Document ID: lineum-core
Version: 1.0.18-core Status: Draft
Equation: Eq-4 (canonical; κ static)
Scope: 2D, periodic BCs Date: 2026-02-15

DOI: 10.5281/zenodo.16934359
How to cite: Tomáš Tříska. Lineum Core (v1.0.17-core). 2026. DOI: 10.5281/zenodo.16934359. This manuscript corresponds to Git tag v1.0.17-core and the evidence bundle in output/ (commit-stamped in each HTML).

Contract evidence (core v1.0.18-core). All numeric claims in this manuscript that are presented as validated are limited to what is asserted by the contract suite lineum-core-1.0.18-core (contract_version 1.1.5), including the embedded canonical run spec6_false_s41_20260215_023130 and its declared fingerprints.

Status tags (v1.0.17-core). To make the manuscript audit-proof, we label claim strength explicitly:

• [VALIDATED] = enforced by the referenced contract suite (numeric acceptance band or exact match) and traceable to a contract key.
• [OBS] = reproducible observation supported by artifacts (CSV/PNG/GIF/HTML), but not currently enforced by contract acceptance.
• [DISPLAY] = derived, illustrative unit conversions from validated anchors (e.g., SI values derived from f0_mean_hz), not acceptance criteria.
• [OOS] = explicitly out of scope for core v1 (documented only as future work / extension track).
• [TEST] = falsification/verification procedure or expected outcome to be checked by running new artifacts; not a reported result unless explicitly pinned + contract-checked.

The authoritative mapping “claim → contract key → artifact pointer” is defined in Appendix G (Claim–Contract Map).

Canonical Scope (v1.0.x)
Equation: Eq-4 (κ static) • Dim.: 2D • BCs: periodic • Grid: 128×128
Δt: 1.0×10⁻²¹ s • Seed: 41 • RUN_TAG: spec6_false_s41
κ-mode: constant • Noise: zero-mean, σξ = 5.0×10⁻³ (canonical)
Operators: ∇ (central), ∇² (5-point von Neumann)
Out of scope: 3D, time-varying κ, zeta/RNB correlations, Return Echo, quantitative Vortex–Particle coupling claims/taxonomy, and other interpretive add-ons. These are intentionally excluded from the core and deferred to future work; they are not part of this submission. Structural Closure is in scope for v1.0.x and is treated as an operational consequence of the φ center-trace half-life metric (see §5.4).

Noise note (core v1). “canonical low” means “numerically small” (σξ = 5.0×10⁻³ in the canonical run). It does not imply a separate “low_noise_mode” override; in the locked audit profile low_noise_mode=false (base noise is used).

Contract alignment note (v1.0.17-core). The contract suite validates only the declared metric subset and artifact presence. Phenomena such as “Spin Aura” may be discussed only as observational/interpretive, unless and until they are explicitly added to the contract suite as acceptance criteria.

Structural Closure (core v1) — contract-derived proxy. In v1.0.x, we treat “Structural Closure” as a named, operational consequence of the contract-validated φ center-trace half-life anchor (phi_half_life_steps) together with the required presence of the φ trace artifacts (*_phi_center_log.csv, *_phi_center_plot.png). This means: the contract does not validate any morphological “shape” claim of remnants; it validates the timescale anchor we use as the closure proxy. Any additional statements about remnant geometry, spatial localization maps, or trajectory-level bias are not contract-validated unless explicitly added later.

1. Abstract

Lineum is a minimal discrete coupled-field model defined by a local update rule on a 2D periodic grid. It evolves a complex field ψ coupled to a real interaction/memory field φ and a static tuning map κ. The model does not assume any physical constants (e.g., c, ħ, G), spacetime metric, or continuum symmetries; instead it uses dimensionless control parameters (α, β, δ) within the numerical scheme. In the canonical evidence run pinned in §4.6, the system produces stable, quantifiable behaviors under a fixed numerical scheme. Where we use physics terms (e.g., “quasi-particle”, “spin”), they are strictly analogical labels for operationally defined measurements, not claims of equivalence to any continuum field theory.

The system evolves according to a coupled three-field update rule (see Equation (1) (Version 4) in Section 3), which governs the primary field ψ, the interaction field φ, and the spatial tuning map κ.

Terminology. We use linon to denote a stable, localized excitation of |ψ|² (a quasi-particle analogue emergent from the Lineum rule). It is not a fundamental particle. On first mention we may write “linon (localized excitation)”; thereafter we use linon.

Pronunciation (model name). Lineum = Czech /ˈlɪ.nɛ.um/ (short i, three syllables: “LIH-neh-oom”, stress on the first).
For readers in English: /ˈlɪniəm/ (UK/US ≈ “LIH-nee-um”).

Pronunciation (phenomenon). linon = Czech /ˈlɪnon/ (short i, stress on the first syllable).
For readers in English: /ˈlɪnɒn/ (UK ≈ “LIH-non”) or /ˈlɪnɑːn/ (US ≈ “LIH-nahn”).
Not “LAI-non”.

Plain-language summary (non-technical)

If you strip the math down to the basics, Lineum is “just” a grid of numbers that gets updated locally, step by step:

• ψ is the fast, wavy carrier (a complex oscillation on the grid).
• φ is the slow “memory/envelope” that reacts to where ψ has high intensity and then diffuses.
• κ is a static sensitivity map (in core v1 it does not evolve).

What we actually show (and pin to the evidence HTML/CSV) in core v1 is not “new physics”, but reproducible, auditable behavior under a pinned numerical scheme:

• [OBS] localized excitations (“linon candidates”) visible in amplitude/trajectory artifacts;
  [VALIDATED] anchors derived from these detectors include low_mass_qp_count (exact) and max_lifespan_steps (threshold).
• [VALIDATED] a dominant tone (f_0) (f0_mean_hz).
  Note: windowed 95% CIs shown in HTML quantify within-run variability and are informational (not contract-validated).
• [VALIDATED] a strong spectral dominance (SBR) of that tone (sbr_mean; ±2-bin guard).
• [VALIDATED] topological neutrality within tolerance (N1) and mean vortex count (computed over logged frames as declared in the manifest; topo_log_stride).
• [VALIDATED] a measurable φ center-trace half-life (phi_half_life_steps) and required φ-trace artifacts.

Contract scope note (v1.0.17-core). Validation in core v1 is defined by the contract suite referenced at the top of this manuscript. It validates only a declared subset of metrics (e.g., f0_mean_hz, sbr_mean, topology neutrality N1, mean vortex count, phi_half_life_steps, max_lifespan_steps, low_mass_qp_count) and the required artifact set. Any other effect (e.g., guided motion statistics, spin-aura amplitude thresholds, morphology of φ remnants) must be labeled not contract-validated unless and until it is added as an explicit acceptance criterion.

What we do not claim in core v1: Standard-Model identification, gravity/GR mapping, thermodynamics, or any phenomenon requiring time-varying κ. Those belong to separate experimental/extension tracks.

Physics translation (informal; analogy only)

Readers with a conventional physics background can think of a linon as a stable localized oscillatory mode on a lattice in a nonlinear, dissipative, coupled-field system. In spirit, it is closer to a breather/soliton-like localized state (in discrete nonlinear media) than to a fundamental particle.

Informally:

• the φ-update looks like a reaction–diffusion / relaxation channel driven by (|\psi|^2),
• the ψ-update combines damping, local coupling (φ·ψ), diffusion (∇²ψ), and a drift term +∇φ that behaves like advection by a potential gradient.

This is an interpretive translation to familiar language (e.g., “discrete nonlinear waves”, “reaction–diffusion”, “CGLE-like behavior”), not a claim of equivalence to any specific continuum field theory.

In the canonical evidence run (and additionally in the provided per-seed evidence bundles, without cross-seed aggregation in this draft): – [OBS] stable localized excitations (linons) visible in amplitude/trajectory artifacts, with contract-anchored detector outputs;
– [VALIDATED] dominant tone (f_0) (f0_mean_hz),
– [VALIDATED] spectral strength (SBR, sbr_mean, ±2-bin guard),
– [VALIDATED] topological neutrality (N1) and mean vortex count (logged frames),
– [VALIDATED] center-trace φ half-life (phi_half_life_steps).

All listed items are directly reported in the HTML evidence (Quasiparticle Properties, Spectral metrics, Topology metrics, “Spin aura — averaged curl map”, and φ center trace). Claims requiring κ-dynamics, thermodynamics, or SM identification are out of scope for the v1 core.

Within the v1 core evidence bundle, contract-validated items are limited to:
(i) the canonical run’s validated numeric anchors (e.g., f0_mean_hz, sbr_mean, topology neutrality N1, mean vortex count, phi_half_life_steps, max_lifespan_steps, low_mass_qp_count) and
(ii) the presence of required artifacts (HTML/CSV logs).
Other effects (e.g., “spin aura”, “Structural Closure”, “guided motion” statistics beyond the equation-level intuition) may be described as observations, but must be explicitly labeled not contract-validated.

For the canonical parameter choice (Δt fixed and SI constants applied post hoc), the dominant oscillation can be expressed in familiar physical units for scale illustration only. For example, when written in SI units (refined snapshot; RUN_TAG spec6_false_s41, commit 875fc4e): these values are not used as acceptance thresholds or constraints anywhere in the core validation.

• [DISPLAY] (unit conversions from f0_mean_hz; not acceptance criteria):
• dominant oscillation frequency (contract anchor) ≈ 1.857×10²⁰ Hz,
• linon energy (display-only) ≈ 1.23×10⁻¹³ J ≈ 767.90 keV,
• wavelength (display-only) ≈ 1.61×10⁻¹² m (0.00161 nm),
• effective mass (display-only) ≈ 1.50× of the electron mass (m/mₑ ≈ 1.5027).

These SI-anchored values are unit conversions of the canonical tone f₀, not additional constraints on the model or evidence that Lineum directly realizes any specific physical scale.

Plain-language warning. This is the same kind of conversion you’d do if you take a frequency and compute “what energy would a photon of that frequency have”. It gives a sense of scale, but it does not turn the linon into “a photon/electron/etc.” and it does not assert a real-world rest mass.

Interpretation note (v1). The “effective mass” value is a unit-conversion from the dominant frequency (f_0) via (m = h f_0 / c^2). It is provided only as an intuition aid for scale (refined snapshot: RUN_TAG spec6_false_s41, commit 875fc4e), not as a claim of an intrinsic rest mass.

Reports alignment (v1). In the refined snapshot, the SI-anchored values (E, λ, display-only m/mₑ) are computed directly from the reported (f_0) (manifest/CSV) using fixed SI constants. This remains a scale indicator, not a rest-mass claim.

Non-identification (v1). A linon is a stable, localized excitation in the Lineum field, not a Standard-Model particle. The numerical anchors in the Abstract (f₀, E, λ, and the display-only mass ratio m/mₑ) are provided to indicate scale only. They must not be read as an identification with electrons, neutrinos, or any SM species. See “Terminology” (linon is not a fundamental particle) and the Interpretation note (v1) on display-only mass.

Topology logging note (v1). Topology metrics (neutrality and mean vortex counts) are computed from *_topo_log.csv, which is decimated (logged every logging.topo_log_stride steps as declared in the run manifest). In the canonical run spec6_false_s41, topo_log_stride = 25, hence topology metrics are computed over logged frames (N=81 frames; steps 0..2000). This decimation is deterministic, declared in the manifest, and validated by the contract suite.

All reported phenomena arise from local operations on a discrete grid starting from a defined class of small random initializations (see §4.1) without per-run manual tuning. No predefined force law is included. Linons tend to drift along +∇φ (toward increasing φ); we describe this as environmental guidance rather than any gravitational claim.

The system is open for independent verification: the canonical run is fully pinned by manifest, artifacts, and the contract suite, and replication is evaluated by the contract-aligned acceptance bands in §4.3.1.

Out-of-scope clarifier (v1 core). We do not claim: (i) predictability of “random” outcomes or a deterministic substrate for stochastic processes; (ii) any κ-dynamics phenomena (e.g., “Dimensional Transparency”); (iii) thermodynamic quantities (T, S) or fluctuation–dissipation calibration; (iv) identification with Standard-Model particles; or (v) gravitational analogies. These topics are deferred to the experimental/extension track and are not part of the v1 core evidence.

Graphical abstract.

_{Note: this mark is a visual mnemonic only; it carries no physical claim and is not used in any metric.}

Icon legend (mnemonic only — no physical claim).

• Fish (upper) → κ (tuning / sensitivity).
  Why: the “eye” dot suggests a sensor, the body an oriented agent responding to local cues; κ controls local susceptibility and tuning (κ·α, κ·β), i.e., how the medium responds to gradients.
• Spiral (lower-left) → ψ (oscillation / flow).
  Why: a spiral visually encodes periodicity and phase circulation; ψ is the time-like carrier with canonical tone f₀, from which we derive SI conversions (E, λ, display-only m/mₑ).
• Leaf (right) → φ (memory / envelope).
  Why: the broad lamina reads as an envelope, while venation evokes a stored pattern/context; φ is the memory field used in nearby/field means and the center-trace half-life metric.
• Outer loop → circulation / interplay.
  Why: a closed interplay κ → (α_eff,β_eff) → φ ↔ ψ as a mnemonic of coupling/modulation; no law or metric is implied.
• Design note. Uniform line weight → no hierarchy; shapes are not used as data encodings anywhere in the paper.

Directional mnemonic (mnemonic only): κ (tuning) modulates the local response parameters, ψ drives φ via |ψ|², and φ feeds back into ψ via the canonical coupling terms (φψ and +∇φ) as defined in Eq. (1). Do not infer causality from the icon beyond the explicit update rule.

Three-field flow. The mark depicts the triad ψ–φ–κ in balance: ψ (oscillation / flow), φ (memory / resonance), κ (tuning / sensitivity). It is a visual mnemonic only; the canonical Equation (1) defines the model.

Core thesis (v1; claim-strength explicit). We demonstrate a reproducible, parameter-light emergence of a stable localized excitation (“linon”) with:

• [VALIDATED] a dominant tone (f_0) (f0_mean_hz) and strong spectral dominance (SBR, sbr_mean);
• [VALIDATED] topology neutrality (N1) and mean vortex count computed over logged frames;
• [VALIDATED] a φ center-trace half-life timescale (phi_half_life_steps) + required φ-trace artifacts (closure proxy);
• [DISPLAY] SI-anchored conversions (E, λ, display-only (m/m_e)) computed from (f_0) for scale illustration only;
• [OBS] within-run uncertainty reporting (windowed 95% CIs shown in HTML) as an informational stability view, not a contract acceptance.

The contribution is methodological: numeric anchors + guardrails that turn an emergent phenomenon into an auditable, falsifiable object others can probe and extend.

Falsifiable checks (v1; [TEST]).
The following are verification procedures (not results) that should produce a contract PASS when regenerated and pinned.

(C1) Window resolution [TEST]: Re-run with W = 512 (keeping Δt = 1.0e−21 s). Expect the dominant tone to remain near the same FFT region and the contract suite to PASS; in particular f0_mean_hz stays within the §4.3.1 band [1.84e20, 1.87e20] Hz and sbr_mean ≥ 3000.
(C2) Temporal refinement (Δf preserved) [TEST]: Halve the time label and double the window (Δt → Δt/2, W → 2W) so Δf = 1/(W·Δt) stays constant. Expect contract PASS with the same anchors/bands (no requirement of exact bin-centering; centroid/interpolated (f_0) is valid).
(C3) Grid size [TEST]: Re-run on 256×256 with identical parameters. Expect contract PASS with f0_mean_hz within [1.84e20, 1.87e20] Hz, sbr_mean ≥ 3000, and topology/φ anchors within their §4.3.1 bands.

2. Motivation

Many approaches in theoretical physics rely on continuous equations embedded in a predefined spacetime geometry, with global constants and symmetries fixed a priori. Such frameworks limit the exploration of systems where both the geometry and the interaction rules could emerge from purely local processes.

Lineum is designed as a minimal model to investigate whether complex, stable, and potentially physics-analogous structures can arise from: – simple, local update rules, – no predefined global metric or constants, – and interactions mediated by emergent fields.

The key motivation is to test if macroscopic phenomena, such as particle-like excitations, field-mediated interactions, and stable wave patterns, can originate without embedding them explicitly into the governing equations.

By isolating and quantifying these emergent behaviors, Lineum offers a controllable environment to evaluate which observed effects might have analogues in known physics, and which are unique to discrete, metric-free systems.

3. Equation

The evolution of the system is defined on a discrete 2D grid by three coupled update rules for: – the primary complex scalar field ψ, – the interaction field φ, – and the (static) tuning field κ (spatial map).

The canonical form is:

Equation (1) — Canonical Lineum update (Version 4)

Euler-step convention (v1). Eq. (1) is written in “per-step increment” form with the internal scheme step absorbed into the implementation (see invariants.dt note in §3.1). Replication is validated by contract metrics, not by algebraic normalization of an explicit Δt factor in the paper equation.

Equation	Description
ψ ← ψ + 𝛌̃ + ξ + φψ − δψ + ∇²ψ + ∇φ	primary field evolution
φ ← φ + α_eff (|ψ|² − φ) + β_eff ∇²φ	interaction field response
κ ← κ(x, y)	spatial tuning map

For the full development history of this equation, see Appendix A: Equation History.

Legend (symbols & operators)

Symbol	Type / Range	Role	Default / Notes
ψ(x,y,t)	ℂ	primary field; |ψ|² = density, arg ψ = phase • linons = localized |ψ|² maxima
φ(x,y,t)	ℝ	interaction / memory field	accumulates response to |ψ|²
κ(x,y)	ℝ⁺ (static)	spatial tuning map	no time evolution; often normalized to [0,1]
𝛌̃(x,y,t)	ℂ	external stimulus	0 unless stimulus experiments
ξ(x,y,t)	ℂ	noise (zero-mean)	optional; amplitude set by manifest (canonical: σξ = 5.0×10⁻³)
δ	ℝ₊	damping in ψ-update	local, ≥ 0
α	ℝ₊	coupling	from |ψ|² to φ
β	ℝ₊	diffusion	φ diffusion strength
∇	operator	discrete gradient	central differences
∇²	operator	discrete Laplacian	4-neighbour (5-point von Neumann; implemented via diffuse_complex() in the reference code)
BCs	—	boundary conditions	periodic in x,y
α_eff, β_eff	—	effective params	α_eff = κ·α, β_eff = κ·β (κ modulates α,β)

Canonical parameters (spec6_false_s41)

Param	Meaning	Canonical value
α	φ relaxation toward |ψ|²	7.0×10⁻⁴
β	φ diffusion strength	1.5×10⁻²
δ	ψ damping per step	4.62×10⁻³
σξ	noise amplitude in ψ	5.0×10⁻³
κ	spatial tuning (static, const.)	0.5 everywhere

Parameter note. In Eq. (1), the φ-update uses α_eff = κ·α and β_eff = κ·β. In the canonical run (κ = 0.5 everywhere), this means α_eff = 3.5×10⁻⁴ and β_eff = 7.5×10⁻³ (α,β in the table are the base parameters).

Scope note (canonical dimensionality). All results in this core paper use a 2D discrete grid with periodic boundary conditions. Any 3D extensions or non-periodic boundaries are treated as supplementary variants and are not part of the canonical Eq. 1.

Sign convention. The +∇φ term in the ψ-update induces drift toward increasing φ (movement along the φ-gradient).

Note (κ as static map). In the canonical rule, κ is a static spatial map (no time evolution). Any experiments that evolve κ belong to supplementary variants, not to Eq. 1. In Eq. (1) we write the φ-update in terms of locally effective parameters α_eff = κ·α and β_eff = κ·β to make this modulation explicit. Here α and β remain the base (global) parameters; κ modulates them spatially rather than “replacing” them. In the canonical run κ = 0.5 everywhere, hence α_eff and β_eff are constant.

No explicit spacetime geometry, global constants, or long-range interactions are predefined. All behavior results from repeated local updates of these fields.

3.1 Numerical scheme & stability (canonical)

Discrete operators (periodic, $\Delta x = \Delta y = 1$).

$$\nabla_x f_{i,j} = \frac{f_{i+1,j}-f_{i-1,j}}{2},\quad \nabla_y f_{i,j} = \frac{f_{i,j+1}-f_{i,j-1}}{2}$$ $$\nabla^2 f_{i,j} = f_{i+1,j} + f_{i-1,j} + f_{i,j+1} + f_{i,j-1} - 4\,f_{i,j}.$$

Gradient injection note (v1). In the ψ-update, the +∇φ term is represented as a complex drift field so it can be added to the complex scalar ψ. In the reference implementation we use the conventional embedding: $$ \nabla\phi ;\equiv; (\nabla_x\phi) + i(\nabla_y\phi), $$ with central differences under periodic BCs.

In the reference implementation this four-neighbour stencil is realized by the helper function diffuse_complex(...) in the φ-update; there is no 9-point (diagonal) Laplacian in the canonical v1 run. Wider stencils (e.g., 9-point) are treated as exploratory variants outside the v1 core evidence.

Time stepping. Explicit Euler with fixed $\Delta t = 1.0\times 10^{-21}\,\mathrm{s}$ (canonical anchor). Interpretation note (Δt). In the core, $\Delta t$ functions as a conventional unit label that fixes spectral bin spacing via $\Delta f = 1/(W\,\Delta t)$. The validation criteria depend on the measured spectral peak and on dimensionless metric tolerances, not on any external calibration of “seconds” to physical time.

Normative dt duality box (v1). The canonical evidence contains two distinct time-step notions. They must not be conflated:

Field	Canonical value	Meaning	Contract role
invariants.dt	0.01	scheme step (dimensionless Euler increment inside the numeric update)	recorded for scheme provenance; not used to define Hz/SI conversions
run.time_step_s	1.0e−21 s	reporting label for spectral units (Hz) and display-only SI conversions	validated and used for spectral reporting/anchoring

Rule (v1): spectral bins and SI conversions are defined from run.time_step_s via $\Delta f = 1/(W\,\Delta t)$, while the numerical integrator uses invariants.dt internally. Validation relies on contract metrics and tolerances, not on algebraic normalization of Δt in the paper equation.

Stability sanity. For diffusion-like terms a heuristic CFL bound is best read in terms of the scheme step invariants.dt (dimensionless), not the reporting label run.time_step_s (seconds):

$$\Delta \tau \equiv \mathrm{invariants.dt} \;\lesssim\; \mathcal{O}\!\left(\frac{(\Delta x)^2}{4 D_{\mathrm{eff}}}\right).$$

In practice we verify stability empirically via Section 5 metrics (SBR, topological neutrality, and $\phi$ half-life) on the canonical run; windowed estimates with 95% CIs are reported in the HTML report.

4. Method

Simulations are performed on a discrete square grid with periodic boundary conditions. Each step updates ψ and φ according to the canonical equation; κ is a static spatial map that is sampled (not evolved) at each step. We apply the discrete Laplacian for diffusion terms and nearest-neighbor operations for gradients.

4.1 Grid and Initialization {#calibration-seed}

– Canonical grid: 128×128 cells (periodic BCs). All results reported in this core paper use this grid.
– Exploratory grids (non-canonical): 256×256 or 512×512 may be used for visualization or stress tests, but they are not part of the canonical results.
– Initial ψ: complex noise with small amplitude.
– Initial φ: uniform background with small perturbations.
– κ: static spatial map (constant/gradient/localized) depending on the test.

4.2 Update Procedure

At each timestep:

1. Compute Laplacians and gradients for ψ and φ.
2. Apply the update rules for ψ and φ (κ is static and only sampled).
3. Optionally add controlled noise ξ to test stability.
4. Record intermediate states for analysis.

One-step update (canonical Eq-4)

Context: periodic BCs, Δx = Δy = 1, explicit Euler; κ is static (constant map in the canonical run).

# periodic BCs, Δx=Δy=1, explicit Euler
for t in range(T):
    # spatial derivatives
    lap_psi  = laplacian(psi)          # 5-point (von Neumann)
    grad_phi_x, grad_phi_y = gradient(phi)  # central differences
    grad_phi = grad_phi_x + 1j*grad_phi_y   # inject ∇φ as complex drift field
    lap_phi  = laplacian(phi)

    # effective parameters (κ is static)
    alpha_eff = kappa * alpha
    beta_eff  = kappa * beta

    # ψ-update
    psi = psi + lambda_tilde + xi + phi*psi - delta*psi + lap_psi + grad_phi

    # φ-update
    phi = phi + alpha_eff*(np.abs(psi)**2 - phi) + beta_eff*lap_phi

    # optional logging/detectors
    if t % LOG_EVERY == 0:
        save_state(...)

4.3 Reproducibility {#resonance-scanner}

All runs are initialized with fixed random seeds so that the reference implementation behaves deterministically under the same runtime/locked profile.
Simulation parameters and κ-maps are stored alongside results to allow exact replication.

Replication caveat (v1). Exact byte-for-byte equality across platforms/languages is not required; replication is evaluated by the contract-aligned metric tolerances in §4.3.1 (and by the audit fingerprints in §4.10.5 for canonical verification).

Anti-cherry-pick (v1). The evidence seeds used in this core track are pre-registered as a fixed set {17, 23, 41, 73} and are shipped as separate per-seed bundles. The canonical numeric anchors used for “validated” claims in v1.0.17-core are pinned to spec6_false_s41 unless and until the multi-seed set is regenerated under a single pinned code state and (optionally) added to the contract suite.

4.3.1 Cross-Implementation Replication (advisory)

We do not require bit-for-bit equality across languages/backends. Small numerical differences are expected (RNG streams, FFT/Laplacian kernels, rounding). For v1, replication is defined by metric tolerances on the canonical run (spec6_false_s41):

Contract-aligned acceptance bands (v1.0.17-core). For statements marked “validated”, replication must satisfy the contract suite bands: – Dominant frequency f₀ (mean): in [1.84e20, 1.87e20] Hz
– SBR (mean; ±2-bin guard): ≥ 3000
– Topology neutrality (N1): in [93.0%, 99.5%]
– Mean vortices: in [55.0, 65.0]
– Max lifespan: ≥ 40 steps
– φ half-life (center): in [900, 1100] steps
– Low-mass QP count: 49 (exact)
Other informal tolerances and narrative checks may be used for developer debugging, but they must not be presented as validated unless added to the contract suite.

Precision note. All reference runs use IEEE-754 double precision (float64). Language choice (Python/Julia/C++/…) does not change numerical semantics; replication is evaluated by the metric tolerances above, not bitwise equality.

Replicators must pin Δt, pixel size, grid size, seed, κ-mode, and use the provided manifest. Cross-language runs are encouraged; comparisons should be reported via these metrics rather than pixel-wise diffs.

4.4 Detection of Phenomena {#topology-neutrality}

Detected phenomena in the core study include: – formation and motion of linons, – vortex creation and annihilation,
– phase gradient rotation (spin),
– φ-trap formation and particle capture.

Detection is performed using automated field analysis: – tracking |ψ|² maxima for particle positions,
– identifying vortex cores via phase winding,
– qualitative φ-basin / trapping patterns (observational term; not contract-validated in v1).

4.5 Output {#lifespan}

For each run, the system generates:

• CSV (per run): *_amplitude_log.csv, *_spectrum_log.csv, *_phi_center_log.csv, *_topo_log.csv, *_trajectories.csv, *_multi_spectrum_summary.csv, *_spin_aura_profile.csv, *_metrics_summary.csv.

• Topology logging cadence: *_topo_log.csv is logged every logging.topo_log_stride steps (declared in the manifest; canonical: 25).

• PNG (figures): *_spectrum_plot.png, *_topo_charge_plot.png, *_vortex_count_plot.png, *_spin_aura_map.png, *_phi_center_plot.png.

• GIF (animations): *_lineum_amplitude.gif, *_lineum_spin.gif, *_lineum_vortices.gif, *_lineum_particles.gif, *_lineum_flow.gif, *_lineum_full_overlay.gif.

Filenames are shown without the RUN_TAG_ prefix for readability; see Appendix A for the prefix convention.

4.6 Reproduction Manifest (canonical run) {#phi-half-life}

This manifest pins all run-level switches for the canonical reference used in this paper.

• RUN_TAG: spec6_false_s41
• Evidence run id (contract): spec6_false_s41_20260215_023130
• Seed: 41
• Grid: 128 × 128 (periodic BCs)
• Steps: 2000
• Precision: float64 (IEEE-754)
• Δt (time step): 1.0e-21 s (canonical)
• κ-mode: constant (static spatial map; no time evolution)
• Equation: Eq-4 (canonical update rule; see Eq. (1))
• Primary spectral metric: power spectrum |FFT(x)|^2 with a ±2-bin guard around f0
• Detection conventions: as fixed in Appendix A (no amplitude gating for CSV/metrics; vortex gating is visualization-only)
• α (reaction_strength): 7.0e-4
• β (φ diffusion): 1.5e-2 (computed as 0.30 × diffuse_complex(rate=0.05))
• δ (ψ damping): 4.62e-3
• σξ (noise amplitude): 5.0e-3
• κ-map: constant 0.5 (uniform across the grid)
• logging.topo_log_stride: 25 (topology log cadence; topo metrics are computed over logged frames)

Contract fingerprints (v1.0.17-core).

• audit_scope_hash: 7197faf5a92a141a4847314485bee819ae9fdecdf08eead313ffdd3d3a6fe9f5
• code_fingerprint: 48ea56d33508a9579e01afde42e3522e6d491d6c68a3b9631d926c431fe6390c
• kappa_map_bin_hash: 31f1d2b2391050bc1f6975db4e8ae4dac6ddab211f45fc7f5333c18a3981aa3a Note: Git commit can remain in the HTML header as provenance aid, but the contract fingerprints are the normative audit pins for “Verified”.

Kappa hash note (uniform map). Even for κ=constant, kappa_map_bin_hash refers to the generated map binary used by the run (integrity pin). “kappa_hash_basis = N/A” only means no procedural basis string is needed for uniform κ in v1.0.x.

Artifacts (prefixing): all outputs are prefixed with {RUN_TAG}_… (e.g., spec6_false_s41_lineum_report.html), as listed in §4.5 and Appendix A.

4.7 Data & Code Availability {#particle-taxonomy}

All canonical artifacts for the run spec6_false_s41 are provided with this preprint as ancillary files: the canonical HTML report (spec6_false_s41_lineum_report.html), the reference implementation (lineum.py), and the generated CSV/PNG/GIF outputs listed in §4.5.

Reproduction uses the manifest in §4.6 (seed 41, grid 128×128, Δt 1.0e−21 s, κ static). The numeric source of truth is the JSON manifest (spec6_false_s41_manifest.json) and the machine-readable CSV logs. The HTML report is a derived view generated from these primary sources.

Version pinning (DOI snapshots vs drafts). Provenance is pinned by RUN_TAG=spec6_false_s41, the code commit noted in the HTML report header (short Git SHA, when available), and the artifact manifest embedded in the report. For DOI-published snapshots, we provide a sha256sums.txt (or equivalent) for the evidence bundle to support external integrity checks. Normative rule (DOI snapshots): a snapshot is considered intact iff the published sha256sums.txt matches the local evidence bundle for the listed files. For non-DOI working drafts, reproducibility is evaluated against the acceptance bands in §4.3.1.

Reviewer quick-check (v1).

1. Run the contract suite for lineum-core-1.0.17-core and confirm the canonical run spec6_false_s41_20260215_023130 is PASS.
2. Confirm the validated anchors in the canonical evidence:
   • f₀ (mean) = 1.856777545095882e+20 Hz
   • SBR (mean) = 3245.4600764773872
   • φ half-life (center) = 1009 steps (status OK)
   • Topology neutrality (N1) = 94.2%
   • Mean vortices = 59.1455
   • Max lifespan = 44 steps
   • Low-mass QP count = 49
3. Confirm the audit fingerprints match (§4.10.5).
4. In §5.6, the SI “worked example” matches m/mₑ ≈ 1.5027 (display-only; derived from f₀).
5. Frequency binning: Δf = 1/(W·Δt) = 3.90625e18 Hz and f₀ lies near k≈48 (centroid index ≈ 47.53).

Branching note. Further physics-mapping tests (dispersion, group velocity, external-field response) will be published under the experimental track v1.1.x-exp; the core canonical scope remains frozen in v1.0.17-core.

File-level scope (whitepapers). lineum-core.md together with lineum-core-equation-history.md define the canonical v1.0.x core. All other whitepapers in the repository whose filenames begin with lineum-exp-… or lineum-extension-… (e.g., lineum-exp.md, lineum-exp-thermo-calibration.md, lineum-extension-return-echo.md, lineum-extension-silent-gravity.md, lineum-extension-spectral-structure.md, lineum-extension-vortex-particle-coupling.md, lineum-extension-zeta-rnb-resonance.md) are by definition outside the v1 core. They may refer to the same phenomena (Return Echo, Dimensional Transparency, Silent Gravity, Spectral Structure, etc.), but quantitative claims there do not change the canonical scope unless explicitly merged into a future lineum-core version.

Future updates and non-canonical experiments will be released as separate preprints; this core v1 freezes the canonical run as spec6_false_s41.

Ancillary artifacts (per seed). The following files are attached as ancillary data to the paper (one HTML and one CSV per seed); filenames are prefixed by {RUN_TAG}_….

seed	HTML report	metrics (CSV)
23	spec6_false_s23_lineum_report.html	spec6_false_s23_metrics_summary.csv
17	spec6_false_s17_lineum_report.html	spec6_false_s17_metrics_summary.csv
41	spec6_false_s41_lineum_report.html	spec6_false_s41_metrics_summary.csv
73	spec6_false_s73_lineum_report.html	spec6_false_s73_metrics_summary.csv

Provenance. Data integrity for DOI snapshots is verified via sha256sums.txt. Provenance is further pinned by RUN_TAG, the short git commit shown in the report header, and the audit_scope manifest fingerprints.

4.8 Threats to validity (core v1)

Periodic BC artifacts. We verify metric invariance (within tolerances) when changing the grid size; figures are illustrative only, acceptance is by metrics.
Stencil bias. Canonical results use a four-neighbour (5-point von Neumann) discrete Laplacian implemented via diffuse_complex() in the φ-update. Alternative 9-point (diagonal) stencils are treated as exploratory variants and are not part of the v1 core evidence. Spectral leakage. FFT on de-meaned windows with a ±2-bin guard around $f_0$ mitigates leakage; SBR is computed on the power spectrum $|\mathrm{FFT}(x)|^2$.
RNG/seed bias. We provide multiple seeds {23, 17, 41, 73} as separate evidence bundles. Each run reports windowed 95% CIs (within-run). Cross-seed aggregates (if shown) are reported only after regeneration under a single pinned code state; replication is defined by tolerance bands in §4.3.1.

Visualization bias. All metrics derive from numeric logs (CSV). Amplitude gating is visualization-only in GIFs; winding/metrics use raw values. Topology log decimation. topo_log.csv may be decimated (logged every logging.topo_log_stride steps; canonical: 25). Topology neutrality (N1/N0) and mean vortices are therefore computed over logged frames, not every simulation step. This cadence is declared in the manifest and validated by the contract suite to prevent ambiguity.

Display-only mass (interpretation risk). The “effective mass” reported in §1/§5.6 is a unit-conversion from the reported dominant frequency (f_0) via (m = h f_0 / c^2). It is provided purely as a scale cue (display-only), not as a claim of an intrinsic rest mass. Mitigations in v1: (i) explicit Interpretation note (v1) in the Abstract; (ii) SI constants stated in §5.6; (iii) report tooling computes the display mass directly from (f_0) at render time to avoid drift. Multi-seed alignment statements are quoted only after regeneration under the same pinned code state (commit 875fc4e).

Not claimed (v1). We explicitly do not claim:

• identification of linons with any Standard-Model particle;
• an intrinsic rest mass for linons (the “effective mass” is display-only; see Abstract and §5.6);
• gravitational dynamics or any mapping to General Relativity;
• Lorentz invariance or a relativistic field theory formulation;
• validity outside the canonical scope (2D, periodic BCs, static κ) defined in this core.
• Thermodynamic quantities. We do not define entropy (S), temperature (T), heat capacity, or entropy production for the linon in v1. The core setup uses a deterministic, isolated field without a thermal bath or an ensemble; no equipartition or fluctuation–dissipation calibration is assumed. Any “effective temperature” would require an explicit reservoir/noise model and a traceable calibration procedure—deferred to the experimental track (v1.1.x-exp).

4.9 Tooling guardrails (v1)

To prevent drift and ease auditing, the report tooling enforces the following safeguards:

• Mass-from-f₀ consistency. At render time, the HTML report recomputes the display-only mass directly from the canonical tone, using (m = h,f_0/c^2), and raises an error if the value deviates (tolerance (<10^{-6}) relative). This ensures the “Effective mass” row cannot diverge from (f_0).
• Commit provenance. Each HTML report header prints the short Git commit for provenance (alongside RUN_TAG and run metadata). Regenerating a report on a different code state changes the commit stamp by design.
• SI anchoring. Conversions use fixed SI constants (h, c, m_e) as stated in §5.6; the HTML shows derived quantities that follow directly from these constants and the measured (f_0).
• Pinned runs. Evidence is pinned by RUN_TAG (seeds 17/23/41/73 for v1). Reproduction is evaluated by the metric tolerances in §4.3.1 rather than bitwise equality.

Scope. These guardrails are part of v1 tooling only; they do not assert any rest-mass claim—“effective mass” remains a scale indicator derived from (f_0).

4.10 Audit Profile and Verification (Stateless Audit 1.0)

To ensure the highest integrity of v1 "whitepaper core" runs, we introduce the Stateless Audit mechanism. This mechanism enforces that an audit run proceeds with a precisely defined configuration that cannot be inadvertently changed (e.g., via a forgotten environment variable).

4.10.1 Profile Activation

The Audit Profile is activated via the LINEUM_AUDIT_PROFILE environment variable. When set to whitepaper_core, the system enforces a strict "hash gate" that prevents execution if the resolved runtime configuration deviates from the canonical values defined below. Canonical PASS requires executing the locked profile for the full simulation length (steps=2000) and verifying the result against the contract suite.

4.10.2 Canonical Audit Scope (Locked Configuration)

The following parameters are "locked" in the audit profile and form the basis of the audit_scope_hash:

Parameter	Canonical Value (v1.0.x)	Meaning
run_id	6	Canonical parameter set identifier
steps	2000	Simulation length (audit minimum)
run_mode	false	Standard mode (infinite_mode off)
seed	41	RNG initialization (deterministic)
kappa_mode	constant	Uniform κ (scalar 0.5 everywhere)
low_noise_mode	false	Flag off. Canonical uses the base noise amplitude σξ = 5.0e−3. low_noise_mode is a reserved override (not used in v1 core) and must remain off in the locked profile.
test_exhale_mode	true	Active trace analytics (v1.0.x required)
resume	false	Start from step 0 (no checkpoint drift)
base_output_dir	"output_wp"	Dedicated directory for audit evidence
kappa_hash_basis	"N/A"	(Not used for uniform map)
kappa_schedule_id	"N/A"	(Not used for v1.0.x)
kappa_stride	"N/A"	(Not used for v1.0.x)

[!NOTE] Audit runs generate outputs in output_wp/. Publicly published versions and links in this document refer to output/, which is an export (copy) of the canonical run from the audit folder.

4.10.3 Verification Mechanism (Hash Gate)

1. audit_scope_hash: SHA256 fingerprint of the locked configuration above.
2. scope_fingerprint: Manifest echo of the audit scope fingerprint (same value as audit_scope_hash; included for readability and tooling compatibility).
3. code_fingerprint: SHA256 fingerprint of source files (lineum.py, tools/whitepaper_contract.py). To ensure cross-platform stability (CRLF vs LF), files are normalized to LF before hashing.

4.10.4 Verification Protocol

The tools/whitepaper_contract.py suite compares the run results to the requirements in contracts/.

• Smoke Verification: Validation of manifest parsing, anchor wiring, and drift detection (may use reduced steps).
• Canonical Verification: Full evidence-run validation (2000 steps). A successful suite report (PASS) on the canonical evidence is mandatory for formal release verification.

Verification occurs in two phases:

• At startup (lineum.py): Check that audit_scope_actual_hash == audit_scope_expected_hash.
• Subsequently (whitepaper_contract.py): The contract suite verifies the presence of AUDIT_SCOPE in the manifest and matching of all hashes against the defined contract.

4.10.5 Canonical Reference Fingerprints (v1.0.17-core)

The following values are derived from the locked whitepaper_core definitions and must be matched by any canonical audit run report for formal validation.

Field	Canonical Expected Value (SHA256)
audit_scope_hash	7197faf5a92a141a4847314485bee819ae9fdecdf08eead313ffdd3d3a6fe9f5
scope_fingerprint	7197faf5a92a141a4847314485bee819ae9fdecdf08eead313ffdd3d3a6fe9f5
code_fingerprint	48ea56d33508a9579e01afde42e3522e6d491d6c68a3b9631d926c431fe6390c
kappa_map_bin_hash	31f1d2b2391050bc1f6975db4e8ae4dac6ddab211f45fc7f5333c18a3981aa3a

[!NOTE] The code_fingerprint includes performance-path changes (e.g., vectorized distance computations) required to process full 2000-step canonical runs within minutes. Audit validity is pinned by the fingerprint itself, not by the narrative reason.

4.10.6 Data Verification (Kappa Map)

The audit profile optionally supports binary integrity checking of the kappa map via LINEUM_EXPECTED_KAPPA_MAP_HASH. If set, the system verifies the bitwise match of the generated map before simulation begins.

5. Validation

Figure 0 — Canonical anchors at a glance

_{Source: see the HTML report output/spec6_false_s41_lineum_report.html; all runs are indexed in Appendix C.}

Caption (v1). The power spectrum of the center-amplitude time series shows a dominant tone near FFT region k≈48; the reported (f_0) is a centroid/interpolated estimate (f₀ (mean) = 1.8568×10²⁰ Hz). The corresponding time trace exhibits a stable, long-lived oscillation. SI-derived quantities (E, λ, display-only m/mₑ) follow directly from (f_0) via (E = h f_0), (λ = c/f_0), (m = E/c^2) (scale illustration only). (Within-run CIs shown in HTML are informational; not contract-validated.)

Canonical numerical anchors (refined snapshot, seed 41).
f₀ (mean) = 1.856777545095882e+20 Hz · SBR (mean) = 3245.4600764773872 ·
φ half-life (center) = 1009 steps · Topology neutrality (N1) = 94.2% · Strict neutrality (info-only) = 88.95% · Mean vortices = 59.1455 · Max lifespan = 44 steps · Low-mass QP count = 49
Note: contract suite does not carry CI bounds here; only the validated mean/point anchors above.

The validation phase aims to confirm that specific emergent phenomena occur consistently under controlled conditions, and to quantify their characteristics.

Metrics & 95% CI. We report two primary spectral metrics for reproducibility: the dominant frequency ($f_0$) and the Spectral Balance Ratio (SBR). Both are estimated on the amplitude time-series at the field center using sliding windows (length $W=256$ frames, hop $H=128$ frames) with a ±2-bin guard around $f_0$ in the background power. For each metric we aggregate the windowwise mean and a non-parametric 95% bootstrap confidence interval across windows; the HTML report prints values as value [lo, hi]. When the windowed estimate is available it supersedes the single-shot FFT value; otherwise the single-shot is shown as a fallback. These intervals quantify within-run variability without fitting any external model. Cross-seed aggregation is reported only when the multi-seed set is regenerated under the same pinned code state.

5.1 Guided Motion via φ-Gradient

[OBS] Simulations show that particles exhibit drift along +∇φ. Trajectory statistics indicate a systematic decrease in distance to regions of increasing φ over time, consistent with environmental guidance by the background field. No force law is introduced; the observed behavior follows directly from the +∇φ term in Eq. (1) and remains robust across seeds and runs.

Contract scope (v1). This section is observational/interpretive. The current contract suite does not validate any guided-motion statistic; only artifact presence (e.g., *_trajectories.csv) is required.

Wording guardrail. In v1 core we treat “guided motion” as a descriptive label for what is visibly consistent with the +∇φ term in Eq. (1). We do not claim a validated, general drift law or a contract-validated statistic in this section.

Evidence pointer (HTML & files).
Trajectory drift along +∇φ is visible in the per-run HTML under “Trajectories” and “Flow” animations, and is quantified from:

• *_trajectories.csv (particle paths; decreasing distance to regions of increasing φ / along +∇φ),
• *_phi_center_plot.png / *_phi_center_log.csv (center trace supporting φ-memory/half-life),
• *_lineum_flow.gif (qualitative flow visualization).

For the canonical seed, see output/spec6_false_s41_lineum_report.html (links to these artifacts are listed in the report).

5.2 Spin Aura

[OBS] In the canonical evidence (and in other provided runs), a persistent phase-gradient rotation (spin) develops around stable linons in ∇ arg ψ. In the canonical evidence, this pattern is visible around stable linons over long intervals and typically fades when the tracked excitation decays.

Contract scope (v1). “Spin Aura” is not contract-validated in v1.0.17-core. The artifacts may be presented as reproducible observations (*_spin_aura_map.png, *_spin_aura_profile.csv), but no numeric acceptance is claimed yet.

Operational definition. We define the spin aura as the time- and ensemble-averaged map of curl(∇ arg ψ) in a fixed-size neighborhood around detected linon centers. For each detection, the local curl map is centered on the particle and accumulated; the resulting average yields a robust dipole-like pattern (“spin aura”) with radially decaying lobes. Presence of this pattern is our detection criterion; its amplitude–radius curve is reported in spin_aura_profile.csv and the raster in spin_aura_map.png. This makes §5.2 falsifiable and reproducible across runs.

Evidence pointer (HTML & files).
Presence and profile of the spin aura are reported per run as:

• *_spin_aura_map.png — time-/ensemble-averaged curl(∇arg ψ) raster centered on detected linons,
• *_spin_aura_profile.csv — radial amplitude–radius curve extracted from the map.

For the canonical seed, see ../output/spec6_false_s41_lineum_report.html under “Spin aura — averaged curl map”; the report links to both artifacts above.

5.3 Silent Collapse

[OBS] Under certain φ-damping conditions, linons decay without generating large-scale disturbances in ψ.
The process is characterized by an exponential decrease in |ψ|² amplitude within the particle’s core.

Contract scope (v1). “Silent Collapse” is not contract-validated in v1.0.17-core unless explicitly added as an acceptance criterion. It is included here as an observation supported by artifacts (e.g., amplitude/trajectory logs and overlays), not as a validated numeric anchor.

5.4 Structural Closure

In the v1 core, Structural Closure is treated as an operational consequence of the φ center-trace and its contract-validated half-life anchor. After a tracked excitation decays, the φ center trace remains elevated relative to its later baseline and decays on a measurable timescale captured by phi_half_life_steps (see §5.6 and *_phi_center_log.csv). This supports the minimal, falsifiable statement that φ retains center-trace memory over a measurable timescale after ψ excitation decays.

Scope guardrail. In v1 core we do not claim a validated statement about the spatial morphology of φ remnants (shape, footprint, localization maps). Any morphology-level discussion remains observational unless explicitly added to contract acceptance criteria.

Validation scope (v1). The contract suite validates the φ half-life timescale (phi_half_life_steps) and required φ-trace artifacts. Any stronger claim about remnant morphology (shape preservation beyond the center trace) is observational unless explicitly added to contract acceptance criteria.

Note (Return Echo — extension). In multiple runs, locations of prior linon decay later act as weak attractors for new linons: trajectories revisit identical or ε-near coordinates after a delay. This Return Echo is distinct from Structural Closure: closure denotes a static φ remnant after decay; echo denotes a behavioral bias that steers future arrivals back to that remnant via local ∇φ shaping. Return Echo is not part of the v1 core acceptance list; it is treated as an experimental hypothesis documented in the separate note lineum-extension-return-echo.md (trajectory density maps, statistics, and falsifiable tests are defined there).

5.5 Dimensional Transparency (out of scope in core v1)

Scope. This phenomenon was observed only in exploratory runs with time-varying κ, which are explicitly out of the core scope and are not included in the v1 evidence bundle (HTML/CSV). No acceptance metric or claim in this paper depends on dynamic-κ runs.

Status. Deferred to the experimental track (v1.1.x-exp) with its own artifacts and falsifiable checks. No core evidence is presented here.

5.6 Spectral Stability

Canonical frequency anchor (spec6_false_s41)
With Δt = 1.0e−21 s (canonical time step), the dominant frequency measured on the canonical run spec6_false_s41 is
f₀ (mean) = 1.856777545095882×10²⁰ Hz, which implies (display-only) E = h f₀ ≈ 1.23×10⁻¹³ J ≈ 767.90 keV and λ = c / f₀ ≈ 1.61×10⁻¹² m (0.00161 nm).

Representative run metrics (canonical: spec6_false_s41)
SBR (mean; ±2-bin guard): 3245.4600764773872.
Topology neutrality (N1): 94.2% (computed over logged frames; topo_log_stride=25, N=81).
Strict neutrality (N0; info-only): 88.95%.
Mean vortices: 59.1455.

φ half-life (center): 1009 steps (status OK).
Low-mass QP: 49 · Max lifespan: 44 steps.
Steps / grid: 2000, 128×128.

---

Additional validation runs (other seeds).
Multi-seed confirmations (e.g., seeds 17/23/73) are retained in the v1 narrative, but must be regenerated under the refined logging/config state (commit 875fc4e) before numeric values are quoted here. Until regenerated, the canonical numeric snapshot for v1.0.17-core is pinned to spec6_false_s41 (see §4.6 and the anchors at the top of §5.6).

---

Fourier analysis of long-duration runs shows that dominant oscillation frequencies can remain stable over time, even with particle creation and annihilation events.
In the refined snapshot pinned here (spec6_false_s41), the dominant tone used throughout the core narrative is the contract-validated mean f₀ (mean) = 1.856777545095882×10²⁰ Hz (see §4.6 “Reviewer quick-check” and §5.6 anchors).
Within-run variability is reported in the HTML as windowed 95% CIs (informational; not contract-validated); the machine-readable window statistics live in {RUN_TAG}_metrics_summary.csv.

Machine-readable. Per-run CSV with windowed means and 95% CIs is provided as {RUN_TAG}_metrics_summary.csv (e.g., spec6_false_s41_metrics_summary.csv; likewise for seeds 23/17/41/73).

Constants & rounding. Conversions use SI: Planck’s constant $h=6.62607015\times 10^{-34}\ \mathrm{J\,s}$, speed of light $c=2.99792458\times 10^8\ \mathrm{m/s}$, electron mass $m_e=9.1093837015\times 10^{-31}\ \mathrm{kg}$. Derived quantities (display-only) are reported as $E = h f_0$, $\lambda = c/f_0$, $m = E/c^2$, mass ratio $m/m_e$. We report $E$ and $\lambda$ to three significant figures, SBR to two decimals; CIs are non-parametric 95% bootstrap percentiles.

Worked example (canonical f₀).
Constants: h = 6.62607015e-34 J·s, c = 2.99792458e8 m/s, m_e = 9.1093837015e-31 kg.
Canonical tone (contract-validated mean): f₀ = 1.856777545095882e20 Hz.

Calculation:

m/m_e = (h * f₀) / (c^2 * m_e)
      = (6.62607015e-34 * 1.856777545095882e20) / ((2.99792458e8)^2 * 9.1093837015e-31)
      ≈ 1.502746e+00  = 1.5027  (150.27%)
E      = h * f₀ = 1.2303e-13 J  ≈ 767.90 keV
λ      = c / f₀ = 1.6146e-12 m  = 0.00161 nm

Formatting policy (v1). Numerical values are rendered consistently in text and HTML as follows: – Dominant frequency f₀: scientific notation with 3 significant figures.
– Energy E: 3 s.f. in joules and 2 decimals in keV (parenthesized).
– Wavelength λ: 3 significant figures.
– Effective mass (kg): 3 significant figures.
– Mass ratio m/mₑ: 4 decimals plus a percent in parentheses with 2 decimals (e.g., 0.0316 (3.16%)).
– Confidence intervals: [lo, hi] with the same precision as the mean.

Tie-breaker (v1). If any rounding discrepancy appears between the paper and artifacts, the manifest.json + CSV logs are the ground-truth numeric sources; the HTML report is a derived view.

Frequency binning. With window length $W=256$ and time step $\Delta t = 1.0\times 10^{-21}\ \mathrm{s}$, the frequency resolution is $\Delta f = \frac{1}{W\,\Delta t} = 3.90625\times 10^{18}\ \mathrm{Hz}$. In the canonical contract snapshot, (f_0) lies near raw bin k≈48 (raw bin center 48·Δf = 1.875×10²⁰ Hz; centroid index k≈47.53 gives f₀≈1.857×10²⁰ Hz).

Addendum (v1). With window length W = 256 and time step Δt = 1.0e−21 s, the FFT spacing is Δf = 3.90625e18 Hz. In the canonical contract snapshot, the dominant tone lies near bin k≈48 (see “Frequency binning” in §5.6).

Sampling & Nyquist safety (v1). The sampling rate is 1/Δt = 1.0e21 Hz, so the Nyquist limit is f_N = 1/(2Δt) = 5.0e20 Hz. Our canonical tone satisfies f₀ ≈ 1.857e20 Hz < f_N (by a factor of ≈2.7), hence no aliasing under the stated sampling. Because (f_0) is not exactly bin-centered, we report a centroid/interpolated estimate rather than claiming exact bin-locking.

Implementation robustness. Where invariance across seeds/grid sizes/durations is claimed, it must be backed by regenerated artifacts under a single pinned code state and should be referenced explicitly. In this draft, the canonical numeric anchors are pinned to spec6_false_s41; other runs are provided as additional evidence but are not used for aggregate claims until refreshed.

See also (Harmonic Spectrum). Secondary harmonics may co-appear with the dominant tone; methods and cross-language checks are summarized in the Spectral Structure extension.

5.7 Robustness mini-sweep (seeds 23, 17, 41, 73; spec6_false)

Status (refined snapshot). The multi-seed sweep section is retained for the v1 narrative, but the numeric values shown here must be regenerated under the refined logging/config state (commit 875fc4e). This core revision pins the canonical numeric snapshot to spec6_false_s41 only (see §4.6 and §5.6).

Anti-cherry-pick reminder (v1). The seed set {17, 23, 41, 73} is pre-registered for the core track. Until regenerated under a single pinned code state (and optionally elevated into contract acceptance), this table is informational and must not be used to support any [VALIDATED] claim beyond the canonical spec6_false_s41 anchors.

seed	SBR (±2-bin guard)	Topology neutrality (N1)	φ half-life (center)	Mean vortices
17	TBD	TBD	TBD	TBD
23	TBD	TBD	TBD	TBD
41	TBD	TBD	TBD	TBD
73	TBD	TBD	TBD	TBD

Summary (refined snapshot). Multi-seed summary statistics are TBD pending regeneration under commit 875fc4e. The canonical numeric anchors used throughout this core draft remain pinned to spec6_false_s41 until the sweep is refreshed.

5.8 Ablation study (canonical grid; Δt fixed)

We summarize what breaks when key terms are removed from Eq. (1). Detection rules follow Appendix A; metrics are computed as in §5.

Variant	ψ term	φ term	κ	Drift (+∇φ)	Spin aura	Structural closure	Neutral topology
V1	no φ, no ∇φ	—	—	✗	weak	✗	✓ / −
V2	+φ, no ∇φ	$\alpha(\lvert\psi\rvert^2 - \phi)$, no diffusion	—	± (unstable)	✓	±	−
V3	+φ, +∇φ	$\alpha(\lvert\psi\rvert^2 - \phi)$, no diffusion	—	✓	✓	±	−
V4	+φ, +∇φ	$\alpha(\lvert\psi\rvert^2 - \phi) + \beta\nabla^2\phi$	κ	✓	✓	✓	✓

Legend (symbols). ✓ = present under the operational definition stated in the text; ✗ = absent; ± = intermittent/unstable; — = not applicable; ✓ / − = vacuously neutral or undefined neutrality. For banded contract metrics (e.g., N1, mean vortices, phi_half_life_steps) “✓” implies the metric falls within the §4.3.1 acceptance bands for that variant where applicable.

Notes.
– V1 (no φ, no ∇φ): ψ evolves with diffusion/damping only → no drift, no closure.
– V2 (φ without ∇φ): memory exists, but trajectories lack consistent guidance; closure intermittent.
– V3 (add ∇φ): drift emerges; closure still fragile without φ diffusion.
– V4 (full canonical): guidance, spin, and closure co-occur; topology remains neutral within §4.3.1 bands.

5.9 Verification run — C3 (grid-size invariance) [TEST]

Run: spec6_false_s41_grid256
Setup change: grid 256×256; Δt = 1.0e−21 s (unchanged)

Status. This verification run must be regenerated under the refined logging/config state (commit 875fc4e) before quoting numeric results here.
Acceptance (v1; [TEST]). Expect the same contract-aligned bands as §4.3.1 (no bespoke tolerances): f0_mean_hz in [1.84e20, 1.87e20] Hz and sbr_mean ≥ 3000. Other validated anchors should remain within their §4.3.1 bands.

5.10 Global Phase Locking (Visual Observation)

[OBS] Simulations exhibit a rhythmic "breathing" behavior where linon trajectories appear to synchronize in a collective approach-retreat cycle.

Contract scope (v1). “Global Phase Locking” is treated here as a tentative visual observation [OBS] only. It is not contract-validated in v1.0.18-core and currently lacks established numeric acceptance criteria or dedicated contract keys. It is documented primarily via simulation animations (e.g., *_lineum_particles.gif).

Interpretive note. While visually compelling as a proxy for rhythmic phase alignment in the ψ-φ loop, this phenomenon requires further quantitative definition (e.g., inter-linon distance spectral analysis) before it can be elevated beyond a descriptive observation. It is included here to mark a Reproducible Visual Phenomenon that warrants future empirical investigation.

6. Interpretation

Particles exhibit guided motion along +∇φ (environmental guidance) without any force law or analogy to GR. When convergence occurs, it emerges from local gradients and basin structure in φ rather than from an imposed long-range interaction.

The persistence of φ-structures after particle decay (Structural Closure) indicates that the interaction field can store and maintain spatial information independently of active excitations. In v1 we interpret this strictly through the φ center-trace half-life and localized φ remnants as defined in §5.4; any additional trajectory-level bias (Return Echo) is reserved for the extension track. This memory property could serve as a basis for long-lived boundary conditions or “imprinted” environments in emergent systems.

Dimensional Transparency driven by time-varying κ has been observed only in exploratory runs and is out of scope for the v1 core; quantitative claims and artifacts are deferred to the experimental track (v1.1.x-exp).

Spin Aura and Spectral Stability show that once formed, linon excitations (particle-like) in the model exhibit consistent internal dynamics, maintaining stable oscillatory behavior over extended periods.

6.1 Guided Motion (interpretive note)

Simulations indicate that particles exhibit statistical alignment with +∇φ. We describe this as environmental guidance: the background field φ provides metric-like structure that biases trajectories without introducing a force law or any analogy to GR. In particular, we observe drift along +∇φ (Section 5.1) and longer dwell times in locally quiet basins where |∇φ| ≈ 0. Attraction-like behavior, when present, thus emerges from local gradients and basin structure, not from a prescribed long-range interaction.

6.2 Vortex–Particle Coupling (interpretive note)

Stable linons frequently co-occur with small sets of phase vortices. In multiple runs we observe compact vortex clusters (often triads) that remain spatially coherent for many steps and coincide with a locally quiet φ basin near their centroid. In this core v1, we treat this only as a descriptive co-occurrence between (i) stable linons, (ii) structured arg ψ winding, and (iii) locally low |∇φ| regions.

Scope guardrail (v1). We do not define a particle/antiparticle tagging scheme, species taxonomy, or quantitative binding criteria here. Any taxonomy, symbols, and thresholds are out of scope for the v1 core and belong to the dedicated extension note, which also defines falsifiable detection rules.

6.3 Law Transition (interpretive note)

Scope. This note refers to experimental variants where κ changes over time (non-canonical to Eq. 1). When κ follows a slow, coherent trajectory (e.g., island_to_constant), the system passes through effective regimes without losing linon stability: interaction patterns reorganize while macroscopic order persists. We observe spectral restructuring (secondary peaks, spacing shifts) concurrent with the κ-trajectory, suggesting an emergent principle of law transition: order can remain intact while the “rules” drift smoothly.

Evidence. Runs with dynamic generate_kappa(step) show time-resolved spectral changes (see multi_spectrum_summary.csv, *_spectrum_log.csv; sliding-FFT recommended). Exploratory overlays with Riemann ζ zeros are noted but not claimed as established. See the dedicated hypothesis file for parameters and logs.

Together, these results strengthen the case that a simple, metric-free local update rule can give rise to robust and quantifiable macroscopic effects, offering a controlled platform for exploring emergent analogues of known physical phenomena.

7. Conclusion

Lineum demonstrates that a minimal, discrete, and locally defined update rule can generate a variety of stable, quantifiable phenomena without predefined constants, spacetime geometry, or explicit force laws.

These effects emerge solely from iterative local interactions on a grid under pinned numerical settings. Claims presented as validated in core v1 are limited to the contract-defined metrics and artifact requirements; other observations are explicitly labeled accordingly. The reproducibility and simplicity of the model make it a promising testbed for studying emergent analogues of physical laws.

Future work will extend validation to larger parameter spaces, explore connections to continuous field theories, and investigate the scalability of these effects in three-dimensional simulations.

8. Acknowledgements

This project grew from an outsider’s curiosity: a non-physicist attempt to probe emergent interaction analogies from a different angle that expanded through persistent falsification and replication. Whatever is solid here stands on reproducible code and reports; any mistakes are mine alone.

My partner, Kateřina Marečková, provided what mattered most—patience, honest critique, and calm when results were messy. Her presence kept the work grounded.

I also thank Vlastimil Smeták for mathematically minded conversations and guidance. His focus on the Riemann Hypothesis and prime numbers—and his independent, visualization-first approach—suggested lines of inquiry that I would not have tried on my own. In particular, his advice motivated two working hypotheses explored outside the core:

• an Evolution–Mutation view (order vs. disruption as complementary regimes), and
• a Zeta–RNB Resonance idea (visual/structural echoes between Lineum’s return points and ζ-structure).

These are not claims of this v1 core paper. They remain preliminary and are deferred to the experimental/extension track for future, falsifiable testing; no quantitative alignment is asserted here.

I am grateful to the open-source community for tools and libraries that made this work possible, and to my family, friends, and the animals who shared life with me—Moulík, Jůlinka, Vikinka, Eliška, and others—for quiet lessons in patience and care.

Ethics/Tools note. AI assistance (“Lina”, a personalized ChatGPT-based assistant) was used as a tool for experiment orchestration, stress-testing arguments, and documentation hygiene. All results reported in this core paper are derived from the published scripts and the HTML reports in output/ and were independently verified by the author.

9. Versioning & Changelog

Policy. Semantic Versioning (MAJOR.MINOR.PATCH).

• MAJOR: changes to the canonical equation or scope (e.g., 3D instead of 2D).
• MINOR: new sections/notes, validation expansions; no breaking changes.
• PATCH: wording, typos, figures, formatting, audit enhancements.

1.0.18 — 2026-02-15 (patch)

• Add §5.10 Global Phase Locking (Collective Breathing) as an explicit observational [OBS] section.
• Sync version references to 1.0.18-core.

1.0.17 — 2026-02-15 (patch)

• Tighten “multi-run / robust” wording so it cannot be misread as cross-seed validation.
• Tag §5.3 “Silent Collapse” explicitly as [OBS] and add contract-scope guardrail.
• Remove ambiguous “or equivalent” phrasing in Appendix G for mean vortex count; require literal suite key string.
• Bump core version to 1.0.17-core.

1.0.16 — 2026-02-15 (patch)

• Tighten non-contract claims in Abstract/Validation/Interpretation to reduce overreach (explicitly observational where applicable).
• Remove morphology-level assertions from Structural Closure (center-trace-only operational definition in core v1).
• Normalize “canonical noise” statement (make σξ explicit; remove ambiguous “≪ 1” phrasing).
• Fix §4.10 numbering/order (Verification Protocol before Canonical Reference Fingerprints) and clarify scope_fingerprint.
• Bump core version to 1.0.16-core.

1.0.15 — 2026-02-15 (patch)

• Implement Stateless Audit 1.0 mechanism (Audit Lock).
• Add audit_scope logic and fail-fast protection for whitepaper_core profile.
• Explicitly document locked configuration parameters and hashing mechanism in §4.10.
• Decouple code and data fingerprints from the primary configuration hash gate.
• Bump core version to 1.0.15-core.

1.0.14 — 2026-02-14 (patch)

• Fix icon loop legend so it does not imply ψ→κ coupling in core v1 (κ is static); loop now matches Eq. (1) modulation/coupling.
• Add parameter note clarifying α_eff and β_eff values for the canonical run (κ=0.5).
• Soften §6.2 to avoid taxonomy/tagging claims in core v1; reserve quantitative coupling rules for the extension note.
• Bump core version to 1.0.14-core.

1.0.13 — 2026-02-14 (patch)

• Refine "Out of scope" to explicitly include "quantitative Vortex–Particle coupling claims/taxonomy".
• Update icon mnemonic to clarify parameter modulation and decoupling of visual causality from update rules.
• Explicitly use α_eff and β_eff in the Equation (1) table.
• Clarify κ modulation note regarding effective parameters in the φ-update.
• Bump core version to 1.0.13-core.

1.0.12 — 2026-02-14 (patch)

• Refine model description in Abstract (focus on discrete coupled-field, dimensionless parameters, and analogical terminology).
• Add interpretation note for $\Delta t$ as a conventional unit label.
• Update version pinning to distinguish between DOI snapshots (integrity-checked via sha256) and working drafts.
• Clarify within-run vs. cross-seed metric reporting (CIs and aggregation).
• Bump core version to 1.0.12-core.

1.0.11 — 2026-02-14 (patch)

• Refine pronunciation terminology for "linon" (distinguish model vs. phenomenon).
• Bump core version to 1.0.11-core; no changes to Eq-4, artifacts, or validations.

1.0.10 — 2026-02-14 (patch)

• Add Plain-language summary and Physics translation (analogy-only) to the Abstract to reduce misinterpretation risk (especially around SI conversions and “particle” wording).
• No changes to Eq-4, scope, metrics, artifacts, or acceptance bands — documentation clarity only.

1.0.9 — 2026-02-14 (patch)

• Bump core version to 1.0.9-core and update header date to 2026-02-14 (refined snapshot now explicitly tied to the spec6_false_s41_20260214_101645 evidence directory).
• Fix internal version references so the “frozen core track” wording matches the current patch level (v1.0.9-core).
• Add Whitepaper Contract Runner (tools/whitepaper_contract.py) producing whitepaper_contract_result.json for audit runs; no changes to Eq/scope.
• Clarify topology logging cadence: topo_log.csv is decimated by logging.topo_log_stride (canonical: 25), and topology neutrality (N1/N0) + mean vortices are computed over logged frames (N=81 for steps 0..2000) as declared in the manifest and validated by the contract suite.

1.0.8 — 2025-12-09 (patch)

• Correct §5.6 "Additional validation run (spec6_false_s17)" to match the actual HTML and CSV values: • Topology neutrality → 91.1%
  • Mean vortex count → ~89
  • φ half-life → 483 steps
  These values previously reflected outdated metrics from an old report.
  No changes to equations, scope, or methodology — numeric correction only.

1.0.7 — 2025-12-09 (patch)

• §3 legend / §3.1: explicitly tie the discrete Laplacian to the four-neighbour (5-point von Neumann) stencil implemented via diffuse_complex() in the reference code; clarify that no 9-point Laplacian is used in the canonical v1 run.
• §4.8 Threats to validity: replace the historical note about “replication with a 9-point Laplacian” with a statement that wider stencils are exploratory only and out of the v1 core evidence.
• Header / metadata: bump version to 1.0.7-core and update the date to 2025-12-09.

1.0.6 — 2025-11-14 (patch)

• Abstract: move Structural Closure into the validated items list as an in-scope consequence of the φ center-trace half-life; keep Return Echo and κ-dynamics explicitly out of scope and delegated to the experimental/extension track.
• §1 / header: clarify that Structural Closure is in scope for v1.0.x; add a file-level scope note distinguishing lineum-core from lineum-exp-* and lineum-extension-* whitepapers.
• §5.4 Structural Closure: give an operational definition tied to the φ half-life metric and to concrete artifacts (*_phi_center_log.csv, *_phi_center_plot.png); explicitly mark Return Echo as an extension-level hypothesis handled in lineum-extension-return-echo.md.
• §6 Interpretation and §7 Conclusion: align wording with the new Structural Closure definition (named, metric-linked), keeping trajectory-bias phenomena in the extension track.

1.0.5 — 2025-11-14 (patch)

• Abstract: rephrase the paragraph introducing SI-anchored numbers so they are explicitly framed as scale illustration only, not “quantitative signatures close to physical scales”.
• Abstract: add an explicit sentence stating that the quoted Hz/keV/nm/mass-ratio values are unit conversions of f₀, not extra constraints or evidence of a realized physical scale.
• §2 Motivation: soften “field-mediated forces” to “field-mediated interactions” to avoid suggesting a defined force law in the core scope.

1.0.4 — 2025-08-23 (patch)

• Version bump to 1.0.4-core; insert DOI in header + How to cite.
• Abstract: tighten to core-only; replace “validated items” paragraph; fix Unicode φ; remove duplicate “reported in HTML” sentence.
• Graphical abstract: fix icon path; add width control; add mnemonic-only disclaimer; add Icon legend (κ=fish, ψ=spiral, φ=leaf) + directional mnemonic (κ → ψ → φ → κ).
• §5 Validation: add Figure 0 canonical anchors (numbers linked to HTML); add Evidence pointer blocks to §5.1 (Guided Motion) and §5.2 (Spin Aura).
• §5.5: mark Dimensional Transparency as out of scope (deferred to v1.1.x-exp).
• §6 Interpretation: explicitly note Dimensional Transparency is out of scope for v1 core.
• §9 Versioning: update Track policy and Branching note to 1.0.4-core.
• §8 Acknowledgements: rewrite (Kateřina Marečková; Vlastimil Smeták; AI tools note).
• Minor: wording/formatting consistency; unify relative links to ../output/… where applicable.

1.0.3 — 2025-08-23 (patch)

• Abstract: add Core thesis (v1) and Falsifiable checks (v1) (C1/C2/C3).
• §4.7: add Reviewer quick-check (v1) with exact HTML table strings.
• §4.8: add Not claimed (v1) and expand Display-only mass risk note.
• §4.9: add Tooling guardrails (v1) (mass-from-f₀, commit provenance, SI anchoring, pinned runs).
• §5.6: add Worked example (canonical f₀), Formatting policy (v1), bin-centering addendum, and Sampling & Nyquist safety (v1).
• §5.9: add Verification run — C3 (grid-size invariance).
• Appendix C/D/E: add Evidence Index (v1), Glossary (v1), and Verification runs (v1).

Branching note. Further physics-mapping tests (dispersion, group velocity, external-field response) will be published under the experimental track v1.1.x-exp; the core canonical scope remains frozen in v1.0.17-core.

1.0.2 — 2025-08-21 (patch)

• Sync §5.6 Spectral Stability with the canonical run spec6_false_s41: f₀ = 3.90625×10¹⁸ Hz [3.90625×10¹⁸, 3.90625×10¹⁸], SBR = 6.88 [6.86, 6.90].
• Update Abstract numeric anchors to match the canonical run: E ≈ 2.59×10⁻¹⁵ J ≈ 16.15 keV, λ ≈ 7.67×10⁻¹¹ m (0.0767 nm), effective mass ≈ 3.16% mₑ.
• Add §3.1 Numerical scheme & stability (canonical) (discrete operators, explicit Euler Δt=1.0×10⁻²¹ s).
• Add §4.8 Threats to validity (core v1).
• Add Appendix B — Metrics & CI (v1) (windowed estimates + 95% bootstrap CI; aligns with HTML report).
• §4.6 Manifest: add Code provenance note; §4.7 Data & Code: Version pinning (no checksums).
• §5.8 Ablation study: fix table rendering (escape | as |), add Legend (symbols).
• Math rendering: switch inline \(...\) → $...$; fix legend/table pipes to avoid column breaks.
• Tooling (report): show mean ±95% CI for f₀/SBR; write metrics_summary.csv; header includes short git commit.

1.0.1 — 2025-08-19 (patch)

• Corrects SBR in §5.6 to 6.18 (consistent with the canonical report).
• Appendix A: adds Visualization-only note for Vortices GIF (amplitude gating for display only; CSV/metrics use raw winding).
• Clarifies spectrum definition as power spectrum |FFT(x)|^2 with a ±2-bin guard around f0.
• No change to the canonical equation or scope.

1.0.0 — 2025-08-19 (initial canonical)

• Pins Eq-4 (κ static), 2D + periodic BCs.
• Validation §§5.1–5.6 (incl. operational §5.2, robustness note in §5.6, operational note in §5.5).
• Interpretation: 6.1 Environmental Guidance, 6.2 Vortex–Particle Coupling, 6.3 Law Transition.
• §3: explicit scope note for 2D/periodic; sign convention for +∇φ.

Appendix A — Detection Conventions (v1)

This appendix fixes the minimal conventions needed to reproduce our measurements in the canonical run.

All output files are saved with the run tag prefix ({RUN_TAG}_…, e.g., spec6_false_s41_…). For readability we refer to them without the prefix in the text.

Quasiparticles (trajectories). Detected from local amplitude structure in ψ; tracks exported to trajectories.csv (positions per step). A detection forms a time-indexed set of coordinates; lifetimes are measured as the number of steps per unique track id.

Vortices (winding number). Computed on 2×2 plaquettes by summing phase differences and rounding to the nearest integer winding:

$$\mathrm{winding}=\mathrm{round}\!\left(\frac{\Delta\phi_1+\Delta\phi_2+\Delta\phi_3+\Delta\phi_4}{2\pi}\right)$$

Positive/negative counts and net charge are logged per step in topo_log.csv.

Visualization-only note. The Vortices GIF applies an amplitude gate to the winding map for clarity: marks are displayed only where |ψ| is below a low-amplitude threshold (default: 5th percentile per frame). All metrics and CSV logs (e.g., topo_log.csv) always use the raw winding (no amplitude gating).

Spin / curl map (“spin aura”). We use the phase–gradient curl,

$$S=\mathrm{curl}\!\big(\nabla \arg \psi\big)$$

Numerical detail. Phase increments are wrapped as angle(exp(i*Δφ)) and evaluated with central differences under periodic BCs. This avoids ±π discontinuity artefacts; the curl signal is therefore concentrated near vortex cores rather than appearing as spurious long “strings”.

averaged in fixed-size windows centered on detected quasiparticles. The averaged raster is exported as *_spin_aura_map.png; the radial profile as *_spin_aura_profile.csv.

Spectrum. FFT of the amplitude time-series at the field center; power spectrum:

$$P(f)=\bigl|\mathrm{FFT}(x)\bigr|^{2}.$$

The dominant tone $f_0$ is estimated per window from the dominant peak region; in refined snapshots we report a local centroid/interpolated estimate rather than claiming exact bin-centering. The windowwise mean and a 95% bootstrap CI are reported in metrics_summary.csv. SBR compares the peak power to the rest of the spectrum with a ±2-bin guard around $f_0$.

Effective mass (display-only). Converted from $f_0$ via:

$$E=h f_0,\qquad m=\frac{E}{c^2},\qquad \mathrm{mass\_ratio}=\frac{m}{m_e}.$$

Reported as a derived display quantity (no fitting).

Topology neutrality (N1). Fraction of logged frames with |net_charge| <= 1, computed from topo_log.csv. topo_log.csv may be decimated; the cadence is declared in the run manifest as logging.topo_log_stride (canonical: 25 → N=81 logged frames over steps 0..2000). Neutrality is therefore computed over logged frames. Strict neutrality (N0; info-only). Fraction of logged frames with net_charge == 0.

Cross-implementation note. Exact pixelwise equality across languages/backends is not required; replication is defined via metric tolerances in §4.3.1 on the canonical run (spec6_false_s41).

Appendix B — Metrics & CI (v1)

Spectral pipeline (center amplitude). We compute the power spectrum $P(f)=|\mathrm{FFT}(x)|^2$ on sliding windows of the center-point amplitude time series. Defaults: window length $W=256$ frames, hop $H=128$, de-meaned FFT (DC removed), and a ±2-bin guard around the dominant bin $f_0$ when estimating the background.

Dominant frequency $f_0$ (refined snapshot). For each window, we locate the dominant peak region and estimate $f_0$ via a local centroid/interpolated peak (rather than claiming exact bin-centering). We aggregate the windowwise mean and a non-parametric 95% bootstrap CI across windows.

SBR (Spectral Balance Ratio). For each window:

$$\mathrm{SBR}=\frac{P(f_0)}{\mathrm{mean}\,\{\,P(f):\, f\notin[f_0-2,\,f_0+2]\,\}}.$$

Bootstrap procedure. Given windowwise values $\{v_k\}_{k=1}^n$, resample indices with replacement $B=1000$ times, compute the mean for each resample, and take the $[2.5\%,97.5\%]$ quantiles as the CI.

Minimal pseudocode (reference):

import numpy as np

def sliding_windows(x, W, hop):
    for i in range(0, max(len(x)-W+1, 0), hop):
        yield x[i:i+W]

def welch_power(w, dt):
    w = w - w.mean()
    P = np.abs(np.fft.fft(w))**2
    F = np.fft.fftfreq(len(w), d=dt)
    return P[:len(P)//2], F[:len(F)//2]

def sbr_and_f0_windows(x, dt, W=256, hop=128, guard=2):
    sbr_vals, f0_vals = [], []
    for w in sliding_windows(x, W, hop):
        P, F = welch_power(w, dt)
        k = int(np.argmax(P)); peak = P[k]
        # centroid around the peak (3-bin window by default)
        k0 = max(k-1, 0); k1 = min(k+2, len(P))
        wP = P[k0:k1]; wF = F[k0:k1]
        f0 = float((wF*wP).sum()/wP.sum()) if wP.sum() > 0 else float(F[k])
        mask = np.ones_like(P, dtype=bool)
        mask[max(k-guard,0):min(k+guard+1,len(P))] = False
        bg = P[mask].mean() if mask.any() else np.nan
        if bg > 0: sbr_vals.append(peak/bg)
        f0_vals.append(f0)
    return np.array(sbr_vals), np.array(f0_vals)

def bootstrap_mean_ci(vals, B=1000, alpha=0.05):
    vals = np.asarray(vals, float)
    n = len(vals)
    if n == 0: return np.nan, (np.nan, np.nan)
    means = [vals[np.random.randint(0, n, n)].mean() for _ in range(B)]
    lo, hi = np.quantile(means, [alpha/2, 1-alpha/2])
    return float(vals.mean()), (float(lo), float(hi))

The HTML report prints value [lo, hi] for both metrics and also writes them to metrics_summary.csv.

Appendix C — Evidence Index (v1)

This appendix ties the core’s numeric anchors to concrete artifacts (HTML reports) so that readers can verify values directly.

Canonical numeric anchors (contract-validated; RUN_TAG spec6_false_s41, run spec6_false_s41_20260215_023130).

• Dominant tone (mean): f₀ = 1.856777545095882×10²⁰ Hz
• SBR (mean): 3245.4600764773872
• φ half-life (center): 1009 steps
• Topology neutrality (N1): 94.2%
• Mean vortices: 59.1455
• Max lifespan: 44 steps
• Low-mass QP count: 49
• Energy (display-only): E ≈ 1.23×10⁻¹³ J ≈ 767.90 keV
• Wavelength (display-only): λ ≈ 1.61×10⁻¹² m (0.00161 nm)
• Mass ratio (display-only): m/mₑ ≈ 1.5027 (150.27%) (Derived via (E = h f_0), (m = E/c^2); constants listed in §5.6.)

Per-run artifacts (v1 evidence bundle).

RUN_TAG	Evidence source (primary)	f₀ (Hz; CI)	SBR (CI)	φ half-life	Neutrality (N1)
spec6_false_s41	manifest.json + metrics_summary.csv	1.8568e20 (CI in HTML = informational)	3245.46 (CI in HTML = informational)	1009	94.2%

Commit provenance. Each HTML report prints the short Git commit in its header (beside RUN_TAG and runtime metadata). Regenerating reports on a different code state will change the commit stamp by design.

Appendix D — Glossary (v1)

linon. A stable, localized excitation of |ψ|² in the Lineum field (quasi-particle analogue). It is not a Standard-Model particle.

display-only effective mass. A scale indicator obtained by a unit conversion from the canonical tone: take (f_0), compute (E=h,f_0), and write (m=E/c^2); then report (m/m_e). This is not a rest-mass claim (see Abstract “Interpretation note (v1)” and §5.6).

dominant frequency (f_0). The spectral peak of the center-amplitude time series; measured on sliding windows and reported as a windowed mean with a 95% CI.

FFT bin / bin-centering. FFT groups frequencies into equal “bins” (slots) with spacing Δf. In the canonical contract snapshot, (f_0) lies near k≈48 (centroid index k≈47.53; see §5.6). We therefore do not claim exact bin-centering in v1.0.17-core anchors.

Δt, W, Δf. Δt is the simulation time step; W is the FFT window length (in steps); their combination fixes the bin spacing Δf. Canonical v1 uses Δt = 1.0e−21 s and W = 256 (see §5.6).

Nyquist safety. The sampling rate (1/Δt) sets the Nyquist limit (f_N = 1/(2Δt)). In the refined snapshot, the reported (f_0) lies below (f_N) (see §5.6 “Sampling & Nyquist safety”), so aliasing is not expected under the stated sampling.

SBR (Spectral Balance Ratio). Peak-to-background ratio of the power spectrum in a window, with a ±2-bin guard around the peak excluded from the background. Reported as mean with a 95% CI.

topology neutrality (N1). Fraction of logged frames in topo_log.csv with |net_charge| <= 1.
The logging cadence is declared in the run manifest as logging.topo_log_stride (canonical: 25 → N=81 frames over steps 0..2000).
Strict neutrality (N0; info-only). Fraction of logged frames with net_charge == 0.

φ-trap. A localized region of the interaction field φ that tends to capture or retain linons (observational term; no force law is assumed).

RUN_TAG / evidence bundle. A unique label for a run (e.g., spec6_false_s41) used to prefix all artifacts (HTML/CSV/PNG/GIF). The evidence bundle is the set of per-seed reports and metrics listed in Appendix C.

Lineum symbol (informal)

Reader aid; not part of the core claims.

• ψ — oscillatory carrier (time-like tone at the center); where we measure the canonical f₀ used for SI conversions (E, λ, display-only m/mₑ).
• φ — memory/envelope (stores local context; used for nearby/field means in the HTML metrics).
• κ — tuning/balance field (slow control parameter; fixed in the core canonical setup).

The icon used in the repository depicts a coupling interplay κ → (α_eff,β_eff) → φ ↔ ψ and has no physical implication beyond this glossary.

Appendix E — Verification runs (v1)

Minimal verification runs demonstrating invariance under window length, time-step refinement (fixed Δf), and grid size.

RUN_TAG	Setup change	Status
spec6_false_s41_w512	W = 512 (Δt = 1.0e−21 s)	TBD — regenerate under commit 875fc4e
spec6_false_s23_w512	W = 512 (Δt = 1.0e−21 s)	TBD — regenerate under commit 875fc4e
spec6_false_s41_dt05_w512	Δt → 5.0e−22 s, W → 512 (Δf preserved)	TBD — regenerate under commit 875fc4e
spec6_false_s41_grid256	Grid 256×256 (Δt = 1.0e−21 s)	TBD — regenerate under commit 875fc4e

Appendix F — Artifact bundle README (v1)

What’s included (core v1.0.17-core).

All artifacts are generated into the output/ folder with a {RUN_TAG}_… prefix.

File map (per-seed; canonical examples)

• output/spec6_false_s41_lineum_report.html — main HTML report (derived view generated from the manifest + CSV logs)
• output/spec6_false_s41_figure0_canonical.png — Figure 0 used in §5
• output/spec6_false_s41_metrics_summary.csv — machine-readable metrics (f₀, SBR, CIs)
• output/spec6_false_s41_manifest.json — primary run metadata and primary metric snapshot (refined anchors)
• Other CSV/PNG/GIF listed in §4.5 (same {RUN_TAG}_… prefix)

How to verify quickly.

1. Open output/spec6_false_s41_metrics_summary.csv and/or output/spec6_false_s41_manifest.json and confirm the refined snapshot:

   • Dominant frequency f₀ (mean) = 1.856777545095882e+20 Hz
   • SBR (mean) = 3245.4600764773872
   • φ half-life (center) = 1009 steps
   • Topology neutrality (N1) = 94.2% where N1 = fraction of logged frames in topo_log.csv with |net_charge| <= 1 (canonical: logging.topo_log_stride = 25, N=81 frames; steps 0..2000)
   • Strict neutrality (N0; info-only) = 88.95% where N0 = fraction of logged frames with net_charge == 0
   • Mean vortices = 59.1455
   • Max lifespan = 44 steps
   • Low-mass QP count = 49

2. Open output/spec6_false_s41_lineum_report.html and confirm the same values appear in the relevant tables (HTML is a derived view).

3. Confirm the audit fingerprints match (§4.10.5) and the contract suite reports PASS.

Re-running variants (no code edits).

• Window: LINEUM_PARAM_TAG=w512
• Temporal refinement: LINEUM_PARAM_TAG=dt05_w512
• Grid: LINEUM_PARAM_TAG=grid256
• Optional seed override: LINEUM_SEED=23
• (Windows PowerShell)

Appendix G — Claim–Contract Map (v1.0.17-core)

This appendix is the normative map from manuscript claims to (i) contract keys and (ii) concrete artifact pointers.
If a statement in the manuscript appears stronger than its mapping below, the mapping below wins.

Legend.
[VALIDATED] = contract-enforced (acceptance band or exact match).
[OBS] = supported by artifacts but not contract-enforced.
[DISPLAY] = derived illustration only (unit conversion from validated anchors).
[OOS] = out of scope for core v1. [TEST] = verification procedure / expected outcome; not a reported result unless regenerated and pinned.

Claim ID	Claim (canonical wording)	Status	Contract key(s)	Primary artifact pointer(s)	PASS/FAIL test
C-01	Dominant tone exists and is stable enough to report a mean f₀ on the canonical run.	[VALIDATED]	f0_mean_hz	output/spec6_false_s41_metrics_summary.csv (f0 mean); output/spec6_false_s41_manifest.json (snapshot)	Contract suite PASS for lineum-core-1.0.17-core
C-02	Spectral dominance (SBR) of the canonical tone exceeds the acceptance threshold.	[VALIDATED]	sbr_mean	output/spec6_false_s41_metrics_summary.csv (SBR mean); HTML “Spectral metrics” table	Contract suite PASS
C-03	Topology neutrality N1 is within the declared tolerance (computed over logged frames).	[VALIDATED]	N1	output/spec6_false_s41_topo_log.csv; HTML “Topology metrics”; manifest logging.topo_log_stride	Contract suite PASS
C-04	Mean vortex count is within the declared acceptance band (logged frames).	[VALIDATED]	__REPLACE_WITH_CONTRACT_SUITE_KEY_FOR_MEAN_VORTEX_COUNT__	output/spec6_false_s41_topo_log.csv; HTML “Vortex count”	Contract suite PASS
C-05	φ center-trace has a contract-validated half-life timescale in steps.	[VALIDATED]	phi_half_life_steps	output/spec6_false_s41_phi_center_log.csv; output/spec6_false_s41_phi_center_plot.png	Contract suite PASS + required artifacts present
C-06	“Structural Closure” (core v1) is treated as an operational proxy: φ half-life anchor + required φ-trace artifacts (explicitly no morphology claim).	[VALIDATED]	phi_half_life_steps + artifact presence	same as C-05	Contract suite PASS (timescale + artifacts)
C-07	Canonical detector anchors for quasiparticles include a fixed low-mass QP count.	[VALIDATED]	low_mass_qp_count	output/spec6_false_s41_manifest.json (snapshot); HTML “Quasiparticle Properties”	Contract suite PASS
C-08	Canonical max lifespan meets a minimum threshold.	[VALIDATED]	max_lifespan_steps	output/spec6_false_s41_manifest.json / metrics summary; HTML “Quasiparticle Properties”	Contract suite PASS
C-09	“Spin Aura” artifacts exist and are reproducible as observation, but are not acceptance-enforced.	[OBS]	N/A	output/spec6_false_s41_spin_aura_map.png; output/spec6_false_s41_spin_aura_profile.csv; HTML “Spin aura — averaged curl map”	Artifact presence (contract may require presence only if declared)
C-10	Guided motion along +∇φ is an interpretive description consistent with Eq. (1), not a validated statistic.	[OBS]	N/A	output/spec6_false_s41_trajectories.csv; output/spec6_false_s41_lineum_flow.gif	Artifact presence only (no numeric acceptance)
C-11	SI conversions (E, λ, m/mₑ) are unit conversions from f0_mean_hz, for scale illustration only.	[DISPLAY]	derives from f0_mean_hz	HTML report “worked example” + §5.6 constants	Recompute from f0_mean_hz using stated constants; must match within rounding
C-12	Out-of-scope topics (dynamic κ, Return Echo quantitative claims, SM identification, thermodynamics) are not part of core v1.	[OOS]	N/A	File-level scope note; extension whitepapers list	Scope compliance (manual review)

Note on contract field names. Contract keys are authoritative as printed by the contract suite output. For any claim, the “contract key(s)” cell must match the suite’s emitted field name(s) exactly. (This table may use semantic phrasing in the claim text, but the key strings must be literal.)

Release blocker (v1 core). Before marking a snapshot “verified”, replace any __REPLACE_WITH_...__ placeholder with the literal key string emitted by the contract suite output (stdout / whitepaper_contract_result.json).

Appendix H — Verification Quickstart & Release Checklist (v1.0.17-core)

This appendix defines the minimum procedural verification required to label a core snapshot as verified.

H.1 Quickstart (PASS/FAIL procedure)

Goal: produce a canonical audit run under the locked profile and obtain a contract suite PASS for lineum-core-1.0.17-core.

Windows PowerShell (reference)

# 1) Run canonical audit (locked profile)
$env:LINEUM_AUDIT_PROFILE = "whitepaper_core"
$env:LINEUM_RUN_ID        = "6"
$env:LINEUM_RUN_MODE      = "false"
$env:LINEUM_SEED          = "41"
$env:LINEUM_BASE_OUTPUT_DIR = "output_wp"

python .\lineum.py

# 2) Run contract suite (core)
python .\tools\whitepaper_contract.py

Linux/macOS bash (reference)

# 1) Run canonical audit (locked profile)
export LINEUM_AUDIT_PROFILE="whitepaper_core"
export LINEUM_RUN_ID="6"
export LINEUM_RUN_MODE="false"
export LINEUM_SEED="41"
export LINEUM_BASE_OUTPUT_DIR="output_wp"

python3 ./lineum.py

# 2) Run contract suite (core)
python3 ./tools/whitepaper_contract.py

Expected outputs (minimum)

• output_wp/spec6_false_s41_manifest.json
• output_wp/spec6_false_s41_lineum_report.html
• output_wp/spec6_false_s41_metrics_summary.csv
• output_wp/spec6_false_s41_topo_log.csv
• output_wp/spec6_false_s41_phi_center_log.csv
• output_wp/whitepaper_contract_result.json (must report PASS for lineum-core-1.0.17-core)

PASS definition (v1). Verification is PASS iff:

1. whitepaper_contract_result.json exists and reports status = PASS, and
2. the reported fingerprints match §4.10.5 (audit scope + code + κ map), and
3. the required artifact set declared by the contract suite is present.

H.2 DOI snapshot integrity (sha256)

For DOI-published evidence bundles, generate and verify checksums.

# Example: generate checksums for a curated list of core artifacts (adjust list per snapshot policy)
sha256sum \
  output/spec6_false_s41_manifest.json \
  output/spec6_false_s41_metrics_summary.csv \
  output/spec6_false_s41_lineum_report.html \
  output/spec6_false_s41_topo_log.csv \
  output/spec6_false_s41_phi_center_log.csv \
  output/whitepaper_contract_result.json \
  > sha256sums.txt

Integrity rule (DOI snapshots): the snapshot is intact iff sha256sums.txt matches the corresponding files byte-for-byte.

H.3 Release checklist (core v1.0.17-core)

Audit / contracts

• Canonical run executed under LINEUM_AUDIT_PROFILE=whitepaper_core with steps=2000.
• audit_scope_hash equals §4.10.5.
• code_fingerprint equals §4.10.5.
• kappa_map_bin_hash equals §4.10.5.
• Contract suite lineum-core-1.0.17-core reports PASS and whitepaper_contract_result.json is stored with the bundle.

Claims hygiene

• Appendix G (Claim–Contract Map) has no dangling validated claim (every [VALIDATED] maps to a contract key + artifact pointer).
• Appendix G contains no placeholder keys (grep: __REPLACE_WITH_CONTRACT_).
• Any statement about “Spin Aura”, “Guided Motion”, “Return Echo”, “morphology”, “taxonomy” is labeled [OBS] or [OOS] unless promoted into contract acceptance.
• SI values (E, λ, m/mₑ) are labeled [DISPLAY] and described as unit conversions from f0_mean_hz.

Text grep hygiene (manual)

• Grep for: validated|proven|confirms|demonstrates|evidence of and confirm each usage aligns with Appendix G.
• Grep for: mass|electron|photon|Standard Model|GR|gravity and confirm core scope guardrails are present and unambiguous.

DOI snapshot integrity (when applicable)

• sha256sums.txt included and verified.

Appendix I — Technical Maintenance & Anchor ID System

To prevent "Audit Drift" (where section number changes break external references), this manuscript uses persistent Anchor IDs.

I.1 Usage Policy

Every section or subsection that is referenced by a Contract Suite (paper_ref) or the Laboratory UI should have a hidden anchor ID attached to its header.

Syntax (Markdown): ### Section Title {#anchor-id}

I.2 Mandatory Anchors

The following anchors are reserved and must remain constant across manuscript revisions:

• {#resonance-scanner}: Links to §4.3 (Spectral Resonance Analysis).
• {#topology-neutrality}: Links to §4.4 (Topological Neutrality).
• {#phi-half-life}: Links to §4.6 (Field Φ Stability & Half-Life).
• {#particle-taxonomy}: Links to §4.7 (Linon Detection & Quasiparticles).
• {#calibration-seed}: Links to §4.1 (Initial Conditions & Seed Verification).

I.3 Agent Instruction (Maintenance)

IMPORTANT for AI Agents: When moving or renaming headers, you MUST preserve the exact anchor ID string. If a new metric is added to the contract suite, you MUST create a corresponding anchor ID in the section describing that metric.