EMAN: Optimization-Driven Capacity Growth through Path Emergence in Multi-Task Learning

# EMAN: Optimization-Driven Capacity Growth through Path Emergence in Multi-Task Learning
Source: [https://arxiv.org/html/2608.16930](https://arxiv.org/html/2608.16930)
Chenlei Fang1,2,3,4,5, Jingchen Li1, Hongzong LI1, Qingyao Li2,3,4,5, Yixuan Zhang2,3,4,5, Huarui Wu2,3,4,5, Haobin Shi1, Chunjiang Zhao2,3,4,5\\corresponding

###### Abstract

Existing multi\-task learning methods rely on hard sharing, multiple paths or experts, adaptive sharing, and dynamic expansion\. However, their capacity changes are usually constrained by predefined structures or triggered by task boundaries and conflict signals\. This raises a fundamental question: can a network start from exact single\-path computation and grow a new independent path only when persistent optimization evidence appears? We propose theEmergentModularAtomicNetwork \(EMAN\), an optimization\-driven framework for exposing an antisymmetric growth direction through latent relative phases without instantiating a second path, and for monitoring multiple decision signals during the training process through a set of optimization evidence to transform local optimization evidence into a structural decision\. EMAN materializes two equal\-capacity independent paths only after certification\. EMAN adaptively allocates shared and task\-specific representation capacity to accommodate varying task requirements\. Extensive experiments on controlled rank settings, PASCAL\-Context, and NYUv2 validate its effectiveness, achieving improved performance at a competitive computational cost\.

## 1Introduction

Hard parameter sharing remains one of the simplest and most efficient designs for multi\-task learning \(MTL\)\(Caruana[1997](https://arxiv.org/html/2608.16930#bib.bib1)\)\. A dense shared backbone is fully differentiable and inexpensive to optimize\. Its fixed representation capacity can also create negative transfer or a hard bottleneck under heterogeneous task demands\. Soft\-sharing architectures, task\-interaction modules, and adaptive sharing policies provide greater flexibility\(Misraet al\.[2016](https://arxiv.org/html/2608.16930#bib.bib2); Ruderet al\.[2019](https://arxiv.org/html/2608.16930#bib.bib3); Liuet al\.[2019](https://arxiv.org/html/2608.16930#bib.bib4); Vandenhendeet al\.[2020](https://arxiv.org/html/2608.16930#bib.bib5); Xuet al\.[2018](https://arxiv.org/html/2608.16930#bib.bib6); Sunet al\.[2020](https://arxiv.org/html/2608.16930#bib.bib7)\)\. Mixture\-of\-Experts \(MoE\) models take another route\. They direct examples or tasks through specialized experts\(Shazeeret al\.[2017](https://arxiv.org/html/2608.16930#bib.bib14); Maet al\.[2018](https://arxiv.org/html/2608.16930#bib.bib15); Lepikhinet al\.[2021](https://arxiv.org/html/2608.16930#bib.bib16); Feduset al\.[2022](https://arxiv.org/html/2608.16930#bib.bib17)\)\. Both families improve flexibility\. Both also commit to multiple modules, experts, routing rules, or sharing policies before optimization reveals the need for extra capacity\.

Recent MTL research approaches this tension from several directions\. Current studies examine spurious inter\-task correlations, shared and task\-specific optimization, and task\-adaptive parameter\-efficient sharing\(Chaiet al\.[2025](https://arxiv.org/html/2608.16930#bib.bib32); Qinet al\.[2025](https://arxiv.org/html/2608.16930#bib.bib33); Baeket al\.[2025](https://arxiv.org/html/2608.16930#bib.bib34); Mantriet al\.[2025](https://arxiv.org/html/2608.16930#bib.bib35)\)\. Dense\-scene MTL also revisits cross\-task structure on NYUv2 and PASCAL\-Context\(Wanget al\.[2026](https://arxiv.org/html/2608.16930#bib.bib36)\)\. Recent expert systems adapt expert allocation or expand expert pools during training\(Park and Park[2026](https://arxiv.org/html/2608.16930#bib.bib37); Zhanget al\.[2026](https://arxiv.org/html/2608.16930#bib.bib38)\)\. These advances improve interaction, allocation, and expansion\. The number of independent modules still depends on fixed task\-specific structures, explicit routing, or declared task boundaries\. Two paths can add capacity\. The open question is when their additional cost becomes justified\.

![Refer to caption](https://arxiv.org/html/2608.16930v1/x1.png)Figure 1:Capacity mismatch across training\. Hard sharing is inexpensive early but can become insufficient later\. Always\-dual paths avoid the bottleneck but maintain excess capacity throughout training\. EMAN starts from one exact shared path and materializes a second path only after persistent evidence\.Figure[1](https://arxiv.org/html/2608.16930#S1.F1)illustrates this mismatch\. Hard sharing fixes one\-path capacity, so later demand can exceed the available representation capacity\. Always\-dual paths avoid this bottleneck, but they maintain two\-path computation from the start\. A more selective design should retain exact sharing while one path is sufficient and add independent capacity after persistent demand appears\. The growth decision should emerge from optimization itself\.

We introduce the Emergent Modular Atomic Network \(EMAN\), an optimization\-driven dense\-to\-modular framework\. EMAN starts with one physical forward path and exact shared computation\. A latent relative phase then probes the local optimization geometry without changing physical capacity\. The fixed odd phase interaction exposes a candidate antisymmetric growth direction\. Optimization Evidence Certification \(OEC\) evaluates its strength, cross\-probe consistency, finite\-step descent, dynamic safety, and temporal persistence\. Insufficient evidence retains the shared path\. A successful certificate materializes two isomorphic and independently trainable paths through opposite perturbations\. Training data drive the entire transition, and the certification rule remains fixed throughout training\.

The local analysis describes this transition through the antisymmetric path coordinateddand the relative phaseϕ\\phi\. Near the shared solution, a phase\-symmetric branch can lose transverse stability before the path coordinates separate\. After phase release, the cross termϕ​c⊤​d\\phi c^\{\\top\}dinduces the antisymmetric forcec​ϕc\\phi\. The second physical path is still absent, yet the local geometry already exposes a candidate descent direction\. Rank\-controlled experiments connect this mechanism to explicit capacity demand\. One physical path has rank\-three capacity, and the joint task structure requires ranks of three, four, and six\. EMAN remains shared at joint rank three\. It materializes a second path under the rank\-six bottleneck and recovers the missing latent factors and predictive performance\. The rank\-four condition characterizes OEC near the single\-path capacity limit\. The released paths also span complementary latent subspaces\.

Natural\-image experiments examine the timing of this structural decision\. In the static PASCAL\-Context setting, all four tasks appear from the beginning, and EMAN releases early\. Little opportunity remains for delayed\-computation savings\. Under task arrival, EMAN preserves one\-path sharing during the initial two\-task stage and releases after the saliency and surface\-normal tasks enter training\. Across three seeds, it reaches final performance comparable to the always\-dual baseline, shortens the active dual\-path stage to about half, and uses about75%75\\%of its path\-training compute\. Earlier demand moves the same frozen decision from epoch 21 to epoch 11\. Gradual demand provides a further test under progressive capacity requirements\. On NYUv2, EMAN achieves the best mean semantic\-segmentation and depth results and the highest aggregate score\. Recon retains the strongest surface\-normal result and joint loss\. PASCAL\-Context shows less uniform task\-level gains\. The overall pattern is consistent: the timing and strength of optimization demand shape the value of path emergence\.

Our contributions are summarized as follows:

1. 1\.Optimization\-driven capacity growth from exact sharing\.We propose EMAN, which starts with one physical path and treats independent capacity as a structural decision during optimization\. The model retains exact sharing while one path remains sufficient\.
2. 2\.Phase\-first evidence\-gated path emergence\.EMAN exposes an antisymmetric growth direction through latent relative phase before adding an independent parameter block\. A fixed evidence rule evaluates the direction and triggers a zero\-sum materialization into two independently trainable paths\.
3. 3\.Controlled and real\-data validation\.Rank\-controlled experiments establish capacity preservation, conditional rank recovery, and complementary released representations\. NYUv2 and PASCAL\-Context further demonstrate adaptive release, competitive performance, and reduced path\-training computation\.

## 2Related Work

##### Sharing in multi\-task learning\.

Hard parameter sharing provides an efficient foundation for multi\-task learning, but one representation can entangle incompatible objectives\(Caruana[1997](https://arxiv.org/html/2608.16930#bib.bib1)\)\. Cross\-stitch networks, sluice networks, attention modules, and multi\-scale interaction mechanisms learn softer sharing patterns\(Misraet al\.[2016](https://arxiv.org/html/2608.16930#bib.bib2); Ruderet al\.[2019](https://arxiv.org/html/2608.16930#bib.bib3); Liuet al\.[2019](https://arxiv.org/html/2608.16930#bib.bib4); Vandenhendeet al\.[2020](https://arxiv.org/html/2608.16930#bib.bib5); Xuet al\.[2018](https://arxiv.org/html/2608.16930#bib.bib6)\)\. AdaShare learns task\-dependent sharing policies\(Sunet al\.[2020](https://arxiv.org/html/2608.16930#bib.bib7)\)\. Gradient\-based methods address loss imbalance and task conflict\(Chenet al\.[2018](https://arxiv.org/html/2608.16930#bib.bib8); Yuet al\.[2020](https://arxiv.org/html/2608.16930#bib.bib9); Liuet al\.[2021](https://arxiv.org/html/2608.16930#bib.bib10); Navonet al\.[2022](https://arxiv.org/html/2608.16930#bib.bib11); Javaloy and Valera[2022](https://arxiv.org/html/2608.16930#bib.bib12)\), and Recon converts highly conflicting shared layers into task\-specific ones\(Shiet al\.[2023](https://arxiv.org/html/2608.16930#bib.bib13)\)\. Recent work further explores the balance between shared and task\-specific representations\. ConsMTL optimizes both parameter groups\(Qinet al\.[2025](https://arxiv.org/html/2608.16930#bib.bib33)\)\. TADFormer uses input\-conditioned task adaptation, and DiTASK applies parameter\-efficient task\-specific transformations\(Baeket al\.[2025](https://arxiv.org/html/2608.16930#bib.bib34); Mantriet al\.[2025](https://arxiv.org/html/2608.16930#bib.bib35)\)\. Other studies identify spurious inter\-task correlations or model geometric cross\-task relations\(Chaiet al\.[2025](https://arxiv.org/html/2608.16930#bib.bib32); Wanget al\.[2026](https://arxiv.org/html/2608.16930#bib.bib36)\)\. This line of work makes sharing more flexible, yet the number of independent paths remains fixed before training\.

##### Experts, modularity, and expansion\.

Sparse MoE and multi\-gate expert models use routers to select specialized computation\(Shazeeret al\.[2017](https://arxiv.org/html/2608.16930#bib.bib14); Maet al\.[2018](https://arxiv.org/html/2608.16930#bib.bib15); Lepikhinet al\.[2021](https://arxiv.org/html/2608.16930#bib.bib16); Feduset al\.[2022](https://arxiv.org/html/2608.16930#bib.bib17)\)\. Modularity creates room for specialization, but data structure and optimization also shape the resulting functions\(Csordáset al\.[2021](https://arxiv.org/html/2608.16930#bib.bib18); Mittalet al\.[2022](https://arxiv.org/html/2608.16930#bib.bib19); Jarviset al\.[2023](https://arxiv.org/html/2608.16930#bib.bib20); Zhanget al\.[2023](https://arxiv.org/html/2608.16930#bib.bib21); Schuget al\.[2024](https://arxiv.org/html/2608.16930#bib.bib22)\)\. Progressive networks, dynamically expandable networks, and PathNet add or select capacity across tasks\(Rusuet al\.[2016](https://arxiv.org/html/2608.16930#bib.bib23); Yoonet al\.[2018](https://arxiv.org/html/2608.16930#bib.bib24); Fernandoet al\.[2017](https://arxiv.org/html/2608.16930#bib.bib25)\)\. Net2Net preserves the learned function during architectural growth\(Chenet al\.[2016](https://arxiv.org/html/2608.16930#bib.bib26)\)\. More recent systems make expert allocation adaptive: MASS detects semantic drift and expands a routed expert pool\(Park and Park[2026](https://arxiv.org/html/2608.16930#bib.bib37)\), and GoD\-MoE estimates expert demand and adds LoRA experts during continual instruction tuning\(Zhanget al\.[2026](https://arxiv.org/html/2608.16930#bib.bib38)\)\. These methods begin with an explicit expert structure or a declared task sequence\. EMAN begins with dense single\-path computation\. Phase\-structured evidence determines whether a second independent path should emerge\. Capacity growth thus becomes an endogenous event in optimization\.

##### Symmetry and local curvature\.

Exchange\-symmetric components admit symmetric and antisymmetric coordinates, a standard basis for symmetry\-breaking analysis\(Golubitskyet al\.[1988](https://arxiv.org/html/2608.16930#bib.bib27); Kuznetsov[2004](https://arxiv.org/html/2608.16930#bib.bib28)\)\. Hessian\-vector products provide local curvature information without constructing the full Hessian\(Pearlmutter[1994](https://arxiv.org/html/2608.16930#bib.bib29)\)\. We draw on these ideas near the exactly shared state\. In EMAN, this information characterizes the transverse geometry of the shared branch\. Relative phase exposes a coupled antisymmetric direction, and OEC evaluates its significance during training\. The analysis connects local geometry to the decision to create independent capacity\.

## 3EMAN: Optimization\-Driven Path Emergence

### Overview

EMAN treats multi\-task capacity growth as a structural decision during training\. Training begins with exact single\-path computation\. A relative\-phase probe then exposes the local optimization geometry and a candidate antisymmetric growth direction\. Optimization Evidence Certification \(OEC\) evaluates direction strength, cross\-probe consistency, finite\-step descent, dynamic safety, and temporal persistence\. Persistent evidence triggers path materialization; otherwise, the model retains exact sharing\.

The complete procedure contains four stages:

exact sharing→\\rightarrowphase probing→\\rightarrowOEC

→\\rightarrowconditional path materialization

Figure[2](https://arxiv.org/html/2608.16930#S3.F2)focuses on the local transition\. The initial state contains one physical path, zero relative phase, and zero antisymmetric displacement\. Phase probing reveals a candidate direction without adding physical capacity\. OEC then examines the full evidence set\. A successful certificate produces a zero\-sum split, followed by independent optimization of the two materialized paths\.

![Refer to caption](https://arxiv.org/html/2608.16930v1/x2.png)Figure 2:Phase\-first certification and path materialization\. EMAN starts from one tied physical path\. Phase probing exposes a candidate antisymmetric direction without creating a new path\. OEC evaluates strength, consistency, descent, safety, and persistence\. Persistent evidence triggers a zero\-sum split, and the released paths then optimize independently\.Figure[3](https://arxiv.org/html/2608.16930#S3.F3)gives the architecture\-level view\. The shared encoder first performs exact single\-path computation\. The lower central block contains phase\-coupled probing and OEC\. A successful certificate materializes two equal\-capacity paths through opposite perturbations\. The upper central block then performs two\-path computation, feature fusion, and task\-specific prediction\. The right block shows the phase\-coupled pair\-interaction matrix: off\-diagonal entries encode pair interactions, and diagonal entries encode self terms\.

![Refer to caption](https://arxiv.org/html/2608.16930v1/x3.png)Figure 3:Overview of EMAN\. The shared encoder performs exact single\-path computation\. Phase\-coupled probing introduces latent relative phases and exposes an antisymmetric direction without adding independent path capacity\. OEC evaluates direction strength, cross\-probe consistency, finite\-step descent, dynamic safety, and temporal persistence\. A successful certificate materializes two equal\-capacity paths through opposite perturbations\. Their representations are fused and passed to the task\-specific heads\. The right panel shows the phase\-coupled pair\-interaction matrix\. Off\-diagonal entries represent pair interactions, and diagonal entries represent self terms\.
### Exact Shared Computation

Letxxdenote the input feature\. The shared path produces

h=f​\(x;ω¯\),h=f\(x;\\bar\{\\omega\}\),\(1\)with the shared parameter tensorω¯\\bar\{\\omega\}\. Before materialization, EMAN stores only this tensor and performs one forward evaluation\. Its output is reused in two logical path coordinates:

p1=p2=h\.p\_\{1\}=p\_\{2\}=h\.\(2\)The logical duplication exposes symmetric and antisymmetric coordinates for structural probing\. It allows a zero\-sum variation aroundω¯\\bar\{\\omega\}without storing or training a second parameter block\. The coordinates remain exchangeable and exactly tied, while the executed network still contains one physical path\. Parameter count, memory cost, and path computation therefore remain equal to those of a shared model\.

### Relative\-Phase Interaction

EMAN introduces two latent phase variablesθ1\\theta\_\{1\}andθ2\\theta\_\{2\}\. Their relative phase is

ϕ=θ1−θ2\.\\phi=\\theta\_\{1\}\-\\theta\_\{2\}\.\(3\)The shared component and the phase\-coupled pair component are

u\\displaystyle u=12​\[A​\(p1\)\+A​\(p2\)\],\\displaystyle=\\frac\{1\}\{2\}\\left\[A\(p\_\{1\}\)\+A\(p\_\{2\}\)\\right\],\(4\)qc\\displaystyle q\_\{c\}=β​cos⁡\(ϕ\)​\(p1⊙p2\),\\displaystyle=\\beta\\cos\(\\phi\)\\left\(p\_\{1\}\\odot p\_\{2\}\\right\),\(5\)qs\\displaystyle q\_\{s\}=κ​sin⁡\(ϕ\)​\(p1−p2\),\\displaystyle=\\kappa\\sin\(\\phi\)\\left\(p\_\{1\}\-p\_\{2\}\\right\),\(6\)q\\displaystyle q=qc\+qs\.\\displaystyle=q\_\{c\}\+q\_\{s\}\.\(7\)
The cosine term captures symmetric pair interaction\. The sine term captures the antisymmetric response\. At the tied point,p1−p2=0p\_\{1\}\-p\_\{2\}=0andqs=0q\_\{s\}=0, but the derivative along the antisymmetric coordinate remains non\-zero\. EMAN uses the fixed odd map

This map provides unit local gain atd=0d=0\. Relative\-phase release can thus expose an antisymmetric path force before path materialization\. At this stage, the phase variable changes the local response along the tied antisymmetric coordinate\. Physical capacity remains unchanged, and the forward model still executes one path\.

High\-dimensional product interactions can dominate the shared component\. EMAN controls their magnitude through norm calibration:

q^=q​sg⁡\(RMS⁡\(u\)RMS⁡\(q\)\+ε\),z=u\+ρ​q^\.\\widehat\{q\}=q\\,\\operatorname\{sg\}\\\!\\left\(\\frac\{\\operatorname\{RMS\}\(u\)\}\{\\operatorname\{RMS\}\(q\)\+\\varepsilon\}\\right\),\\qquad z=u\+\\rho\\widehat\{q\}\.\(9\)The operatorsg\\operatorname\{sg\}denotes stop\-gradient\. It blocks gradients through the calibration ratio while controlling the forward magnitude of the pair term\. Norm calibration preserves the phase\-dependent sign structure and coefficient geometry\. The same representationzzenters the task heads in every structural state\.

### Optimization Evidence Certification

Capacity growth is irreversible, so one local descent observation is not enough\. Numerical noise, batch variation, and subset\-specific behavior can all produce an isolated signal\. OEC combines multiple criteria and requires persistent evidence before a structural transition\.

LetPdP\_\{d\}project the logical path gradients onto the zero\-sum antisymmetric coordinate\. The projected gradient at the tied state is

gd=Pd⊤​∇ω1,ω2ℒ\|ω1=ω2=ω¯\.g\_\{d\}=P\_\{d\}^\{\\top\}\\nabla\_\{\\omega\_\{1\},\\omega\_\{2\}\}\\mathcal\{L\}\\bigg\|\_\{\\omega\_\{1\}=\\omega\_\{2\}=\\bar\{\\omega\}\}\.\(10\)The normalized candidate growth direction is

danti=−gd‖gd‖2\+ε\.d\_\{\\mathrm\{anti\}\}=\-\\frac\{g\_\{d\}\}\{\\\|g\_\{d\}\\\|\_\{2\}\+\\varepsilon\}\.\(11\)
Direction strength and consistency\.The antisymmetric force must first exceed the frozen numerical\-noise threshold:

St=‖gd\(t\)‖2\.S\_\{t\}=\\\|g\_\{d\}^\{\(t\)\}\\\|\_\{2\}\.\(12\)EMAN also divides the training population into deterministic subsets and constructs one candidate direction from each subset\. Their agreement is measured by

Kthalf\\displaystyle K\_\{t\}^\{\\mathrm\{half\}\}=\|cos⁡\(danti\(t,a\),danti\(t,b\)\)\|,\\displaystyle=\\left\|\\cos\\\!\\left\(d\_\{\\mathrm\{anti\}\}^\{\(t,a\)\},d\_\{\\mathrm\{anti\}\}^\{\(t,b\)\}\\right\)\\right\|,\(13\)Ktshard\\displaystyle K\_\{t\}^\{\\mathrm\{shard\}\}=mediani<j\|cos⁡\(danti\(t,i\),danti\(t,j\)\)\|\.\\displaystyle=\\operatorname\*\{median\}\_\{i<j\}\\left\|\\cos\\\!\\left\(d\_\{\\mathrm\{anti\}\}^\{\(t,i\)\},d\_\{\\mathrm\{anti\}\}^\{\(t,j\)\}\\right\)\\right\|\.\(14\)The absolute value removes the orientation ambiguity of the antisymmetric axis\. Both consistency scores must exceed their frozen thresholds\.

Descent compatibility and dynamic safety\.OEC applies a finite zero\-sum step along the candidate direction:

Dt=ℒ​\(ω¯\+η​danti,ω¯−η​danti\)−ℒ​\(ω¯,ω¯\)\.\\begin\{split\}D\_\{t\}=\\mathcal\{L\}\\left\(\\bar\{\\omega\}\+\\eta d\_\{\\mathrm\{anti\}\},\\bar\{\\omega\}\-\\eta d\_\{\\mathrm\{anti\}\}\\right\)\-\\mathcal\{L\}\\left\(\\bar\{\\omega\},\\bar\{\\omega\}\\right\)\.\\end\{split\}\(15\)The frozen test keeps the calibration state fixed\. The dynamic test recomputes it after the finite step\. Both loss changes must be negative\. OEC also checks numerical stability, NormCal stability, path\-exchange symmetry, path\-swap loss residuals, and output residuals\. All values must remain inside their frozen tolerances\.

Temporal persistence and certification\.The instantaneous criteria must pass for a fixed number of consecutive probe epochs\. The evidence vector is

et=\(St,Kthalf,Ktshard,Dtfrozen,Dtdynamic,Ut\),e\_\{t\}=\\left\(S\_\{t\},K\_\{t\}^\{\\mathrm\{half\}\},K\_\{t\}^\{\\mathrm\{shard\}\},D\_\{t\}^\{\\mathrm\{frozen\}\},D\_\{t\}^\{\\mathrm\{dynamic\}\},U\_\{t\}\\right\),\(16\)whereUtU\_\{t\}collects the dynamic safety indicators\. The certification state is

Ct=𝕀​\[eτ∈ℰsafe​for all​τ∈𝒲t\]\.C\_\{t\}=\\mathbb\{I\}\\left\[e\_\{\\tau\}\\in\\mathcal\{E\}\_\{\\mathrm\{safe\}\}\\text\{ for all \}\\tau\\in\\mathcal\{W\}\_\{t\}\\right\]\.\(17\)The setℰsafe\\mathcal\{E\}\_\{\\mathrm\{safe\}\}contains the frozen thresholds, and𝒲t\\mathcal\{W\}\_\{t\}contains the required consecutive probe epochs\. The criteria address distinct failure modes\. Strength separates the signal from the numerical floor; cross\-probe agreement rejects subset\-specific directions; descent tests local utility; safety rejects unstable probes; and persistence removes transient events\. Certification uses training data only\.

### Conditional Path Materialization

EMAN moves through three structural states\. It starts in the exact shared state:

ω1=ω2=ω¯,θ1=θ2=0\.\\omega\_\{1\}=\\omega\_\{2\}=\\bar\{\\omega\},\\qquad\\theta\_\{1\}=\\theta\_\{2\}=0\.\(18\)Only the mother tensorω¯\\bar\{\\omega\}is physically stored, and each input uses one path evaluation\.

At scheduled probe epochs, EMAN evaluates the deterministic phase certificate\. Persistent and numerically safe evidence releases the latent relative phase\. The path parameters remain tied\. A fixed adaptation window then allows the phase to evolve, and OEC begins after this window\. Insufficient evidence retains exact shared computation\.

A successful OEC decision givesCt=1C\_\{t\}=1\. EMAN copies the mother tensor into two equal\-capacity paths and applies the zero\-sum perturbation

ω1=ω¯\+ϵd​danti,ω2=ω¯−ϵd​danti\.\\omega\_\{1\}=\\bar\{\\omega\}\+\\epsilon\_\{d\}d\_\{\\mathrm\{anti\}\},\\qquad\\omega\_\{2\}=\\bar\{\\omega\}\-\\epsilon\_\{d\}d\_\{\\mathrm\{anti\}\}\.\(19\)The arithmetic mean of the two initialized paths remainsω¯\\bar\{\\omega\}\. The transition opens an antisymmetric degree of freedom and preserves the shared component at the release point\. The perturbation uses normalized parameter coordinates\. The optimizer state is copied symmetrically to both paths, so the transition preserves the accumulated optimization history\. The two paths then become independently trainable\. The task heads, task losses, optimizer, and representation interface remain unchanged\. The original multi\-task objective drives all later path differentiation\.

### Local Phase\-First Dynamics

To describe phase\-first path emergence, we consider a local expansion around the shared state\. Letdddenote the antisymmetric path coordinate, and letϕ\\phidenote the relative phase\. The local risk near an exchange\-symmetric stationary branch is

ℛ​\(d,ϕ\)=ℛ0\+aϕ2​ϕ2\+bϕ4​ϕ4\+12​d⊤​Ad​d\+ϕ​c⊤​d\+h\.o\.t\.\.\\begin\{split\}\\mathcal\{R\}\(d,\\phi\)=\\mathcal\{R\}\_\{0\}&\+\\frac\{a\_\{\\phi\}\}\{2\}\\phi^\{2\}\+\\frac\{b\_\{\\phi\}\}\{4\}\\phi^\{4\}\\\\ &\+\\frac\{1\}\{2\}d^\{\\top\}A\_\{d\}d\+\\phi c^\{\\top\}d\+\\mathrm\{h\.o\.t\.\}\.\\end\{split\}\(20\)
The coefficientbϕb\_\{\\phi\}is positive, and the matrixAdA\_\{d\}is positive definite\. These conditions describe local stability along the tied path coordinate\. The cross termϕ​c⊤​d\\phi c^\{\\top\}dcouples relative phase and antisymmetric path separation\.

###### Proposition 1\(Phase\-First Antisymmetric Exposure\)\.

Assumeaϕ<0a\_\{\\phi\}<0\. Restriction of Eq\. \([20](https://arxiv.org/html/2608.16930#S3.E20)\) tod=0d=0gives two local stationary phase offsets:

ϕ±=±−aϕ/bϕ\+o​\(\|aϕ\|\)\.\\phi\_\{\\pm\}=\\pm\\sqrt\{\-a\_\{\\phi\}/b\_\{\\phi\}\}\+o\\\!\\left\(\\sqrt\{\|a\_\{\\phi\}\|\}\\right\)\.\(21\)These offsets generate opposite antisymmetric path forces:

∇dℛ​\(0,ϕ±\)=c​ϕ±\+o​\(\|ϕ±\|\)\.\\nabla\_\{d\}\\mathcal\{R\}\(0,\\phi\_\{\\pm\}\)=c\\phi\_\{\\pm\}\+o\(\|\\phi\_\{\\pm\}\|\)\.\(22\)Path release gives the conditional local branch

d​\(ϕ\)=−Ad−1​c​ϕ\+o​\(\|ϕ\|\)\.d\(\\phi\)=\-A\_\{d\}^\{\-1\}c\\phi\+o\(\|\\phi\|\)\.\(23\)

The two phase offsets select opposite orientations of the same antisymmetric branch\. The conditional branch links the exposed force to the initial direction of path separation\. This ordering motivates phase\-first probing: relative phase can move before the physical paths separate\.

## 4Experiments

The experiments first isolate capacity demand under controlled rank constraints\. We then examine certification dynamics, released representations, and the performance–computation trade\-off on natural\-image multi\-task benchmarks\.

![Refer to caption](https://arxiv.org/html/2608.16930v1/x4.png)

\(a\) Internal phase and OEC dynamics\.

![Refer to caption](https://arxiv.org/html/2608.16930v1/x5.png)

\(b\) Population\-level Capacity\-Stress certification\.

![Refer to caption](https://arxiv.org/html/2608.16930v1/x6.png)

\(c\) Released\-path representation geometry\.

Figure 4:Certification and controlled\-capacity diagnostics\. \(a\) Phase curvature, antisymmetric strength, and cross\-half consistency; dashed lines mark frozen thresholds\. \(b\) Population\-level evidence satisfies the frozen certificate under Capacity\-Stress\. \(c\) Released paths span complementary subspaces and recover all latent factors\.### Experimental Setup

##### Benchmarks\.

The controlled benchmark contains six latent factors and three regression tasks\. Joint target ranks are 3, 4, and 6 forHigh,Medium, andLow\. Each physical path has rank 3\. NYUv2 contains semantic segmentation, depth estimation, and surface\-normal prediction\(Silbermanet al\.[2012](https://arxiv.org/html/2608.16930#bib.bib30)\)\. Standard capacity uses 64 channels\. Capacity\-Stress uses 8 channels, or0\.125×0\.125\\timespath width\. PASCAL\-Context provides a second dataset and a different task composition\(Mottaghiet al\.[2014](https://arxiv.org/html/2608.16930#bib.bib31)\)\. Task\-arrival protocols delay part of the task set and leave the transition rule unchanged\. Earlier\-arrival and gradual\-demand variants change the demand onset, so they isolate timing sensitivity without threshold retuning\.

Table 1:Controlled capacity selection\. Mean±\\pmsample standard deviation over three seeds\. Phase/Path rel\. reports the numbers of seeds with phase release and path materialization, respectively\. Best results and key EMAN release outcomes are shown in bold\.
##### Metrics and baselines\.

We report native task metrics and the frozen directional aggregate

Dt​\(M\)=st​Mt−MtHS\|MtHS\|\+εt,G​\(M\)=1T​∑t=1TDt​\(M\),D\_\{t\}\(M\)=s\_\{t\}\\frac\{M\_\{t\}\-M\_\{t\}^\{\\mathrm\{HS\}\}\}\{\|M\_\{t\}^\{\\mathrm\{HS\}\}\|\+\\varepsilon\_\{t\}\},\\qquad G\(M\)=\\frac\{1\}\{T\}\\sum\_\{t=1\}^\{T\}D\_\{t\}\(M\),\(24\)wherest=1s\_\{t\}=1for higher\-is\-better metrics andst=−1s\_\{t\}=\-1for lower\-is\-better metrics\. The constantεt\\varepsilon\_\{t\}prevents division by zero, and HS denotes Hard\-sharing\. PositiveGGdenotes an average directional improvement over Hard\-sharing\. The native task metrics show which tasks contribute to that change\. Controlled comparisons include Hard\-sharing, PhaseOnly, MeanPaths, and EMAN\. Natural\-image comparisons also include AdaShare, Recon, Conflict Trigger, and Scheduled split\. Every method uses the same data split, augmentation, heads, losses, optimizer, scheduler, and training horizon\. Structural decisions use training data only\. Full settings and per\-seed results appear in the supplementary material\.

### Controlled Capacity and Certification

Figure[4](https://arxiv.org/html/2608.16930#S4.F4)\(a,b\) connects internal optimization evidence to the structural decision\. In panel \(a\), phase\-curvature crossings are transient\. Antisymmetric strength remains above the numerical floor, but cross\-half consistency does not persist long enough for certification\. Panel \(b\) shows a population\-level certificate under Capacity\-Stress\. Table[1](https://arxiv.org/html/2608.16930#S4.T1)evaluates the same decision principle under explicit rank constraints\. InHigh, the joint task rank equals the rank\-3 path capacity\. EMAN remains shared in all seeds and matches Hard\-sharing at0\.9021±0\.08500\.9021\\pm 0\.0850macro\-R2R^\{2\}\. InLow, the joint rank rises to 6\. Hard\-sharing reaches0\.4838±0\.02070\.4838\\pm 0\.0207\. PhaseOnly releases phase in all seeds, retains one physical path, and reaches0\.4834±0\.02080\.4834\\pm 0\.0208\. EMAN releases both phase and path in all seeds and reaches0\.8661±0\.12520\.8661\\pm 0\.1252\. Phase release reveals the growth direction; path materialization supplies the missing representation rank\.

The rank\-4Mediumregime lies near the one\-path capacity limit\. EMAN remains shared and matches Hard\-sharing at0\.70860\.7086, while PhaseOnly reaches0\.70810\.7081\. Its evidence does not satisfy the frozen persistence criterion\. Figure[4](https://arxiv.org/html/2608.16930#S4.F4)\(c\) examines the representation after release under rank\-6 demand\. Principal angles range from58\.3∘58\.3^\{\\circ\}to72\.0∘72\.0^\{\\circ\}\. The concatenated representation recovers all six latent factors withR2≥0\.999892R^\{2\}\\geq 0\.999892\. The permutation control reduces mean recovery to0\.02430\.0243\. The released paths therefore add complementary representation capacity\.

### Real\-Data Performance and Computation

Table[2](https://arxiv.org/html/2608.16930#S4.T2)\(a\) reports standard\-capacity NYUv2 results\. EMAN has the best mean mIoU, depth error, andGG; Recon has the best normal error and joint loss\. EMAN reachesG=0\.0078±0\.0152G=0\.0078\\pm 0\.0152, with seed values0\.02530\.0253,0\.00050\.0005, and−0\.0023\-0\.0023\. Its clearest gains appear in semantic segmentation and depth estimation\. Table[2](https://arxiv.org/html/2608.16930#S4.T2)\(b\) reports structural selection and computation under Capacity\-Stress\. EMAN achieves the highestGGat0\.0306±0\.00490\.0306\\pm 0\.0049, compared with0\.0237±0\.00860\.0237\\pm 0\.0086for MeanPaths and0\.0115±0\.00480\.0115\\pm 0\.0048for PhaseOnly\. It materializes the second path in two of three seeds and remains shared in the other\. Its dual\-active ratio is0\.3920\.392, and it uses69\.6%69\.6\\%of MeanPaths path\-training FLOPs\.

With task arrival at epoch 21, EMAN reachesG=0\.0226±0\.0081G=0\.0226\\pm 0\.0081, compared with0\.0202±0\.00760\.0202\\pm 0\.0076for Scheduled split and0\.0114±0\.01210\.0114\\pm 0\.0121for MeanPaths\. EMAN and Scheduled split use75%75\\%of MeanPaths path\-training FLOPs\. Release occurs near the demand change, and active dual\-path training covers roughly the latter half of the schedule\. Their wall\-clock times remain close to MeanPaths\.

Table 2:NYUv2 performance and computation\. Panel \(a\) reports standard\-capacity task performance and parameter counts\. Panel \(b\) reports structural selection and computation under Capacity\-Stress\. Task metrics andGGare reported as mean±\\pmsample standard deviation over three seeds\. Phase/Path rel\. reports the number of seeds with phase release and path materialization\. Best mean performance values and EMAN release counts are shown in bold\. Capacity\-Stress uses0\.125×0\.125\\timespath width; FLOPs are normalized by MeanPaths\.##### Cross\-dataset evaluation\.

PASCAL\-Context tests the same structural rule under a different dataset and task composition\. In the epoch\-21 task\-arrival setting, EMAN releases at the demand change in all three seeds and reachesG=0\.0043±0\.0042G=0\.0043\\pm 0\.0042\. MeanPaths reaches0\.0027±0\.00180\.0027\\pm 0\.0018, and Hard\-sharing reaches0\. EMAN uses75%75\\%of the path\-training compute of always\-dual MeanPaths\. Earlier task arrival moves the same frozen release from epoch 21 to epoch 11\. EMAN reachesG=0\.0037G=0\.0037, while the matched Scheduled split reaches0\.00200\.0020\. The release time thus follows the observed demand onset without access to the task schedule\. Static demand tests immediate capacity pressure from the first epoch and therefore favors an early structural response\. Gradual demand asks whether the evidence can accumulate under a smoother onset instead of one abrupt task\-arrival event\. Detailed task\-wise results, per\-seed traces, wall\-clock profiles, and replay manifests appear in the supplementary material\.

## 5Discussion and Limitations

The central result concerns the timing of capacity, not only its final amount\. EMAN separates structural diagnosis from structural commitment\. A useful local direction does not yet establish a persistent capacity deficit\. Phase probing identifies a possible direction of growth; OEC decides whether its strength, consistency, descent, safety, and persistence justify new parameters\.

The controlled experiments expose this distinction clearly\. One path remains sufficient at joint rank three\. Joint rank six creates a clear deficit, and materialization recovers the missing factors\. The released paths also occupy complementary latent subspaces rather than duplicate the shared representation\. Natural\-image experiments show the same timing principle\. Delayed demand produces delayed release, earlier demand shifts the same frozen decision from epoch 21 to epoch 11, and static demand produces an early release\. The value of path emergence therefore depends on when additional capacity becomes useful and how long that demand persists\. Delayed and persistent demand provides the clearest setting for EMAN: a meaningful shared\-training stage precedes materialization\.

The current certificate favors reliable evidence over aggressive expansion\. TheMediumregime exposes this trade\-off\. A strict persistence rule avoids premature growth, but it can miss a mild bottleneck\. The reported savings measure path\-training evaluations before release\. Both paths remain active after materialization and during inference\. Future work can address these two limits separately: improve sensitivity near the capacity boundary and reduce post\-release cost through compression, path reuse, or conditional execution\.

## 6Conclusion

Model capacity need not be fixed before optimization begins\. EMAN starts from exact single\-path computation and uses phase\-first evidence to decide whether independent capacity should emerge\. OEC turns a local antisymmetric direction into a persistent structural decision, and the zero\-sum transition preserves the accumulated optimization state at release\. Controlled experiments show exact sharing under sufficient rank, capacity growth under a clear deficit, and complementary representation recovery\. The PhaseOnly comparison also shows that direction discovery alone does not supply missing representation capacity\. NYUv2 and PASCAL\-Context demonstrate adaptive release under changing task demand, competitive final performance, and reduced path\-training computation\.

These results position capacity growth as part of optimization itself\. EMAN offers a practical route from fixed hard sharing to evidence\-driven structural expansion\.

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Supplementary Material for EMAN: Optimization\-Driven Capacity Growth through Path Emergence in Multi\-Task Learning Anonymous Submission

## Appendix AExtended Theory and Method Details

### Physical Paths, Logical Coordinates, and Structural States

EMAN separates a physical computation graph from a local coordinate system\. Before materialization, the model stores one trainable path with parametersω¯\\bar\{\\omega\}\. It evaluates this path once:

h=f​\(x;ω¯\)\.h=f\(x;\\bar\{\\omega\}\)\.\(25\)The model then defines two logical coordinates with the same output:

p1=p2=h\.p\_\{1\}=p\_\{2\}=h\.\(26\)These coordinates support symmetry analysis and directional probes\. They do not create two encoders\. They do not double path memory\. They do not add an independent representation parameterization\.

The structural state belongs to

𝒮t∈\{Exact\-Shared,Phase\-Released,Path\-Released\}\.\\mathcal\{S\}\_\{t\}\\in\\left\\\{\\begin\{array\}\[\]\{l\}\\textsc\{Exact\-Shared\},\\;\\textsc\{Phase\-Released\},\\\\ \\textsc\{Path\-Released\}\\end\{array\}\\right\\\}\.\(27\)The state sequence is monotone:

Exact\-Shared⟶Phase\-Released\\displaystyle\\longrightarrow\\textsc\{Phase\-Released\}\(28\)⟶Path\-Released\.\\displaystyle\\longrightarrow\\textsc\{Path\-Released\}\.A failed certificate keeps the current state\. The implementation has no fallback split\. It has no validation\-driven split\. It has no task\-ID trigger\.

For two logical outputs, the representation map is

u\\displaystyle u=12​\[A​\(p1\)\+A​\(p2\)\],\\displaystyle=\\tfrac\{1\}\{2\}\\left\[A\(p\_\{1\}\)\+A\(p\_\{2\}\)\\right\],\(29\)qc\\displaystyle q\_\{c\}=β​cos⁡\(ϕ\)​\(p1⊙p2\),\\displaystyle=\\beta\\cos\(\\phi\)\(p\_\{1\}\\odot p\_\{2\}\),\(30\)qs\\displaystyle q\_\{s\}=κ​sin⁡\(ϕ\)​\(p1−p2\),\\displaystyle=\\kappa\\sin\(\\phi\)\(p\_\{1\}\-p\_\{2\}\),\(31\)q\\displaystyle q=qc\+qs,\\displaystyle=q\_\{c\}\+q\_\{s\},\(32\)q^\\displaystyle\\widehat\{q\}=q​sg⁡\(RMS⁡\(u\)RMS⁡\(q\)\+ε\),\\displaystyle=q\\,\\operatorname\{sg\}\\\!\\left\(\\frac\{\\operatorname\{RMS\}\(u\)\}\{\\operatorname\{RMS\}\(q\)\+\\varepsilon\}\\right\),\(33\)z\\displaystyle z=u\+ρ​q^,\\displaystyle=u\+\\rho\\widehat\{q\},\(34\)with relative phaseϕ=θ1−θ2\\phi=\\theta\_\{1\}\-\\theta\_\{2\}\. The odd branch uses the fixed mapB​\(d\)=dB\(d\)=d\. The same task heads consumezzin every structural state\.

Table 3:Structural states\. Logical duplication never implies physical duplication\.Table[3](https://arxiv.org/html/2608.16930#A1.T3)separates logical coordinates from trainable parameter blocks and physical path evaluations\. The exact\-shared and phase\-released states both use one encoder evaluation\. They differ only in the availability of the latent relative phase\. Independent representation capacity appears only after path release\. PhaseOnly can therefore change the probing geometry, but it cannot instantiate a second encoder\.

#### State invariants and accounting

LetNtphysN\_\{t\}^\{\\mathrm\{phys\}\}denote the number of stored physical paths,NtevalN\_\{t\}^\{\\mathrm\{eval\}\}the number of path evaluations per input, andNtindN\_\{t\}^\{\\mathrm\{ind\}\}the number of independently trainable path parameter blocks\. The three states satisfy

𝒮tNtphysNtevalNtindExact\-Shared111Phase\-Released111Path\-Released222\.\\begin\{array\}\[\]\{c\|ccc\}\\mathcal\{S\}\_\{t\}&N\_\{t\}^\{\\mathrm\{phys\}\}&N\_\{t\}^\{\\mathrm\{eval\}\}&N\_\{t\}^\{\\mathrm\{ind\}\}\\\\ \\hline\\cr\\textsc\{Exact\-Shared\}&1&1&1\\\\ \\textsc\{Phase\-Released\}&1&1&1\\\\ \\textsc\{Path\-Released\}&2&2&2\\end\{array\}\.\(35\)The logical pair\(p1,p2\)\(p\_\{1\},p\_\{2\}\)exists in all states, but it does not change these physical counts\. Consequently, the phase\-probing stage can inspect a candidate split direction without paying the path\-training cost of an always\-dual model\. After materialization, the two physical paths remain active for both training and inference\. The reported savings therefore concern the pre\-release training interval rather than sparse post\-release execution\.

### Exchange Symmetry and Coordinate Decomposition

Define the exchange operator

Π​\(p1,p2,θ1,θ2\)=\(p2,p1,θ2,θ1\)\.\\Pi\(p\_\{1\},p\_\{2\},\\theta\_\{1\},\\theta\_\{2\}\)=\(p\_\{2\},p\_\{1\},\\theta\_\{2\},\\theta\_\{1\}\)\.\(36\)The symmetric and antisymmetric parameter coordinates are

ω¯=12​\(ω1\+ω2\),d=12​\(ω1−ω2\)\.\\bar\{\\omega\}=\\tfrac\{1\}\{2\}\(\\omega\_\{1\}\+\\omega\_\{2\}\),\\qquad d=\\tfrac\{1\}\{2\}\(\\omega\_\{1\}\-\\omega\_\{2\}\)\.\(37\)The phase coordinates are

θ¯=12​\(θ1\+θ2\),ϕ=θ1−θ2\.\\bar\{\\theta\}=\\tfrac\{1\}\{2\}\(\\theta\_\{1\}\+\\theta\_\{2\}\),\\qquad\\phi=\\theta\_\{1\}\-\\theta\_\{2\}\.\(38\)Exchange preservesω¯\\bar\{\\omega\}andθ¯\\bar\{\\theta\}\. It mapsddto−d\-dandϕ\\phito−ϕ\-\\phi\.

###### Proposition 2\(Exchange invariance\)\.

The representation in Eq\. \([34](https://arxiv.org/html/2608.16930#A1.E34)\) is invariant underΠ\\Pi\.

###### Proof\.

The shared term is invariant because addition is commutative\. The cosine term is invariant because cosine is even and the product is symmetric\. The sine term is invariant because both factors change sign:

sin⁡\(−ϕ\)​\(p2−p1\)=sin⁡\(ϕ\)​\(p1−p2\)\.\\sin\(\-\\phi\)\(p\_\{2\}\-p\_\{1\}\)=\\sin\(\\phi\)\(p\_\{1\}\-p\_\{2\}\)\.\(39\)The normalization depends only on root\-mean\-square magnitudes\. It is also invariant\. The fused representation is therefore invariant\. ∎

###### Corollary 1\(No accidental path label\)\.

The tied state has no preferred path index\. A path swap leaves every task output unchanged\.

This symmetry matters for certification\. The antisymmetric axis has an orientation ambiguity\. A candidate directiondantid\_\{\\mathrm\{anti\}\}and its negative represent the same separation axis\. OEC therefore uses absolute cosine agreement across probes\.

At the exact tied state, exchange invariance also constrains local derivatives\. For a smooth invariant riskℛ​\(ω¯,d,θ¯,ϕ\)\\mathcal\{R\}\(\\bar\{\\omega\},d,\\bar\{\\theta\},\\phi\),

ℛ​\(ω¯,d,θ¯,ϕ\)=ℛ​\(ω¯,−d,θ¯,−ϕ\)\.\\mathcal\{R\}\(\\bar\{\\omega\},d,\\bar\{\\theta\},\\phi\)=\\mathcal\{R\}\(\\bar\{\\omega\},\-d,\\bar\{\\theta\},\-\\phi\)\.\(40\)Settingϕ=0\\phi=0makes the restriction even indd\. Therefore,

∇dℛ​\(ω¯,0,θ¯,0\)=0\.\\nabla\_\{d\}\\mathcal\{R\}\(\\bar\{\\omega\},0,\\bar\{\\theta\},0\)=0\.\(41\)This zero gradient does not rule out a mixed phase–path instability\. The productϕ​c⊤​d\\phi c^\{\\top\}dis exchange invariant because both factors change sign\. Its mixed derivative can therefore be nonzero at the tied state:

Hd​ϕ\\displaystyle H\_\{d\\phi\}=∂2ℛ∂d​∂ϕ\|d=0,ϕ=0\\displaystyle=\\left\.\\frac\{\\partial^\{2\}\\mathcal\{R\}\}\{\\partial d\\,\\partial\\phi\}\\right\|\_\{d=0,\\phi=0\}\(42\)=c\.\\displaystyle=c\.Phase release activates this mixed block and converts a latent coupling into a measurable antisymmetric force\. This is the local reason for probing phase before allocating a second physical path\.

### What Phase Release Can and Cannot Add

At the tied point,p1=p2=hp\_\{1\}=p\_\{2\}=h\. The odd branch vanishes:

qs=κ​sin⁡\(ϕ\)​\(h−h\)=0\.q\_\{s\}=\\kappa\\sin\(\\phi\)\(h\-h\)=0\.\(43\)The even branch remains a function of the same path output:

qc=β​cos⁡\(ϕ\)​\(h⊙h\)\.q\_\{c\}=\\beta\\cos\(\\phi\)\(h\\odot h\)\.\(44\)Phase release can therefore alter pair\-interaction coefficients\. It can expose local derivatives along the antisymmetric coordinate\. It cannot create a second independently trainable encoder\. The PhaseOnly control isolates this point empirically\. It releases phase but keeps one physical path\.

###### Proposition 3\(One\-path parameter capacity before materialization\)\.

Before path materialization, every trainable encoder parameter belongs toω¯\\bar\{\\omega\}\. Phase release does not add an independent encoder parameter block\.

###### Proof\.

The implementation stores onlyω¯\\bar\{\\omega\}\. Both logical path outputs reusef​\(x;ω¯\)f\(x;\\bar\{\\omega\}\)\. The trainable phase variables affect interaction coefficients\. They do not introduce a second mapf​\(x;ω2\)f\(x;\\omega\_\{2\}\)with independent parameters\. ∎

The proposition concerns physical representation capacity\. It does not claim that phase has no functional effect\. Phase changes the local interaction geometry\. That effect is the probe mechanism\.

### Norm Calibration Preserves the Local Geometry

Define

γ​\(u,q\)=RMS⁡\(u\)RMS⁡\(q\)\+ε\.\\gamma\(u,q\)=\\frac\{\\operatorname\{RMS\}\(u\)\}\{\\operatorname\{RMS\}\(q\)\+\\varepsilon\}\.\(45\)The forward scale is positive whenRMS⁡\(u\)\>0\\operatorname\{RMS\}\(u\)\>0\. The stop\-gradient operator freezes its backward derivative\.

###### Proposition 4\(Forward amplitude control\)\.

Forq≠0q\\neq 0andRMS⁡\(u\)\>0\\operatorname\{RMS\}\(u\)\>0,

RMS⁡\(q^\)=RMS⁡\(u\)​RMS⁡\(q\)RMS⁡\(q\)\+ε<RMS⁡\(u\)\.\\operatorname\{RMS\}\(\\widehat\{q\}\)=\\operatorname\{RMS\}\(u\)\\frac\{\\operatorname\{RMS\}\(q\)\}\{\\operatorname\{RMS\}\(q\)\+\\varepsilon\}<\\operatorname\{RMS\}\(u\)\.\(46\)The calibration preserves the sign of every scalar coefficient inqq\. It also preserves exchange invariance\. IfRMS⁡\(u\)=0\\operatorname\{RMS\}\(u\)=0, the displayed relation is non\-strict rather than strict\.

###### Proof\.

The scaleγ\\gammais positive\. Multiplication by a positive scalar preserves all signs and all pairwise directions\. Root\-mean\-square magnitude is homogeneous\. The stated identity follows directly\. Exchange invariance follows from the invariance ofRMS⁡\(u\)\\operatorname\{RMS\}\(u\)andRMS⁡\(q\)\\operatorname\{RMS\}\(q\)\. ∎

Norm calibration controls amplitude\. It does not create an antisymmetric direction\. It does not change the odd or even phase geometry\. It also does not serve as a growth certificate\.

The stop\-gradient is important for this separation of roles\. During backward propagation, the calibration factor is treated as a constant\. For a local perturbationδ​q\\delta q,

δ​q^=sg⁡\(γ\)​δ​q\.\\delta\\widehat\{q\}=\\operatorname\{sg\}\(\\gamma\)\\,\\delta q\.\(47\)Thus, the backward map applies a positive scalar to the pair\-interaction variation\. It does not add a derivative through the ratio of RMS values\. A fully differentiable normalizer would introduce extra terms proportional to∇qRMS⁡\(q\)\\nabla\_\{q\}\\operatorname\{RMS\}\(q\)and could rotate or attenuate the very direction that OEC is attempting to measure\. NormCal therefore stabilizes forward amplitude while leaving the intended local coefficient geometry intact\.

### Local Phase–Path Expansion

Letdddenote the antisymmetric path coordinate\. Letϕ\\phidenote the relative phase\. Consider the local risk

ℛ​\(d,ϕ\)=ℛ0\+aϕ2​ϕ2\+bϕ4​ϕ4\+12​d⊤​Ad​d\+ϕ​c⊤​d\+r​\(d,ϕ\)\.\\mathcal\{R\}\(d,\\phi\)=\\mathcal\{R\}\_\{0\}\+\\frac\{a\_\{\\phi\}\}\{2\}\\phi^\{2\}\+\\frac\{b\_\{\\phi\}\}\{4\}\\phi^\{4\}\+\\frac\{1\}\{2\}d^\{\\top\}A\_\{d\}d\+\\phi c^\{\\top\}d\+r\(d,\\phi\)\.\(48\)Assumebϕ\>0b\_\{\\phi\}\>0andAd≻0A\_\{d\}\\succ 0\. Letrrbe sufficiently smooth near the origin, with derivatives satisfying the stated higher\-order remainder conditions\. In particular,∇dr​\(d,ϕ\)=o​\(‖d‖\+\|ϕ\|\)\\nabla\_\{d\}r\(d,\\phi\)=o\(\\\|d\\\|\+\|\\phi\|\)\. The cross term couples phase and path separation\.

###### Proposition 5\(Phase\-first antisymmetric exposure\)\.

Asaϕ→0−a\_\{\\phi\}\\to 0^\{\-\}, assumeaϕ<0a\_\{\\phi\}<0\. The tied restrictiond=0d=0has two local nonzero phase stationary points:

ϕ±=±−aϕ/bϕ\+o​\(\|aϕ\|\)\.\\phi\_\{\\pm\}=\\pm\\sqrt\{\-a\_\{\\phi\}/b\_\{\\phi\}\}\+o\(\\sqrt\{\|a\_\{\\phi\}\|\}\)\.\(49\)They produce opposite path forces:

∇dℛ​\(0,ϕ±\)=c​ϕ±\+o​\(\|ϕ±\|\)\.\\nabla\_\{d\}\\mathcal\{R\}\(0,\\phi\_\{\\pm\}\)=c\\phi\_\{\\pm\}\+o\(\|\\phi\_\{\\pm\}\|\)\.\(50\)The conditional path branch satisfies

d​\(ϕ\)=−Ad−1​c​ϕ\+o​\(\|ϕ\|\)\.d\(\\phi\)=\-A\_\{d\}^\{\-1\}c\\phi\+o\(\|\\phi\|\)\.\(51\)

###### Proof\.

Setd=0d=0\. Phase stationarity gives

aϕ​ϕ\+bϕ​ϕ3\+o​\(\|ϕ\|3\)=0\.a\_\{\\phi\}\\phi\+b\_\{\\phi\}\\phi^\{3\}\+o\(\|\\phi\|^\{3\}\)=0\.\(52\)The two nonzero roots give Eq\. \([49](https://arxiv.org/html/2608.16930#A1.E49)\)\. Differentiate Eq\. \([48](https://arxiv.org/html/2608.16930#A1.E48)\) with respect todd\. Evaluation atd=0d=0gives Eq\. \([50](https://arxiv.org/html/2608.16930#A1.E50)\)\. The matrixAdA\_\{d\}is nonsingular, and∇dr​\(d,ϕ\)=o​\(‖d‖\+\|ϕ\|\)\\nabla\_\{d\}r\(d,\\phi\)=o\(\\\|d\\\|\+\|\\phi\|\)\. The implicit function theorem therefore gives a local branch with derivatived′​\(0\)=−Ad−1​cd^\{\\prime\}\(0\)=\-A\_\{d\}^\{\-1\}c\. ∎

###### Corollary 2\(Sign reversal\)\.

The leading path response is odd in phase:

d​\(−ϕ\)=−d​\(ϕ\)\+o​\(\|ϕ\|\)\.d\(\-\\phi\)=\-d\(\\phi\)\+o\(\|\\phi\|\)\.\(53\)The two phase probes therefore expose opposite orientations of the same local separation axis\.

Eliminating the locally stable path coordinate gives an equivalent reduced phase risk\. Substitutingd​\(ϕ\)=−Ad−1​c​ϕ\+o​\(\|ϕ\|\)d\(\\phi\)=\-A\_\{d\}^\{\-1\}c\\phi\+o\(\|\\phi\|\)into Eq\. \([48](https://arxiv.org/html/2608.16930#A1.E48)\) yields

ℛred​\(ϕ\)=ℛ0\+12​\(aϕ−c⊤​Ad−1​c\)​ϕ2\+bϕ4​ϕ4\+o​\(ϕ2\)\.\\mathcal\{R\}\_\{\\mathrm\{red\}\}\(\\phi\)=\\mathcal\{R\}\_\{0\}\+\\frac\{1\}\{2\}\\left\(a\_\{\\phi\}\-c^\{\\top\}A\_\{d\}^\{\-1\}c\\right\)\\phi^\{2\}\+\\frac\{b\_\{\\phi\}\}\{4\}\\phi^\{4\}\+o\(\\phi^\{2\}\)\.\(54\)The coupling therefore lowers the effective phase curvature byc⊤​Ad−1​cc^\{\\top\}A\_\{d\}^\{\-1\}c\. This connects the phase\-first branch calculation to the Schur\-complement condition below\. A stable restricted path block can coexist with a descending coupled mode\.

The result is local\. It provides a sufficient mechanism\. It does not make phase necessary for all growth events\. It does not prove global convergence\. It does not prove persistent specialization after release\.

### Schur\-Complement View of Coupled Curvature

The quadratic Hessian block at the tied state is

H=\[Adcc⊤aϕ\]\.H=\\begin\{bmatrix\}A\_\{d\}&c\\\\ c^\{\\top\}&a\_\{\\phi\}\\end\{bmatrix\}\.\(55\)
###### Proposition 6\(Coupled negative mode\)\.

AssumeAd≻0A\_\{d\}\\succ 0\. The matrixHHhas a negative eigenvalue if and only if

aϕ−c⊤​Ad−1​c<0\.a\_\{\\phi\}\-c^\{\\top\}A\_\{d\}^\{\-1\}c<0\.\(56\)

###### Proof\.

Apply the congruence transform

\[I−Ad−1​c01\]⊤​H​\[I−Ad−1​c01\]=\[Ad00aϕ−c⊤​Ad−1​c\]\.\\begin\{bmatrix\}I&\-A\_\{d\}^\{\-1\}c\\\\ 0&1\\end\{bmatrix\}^\{\\\!\\top\}H\\begin\{bmatrix\}I&\-A\_\{d\}^\{\-1\}c\\\\ 0&1\\end\{bmatrix\}=\\begin\{bmatrix\}A\_\{d\}&0\\\\ 0&a\_\{\\phi\}\-c^\{\\top\}A\_\{d\}^\{\-1\}c\\end\{bmatrix\}\.\(57\)Congruence preserves inertia\. The result follows\. ∎

This view separates block stability from coupled stability\. Both restricted blocks can be positive while the coupled system has a negative mode\. OEC does not use this algebraic condition as a direct trigger\. It uses measured training\-time evidence\.

### Finite\-Step Descent and the OEC Descent Test

Letgdg\_\{d\}denote the projected gradient on the zero\-sum coordinate\. The implemented direction is

vd=−gd‖gd‖2\+ε\.v\_\{d\}=\-\\frac\{g\_\{d\}\}\{\\\|g\_\{d\}\\\|\_\{2\}\+\\varepsilon\}\.\(58\)For the local bound, assumegd≠0g\_\{d\}\\neq 0and use the associated unit directionv~d=−gd/‖gd‖2\\widetilde\{v\}\_\{d\}=\-g\_\{d\}/\\\|g\_\{d\}\\\|\_\{2\}\.

###### Lemma 1\(Small\-step descent\)\.

Assume the directional gradient isLdL\_\{d\}\-Lipschitz near the tied point\. Forv~d=−gd/‖gd‖2\\widetilde\{v\}\_\{d\}=\-g\_\{d\}/\\\|g\_\{d\}\\\|\_\{2\},

ℒ​\(η​v~d\)−ℒ​\(0\)≤−η​‖gd‖2\+Ld2​η2\.\\mathcal\{L\}\(\\eta\\widetilde\{v\}\_\{d\}\)\-\\mathcal\{L\}\(0\)\\leq\-\\eta\\\|g\_\{d\}\\\|\_\{2\}\+\\frac\{L\_\{d\}\}\{2\}\\eta^\{2\}\.\(59\)The step is descending for0​<η​<2∥​gd∥2/Ld0<\\eta<2\\\|g\_\{d\}\\\|\_\{2\}/L\_\{d\}\.

###### Proof\.

Apply the descent lemma to the scalar restriction alongv~d\\widetilde\{v\}\_\{d\}\. The first\-order term is−η​‖gd‖2\-\\eta\\\|g\_\{d\}\\\|\_\{2\}\. The remainder is bounded byLd​η2/2L\_\{d\}\\eta^\{2\}/2\. ∎

OEC does not estimateLdL\_\{d\}\. It directly evaluates a finite step\. It performs a frozen\-calibration test and a dynamically recalibrated test\. Both deltas must be negative\. This design guards against a direction with a favorable surrogate but an unfavorable deployed forward map\.

### Formal Optimization Evidence Certification

At probe epochtt, let

gd\(t\)=Pd⊤​∇ω1,ω2ℒt\.g\_\{d\}^\{\(t\)\}=P\_\{d\}^\{\\top\}\\nabla\_\{\\omega\_\{1\},\\omega\_\{2\}\}\\mathcal\{L\}\_\{t\}\.\(60\)The normalized candidate direction is

danti\(t\)=−gd\(t\)‖gd\(t\)‖2\+ε\.d\_\{\\mathrm\{anti\}\}^\{\(t\)\}=\-\\frac\{g\_\{d\}^\{\(t\)\}\}\{\\\|g\_\{d\}^\{\(t\)\}\\\|\_\{2\}\+\\varepsilon\}\.\(61\)OEC evaluates five evidence groups\.

##### Direction strength\.

St=‖gd\(t\)‖2\.S\_\{t\}=\\\|g\_\{d\}^\{\(t\)\}\\\|\_\{2\}\.\(62\)It must exceed a frozen numerical floor\.

##### Cross\-probe consistency\.

Split the training population into two deterministic halves\. Each half yields a candidate direction\. The half score is

Kthalf=\|cos⁡\(danti\(t,a\),danti\(t,b\)\)\|\.K\_\{t\}^\{\\mathrm\{half\}\}=\\left\|\\cos\\\!\\left\(d\_\{\\mathrm\{anti\}\}^\{\(t,a\)\},d\_\{\\mathrm\{anti\}\}^\{\(t,b\)\}\\right\)\\right\|\.\(63\)Fixed shards produce the median score

Ktshard=mediani<j\|cos⁡\(danti\(t,i\),danti\(t,j\)\)\|\.K\_\{t\}^\{\\mathrm\{shard\}\}=\\operatorname\*\{median\}\_\{i<j\}\\left\|\\cos\\\!\\left\(d\_\{\\mathrm\{anti\}\}^\{\(t,i\)\},d\_\{\\mathrm\{anti\}\}^\{\(t,j\)\}\\right\)\\right\|\.\(64\)

##### Finite\-step descent\.

Δtfrozen\\displaystyle\\Delta\_\{t\}^\{\\mathrm\{frozen\}\}<0,\\displaystyle<0,\(65\)Δtdynamic\\displaystyle\\Delta\_\{t\}^\{\\mathrm\{dynamic\}\}<0\.\\displaystyle<0\.\(66\)The second test recomputes the dynamic calibration state\.

##### Dynamic safety\.

Path\-swap loss residuals, output residuals, finite\-value checks, and NormCal checks must stay inside frozen tolerances\.

##### Temporal persistence\.

Letete\_\{t\}collect all instantaneous evidence\. Letℰsafe\\mathcal\{E\}\_\{\\mathrm\{safe\}\}denote the accepted set\. LetWtW\_\{t\}denote the frozen persistence window\. The full decision is

Ct=𝕀​\[eτ∈ℰsafe​for every​τ∈Wt\]\.C\_\{t\}=\\mathbb\{I\}\\\!\\left\[e\_\{\\tau\}\\in\\mathcal\{E\}\_\{\\mathrm\{safe\}\}\\text\{ for every \}\\tau\\in W\_\{t\}\\right\]\.\(67\)
###### Proposition 7\(Orientation\-free consistency\)\.

Absolute cosine consistency is invariant to an independent sign reversal of any probe direction\.

###### Proof\.

For signssi,sj∈\{−1,1\}s\_\{i\},s\_\{j\}\\in\\\{\-1,1\\\},

\|cos⁡\(si​di,sj​dj\)\|=\|si​sj​cos⁡\(di,dj\)\|=\|cos⁡\(di,dj\)\|\.\\left\|\\cos\(s\_\{i\}d\_\{i\},s\_\{j\}d\_\{j\}\)\\right\|=\\left\|s\_\{i\}s\_\{j\}\\cos\(d\_\{i\},d\_\{j\}\)\\right\|=\\left\|\\cos\(d\_\{i\},d\_\{j\}\)\\right\|\.\(68\)∎

###### Proposition 8\(Conservative decision rule\)\.

Path materialization occurs only when every frozen component passes over the full persistence window\. A failed certificate implies no structural action\. It does not imply that one\-path capacity is sufficient\.

###### Proof\.

The first statement follows from Eq\. \([67](https://arxiv.org/html/2608.16930#A1.E67)\)\. The second statement follows becauseCt=0C\_\{t\}=0can result from any failed evidence term\. A false negative is therefore possible\. The controlled rank\-four case is an observed example\. ∎

All probe samples come from the training population\. Validation and test data never enter the structural decision\.

Table 4:Interpretation of the five OEC evidence groups\. Passing one group is not sufficient; the certificate is the conjunction over all groups and the persistence window\.Table[4](https://arxiv.org/html/2608.16930#A1.T4)clarifies why OEC is intentionally conservative\. Direction strength answers whether a nontrivial local force is present\. Consistency asks whether the axis is population\-level rather than subset\-specific\. The two descent tests evaluate the actual finite perturbation under frozen and recomputed calibration\. Safety protects the structural transition itself\. Persistence turns the instantaneous conjunction into a training\-time decision\. No single statistic is interpreted as a capacity certificate in isolation\.

#### End\-to\-end decision sequence

The implemented transition follows a fixed order\.

1. 1\.Train the exact\-shared model with one physical path\.
2. 2\.At a scheduled phase\-probe epoch, evaluate phase curvature and the frozen phase\-safety checks on the training population\.
3. 3\.Release the latent relative phase only after the phase persistence rule passes\. Keep the physical path parameters tied\.
4. 4\.Run the fixed phase\-adaptation window\. This allows the odd interaction to expose an antisymmetric force\.
5. 5\.Form deterministic population halves and shards\. Construct one zero\-sum direction from each partition\.
6. 6\.Evaluate strength, absolute\-cosine agreement, frozen and dynamic finite\-step descent, and all safety residuals\.
7. 7\.Materialize two paths only after the full OEC vector passes for the frozen number of consecutive probe epochs\.
8. 8\.Copy the mother parameters and optimizer state symmetrically\. Continue ordinary first\-order multi\-task optimization with no diversity loss\.

This sequence distinguishes phase release from path release\. A run can pass the first decision and fail the second\. It can also fail the phase decision and never execute a path\-force probe\. Both outcomes are represented explicitly in the per\-seed evidence\.

### Zero\-Sum Materialization and Local Continuity

A successful path certificate creates

ω1=ω¯\+ϵd​danti,ω2=ω¯−ϵd​danti\.\\omega\_\{1\}=\\bar\{\\omega\}\+\\epsilon\_\{d\}d\_\{\\mathrm\{anti\}\},\\qquad\\omega\_\{2\}=\\bar\{\\omega\}\-\\epsilon\_\{d\}d\_\{\\mathrm\{anti\}\}\.\(69\)
###### Proposition 9\(Centroid preservation\)\.

The zero\-sum split preserves the mother parameter exactly:

12​\(ω1\+ω2\)=ω¯\.\\tfrac\{1\}\{2\}\(\\omega\_\{1\}\+\\omega\_\{2\}\)=\\bar\{\\omega\}\.\(70\)

###### Proof\.

Substitute Eq\. \([69](https://arxiv.org/html/2608.16930#A1.E69)\)\. The perturbations cancel\. ∎

###### Proposition 10\(First\-order cancellation for symmetric fusion\)\.

LetF​\(ω1,ω2\)F\(\\omega\_\{1\},\\omega\_\{2\}\)be twice continuously differentiable and exchange symmetric\. Then

F​\(ω¯\+ϵ​d,ω¯−ϵ​d\)=F​\(ω¯,ω¯\)\+O​\(ϵ2\)\.F\(\\bar\{\\omega\}\+\\epsilon d,\\bar\{\\omega\}\-\\epsilon d\)=F\(\\bar\{\\omega\},\\bar\{\\omega\}\)\+O\(\\epsilon^\{2\}\)\.\(71\)

###### Proof\.

Exchange symmetry gives equal partial derivatives at the tied point:∇1F=∇2F\\nabla\_\{1\}F=\\nabla\_\{2\}F\. The first\-order Taylor term is

ϵ​∇1F⊤​d−ϵ​∇2F⊤​d=0\.\\epsilon\\nabla\_\{1\}F^\{\\top\}d\-\\epsilon\\nabla\_\{2\}F^\{\\top\}d=0\.\(72\)The remaining change is second order\. ∎

The proposition does not claim exact function preservation for an arbitrary deep nonlinear network\. It establishes local first\-order continuity under the symmetric interface\. The optimizer state is copied symmetrically\. This keeps the accumulated history aligned at release time\.

### Deep\-Network Local Bridge

The local theory is stated in representation coordinates\. The implemented model uses deep parameters\. Leth​\(ω\)h\(\\omega\)denote a local feature tensor\. LetJ=∂h/∂ωJ=\\partial h/\\partial\\omegabe its Jacobian at the probe checkpoint\. Letghg\_\{h\}be a phase\-exposed feature\-space direction\.

###### Proposition 11\(Local parameter pullback\)\.

The corresponding parameter gradient is

gω=J⊤​gh\.g\_\{\\omega\}=J^\{\\top\}g\_\{h\}\.\(73\)IfJ⊤​gh≠0J^\{\\top\}g\_\{h\}\\neq 0, the feature\-space candidate induces a nonzero local parameter direction\.

###### Proof\.

The result is the chain rule\. ∎

A complementary statement starts from a desired feature perturbation\. Letvhv\_\{h\}lie in the image ofJJ\. Then there exists a parameter perturbationvωv\_\{\\omega\}withJ​vω=vhJv\_\{\\omega\}=v\_\{h\}; the minimum\-norm choice isJ†​vhJ^\{\\dagger\}v\_\{h\}\. For a differentiable task objectiveℓ​\(h\)\\ell\(h\),

dd​η​ℓ​\(h​\(ω¯\+η​vω\)\)\|η=0=∇hℓ​\(h\)⊤​vh\.\\left\.\\frac\{d\}\{d\\eta\}\\ell\\bigl\(h\(\\bar\{\\omega\}\+\\eta v\_\{\\omega\}\)\\bigr\)\\right\|\_\{\\eta=0\}=\\nabla\_\{h\}\\ell\(h\)^\{\\top\}v\_\{h\}\.\(74\)Thus, a descending feature\-space direction that is locally realizable by the network has a parameter\-space representative with the same first\-order loss change\. This statement is useful for interpreting the frozen\-feature and local\-head diagnostics\. It remains checkpoint\-local and makes no global optimization claim\.

This bridge is local\. It applies at the probed checkpoint\. It does not prove that every initialization reaches a separating branch\. It does not prove that the post\-release trajectories remain complementary\. These questions require empirical evidence\.

### Numerical Local Checks

A deterministic float64 implementation checks the local linear algebra\. It is separate from deep training\. Table[5](https://arxiv.org/html/2608.16930#A1.T5)reports the frozen diagnostics from the original theory package\.

Table 5:Numerical checks for the local coupled model\. Positive restricted blocks and a negative coupled mode illustrate the Schur\-complement effect\.The first three rows of Table[5](https://arxiv.org/html/2608.16930#A1.T5)instantiate the coupled\-curvature argument\. Each restricted block has minimum eigenvalue2\.02\.0, yet the complete system has a negative eigenvalue of−1\.1231\-1\.1231\. The minimum mode contains both path and phase energy\. It is therefore a genuinely coupled mode rather than an instability confined to one coordinate\. The analytic, autograd, finite\-difference, HVP, and Lanczos agreements provide independent checks of the same local calculation\. The split\-norm range verifies that the zero\-sum perturbations used in the local continuity tests remain small relative to the mother parameter norm\.

### Theory Claim Boundary

The theory supports a local sufficiency statement\. Relative phase can expose an antisymmetric force before physical separation\. A persistent, descending, and safe force can justify a local zero\-sum release\. Three levels of statement must remain distinct\.

Table 6:Scope of the theoretical statements\. The table separates proved local properties from empirically evaluated consequences and claims that are outside the current result\.Table[6](https://arxiv.org/html/2608.16930#A1.T6)makes the claim hierarchy explicit\. The local result is deliberately narrower than the empirical claim\. It identifies a mechanism by which a tied dense model can expose a meaningful separation direction\. OEC then asks whether this mechanism is present and persistent in a concrete training run\. The experiments test the consequences without converting them into universal theorems\.

## Appendix BExtended Experimental Results

### Reporting Protocol and Evidence Organization

Table 7:Detailed evaluation map\. Wall\-clock values are descriptive and hardware\-specific\.For natural\-image tasks, the directional aggregate is

Dt​\(M\)=st​Mt−MtHS\|MtHS\|\+εt,G​\(M\)=1T​∑t=1TDt​\(M\)\.D\_\{t\}\(M\)=s\_\{t\}\\frac\{M\_\{t\}\-M\_\{t\}^\{\\mathrm\{HS\}\}\}\{\|M\_\{t\}^\{\\mathrm\{HS\}\}\|\+\\varepsilon\_\{t\}\},\\qquad G\(M\)=\\frac\{1\}\{T\}\\sum\_\{t=1\}^\{T\}D\_\{t\}\(M\)\.\(75\)The signsts\_\{t\}is positive for higher\-is\-better metrics and negative for lower\-is\-better metrics\. All native task metrics remain visible\.

Table[7](https://arxiv.org/html/2608.16930#A2.T7)organizes the evidence by the question each protocol can answer\. The controlled benchmark isolates capacity because the joint target rank is known\. Standard\-capacity natural\-image experiments test external task behavior but do not guarantee a capacity bottleneck\. The Capacity\-Stress protocol deliberately narrows the path and records the full population certificate\. Task\-arrival protocols separate the period before and after additional demand\. PASCAL\-Context provides a dataset and task\-composition change\. The early\-arrival and ramp protocols test timing sensitivity and the shape of the demand transition\. This ordering prevents a result from one protocol from being used to answer a question that only another protocol identifies\.

ForGG, Hard\-sharing is the per\-seed reference and therefore has value zero by definition\. A positiveGGdenotes a directional improvement averaged over native metrics\. A negativeGGdenotes a directional decrease\. The aggregate is not a substitute for the task\-level columns\. We report both because a single average can hide compensating improvements and degradations\.

### Implementation and Decision Settings

All values below are transcribed from archived configuration or implementation files\. Missing fields are not inferred\. Table[10](https://arxiv.org/html/2608.16930#A2.T10)separates the standard OEC contract from the population settings used only by Capacity\-Stress\. Table[11](https://arxiv.org/html/2608.16930#A2.T11)records which baselines begin shared, remain shared, or expose two trainable paths\.

\(a\) Controlled and NYUv2 data/model settings

\(b\) PASCAL data/model settings

Table 8:Data and model settings\. Panel \(a\) records controlled and NYUv2 protocols; panel \(b\) records the two PASCAL implementations\.\(a\) Optimization settings

\(b\) Hardware and reproducibility settings

Table 9:Optimization and reproducibility settings\. Panel \(a\) records training contracts; panel \(b\) records hardware and determinism\.\(a\) General frozen decision settings

\(b\) NYUv2 OEC\-specific settings

Table 10:Frozen EMAN decision settings\. Panel \(a\) contains general contracts; panel \(b\) contains NYUv2 OEC thresholds\. Capacity\-Stress population settings are recorded separately in Table[20](https://arxiv.org/html/2608.16930#A2.T20)\.Table 11:Baseline structural states and release rules\.\(a\) Parameter and protocol matching

\(b\) Compute\-accounting contract

Table 12:Matching and compute contracts\. Panel \(a\) records capacity/protocol matching; panel \(b\) defines active\-path and total\-cost accounting\.
### Rank\-Controlled Results for Every Seed

Each physical path has rank\-three capacity\. TheHighdemand has joint rank three\. TheLowdemand has joint rank six\. Table[13](https://arxiv.org/html/2608.16930#A2.T13)reports every task and every seed\. EMAN exactly matches Hard\-sharing in everyHighrun\. It performs no release\. InLow, PhaseOnly releases phase in every run but stays at the Hard\-sharing level\. EMAN materializes in every run and recovers a much larger macro\-R2R^\{2\}\.

Table 13:Complete controlled results\. Phase/Path reports event occurrence in each run\.![Refer to caption](https://arxiv.org/html/2608.16930v1/Figures/controlled_capacity_Seed_1.png)

![Refer to caption](https://arxiv.org/html/2608.16930v1/Figures/controlled_capacity_Seed_2.png)

![Refer to caption](https://arxiv.org/html/2608.16930v1/Figures/controlled_capacity_Seed_3.png)

Figure 5:Per\-seed controlled results\. The three plots expose the variance behind the aggregate main\-paper table\. The structural decision remains stable: allHighruns stay shared, and allLowEMAN runs release\.Table[13](https://arxiv.org/html/2608.16930#A2.T13)and Figure[5](https://arxiv.org/html/2608.16930#A2.F5)show two types of stability\. First, the structural decision is stable across all three seeds: everyHighrun remains exact\-shared, and everyLowEMAN run releases phase and path\. Second, the amount of predictive recovery varies with the seed\. Seed 1 is a near\-ceiling realization\. Seeds 2 and 3 are harder, but EMAN still raises theLowmacro\-R2R^\{2\}from0\.46390\.4639to0\.75180\.7518and from0\.48230\.4823to0\.84660\.8466, respectively\. This distinction matters\. The certificate is evaluated as a structural decision, while final predictive performance remains subject to ordinary optimization variability after release\.

The PhaseOnly rows isolate the role of latent phase\. In eachLowrun, phase is released but the path is not materialized\. The macro\-R2R^\{2\}remains within0\.00080\.0008of Hard\-sharing\. The result rules out the interpretation that phase freedom by itself supplies the missing rank\-six representation\. In contrast, EMAN changes the physical path count and recovers all three tasks simultaneously\. The gains are therefore associated with independent path capacity rather than a task\-specific loss trade\-off\.

TheMediumdemand has joint rank four\. It exceeds one\-path rank by one\. EMAN stays shared and obtains0\.70860\.7086macro\-R2R^\{2\}\. Hard\-sharing obtains0\.70860\.7086, and PhaseOnly obtains0\.70810\.7081\. This setting lies close to the single\-path boundary\. The frozen evidence rule does not certify a persistent benefit from structural growth\. We therefore retain it as a conservative false negative rather than retuning the threshold after observing the result\. The case identifies the sensitivity limit of the current certificate\. It does not affect the stronger rank\-six capacity result, where phase and path release occur in every seed\.

### Archived Structural Events and Diagnostic Boundaries

The Medium joint\-rank\-four result is retained only as the frozen aggregate summary: Hard\-sharing0\.70860\.7086, PhaseOnly0\.70810\.7081, and EMAN0\.70860\.7086macro\-R2R^\{2\}\. The archive does not contain a matching three\-seed table for these numbers\. Per\-seed values are therefore not reconstructed\. A separate single\-run directory namedrank\_controlled\_core/MEDIUMcontains a different final value and is not merged with the frozen summary\.

Table 14:Frozen Medium evidence boundary\. “Not archived” is not replaced by a reconstructed value\.The standard\-capacity NYUv2 external comparison uses the frozen static matched adapter\. Its archived method states record phase release at epoch 1 and path release at epoch 2 for each seed\. These runs do not contain OEC probe traces and therefore cannot support a claim about persistent phase evidence\. They establish a matched static performance comparison only; the separate OEC and task\-arrival protocols supply structural\-decision evidence\.

Table 15:Archived structural states for the static matched adapters\.The static matched adapters reach the dual\-path state at epochs 1 and 2\. These events define the static comparison\. The task\-arrival protocols provide the corresponding evidence for release timing under changing demand\. Static PASCAL uses a dual\-path ratio of39/4039/40and a path\-training ratio of79/8079/80in each archived run\. Machine\-readable per\-seed records, including hardware\-specific wall\-clock values, are provided innyuv2\_standard\_structural\_events\.csvandpascal\_static\_events\.csv\.

### Coalition Diagnostics and Specialization Boundary

The K=2 coalition audit reports path main effects, Shapley contributions, pair synergy, CSS, HPS, and task assignments from the archived controlled checkpoint\. The retained LOW result has near\-zero CSS and assigns all tasks to the same path\. Pair synergy is reported separately and is not interpreted as path specialization\. Consequently, the evidence supports complementary released subspaces, but not task\-wise expert specialization\. Full machine\-readable values appear incoalition\_diagnostics\.csv\.

### Detailed Representation Geometry

The representation analysis uses the releasedLowmodel from Seed 1\. Linear held\-out probes map Path 1, Path 2, and their concatenation to the six latent factors\. Table[16](https://arxiv.org/html/2608.16930#A2.T16)reports all values\. Neither individual path recovers every factor\. The concatenation recovers each factor withR2≥0\.999892R^\{2\}\\geq 0\.999892\.

Table 16:Factor\-wise held\-out recovery for the released Seed 1 model\.Table 17:Canonical\-correlation spectrum between the concatenated path representation and the latent state\.Table 18:Additional latent\-geometry diagnostics for Seed 1\. Effective rank measures spectral uniformity\. It is not algebraic rank\.The three principal angles are58\.2795∘58\.2795^\{\\circ\},59\.3178∘59\.3178^\{\\circ\}, and71\.9900∘71\.9900^\{\\circ\}\. The normalized projector overlap is0\.2108050\.210805\. The unique conditional gains are0\.5008120\.500812and0\.4977770\.497777\.

A permutation control gives mean factorR2=0\.024310R^\{2\}=0\.024310and maximum factorR2=0\.057603R^\{2\}=0\.057603\. These results support complementary capacity\. They do not support one expert per task\.

Table[16](https://arxiv.org/html/2608.16930#A2.T16)provides the most direct factor\-by\-factor view\. Path 1 is strongest ons2s\_\{2\},s3s\_\{3\}, ands6s\_\{6\}\. Path 2 has relatively larger recovery ons4s\_\{4\}ands5s\_\{5\}\. Neither path alone reaches full recovery\. The concatenated representation recovers every factor\. Its minimum factor score is0\.9998920\.999892\. This pattern is incompatible with simple duplication\. Identical path subspaces would not recover the missing factors after concatenation\.

Table[17](https://arxiv.org/html/2608.16930#A2.T17)gives a basis\-invariant check\. Four canonical correlations are numerically one, and the remaining two are0\.9540430\.954043and0\.9400910\.940091\. The complete representation therefore preserves all six latent directions with high linear alignment\. Table[18](https://arxiv.org/html/2608.16930#A2.T18)adds three controls\. The principal angles and low projector overlap show that the individual path subspaces are not identical\. The two unique conditional gains are nearly balanced\. The permutation control collapses mean recovery from0\.9998990\.999899to0\.0243100\.024310, which rules out a probe or evaluation artifact\.

Table 19:LOW controlled coalition diagnostics for EMAN at the archived seed\-2027 final checkpoint\. Pair synergy is task\-wise interaction, not specialization\.A separate coalition diagnostic gives near\-zero task\-wise specialization for EMAN\. Its HPS is1\.27571\.2757, while task\-wise pair synergies are2\.70792\.7079,3\.64723\.6472, and3\.80953\.8095\. All three tasks receive the same path assignment under the main\-effect rule\. The result rejects a task\-expert interpretation\. It does not reject complementary latent information\.

The representation conclusion is consequently narrower and more useful than an expert\-specialization claim\. The two paths become complementary carriers of latent information, but the task heads continue to use their fused representation\. No path is assigned to a predefined task, and the evidence does not require such an assignment\.

### NYUv2 Capacity\-Stress: Complete Per\-Seed Performance

Capacity\-Stress combines an eight\-channel path with the frozenNYUV2\_D11demand profile: segmentation and depth are active during epochs 1–10, and surface\-normal demand is activated from epoch 11\. The training horizon is 80 epochs and the cosine scheduler usesTmax=80T\_\{\\max\}=80\. Each population probe uses all 795 training samples and their horizontal flips\. This yields 1,590 states\. The states are divided into eight deterministic shards\. Seed 1 and Seed 3 release\. Seed 2 remains shared\. Table[21](https://arxiv.org/html/2608.16930#A2.T21)reports every method and seed\.

Table 20:Recorded population\-certificate configuration\. The table reports archived settings rather than tuned post hoc values\.Table[20](https://arxiv.org/html/2608.16930#A2.T20)records the decision protocol rather than a performance result\. Every probe uses the full training population and its registered flip augmentation\. Deterministic shards keep the population partition fixed across probe epochs\. The10−310^\{\-3\}path\-strength floor screens numerically negligible forces\. Separate two\-pass persistence rules govern phase and path release\. Validation and test samples are absent from all decisions\. These settings were frozen before the reported per\-seed outcomes\.

Table 21:Capacity\-Stress performance for all seeds\. Higher is better forGGand mIoU\. Lower is better for the remaining columns\.![Refer to caption](https://arxiv.org/html/2608.16930v1/Figures/capacity_stress_performance_Seed_1.png)

![Refer to caption](https://arxiv.org/html/2608.16930v1/Figures/capacity_stress_performance_Seed_2.png)

![Refer to caption](https://arxiv.org/html/2608.16930v1/Figures/capacity_stress_performance_Seed_3.png)

![Refer to caption](https://arxiv.org/html/2608.16930v1/Figures/capacity_stress_aggregate.png)

Figure 6:Capacity\-Stress per\-seed and aggregateGG\. The first row retains every registered seed; the lower panel reports the three\-seed mean with sample\-standard\-deviation error bars\.Table[21](https://arxiv.org/html/2608.16930#A2.T21)and Figure[6](https://arxiv.org/html/2608.16930#A2.F6)expose the per\-seed task trade\-offs\. EMAN has the highest three\-seed meanGG, while the per\-seed ranking remains mixed\. MeanPaths allocates both paths from epoch 1\. EMAN has a positive directional aggregate in all three seeds, while structural materialization occurs in Seeds 1 and 3 only\. Seed 1 releases late and has its largest directional gain on depth\. Seed 2 does not release and remains in exact single\-path computation\. Its final directional aggregate remains positive under the registered evaluation, with the largest directional gain on surface normals\. Seed 3 releases early and shows positive directional gains on all three registered task metrics, with its largest gain on semantic segmentation\. The native task metrics remain mixed, so the aggregate claim is limited to the registered mean rather than uniform task\-wise superiority\.

The non\-release Seed 2 is informative rather than missing data\. Its phase curvature does not satisfy the frozen confidence rule\. The model therefore keeps exact sharing\. This outcome shows that Capacity\-Stress increases the chance of materialization but does not hard\-code a split\. It also explains why the three\-seed mean trades some performance against fewer active path evaluations\.

Table 22:Capacity\-Stress three\-seed aggregate\. EMAN has the highest registered meanGG; the error column is the sample standard deviation\.Table[23](https://arxiv.org/html/2608.16930#A2.T23)reports event timing and systems measurements\. Seed 1 releases phase at epoch 32 and paths at epoch 49\. Seed 3 releases at epochs 11 and 19\. Seed 2 has no release\. The final path distances are0\.09170\.0917,0, and0\.53030\.5303\. The dual\-active ratios are0\.4000\.400,0, and0\.7750\.775\.

Table 23:Capacity\-Stress event and system details\. Total FLOPs are normalized within each seed by MeanPaths\. Wall\-clock is hardware\-specific\.Table 24:Phase\-certificate snapshots\. Seed 2 reports its terminal probe\. Its upper confidence bound remains positive, so the frozen phase rule does not accept the event\.Table 25:Accepted path\-certificate snapshots\. Both released seeds exceed the recorded10−310^\{\-3\}strength floor, pass consistency, produce negative finite\-step loss changes, and satisfy the two\-probe persistence rule\. Seed 2 never enters path certification because phase release is not accepted\.Table[23](https://arxiv.org/html/2608.16930#A2.T23)separates release timing from final path distance\. Seed 1 materializes at epoch 49\. Its dual\-active ratio is40%40\\%\. Seed 3 materializes at epoch 19 and has a77\.5%77\.5\\%ratio\. The later release in Seed 1 leaves less time for divergence\. Seed 2 has zero distance because no second physical path exists\. MeanPaths is dual from epoch 1 and provides the path\-compute reference\. The wall\-clock columns also expose population\-probe overhead\. Certification can take more time even when it reduces path evaluations\.

Table[24](https://arxiv.org/html/2608.16930#A2.T24)explains the first gate\. Seeds 1 and 3 have negative normalized curvature with negative upper confidence bounds, pass finite\-difference and safety checks, and satisfy the two\-probe persistence rule\. Seed 2 has negative point curvature but a positive upper confidence bound\. OEC therefore treats the evidence as insufficiently stable and does not release phase\. Table[25](https://arxiv.org/html/2608.16930#A2.T25)explains the second gate\. The two accepted seeds exceed the strength floor, show high half\- and shard\-level cosine agreement, and reduce loss under both finite\-step tests\. Seed 2 has no path\-certificate row because the first gate never opens\.

![Refer to caption](https://arxiv.org/html/2608.16930v1/Figures/certificate_evidence_Seed_2.png)

![Refer to caption](https://arxiv.org/html/2608.16930v1/Figures/certificate_evidence_Seed_3.png)

Figure 7:Expanded population certificates under Capacity\-Stress\. Seed 2 never produces a persistent phase certificate and remains in exact single\-path computation\. Seed 3 passes the path gate over the frozen persistence window and releases at epoch 19\.Figure[7](https://arxiv.org/html/2608.16930#A2.F7)contrasts the two decisive certificate patterns\. In Seed 2, only the phase\-curvature history is present; path\-force strength and consistency are absent because the run never enters the OEC path\-probing state\. In Seed 3, the phase event is followed by a stable adaptation interval and then by two consecutive path\-certificate passes\. The comparison demonstrates that a blank path\-force panel denotes an unexecuted probe stage, not a zero\-valued measurement or a plotting failure\.

![Refer to caption](https://arxiv.org/html/2608.16930v1/Figures/capacity_stress_release_events.png)Figure 8:Capacity\-Stress release events\. Seeds 1 and 3 pass the registered phase and path decisions; Seed 2 remains shared\. Final path distances and active\-path ratios are reported in Table[23](https://arxiv.org/html/2608.16930#A2.T23)\.Figure[8](https://arxiv.org/html/2608.16930#A2.F8)provides the event trace and the final structural consequence in one view\. Release markers identify when each state transition occurs\. Path distance remains zero until materialization and grows only after the paths become independently trainable\. Seed 1 and Seed 3 follow this expected ordering\. Seed 2 contains no marker and remains at zero distance\. The figure therefore links the discrete certificate decision to the continuous post\-release separation trajectory\.

EMAN has the highest meanGGunder the registered Capacity\-Stress protocol\. MeanPaths trains two paths throughout\. EMAN uses69\.6%69\.6\\%of MeanPaths path\-training evaluations on average\. Its estimated total FLOPs are97\.7%97\.7\\%because the shared heads, fusion, and other non\-path components remain active\. Population certification raises mean wall\-clock to2\.24×2\.24\\timesMeanPaths in this research implementation\. This gap separates the algorithmic path\-evaluation saving from current systems overhead\. It motivates batched or lower\-frequency certificate implementations; it does not change the recorded path\-compute accounting\.

### Standard\-Capacity NYUv2: Seed Dependence and Systems Cost

The main paper reports the three\-seed aggregate\. Table[27](https://arxiv.org/html/2608.16930#A2.T27)gives every seed\. EMAN has positiveGGin Seed 1, near\-zeroGGin Seed 2, and negativeGGin Seed 3\. The aggregate claim is therefore competitive performance\. It is not uniform superiority\.

Table 26:Mean NYUv2 system measurements over three seeds\. Wall\-clock reports mean±\\pmsample standard deviation\.![Refer to caption](https://arxiv.org/html/2608.16930v1/Figures/NYUv2_external_baselines_Seed_1.png)

![Refer to caption](https://arxiv.org/html/2608.16930v1/Figures/NYUv2_external_baselines_Seed_2.png)

![Refer to caption](https://arxiv.org/html/2608.16930v1/Figures/NYUv2_external_baselines_Seed_3.png)

Figure 9:Task\-level NYUv2 external comparisons for all seeds\. The plots show the positive Seed 1 result and the weaker Seed 2 and Seed 3 results\.Figure[9](https://arxiv.org/html/2608.16930#A2.F9)visualizes the same seed dependence at task level\. The strong Seed 1 result is not hidden by averaging, and the weaker Seed 2 and Seed 3 cases are retained at equal scale\. Across the three plots, EMAN remains close to the leading method on each task\. The variation is mainly in the direction and magnitude of the small differences, not in catastrophic task collapse\. AdaShare is the clear unstable baseline in this matched implementation\.

Table[26](https://arxiv.org/html/2608.16930#A2.T26)separates model size, peak memory, and elapsed time\. EMAN uses 31\.44M parameters, compared with 26\.20M for Hard\-sharing and 28\.82M for MeanPaths\. Its mean peak memory is below AdaShare and Recon\. Its mean wall\-clock matches Hard\-sharing and remains close to MeanPaths, although the larger standard deviation reflects seed\-dependent structural behavior and measurement variability\. Recon and AdaShare have higher runtime in this implementation\. These measurements describe the reported system and do not establish a portable speed ranking across hardware or software stacks\.

Table 27:Standard\-capacity NYUv2 results for every seed\. Boldface marks the strongestGGvalue within each seed\. Since Hard\-sharing defines the reference, itsGGis zero by construction\.Table[27](https://arxiv.org/html/2608.16930#A2.T27)shows why the standard\-capacity claim is stated at the aggregate level\. Seed 1 provides a clear positive EMAN result: it improves segmentation, depth, normal error, andGGrelative to its Hard\-sharing reference\. Seed 2 is near the reference and remains competitive with the external baselines\. Seed 3 is slightly negative inGG, although its task values stay close to Hard\-sharing\. The three\-seed mean is therefore evidence of competitive multi\-task behavior, not a claim that EMAN wins every seed or every metric\.

![Refer to caption](https://arxiv.org/html/2608.16930v1/Figures/NYUv2_task_arrival_Seed_1.png)

![Refer to caption](https://arxiv.org/html/2608.16930v1/Figures/NYUv2_task_arrival_Seed_2.png)

![Refer to caption](https://arxiv.org/html/2608.16930v1/Figures/NYUv2_task_arrival_Seed_3.png)

Figure 10:NYUv2 task\-arrival performance and compute for all seeds\. EMAN and the matched scheduled split use the same estimated path\-training budget\.
### NYUv2 Task Arrival at Epoch 21

The task\-arrival protocol delays part of the demand until epoch 21\. EMAN is not given this epoch\. The frozen transition releases at the demand change in all three runs\. Table[28](https://arxiv.org/html/2608.16930#A2.T28)gives every seed\. EMAN achievesG=0\.0226±0\.0081G=0\.0226\\pm 0\.0081\. Scheduled split obtains0\.0202±0\.00760\.0202\\pm 0\.0076\. MeanPaths obtains0\.0114±0\.01210\.0114\\pm 0\.0121\.

Table 28:NYUv2 task\-arrival results\. The estimated path\-training FLOPs are normalized by always\-dual MeanPaths\. Boldface marks the strongestGGvalues and the matched delayed path budget\.Table[28](https://arxiv.org/html/2608.16930#A2.T28)controls the main compute comparison\. Scheduled split and EMAN materialize at the same epoch and use the same0\.750\.75path\-FLOP ratio\. Their difference is the release direction: Scheduled split uses the matched random split, while EMAN uses the certified antisymmetric direction\. EMAN exceeds Scheduled split in all three seeds by0\.00200\.0020,0\.00300\.0030, and0\.00240\.0024inGG\. The consistent sign supports the reported aggregate, but the sample is too small to establish a universal direction advantage\.

Figure[10](https://arxiv.org/html/2608.16930#A2.F10)shows that the comparison is not produced by a wall\-clock discrepancy\. Scheduled split and EMAN have nearly identical elapsed times, and both remain close to MeanPaths\. The saved path evaluations occur before epoch 21\. After release, all three two\-path methods execute the same number of paths\. The result therefore isolates delayed allocation rather than sparse inference\.

### PASCAL\-Context as a Cross\-Dataset Boundary

The static PASCAL\-Context protocol activates semantic segmentation, human\-part segmentation, saliency, and surface\-normal prediction from the beginning\. It is a cross\-dataset scope check rather than a delayed\-growth setting\. Table[29](https://arxiv.org/html/2608.16930#A2.T29)reports all seeds\. Recon has the highest mean aggregate,G=0\.0158±0\.0069G=0\.0158\\pm 0\.0069, while EMAN hasG=−0\.0098±0\.0007G=\-0\.0098\\pm 0\.0007\. The aggregate is consistently small and negative, but the native task metrics are mixed\. EMAN obtains the strongest semantic score and joint loss in Seed 1, the strongest saliency and joint loss in Seed 2, and near\-reference semantic and joint\-loss values in Seed 3\. The result therefore narrows the performance scope without implying a uniform task failure\.

Table 29:PASCAL\-Context external comparison for every seed\. Higher is better except normal mean and joint loss\.![Refer to caption](https://arxiv.org/html/2608.16930v1/Figures/PASCAL-Context_external_baselines_Seed_1.png)

![Refer to caption](https://arxiv.org/html/2608.16930v1/Figures/PASCAL-Context_external_baselines_Seed_2.png)

Figure 11:PASCAL\-Context external comparisons for Seed 1 and Seed 2\. The panels retain the native task metrics and expose the mixed task\-level pattern behind the aggregate\.![Refer to caption](https://arxiv.org/html/2608.16930v1/Figures/PASCAL-Context_external_baselines_Seed_3.png)Figure 12:PASCAL\-Context external comparison for Seed 3\. EMAN remains close to the shared reference on semantic segmentation, while Recon has the strongest normal result\.Figures[11](https://arxiv.org/html/2608.16930#A2.F11)and[12](https://arxiv.org/html/2608.16930#A2.F12)make the aggregate decomposition explicit\. The negativeGGis mainly associated with human\-part segmentation and normal prediction\. It coexists with competitive semantic, saliency, and joint\-loss values\. Because all four tasks are present from epoch 1, this protocol offers no pre\-demand interval in which delayed materialization can save path training\. It is therefore a demanding test of final multi\-task behavior, but not a direct test of EMAN’s timing advantage\.

The static PASCAL\-Context setting presents all four tasks from the beginning, and EMAN reaches the dual\-path state early\. This leaves little opportunity for delayed path\-training savings\. We therefore use the static protocol mainly as a cross\-dataset performance boundary\. The task\-arrival protocols provide the clearer evidence for release timing under changing demand\.

### PASCAL\-Context Task Arrival at Epoch 21

This protocol activates the later tasks at epoch 21\. EMAN releases at epoch 21 in every seed\. MeanPaths is dual from epoch 1\. Table[30](https://arxiv.org/html/2608.16930#A2.T30)reports the complete result\. EMAN has the highestGGin Seed 3\. It remains close to MeanPaths in Seed 1 and Seed 2\.

Table 30:PASCAL\-Context task\-arrival results\.Table[30](https://arxiv.org/html/2608.16930#A2.T30)separates timing stability from final task variation\. EMAN releases at epoch 21 in every seed, while MeanPaths is dual from epoch 1\. The three\-seed meanGGvalues are0for Hard\-sharing,0\.0027±0\.00180\.0027\\pm 0\.0018for MeanPaths, and0\.0043±0\.00420\.0043\\pm 0\.0042for EMAN\. Seed 1 and Seed 2 are close to MeanPaths\. Seed 3 gives the strongest aggregate and the best semantic, parts, and saliency values among the listed methods\.

Figure[13](https://arxiv.org/html/2608.16930#A2.F13)summarizes the same event timing in all three runs\. The measured MeanPaths times are 14\.1, 15\.8, and 16\.4 minutes\. The EMAN times are 14\.2, 15\.7, and 16\.2 minutes\. These values show that delayed path activation does not introduce a large wall\-clock penalty in this protocol\. They are hardware\-specific and are not used to claim a general speedup\.

### Earlier Arrival at Epoch 11

The earlier\-arrival check uses Seed 1\. Saliency and normal demand arrive at epoch 11\. The transition rule remains frozen\. EMAN releases the path at epoch 11\. Table[31](https://arxiv.org/html/2608.16930#A2.T31)reports all methods\.

Table 31:PASCAL\-Context earlier\-arrival sensitivity\. All values use Seed 1\.![Refer to caption](https://arxiv.org/html/2608.16930v1/Figures/PASCAL_task_arrival_21_Seed_1.png)

![Refer to caption](https://arxiv.org/html/2608.16930v1/Figures/PASCAL_task_arrival_21_Seed_2.png)

![Refer to caption](https://arxiv.org/html/2608.16930v1/Figures/PASCAL_task_arrival_21_Seed_3.png)

![Refer to caption](https://arxiv.org/html/2608.16930v1/Figures/PASCAL_arrival11_sensitivity_Seed_1.png)

Figure 13:PASCAL\-Context timing sensitivity\. The first three panels show the registered arrival\-21 runs; the final panel shows the earlier\-arrival check\. Tables[30](https://arxiv.org/html/2608.16930#A2.T30)and[31](https://arxiv.org/html/2608.16930#A2.T31)distinguish task\-arrival epochs from path\-release events and report the native metrics\.Table[31](https://arxiv.org/html/2608.16930#A2.T31)reports the Seed 1 comparison\. EMAN obtainsG=0\.0037G=0\.0037\. Scheduled split obtains0\.00200\.0020\. MeanPaths obtains0\.00540\.0054\. EMAN has the best semantic, human\-part, and saliency scores\. MeanPaths has the strongest normal error and aggregate\. The experiment therefore supports timing adaptation, not superiority to always\-dual training\.

Figure[13](https://arxiv.org/html/2608.16930#A2.F13)is the key timing\-sensitivity diagnostic\. The later tasks arrive at epoch 11, ten epochs earlier than in the primary protocol, and the path\-release event occurs at epoch 11 under the unchanged rule\. This correspondence does not imply that the certificate observes or identifies the task\-arrival label\. EMAN and Scheduled split use an active dual\-path ratio of0\.750\.75and an estimated path\-training ratio of0\.8750\.875relative to MeanPaths under this shorter horizon\. The movement of the event, not the small performance difference, is the central result\.

Table 32:PASCAL\-Context ramp 11–15\. A missing event is shown as “–”\.![Refer to caption](https://arxiv.org/html/2608.16930v1/Figures/PASCAL_ramp11_15_Seed_1.png)

![Refer to caption](https://arxiv.org/html/2608.16930v1/Figures/PASCAL_ramp11_15_Seed_2.png)

![Refer to caption](https://arxiv.org/html/2608.16930v1/Figures/PASCAL_ramp11_15_Seed_3.png)

Figure 14:Gradual\-demand results for Seeds 1–3\. The archived evidence does not support EMAN superiority over Conflict Trigger in this protocol\.
### Gradual Demand: Ramp from Epochs 11 to 15

The ramp protocol gradually activates the later demand from epochs 11 to 15\. It tests a smoother demand profile than the discrete arrival protocols\. Table[32](https://arxiv.org/html/2608.16930#A2.T32)retains every seed\. EMAN and Scheduled split release at epoch 11\. Conflict Trigger does not release in these archived runs\. The meanGGvalues are0\.0009±0\.00250\.0009\\pm 0\.0025for MeanPaths,−0\.0048±0\.0039\-0\.0048\\pm 0\.0039for Scheduled split,0\.0068±0\.00220\.0068\\pm 0\.0022for Conflict Trigger, and−0\.0039±0\.0053\-0\.0039\\pm 0\.0053for EMAN\.

The strongest aggregate is obtained by Conflict Trigger without a structural release\. This observation suggests that the archived ramp does not create a clear benefit for additional path capacity\. EMAN and the matched Scheduled split have nearly identical means and the same phase/path epochs, so the result does not isolate a disadvantage of the certified direction\. It instead identifies a scope condition: the current phase\-first rule is most clearly supported under discrete demand onset, while a smooth ramp can favor retaining the shared state\.

Figure[14](https://arxiv.org/html/2608.16930#A2.F14)confirms the same pattern in every seed\. Seed 3 is the most favorable EMAN run and remains positive, while Seeds 1 and 2 are negative for both EMAN and Scheduled split\. Retaining all three runs prevents the discrete\-arrival result from being generalized to every temporal demand profile\.

### Replay, Provenance, and Reproducibility Boundaries

The accompanying code package provides checksums, per\-seed result tables, release summaries, figure source data, plotting scripts, and replay summary records\. Large natural\-image checkpoints are not redistributed\.

Archived checkpoint audits reproduced the recorded metrics for the retained task\-arrival run families\. The accompanying code package includes the corresponding replay summaries, but it does not redistribute the large natural\-image checkpoints\. Online and offline evaluations agree within the recorded tolerances\. Exact cross\-platform bitwise identity is not claimed\.

Table 33:Replay status for the retained task\-arrival audits\. PASS denotes agreement within the recorded tolerance\.Table 34:Consolidated claim matrix\. “Supported” denotes direct evidence in the registered protocols\. “Scope condition” denotes an observed boundary that narrows interpretation without invalidating the supported claims\.Table[33](https://arxiv.org/html/2608.16930#A2.T33)reports the run families represented in the replay records included with the code package\. The audits compare stored metrics and event records within the recorded tolerances\. The manifest and checksum layer records the packaged assets\. The replay summaries record the archived audit outcomes\. The per\-seed tables record result inclusion\.

Estimated path\-training FLOPs count path evaluations\. They do not represent a complete accelerator model\. Total training FLOPs include the shared non\-path computation\. Wall\-clock remains hardware\-specific\. Inference remains dual\-path after materialization\.

### Consolidated Evidence Boundary

The complete evidence package supports a layered interpretation\. The strongest conclusions concern structural selection under explicit capacity control and timing adaptation under the registered task\-arrival protocols\. Natural\-image results describe a trade\-off between performance and computation\. They remain task\- and seed\-dependent\.

Table[34](https://arxiv.org/html/2608.16930#A2.T34)consolidates the positive evidence\. It also records the scope of each protocol\. The evidence does not establish task\-wise expert specialization, sparse inference, a universal wall\-clock gain, uniform superiority on every dataset or seed, a generally optimal release direction, or statistical significance from three seeds\. These statements are not required for the central result\. EMAN is evaluated as a mechanism for evidence\-gated capacity growth\. It preserves exact sharing, exposes and certifies a local growth direction, and materializes independent capacity only when the frozen rule accepts the transition\.

#### Evidence hierarchy and interpretation

The experiments form a hierarchy\. The rank\-controlled benchmark identifies whether a capacity deficit exists because the target rank is known in advance\. The certificate traces then identify whether the frozen rule accepts the transition\. The representation diagnostics test whether the released paths carry complementary information\. Finally, the natural\-image protocols evaluate whether the structural decision yields a useful performance–computation trade\-off under realistic task mixtures\. A result at one level is not used as a substitute for evidence at another level\.

The strongest mechanistic conclusion comes from the controlledLowregime\. Hard\-sharing and PhaseOnly remain near the same rank\-limited solution, while EMAN materializes a second path and recovers the missing factors\. TheHighregime provides the complementary non\-growth result\. It shows that the same mechanism can retain exact sharing when one path is sufficient\. Together, these regimes isolate conditional capacity allocation from generic multi\-path overparameterization\.

The natural\-image results answer a different question\. Standard NYUv2 evaluates final task performance under a frozen static matched adapter\. It is not used to assess OEC\-triggered materialization\. Capacity\-Stress and task\-arrival protocols provide the corresponding structural\-decision evidence\. Static PASCAL\-Context and the gradual ramp serve as scope checks\. This separation prevents favorable performance from being treated as proof of a certificate event\.

#### Reading non\-release and weak\-result cases

A non\-release run is an output of the structural rule rather than a missing experiment\. Seed 2 under Capacity\-Stress does not pass the phase certificate, so no path\-force probe is executed and no second path exists\. The empty probe panels record this control flow\. They do not encode zero\-valued force or a failed plotting script\. The result demonstrates that the implementation can retain a shared state when population evidence is not persistent\.

The rank\-four controlled regime is a weak bottleneck\. Its false negative shows that the frozen persistence rule prioritizes conservative growth near the capacity boundary\. This observation narrows sensitivity claims, but it does not contradict the verified rank\-six recovery\. Likewise, static PASCAL\-Context contains several competitive native task values but no delayed compute interval because all tasks are active from the first epoch\. Its mixed aggregate is therefore most informative as a cross\-dataset boundary rather than as a test of delayed materialization\.

The ramp protocol further separates a smooth demand transition from a discrete task arrival\. Conflict Trigger obtains the strongest aggregate without releasing a path\. EMAN and the matched Scheduled split use the same release epochs and have nearly identical means\. The archived result therefore does not isolate a failure of the certified direction\. It indicates that the current rule has its clearest evidence under discrete demand onset and that smooth demand may not require additional capacity in the same way\.

#### Metric\-by\-metric reading guide

Phase curvature is an upstream signal\. A negative point estimate alone is not a release event\. The uncertainty bound, finite\-difference check, safety checks, and persistence rule determine whether phase is released\. Path\-force strength is downstream\. It is evaluated only after phase release and the adaptation window\. A missing path\-force trace therefore means that the run did not enter that state\. It should not be read as a measured force of zero\.

Cross\-half and shard cosines describe the stability of the separation axis\. The absolute value removes its arbitrary sign\. High consistency does not by itself imply that the finite perturbation reduces the deployed objective\. The frozen and dynamic descent tests answer that separate question\. Dynamic safety then checks whether calibration, exchange symmetry, and numerical values remain inside the registered tolerances\. Persistence requires the complete conjunction to survive across probe epochs\.

Path distance is a post\-release diagnostic\. It measures how far the two parameter blocks move apart after the certified split\. A large distance is not a certificate, and a small distance is not a failure\. The value depends on the release epoch and on the remaining optimization horizon\. Representation complementarity is tested separately through latent\-factor recovery, canonical correlation, principal angles, and permutation controls\.

The aggregateGGsummarizes native task directions relative to Hard\-sharing\. It does not replace the task columns\. A positive value can coexist with a weak individual task, and a negative value can coexist with one or more competitive native metrics\. For this reason, every external comparison retains the native metrics and the per\-seed aggregate\. System quantities are also kept separate\. Parameter count, peak memory, estimated path evaluations, total FLOPs, and wall\-clock time answer different questions and should not be interchanged\.