\2,4,8,16\, N∈\512,1024,2048,4096,8192\, 20 coupled seeds. For a fixed (m,seed)(m,seed), the samples are nested across N. The operational capture tube is fixed before all sweeps: qwin≥0.90,|Plose|≤0.10,|θwin|≤0.10.q_win≥ 0.90, |P_lose|≤ 0.10, | _win|≤ 0.10. (D.2) A raw redundant angle is not identifiable after its aggregate functional coefficient vanishes. We therefore compare finite-sample and population trajectories in the functionally identifiable state (t)=(q(t),P(t),q(t)swin(t),P(t)slose(t))∈ℝ6, X(t)= (q(t),\,P(t),\,q(t)s_win(t),\,P(t)s_lose(t) ) ^6, (D.3) using ErrN=[16T∫0T‖N(τ)−pop(τ)‖22dτ]1/2.Err_N= [ 16T _0^T\| X_N(τ)- X_pop(τ)\|_2^2\,\,dτ ]^1/2. (D.4) This prevents an arbitrary direction of a nearly zero-coefficient redundant unit from dominating the validation metric. D.1 Complete trajectory correspondence Figure˜2 compares all four levels for the representative setting (m,D,N)=(8,32,8192)(m,D,N)=(8,32,8192). The curves agree in task loss, winner capture, aggregate functional-mass decay, feature transport, and the reaction–transport dissipation split. The fast theory omits the O(D−2)O(D^-2) redundant-direction motion while accurately tracking every functionally active coordinate. The exact marked and original-parameter population implementations agree over the full population grid to a maximum marked-state error below 1.3×10−71.3× 10^-7; population gauge drift remains below 2.2×10−112.2× 10^-11. A separate full-m versus symmetric-manifold audit at (m,N,D)=(8,2048,32)(m,N,D)=(8,2048,32) finds state and loss differences at machine precision. Duplicate redundant units remain identical, as required by permutation symmetry. D.2 Population scaling across gauge and overparameterization Panels (a)–(b) of figure˜3, panel (b) of figure˜D.1, and table˜D.1 audit the three asymptotic predictions in equation˜D.1. The fitted exponents remain stable from m=2m=2 through m=32m=32. The constants change with m, as the theorem permits, but do not blow up over the tested fixed-multiplicity range. Table D.1: Population scaling laws. All regressions use D∈8,16,32,64,128D∈\8,16,32,64,128\. m capture slope R2R^2 drift slope R2R^2 fast-error slope R2R^2 2 -1.0019 >0.9999>0.9999 -1.9991 >0.9999>0.9999 -1.9981 >0.9999>0.9999 4 -0.9982 >0.9999>0.9999 -1.9965 >0.9999>0.9999 -1.9983 >0.9999>0.9999 8 -0.9967 >0.9999>0.9999 -1.9945 >0.9999>0.9999 -1.9926 >0.9999>0.9999 16 -0.9951 >0.9999>0.9999 -1.9964 >0.9999>0.9999 -1.9941 >0.9999>0.9999 32 -0.9964 >0.9999>0.9999 -1.9907 >0.9999>0.9999 -1.9946 >0.9999>0.9999 The rescaled capture constants DtcapDt_cap are 30.24,24.16,21.84,20.82,20.3430.24,24.16,21.84,20.82,20.34 for m=2,4,8,16,32m=2,4,8,16,32. The corresponding D2D^2-rescaled loser drifts are 1.59×10−2,3.78×10−3,1.44×10−3,6.40×10−4,3.03×10−41.59× 10^-2,3.78× 10^-3,1.44× 10^-3,6.40× 10^-4,3.03× 10^-4. These observations do not constitute a uniform m→∞m→∞ result; they show only that the fixed-m theorem remains numerically well conditioned over the tested range. D.3 Finite-sample trajectory scaling For each m, the mean functionally identifiable trajectory error follows a clean power law over N=512N=512 to 81928192; see table˜D.2. The slopes range from −0.448-0.448 to −0.494-0.494, with R2R^2 between 0.9590.959 and 0.9970.997. Every one of the 4×5×20=4004× 5× 20=400 empirical runs enters the fixed capture tube (D.2). Empirical gauge drift remains below 9.7×10−79.7× 10^-7. Table D.2: Finite-sample full-trajectory scaling at D=32D=32 using 20 coupled seeds. The error is defined in (D.4). m slope R2R^2 mean NErrN N\,Err_N capture rate 2 -0.4908 0.9970 0.2316 20/20 at every N 4 -0.4482 0.9589 0.2724 20/20 at every N 8 -0.4718 0.9753 0.2356 20/20 at every N 16 -0.4942 0.9858 0.2552 20/20 at every N The observed behavior is compatible with the standard fixed-dimensional Monte Carlo scale N−1/2N^-1/2. This is an empirical guide. After capture, the population loss is extremely small and dominated by cancellations, so the identifiable reaction–transport state provides the stable trajectory metric at fixed finite m. A solver audit compares three adaptive-tolerance settings. Relative to the tightest solve, the two looser settings have maximum marked-state errors 3.27×10−53.27× 10^-5 and 1.64×10−51.64× 10^-5, below the empirical trajectory discrepancies shown in figure˜3. The numerical integration error therefore does not explain the finite-sample scaling. D.4 Discrete post-capture transport audit The proof of sections˜C.4 and C.4 retains the residual-dependent factors in every Euler remainder. We independently audit the resulting D−2D^-2 path-length prediction in the raw original-parameter GD implementation. At m=8m=8 and ϕ=π/4φ=π/4, we use D∈16,24,32,48,64,96,128,h∈0.2,0.1,0.05,η=h/D,D∈\16,24,32,48,64,96,128\, h∈\0.2,0.1,0.05\, η=h/D, and sum the common loser-direction increments after the fixed capture condition (D.2). The trajectory is followed to fast time τ=80τ=80; the remaining loss is below 1.6×10−71.6× 10^-7 in every run. Table D.3: Raw discrete-GD post-capture loser path. The fitted exponent is for total directional motion versus D. h fitted D exponent R2R^2 mean D2∑n‖sn+1−sn‖D^2 _n\|s_n+1-s_n\| 0.20 -1.9953 >0.9999>0.9999 1.9029×10−31.9029× 10^-3 0.10 -1.9952 >0.9999>0.9999 1.9294×10−31.9294× 10^-3 0.05 -1.9975 >0.9999>0.9999 1.9305×10−31.9305× 10^-3 The D2D^2-rescaled path varies by less than 0.4%0.4\% across the tested D values for each fixed h. This audit is not used in the proof; it checks that the raw Euler implementation exhibits the summable, loss-weighted behavior required by (C.35). Figure D.1: Additional numerical diagnostics. (a) Both single-unit singular limits converge as D−2D^-2. (b) The finite-gauge marked flow approaches the fast theory as D−2D^-2. (c) Every certified-angle test captures. (d) DtcapDt_ cap remains controlled through m=256m=256 (a finite sweep only). (e) Discrete-GD error is first order in h=ηDh=η D. (f) The raw post-capture path follows D−2D^-2 at every tested h. Dashed lines show predicted orders.