Not new yet
- Interference on a graph.
- Partial which-path visibility loss.
- Quantum eraser behavior.
- Clustered linear modes.
- Effective geometry from connectivity.
- Fixed-Hamiltonian linear graph-QM reformulation.
Main Claim
What the current prototype does and does not claim
Project framing
What is not new yet and what may be new
The page is organized as a control-versus-extension comparison: a frozen linear baseline, then a separate nonlinear layer where the first candidate deviation is explicitly introduced and scanned.
Candidate novelty
- State-dependent weak-link dynamics.
- Occupation-dependent spectral splitting.
- Finite-relaxation memory effects.
- Hysteresis measured by loop area.
- Incomplete-transfer regimes.
- Nonlinear deviation relative to the fixed-Hamiltonian baseline.
Linear baseline
P(Δφ, μ) = 1/2 (1 + μ cos Δφ)
Cluster splitting
Efull = Ecluster ± ε
Backreaction law
εeff(ψ) = ε₀ [1 + α (|ψi|² + |ψj|²)/2]
Frozen Baseline v0.1
Baseline v0.1 as the control group
The baseline is frozen and kept separate from the nonlinear claim. The figures below are the control set: linear fixed-Hamiltonian graph dynamics, generated from the Python scripts and kept unchanged while the nonlinear layer evolves independently.
Nonlinear Results
Nonlinear regime map
The nonlinear layer uses a state-dependent weak-link law with finite relaxation, scanned over 300 parameter combinations. The observables reported here are linear and nonlinear swap time, minimum left-cluster weight, maximum link lag, loop area, and norm drift; the main qualitative result is that only a subset of runs enters a memory-bearing incomplete-transfer regime.
Nonlinear law
dε/dt = γ (ε_target(ψ) - ε)
Scan grid
4 states × 5 α × 5 γ × 3 ε₀ = 300 runs
Measured observables
t* , min W_L, max lag, loop area, norm drift
Main nonlinear figure
The loop-area heatmap is promoted to the main figure because it separates pure renormalization from memory-bearing trajectories more cleanly than a swap-time comparison alone.
Representative regimes
Baseline-equivalent
The nonlinear module reduces to the linear baseline within numerical precision: loop area vanishes, link lag vanishes, and the transfer curve is unchanged.
Representative parameters:
state = pattern1, ε₀ = 0.1, α = 0.0, γ = 0.5
Pure nonlinear renormalization
The transfer time shifts strongly, but the trajectory still behaves as if the weak links were replaced by a different static value rather than by a memory law.
Representative parameters:
state = pattern2, ε₀ = 0.2, α = 12.0, γ = 0.5
Memory regime
Link relaxation and occupation feedback stop the system from reaching full transfer, and the trajectory traces a visible hysteresis loop.
Representative parameters:
state = node0, ε₀ = 0.3, α = 12.0, γ = 4.0
Quantitative verdict
60
Baseline-equivalent runs
164
Pure nonlinear renormalization runs
76
Memory plus incomplete-transfer runs
Strongest loop-area case
state = node0, ε₀ = 0.3, α = 12.0, γ = 4.0
Strongest tunneling-suppression case
state = node0, ε₀ = 0.3, α = 12.0, γ = 2.0
Reproducibility
Canonical public artifacts
These files are the canonical public source of truth for the current nonlinear scan, including the markdown report, the full CSV, the rendered summary, the heatmaps, and the representative trajectories.
Results version 20260414-1 | public site build 2026-04-13
Rendered summary
Loading nonlinear results summary...
Closest Known Model Classes
Closest known model classes
- Linear quantum walks and tight-binding dynamics on graphs.
- Finite-dimensional Hermitian graph models.
- Nonlinear state-dependent hopping or correlated-hopping classes.
- Open question: does the present backreaction law define a genuinely distinct regime, or is it reducible to an already known nonlinear class?
Limitations
Current limits of the evidence
- The frozen baseline is not new physics; it is a graph-language restatement of linear fixed-Hamiltonian quantum dynamics.
- Any novelty in the current page comes only from the explicitly added weak-link backreaction law.
- The current evidence is numerical and still at the toy-model level.
- The next scientific step is a direct comparison against known nonlinear lattice and graph models.
Interactive Lab
Live plots in real time
4-Node Graph
Interference and which-path marking
Quantum Eraser
Total counts and post-selected channels
6-Node Clustered Graph
Spectrum, exact modes, and packet timescales
Emergent Geometry
Effective distances and embedding
State-Dependent Weak Links
Nonlinear extension as a separate module
This panel compares the fixed-Hamiltonian baseline and backreacted weak-link dynamics. For a pure cluster pattern the effect nearly collapses to a renormalization, while a node-localized state already produces genuinely new trajectories.