When does a dynamical structure begin to look particle-like?

Here, “particle-like” is an operational resemblance—not an ontological claim. The question is whether a deterministic, memory-bearing structure can form, persist, escape, carry a spectrum, and respond causally to another structure.

Operational checklist

Five behaviors worth separating from the word “particle.”

None is sufficient on its own. Together they define the current experimental program and, more importantly, expose what remains missing.

01Formation

A low-energy co-ignition window selects an antisymmetric phase-locked sector.

02Persistence

The selected sector holds for a finite, state-dependent interval rather than disappearing immediately.

03Escape

A seeded perturbation produces a reproducible departure time from the symmetric manifold.

04Spectrum

An internal odd-sector frequency tracks accumulated dynamical age and admits an exact linear-mode anchor.

05Interaction

A single switched link produces a differential clock shift in bit-identical paired counterfactuals.

Formation and escape

A finite-lived sector with a measurable route out.

The first two figures concern selection and persistence of an antisymmetric phase-locked sector. They are suggestive of a bound-state picture, but they remain statements about this model family.

The hold decays smoothly with weaker-node energy. — The antisymmetric lock is longest when the weaker node is connected at low energy. The monotone staircase reproduced in two pairs and across all nine threshold/window combinations. Fine attribution to the minimum rather than the maximum node energy remains confounded by pair identity.

Escape time is approximately logarithmic in seed size. — A perturbation of amplitude δ leaves the symmetric manifold at t_esc ≈ 7.33 + 3.06 ln(1/δ). The fit has R²=0.98 and zero censoring, but four points do not distinguish a logarithm from a nearby power law. The result is therefore retained as an effective description, not a universal escape law.

The instability pulse depends on epoch, not scalar energy. — Connections made while the background burn is rising produce a full odd-sector growth pulse. Connections at the peak or on the falling branch remain nearly flat. The t_c=80 and t_c=104 cases revisit comparable scalar energies on opposite sides of the peak: the fold-back rejects instantaneous energy as a state coordinate for λ₋ by more than an order of magnitude.

Spectrum and interaction

An internal clock—and a link that changes it.

The strongest particle-like analogy currently comes from the combination of a state-dependent spectral readout and a paired causal interaction test.

The odd-sector frequency reads accumulated age. — Cold, hold, and post-release measurements follow ω_eff = ω₀√(1 + 0.1b). The low-frequency anchor was independently identified as the exact S2-block quadratic eigenmode: 48.5266, with measured decay(Ψ₋) = Re(QEP) = −0.093. The response to the memory kernel is measured as sublinear; its law remains open.

The connection produces a differential clock shift. — The same-sign shift reproduces across four independent draws. In draws 3 and 4, active and sham branches are bit-identical until the connection is switched on, giving an exactly zero pre-bifurcation baseline. Those paired contrasts attribute the cold splitting to the link itself. Draw 4 also attributes the event-locked bump and release braking to the interaction under controlled surgery.

Where the analogy stops

Interesting dynamics are not yet a particle theory.

Observed in the model

Formation of an odd-parity phase sector
Finite hold, release, and seeded escape
State-dependent spectral readout
Exact linear-mode anchor
Interaction-induced clock splitting

Still required

A transferable interaction or scattering law
Dispersion and scaling beyond the tested networks
A conserved identity across formation and interaction
Independent experimental correspondence
Any connection to relativistic particle ontology

Notation used on this page

S2: Fast internal layer used to define the odd/even pair sectors.
b: Accumulated dressing or age variable in the current model; not physical time itself.
λ₋: Local odd-sector growth rate, λ₋ = ½ d(ln E₋)/dt, estimated on in-life windows.
Δω: Differential frequency, ω̂₋ − ω̂₊; it cancels a common intra-node eigenmode.
draw: One independently seeded deterministic realization—the statistical unit used here.

Source record

Read the figures with their full experimental history.

The versioned artifact includes all seven original charts, retractions, method notes, and the qualitative distance map to known physics. Daily logs preserve preregistration, panel decisions, software receipts, and post-run verdicts.

Open the exact source artifact Browse the Study 06 repository

When does a dynamical structure begin to look particle-like?

Five behaviors worth separating from the word “particle.”

A finite-lived sector with a measurable route out.

The hold decays smoothly with weaker-node energy.

Escape time is approximately logarithmic in seed size.

The instability pulse depends on epoch, not scalar energy.

An internal clock—and a link that changes it.

The odd-sector frequency reads accumulated age.

The connection produces a differential clock shift.

Interesting dynamics are not yet a particle theory.

Read the figures with their full experimental history.