Corpus Semantic Registry

19 symbols 1 documents 0 err / 0 warn 2026-07-05 ·

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Recent (top 20)

2026-07-02 doc Fold_Paper A singular fold model for capacity-constrained dynamics (v67; prior version on Zenodo) v67 provisional
2026-07-02 sym Fold.theorem1_fold_repulsion provisional
2026-07-02 sym Fold.slow_vector_field candidate
2026-07-02 sym Fold.slow_manifold_tube candidate
2026-07-02 sym Fold.prop5_isolating_block provisional
2026-07-02 sym Fold.prop4_local_stability provisional
2026-07-02 sym Fold.prop3_barrier_monotonicity provisional
2026-07-02 sym Fold.prop2_friction_divergence provisional
2026-07-02 sym Fold.prop1_saddle_node provisional
2026-07-02 sym Fold.open_uniform_delta candidate
2026-07-02 sym Fold.open_krupa_szmolyan candidate
2026-07-02 sym Fold.open_blowup_regularisation candidate
2026-07-02 sym Fold.open_basin_overlap candidate
2026-07-02 sym Fold.friction_coupling candidate
2026-07-02 sym Fold.fast_subsystem candidate
2026-07-02 sym Fold.csd_predictions candidate
2026-07-02 sym Fold.criterion1_boundedness provisional
2026-07-02 sym Fold.corollary1_rated_exit provisional
2026-07-02 sym Fold.congestion_regime candidate
2026-07-02 sym Fold.barrier_function candidate

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Preprint Registry (1)

Fold provisional v67 · 19 defines A singular fold model for capacity-constrained dynamics (v67; prior version on Zenodo) PDF ▾

csr.document.Fold_Paper

Display	A singular fold model for capacity-constrained dynamics (v67; prior version on Zenodo)
Version	v67
Status	provisional
Publication mode	external
Last verified	2026-07-02
Defines (19)	`csr.Fold.fast_subsystem`, `csr.Fold.slow_vector_field`, `csr.Fold.friction_coupling`, `csr.Fold.congestion_regime`, `csr.Fold.barrier_function`, `csr.Fold.slow_manifold_tube`, `csr.Fold.prop1_saddle_node`, `csr.Fold.prop2_friction_divergence`, `csr.Fold.prop3_barrier_monotonicity`, `csr.Fold.criterion1_boundedness`, `csr.Fold.prop4_local_stability`, `csr.Fold.theorem1_fold_repulsion`, `csr.Fold.prop5_isolating_block`, `csr.Fold.corollary1_rated_exit`, `csr.Fold.csd_predictions`, `csr.Fold.open_uniform_delta`, `csr.Fold.open_basin_overlap`, `csr.Fold.open_krupa_szmolyan`, `csr.Fold.open_blowup_regularisation`

Symbol Registry

Fold (19)

barrier_function candidate concept · representation L(V) = -F(x*(V)) on the attracting branch; fold-avoidance barrier whose monotonicity carri… PDF ▾

csr.Fold.barrier_function

L(V) = -F(x*(V)) on the attracting branch; fold-avoidance barrier whose monotonicity carries the admissibility argument.

Source: csr.document.Fold_Paper@v67#section.4.2.barrier_function

Display	barrier function L(V) (Definition 1)
Type	concept
Framework layer	`representation`
Owning document	`csr.document.Fold_Paper`
Definition hash	`sha256:92b9166...`
Relations	depends_on: `csr.Fold.friction_coupling` used_by: `csr.Fold.prop3_barrier_monotonicity`, `csr.Fold.theorem1_fold_repulsion`
Verification	declared via prose; last verified 2026-07-02

congestion_regime candidate concept · domain Region of the regime partition of the reduced slow flow on which barrier monotonicity and… PDF ▾

csr.Fold.congestion_regime

Region of the regime partition of the reduced slow flow on which barrier monotonicity and the repulsion results are stated.

Source: csr.document.Fold_Paper@v67#section.4.1.congestion_regime

Display	congestion regime
Type	concept
Framework layer	`domain`
Owning document	`csr.document.Fold_Paper`
Definition hash	`sha256:e35b8dc...`
Relations	used_by: `csr.Fold.prop3_barrier_monotonicity`, `csr.Fold.theorem1_fold_repulsion`
Verification	declared via prose; last verified 2026-07-02

corollary1_rated_exit provisional theorem · domain Every congestion-tube trajectory exits the cooperative face without fold approach in expli… PDF ▾

csr.Fold.corollary1_rated_exit

Every congestion-tube trajectory exits the cooperative face without fold approach in explicit time ~ 1/(gamma0*mu0*delta*eps). Composition of Theorem 1 and Proposition 5.

Source: csr.document.Fold_Paper@v67#section.5.6.corollary1

Display	Cor. 1: rated exit for all tube initial conditions
Type	theorem
Framework layer	`domain`
Owning document	`csr.document.Fold_Paper`
Definition hash	`sha256:2fb900f...`
Relations	depends_on: `csr.Fold.theorem1_fold_repulsion`, `csr.Fold.prop5_isolating_block`
Verification	proved via theorem; last verified 2026-07-02

criterion1_boundedness provisional predicate · domain Structural equivalence on the reduced flow: bounded friction iff bounded-away-from-fold. PDF ▾

csr.Fold.criterion1_boundedness

Structural equivalence on the reduced flow: bounded friction iff bounded-away-from-fold.

Source: csr.document.Fold_Paper@v67#section.4.3.criterion1

Display	Crit. 1: boundedness equivalence
Type	predicate
Framework layer	`domain`
Owning document	`csr.document.Fold_Paper`
Definition hash	`sha256:038c872...`
Relations	depends_on: `csr.Fold.prop3_barrier_monotonicity`
Verification	proved via theorem; last verified 2026-07-02

csd_predictions candidate schema · cross_layer Four distinguishing critical-slowing-down predictions derived from the fold structure; emp… PDF ▾

csr.Fold.csd_predictions

Four distinguishing critical-slowing-down predictions derived from the fold structure; empirically testable, not theorems.

Source: csr.document.Fold_Paper@v67#section.8.3.csd_predictions

Display	fold-consistent CSD predictions
Type	schema
Framework layer	`cross_layer`
Owning document	`csr.document.Fold_Paper`
Definition hash	`sha256:e12e82b...`
Relations	depends_on: `csr.Fold.prop2_friction_divergence`
Verification	argued via prose; last verified 2026-07-02

fast_subsystem candidate structure · domain Fast dynamics in the smooth saddle-node normal form; the smooth half of the fast-slow deco… PDF ▾

csr.Fold.fast_subsystem

Fast dynamics in the smooth saddle-node normal form; the smooth half of the fast-slow decomposition.

Source: csr.document.Fold_Paper@v67#section.3.1.fast_subsystem

Display	fast subsystem (saddle-node normal form)
Type	structure
Framework layer	`domain`
Owning document	`csr.document.Fold_Paper`
Definition hash	`sha256:ef96557...`
Relations	used_by: `csr.Fold.prop1_saddle_node`
Verification	declared via prose; last verified 2026-07-02

friction_coupling candidate structure · domain Coupling under which the friction-scaling law is derived; the exponent -1/2 is specific to… PDF ▾

csr.Fold.friction_coupling

Coupling under which the friction-scaling law is derived; the exponent -1/2 is specific to this choice (F = k/x^a gives -a/2).

Source: csr.document.Fold_Paper@v67#section.3.1.friction_coupling

Display	friction coupling F(x) = k/x
Type	structure
Framework layer	`domain`
Owning document	`csr.document.Fold_Paper`
Definition hash	`sha256:cd7c156...`
Relations	used_by: `csr.Fold.barrier_function`, `csr.Fold.prop2_friction_divergence`
Verification	declared via prose; last verified 2026-07-02

open_basin_overlap candidate open_problem · domain Explicit overlap between Proposition 4's basin and Theorem 1's certified region across V =… PDF ▾

csr.Fold.open_basin_overlap

Explicit overlap between Proposition 4's basin and Theorem 1's certified region across V = V0 + r; needs Lyapunov-function scaling not pursued in v67.

Source: csr.document.Fold_Paper@v67#section.8.2.open_basin_overlap

Display	OPEN: basin overlap of Prop. 4 and Thm. 1 regions
Type	open_problem
Framework layer	`domain`
Owning document	`csr.document.Fold_Paper`
Definition hash	`sha256:ea27122...`
Relations	depends_on: `csr.Fold.prop4_local_stability`, `csr.Fold.theorem1_fold_repulsion`
Verification	declared via prose; last verified 2026-07-02

open_blowup_regularisation candidate open_problem · domain Not attempted in v67; the paper records what a resolution would require. Prerequisite for… PDF ▾

csr.Fold.open_blowup_regularisation

Not attempted in v67; the paper records what a resolution would require. Prerequisite for the uniform-in-delta result.

Source: csr.document.Fold_Paper@v67#section.8.2.open_blowup

Display	OPEN: full-divisor blow-up regularisation of the 1/x singularity
Type	open_problem
Framework layer	`domain`
Owning document	`csr.document.Fold_Paper`
Definition hash	`sha256:ed07f4f...`
Relations	depends_on: `csr.Fold.slow_vector_field`
Verification	declared via prose; last verified 2026-07-02

open_krupa_szmolyan candidate open_problem · domain Standard Krupa-Szmolyan hypotheses fail at x = 0 because of the 1/x singularity; classific… PDF ▾

csr.Fold.open_krupa_szmolyan

Standard Krupa-Szmolyan hypotheses fail at x = 0 because of the 1/x singularity; classification not attempted.

Source: csr.document.Fold_Paper@v67#section.8.2.open_krupa_szmolyan

Display	OPEN: Krupa-Szmolyan jump-point classification
Type	open_problem
Framework layer	`domain`
Owning document	`csr.document.Fold_Paper`
Definition hash	`sha256:e936956...`
Relations	depends_on: `csr.Fold.slow_vector_field`
Verification	declared via prose; last verified 2026-07-02

open_uniform_delta candidate open_problem · domain Joint limit delta -> 0 at fixed eps (the fold neighbourhood): eps0(delta, r) -> 0 as delta… PDF ▾

csr.Fold.open_uniform_delta

Joint limit delta -> 0 at fixed eps (the fold neighbourhood): eps0(delta, r) -> 0 as delta -> 0; requires blow-up resolution.

Source: csr.document.Fold_Paper@v67#section.8.2.open_uniform_delta

Display	OPEN: uniform-in-delta certification at fixed eps
Type	open_problem
Framework layer	`domain`
Owning document	`csr.document.Fold_Paper`
Definition hash	`sha256:11123dd...`
Relations	depends_on: `csr.Fold.theorem1_fold_repulsion`
Verification	declared via prose; last verified 2026-07-02

prop1_saddle_node provisional theorem · domain Saddle-node bifurcation of the fast subsystem. Proved; standard normal-form verification. PDF ▾

csr.Fold.prop1_saddle_node

Saddle-node bifurcation of the fast subsystem. Proved; standard normal-form verification.

Source: csr.document.Fold_Paper@v67#section.3.2.prop1

Display	Prop. 1: saddle-node bifurcation of the fast subsystem
Type	theorem
Framework layer	`domain`
Owning document	`csr.document.Fold_Paper`
Definition hash	`sha256:bc275f8...`
Relations	depends_on: `csr.Fold.fast_subsystem` used_by: `csr.Fold.prop2_friction_divergence`
Verification	proved via theorem; last verified 2026-07-02

prop2_friction_divergence provisional theorem · domain Friction-scaling law along the attracting branch. Proved, CONDITIONAL on the coupling F(x)… PDF ▾

csr.Fold.prop2_friction_divergence

Friction-scaling law along the attracting branch. Proved, CONDITIONAL on the coupling F(x) = k/x; exponent is model-dependent.

Source: csr.document.Fold_Paper@v67#section.4.2.prop2

Display	Prop. 2: conditional friction divergence F ~ (Q-V)^(-1/2)
Type	theorem
Framework layer	`domain`
Owning document	`csr.document.Fold_Paper`
Definition hash	`sha256:51d9464...`
Relations	depends_on: `csr.Fold.friction_coupling`, `csr.Fold.prop1_saddle_node` used_by: `csr.Fold.csd_predictions`
Verification	proved via theorem; last verified 2026-07-02

prop3_barrier_monotonicity provisional theorem · domain Monotonicity of L(V) on the congestion regime of the reduced slow flow, restricted to the… PDF ▾

csr.Fold.prop3_barrier_monotonicity

Monotonicity of L(V) on the congestion regime of the reduced slow flow, restricted to the attracting branch.

Source: csr.document.Fold_Paper@v67#section.4.2.prop3

Display	Prop. 3: barrier monotonicity on the congestion regime
Type	theorem
Framework layer	`domain`
Owning document	`csr.document.Fold_Paper`
Definition hash	`sha256:4e822a0...`
Relations	depends_on: `csr.Fold.barrier_function`, `csr.Fold.congestion_regime` used_by: `csr.Fold.criterion1_boundedness`
Verification	proved via theorem; last verified 2026-07-02

prop4_local_stability provisional theorem · domain Local asymptotic stability of (x0, V0) via Hurwitz linearisation and the Lyapunov indirect… PDF ▾

csr.Fold.prop4_local_stability

Local asymptotic stability of (x0, V0) via Hurwitz linearisation and the Lyapunov indirect method.

Source: csr.document.Fold_Paper@v67#section.5.6.prop4

Display	Prop. 4: local asymptotic stability of the cooperative equilibrium
Type	theorem
Framework layer	`domain`
Owning document	`csr.document.Fold_Paper`
Definition hash	`sha256:c569420...`
Relations	used_by: `csr.Fold.open_basin_overlap`
Verification	proved via theorem; last verified 2026-07-02

prop5_isolating_block provisional theorem · domain Face-flux computation: tube exit only through the cooperative-side face; covers all initia… PDF ▾

csr.Fold.prop5_isolating_block

Face-flux computation: tube exit only through the cooperative-side face; covers all initial conditions in the slow-manifold tube.

Source: csr.document.Fold_Paper@v67#section.5.5.prop5

Display	Prop. 5: isolating-block certificate for the whole tube
Type	theorem
Framework layer	`domain`
Owning document	`csr.document.Fold_Paper`
Definition hash	`sha256:02cbe74...`
Relations	depends_on: `csr.Fold.slow_manifold_tube` used_by: `csr.Fold.corollary1_rated_exit`
Verification	proved via theorem; last verified 2026-07-02

slow_manifold_tube candidate concept · domain Neighbourhood of the attracting slow manifold on which the Lyapunov and isolating-block ce… PDF ▾

csr.Fold.slow_manifold_tube

Neighbourhood of the attracting slow manifold on which the Lyapunov and isolating-block certificates are constructed.

Source: csr.document.Fold_Paper@v67#section.5.1.slow_manifold_tube

Display	slow-manifold tube
Type	concept
Framework layer	`domain`
Owning document	`csr.document.Fold_Paper`
Definition hash	`sha256:7c5ba34...`
Relations	used_by: `csr.Fold.theorem1_fold_repulsion`, `csr.Fold.prop5_isolating_block`
Verification	declared via prose; last verified 2026-07-02

slow_vector_field candidate structure · domain Slow dynamics, singular at the boundary x = 0; the singularity is the model's distinguishi… PDF ▾

csr.Fold.slow_vector_field

Slow dynamics, singular at the boundary x = 0; the singularity is the model's distinguishing structural feature.

Source: csr.document.Fold_Paper@v67#section.3.1.slow_vector_field

Display	slow vector field g(x) = alpha - beta*k/x
Type	structure
Framework layer	`domain`
Owning document	`csr.document.Fold_Paper`
Definition hash	`sha256:9096f97...`
Relations	used_by: `csr.Fold.open_krupa_szmolyan`, `csr.Fold.open_blowup_regularisation`
Verification	declared via prose; last verified 2026-07-02

theorem1_fold_repulsion provisional theorem · domain Fold-repulsion and exit-time bound on strict congestion subsets of the full singular syste… PDF ▾

csr.Fold.theorem1_fold_repulsion

Fold-repulsion and exit-time bound on strict congestion subsets of the full singular system, for every fixed delta > 0 with eps <= eps0(delta, r); weighted Lyapunov certificate removes the delta_min(r) threshold.

Source: csr.document.Fold_Paper@v67#section.5.4.theorem1

Display	Thm. 1: Lyapunov decay and fold-repulsion funnel
Type	theorem
Framework layer	`domain`
Owning document	`csr.document.Fold_Paper`
Definition hash	`sha256:4fdb109...`
Relations	depends_on: `csr.Fold.barrier_function`, `csr.Fold.slow_manifold_tube`, `csr.Fold.congestion_regime` used_by: `csr.Fold.corollary1_rated_exit`, `csr.Fold.open_uniform_delta`, `csr.Fold.open_basin_overlap`
Verification	proved via theorem; last verified 2026-07-02

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