[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86140-en":3,"doc-seo-86140-105":30,"detail-sidebar-cat-0-en-105":96},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},86140,962075114765,"Quinn","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","The Equilibrium Is the Initialization: Lazy Identity Collapse in Physics-Structured Deep Equilibrium Reasoning","Deep equilibrium models aim for input-adaptive implicit computation, where harder problems require more solver iterations and the equilibrium reflects genuine iterative inference. A controlled study examines a port-Hamiltonian DEQ with learned initialization on proof and graph-reachability tasks, finding implicit computation collapses to a silent no-op. Across tasks, seeds, and ablations, the fixed point matches the initialization to numerical precision and improves accuracy by +0.00 pp. Mechanisms point to gradient starvation, plus confounded zeroing ablations and a four-test audit protocol.","arXiv :2607 . 1 1 1 16v 1 [ cs .LG] 13 Jul 2026  \nThe Equilibrium Is the Initialization:  \nLazy Identity Collapse in Physics-Structured Deep Equilibrium  \nReasoning  \nJoyjeet Singh  \nIndependent Researcher  \nORCID: 0009-0005-1512-7439  \nJuly 14, 2026  \nAbstract  \nDeep equilibrium models promise input-adaptive implicit computation: harder problems should demand more solver iterations, and the solved equilibrium should encode the result of genuine iterative inference. We report a cautionary study of a port-Hamiltonian DEQ with a learned initialization on two reasoning tasks — ProofWriter entailment over frozen DeBERTa embeddings and a BFS-verified graph-reachability benchmark—in which the implicit computation is a silent no-op. Across tasks, seeds, and controlled ablation arms, the solved equilibrium equals the solver’s start point to numerical precision, and bypassing the solver entirely changes test accuracy by +0 .00 percentage points in 18 of 19 training runs. Controlled interventions falsify the tempting explanation: removing the anchoring term reproduces every result, and retraining with noise-decoupled starts yields a solver that converges to the noisy start while the decoder learns to ignore it. The single escaping run diverges instead (∥h∗ −z0 ∥ = 171), producing a co-adapted noise channel whose removal improves accuracy. Iteration counts are uncorrelated with ground-truth difficulty (r = 0 .009), and the full apparatus never outperforms a two-layer MLP on either task. We trace the mechanism to gradient starvation along two distinct routes, show that the standard zeroing ablation is confounded and gives wildly seed-dependent answers where the correct substitution test gives a stable zero, and distill a four-test diagnostic protocol for auditing claimed implicit computation. All experiments run on a single free Colab GPU; code, raw logs, and analysis scripts are released.  \n1 Introduction  \nImplicit and recurrent depth is one of the field’s recurring hopes for reasoning: rather than fixing computation at a static layer count, let the model iterate — a deep equilibrium model (DEQ) solving for a fixed point [1], a looped or recurrent-depth transformer unrolling until done [2, 3]—and let harder inputs consume more computation [4] . Physics-structured parameterizations make the proposal more attractive still: port-Hamiltonian and symplectic constraints promise stable, energy-controlled latent dynamics on which an implicit solve can safely iterate [5–7] . The promise is concrete and checkable: the solved equilibrium should carry the result of iterative inference, and solver effort should track problem difficulty.  \nThis paper documents, in controlled detail, a design in which every part of that promise fails silently. We study a port-Hamiltonian DEQ whose state is initialized by a learned, goal-conditioned feed-forward network—a natural pattern whenever an implicit layer is grafted onto a pretrained encoder, and one that (with or without an explicit anchoring term pulling the solve toward the  \ninitialization) appears in many variants across the implicit-models literature. Training converges, accuracies look respectable, and solver telemetry looks plausible. Yet the equilibrium is a copy of its initialization: across two tasks, nineteen training runs, an anchor-strength sweep, and three intervention arms, the fixed-point solve moves the state by at most 10 −6 and contributes exactly +0 .00 percentage points of accuracy, while iteration counts ignore a ground-truth difficulty signal entirely.  \nThe obvious mechanistic story—the anchor both starts and attracts the solver, so the dynamics never receive off-anchor gradients—turns out to be wrong, and we consider its falsification the most useful part of the study. Removing the anchor changes nothing; retraining with noise-decoupled starts produces a solver that faithfully converges to the noisy start while the decoder learns to ignore the channel; and the one run that escapes the","cbCaid9L4li6zmPx","https://ap.wps.com/l/cbCaid9L4li6zmPx","pdf",409147,5,1,14,"English","en",105,"# Introduction\n# Related Work\n# Methodology and Experiments\n## Diagnostic Protocol\n# Results and Ablation Studies\n# Conclusion","[{\"question\":\"What core claim about deep equilibrium models does this paper test?\",\"answer\":\"It tests the promise that a deep equilibrium model’s solved fixed point should reflect iterative inference and that solver effort should track input difficulty.\"},{\"question\":\"Which tasks and model setting are used in the study?\",\"answer\":\"The experiments evaluate a port-Hamiltonian DEQ with learned, goal-conditioned initialization on proof entailment using frozen DeBERTa embeddings and a BFS-verified graph reachability benchmark.\"},{\"question\":\"What is the main failure mode observed for the implicit computation?\",\"answer\":\"The equilibrium becomes a copy of the learned initialization, with the state changing by at most 10^-6 and contributing +0.00 percentage points of accuracy, indicating a silent no-op.\"},{\"question\":\"How does the paper explain why the computation becomes a no-op?\",\"answer\":\"It attributes the behavior to gradient starvation via two distinct gradient routes, and shows that standard zeroing ablations are confounded; a substitution test and a four-test diagnostic protocol are proposed to audit claimed implicit computation.\"}]",1784208862,35,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":91,"head_meta":93,"extra_data":95,"updated_unix":28},"the-equilibrium-is-the-initialization-lazy-identity-collapse-in-physics-structured-deep-equilibrium-reasoning","",{"@graph":36,"@context":90},[37,54,69],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/the-equilibrium-is-the-initialization-lazy-identity-collapse-in-physics-structured-deep-equilibrium-reasoning/86140/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-27","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82,86],{"name":73,"@type":74,"acceptedAnswer":75},"What core claim about deep equilibrium models does this paper test?","Question",{"text":76,"@type":77},"It tests the promise that a deep equilibrium model’s solved fixed point should reflect iterative inference and that solver effort should track input difficulty.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"Which tasks and model setting are used in the study?",{"text":81,"@type":77},"The experiments evaluate a port-Hamiltonian DEQ with learned, goal-conditioned initialization on proof entailment using frozen DeBERTa embeddings and a BFS-verified graph reachability benchmark.",{"name":83,"@type":74,"acceptedAnswer":84},"What is the main failure mode observed for the implicit computation?",{"text":85,"@type":77},"The equilibrium becomes a copy of the learned initialization, with the state changing by at most 10^-6 and contributing +0.00 percentage points of accuracy, indicating a silent no-op.",{"name":87,"@type":74,"acceptedAnswer":88},"How does the paper explain why the computation becomes a no-op?",{"text":89,"@type":77},"It attributes the behavior to gradient starvation via two distinct gradient routes, and shows that standard zeroing ablations are confounded; a substitution test and a four-test diagnostic protocol are proposed to audit claimed implicit computation.","https://schema.org",{"og:url":52,"og:type":92,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":94,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":97},[98,102,106,110,114,119,124,127,132,135,139],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":103,"show_sort_weight":104,"slug":105},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},"Exam",70,"exam",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":111,"show_sort_weight":112,"slug":113},"Comic",60,"comic",{"id":115,"doc_module":4,"doc_module_name":46,"category_name":116,"show_sort_weight":117,"slug":118},6,"Technology",50,"technology",{"id":120,"doc_module":4,"doc_module_name":46,"category_name":121,"show_sort_weight":122,"slug":123},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":125,"slug":126},30,"research-report",{"id":128,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":130,"slug":131},9,"Religion & Spirituality",20,"religion-spirituality",{"id":130,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":130,"slug":134},"World Cup","world-cup",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":136,"slug":138},10,"Lifestyle","lifestyle",{"id":140,"doc_module":4,"doc_module_name":46,"category_name":141,"show_sort_weight":20,"slug":142},19,"General","general"]