Introduction

Directional organization is a persistent feature of particle interactions at high energy. Momentum redistribution in parton cascades produces collimated jets. Interacting particle ensembles expand anisotropically when initial geometry is asymmetric. Produced hadrons exhibit spin alignment reflecting angular momentum transfer during formation.

These phenomena are typically treated as separate topics with specialized methods. However, they share a common structural signature: measurable departure from angular uniformity. This paper treats those departures as instances of angular ordering and proposes a unified quantitative comparison across three regimes:

• cascade topology in multijet production (event-shape ordering),

• collective expansion anisotropy (flow harmonics),

• intrinsic alignment (polarization).

The objective is not to replace established dynamics, but to define a common ordering language, extract ordering magnitudes from published results, and test whether directional organization exhibits consistent scaling behaviour across domains.

Operational claim

The operational claim of this paper is narrow and falsifiable:

Published collider measurements contain enough information to define domain-specific ordering magnitudes that can be compared structurally. If a common ordering principle exists, those magnitudes should exhibit consistent monotonic behaviour with respect to domain-appropriate constraint proxies.

Unified Angular Anisotropy Formalism

Generic angular decomposition

Any \(2\pi\)-periodic angular density can be written as a harmonic expansion: \[p(\phi)=\frac{1}{2\pi}\left(1+2\sum_{n\ge1} a_n\cos(n(\phi-\Psi_n))\right),\] where \(a_n\ge 0\) are harmonic amplitudes and \(\Psi_n\) are corresponding orientations.

This paper uses two levels of description:

1. Orientation-preserving ordering: keep \((a_n,\Psi_n)\) when the experiment provides meaningful event-plane or reference orientation (flow).

2. Orientation-quotiented ordering: keep only ordering magnitudes when the observable removes absolute orientation by construction (event shapes based on maximized axes).

Ordering strength as a scalar

To compare domains, define an ordering strength functional \(O[\cdot]\) that maps an angular object to a nonnegative scalar:

Experimental Observable Mapping

Jets: event-shape ordering (cascade topology)

In multijet events, ordering is probed by event-shape variables computed from charged particles inside jets. These variables quantify collimation versus isotropy and angular dispersion relative to dynamically selected axes. The key structural fact is that some event-shape definitions maximize over an axis, thereby removing absolute orientation information and leaving only ordering magnitude.

Two ordering-relevant classes are:

• Projection ordering: thrust-like variables compress information into a maximized projection.

• Dispersion ordering: broadening-like variables quantify angular spread in \((\eta,\phi)\) about hemisphere axes.

This paper treats the unfolded distributions of these event-shape variables as ordering data. Orientation is quotiented out; only distributional ordering magnitude remains.

Collective systems: flow harmonics (medium ordering)

In nuclear collisions, ordering is measured directly through azimuthal harmonic coefficients \(v_n\) in \[\frac{dN}{d\phi}\propto 1+2\sum_{n\ge1} v_n\cos(n(\phi-\Psi_n)).\] Here the experiment provides both magnitudes and (in the standard formalism) orientations \(\Psi_n\). This is orientation-preserving ordering.

Intrinsic alignment: polarization (spin ordering)

Polarization measurements provide a signed alignment observable, typically corresponding to a dipole-like (\(n=1\)) ordering channel in an appropriate decay-angle basis. This paper uses polarization magnitude as an intrinsic ordering strength and separately tracks sign and charge-asymmetry as additional structure channels.

Directional Ordering Metrics

This section defines concrete metrics used across domains. Each metric is computable from published unfolded histograms or reported coefficients.

Jet ordering metrics from unfolded distributions

Let a published unfolded histogram define probabilities \(p_i\) over bins of some event-shape variable \(x\). Define:

Interpretation: more sharply structured distributions have smaller entropy and larger \(O_H\).

Interpretation: lower tail fraction indicates stronger ordering (more collimated, less isotropic).

Flow ordering metrics

Given reported harmonics \(\{v_n\}_{n=2}^N\) define:

Optionally define shape ratios such as \(R_{23}=v_2/v_3\) when available to compare dominance of elliptic versus triangular ordering.

Polarization ordering metrics

Given polarization \(P\) and total uncertainty \(\sigma\):

When both particle and antiparticle are reported: \[\Delta P = P_{+}-P_{-},\qquad O_{\Delta}=|\Delta P|.\]

Analysis Procedure

Inputs and minimal requirements

This paper requires only:

• unfolded event-shape histograms (jets),

• reported \(v_n\) coefficients (flow),

• reported polarization values and uncertainties (spin).

No detector-level reconstruction is required.

Within-domain monotonicity tests

For each domain define a constraint proxy \(C\):

• jets: a hard-scale binning variable (e.g. leading-jet scale or \(H_{T,2}\)-like binning),

• flow: system size / centrality / multiplicity proxy,

• polarization: kinematic proxy such as \(p_T\) or \(x_F\) bins.

Compute ordering \(O(C)\) and test for monotonic trends using:

• Spearman rank correlation \(\rho_S\),

• Kendall \(\tau\),

• bootstrap confidence intervals on \(O(C)\).

Cross-domain normalization and comparison

Because domains have different units and definitions, comparisons use normalized ordering scores: \[\tilde{O}=\frac{O-\min(O)}{\max(O)-\min(O)}\in[0,1],\] computed within each domain.

A common-ordering hypothesis is supported if:

1. each domain shows consistent monotonic ordering vs its constraint proxy,

2. the normalized ordering curves exhibit similar qualitative behaviour (increase under increasing constraint),

3. the interpretation is stable under alternative ordering metrics (\(O_H\) versus \(O_{\text{tail}}\) for jets, different \(N\) for \(O_F\) for flow).

PFT Structural Interpretation

PFT viewpoint on directional organization

In Pattern Field Theory (PFT), directional ordering is treated as an emergent consequence of constraint-mediated selection in a structured possibility space. The measured ordering metrics are interpreted as empirical projections of a deeper selection process.

We use the following mapping:

• Cascade ordering (jets) corresponds to selection-constrained branching transport in a finite angular budget.

• Flow ordering corresponds to collective response of a constrained ensemble where spatial deformation converts into angular emission preference.

• Spin ordering corresponds to intrinsic alignment channels that retain a signed directional memory.

PAL constraint (operational use)

Persistent directional ordering requires a minimal coherent transport closure, shown in Fig. 1.

Operationally, PAL is supported when ordering metrics remain robust across adjacent scale bins and do not collapse under minor changes of selection criteria (within published systematic uncertainty envelopes).

AOL compatibility statement

This paper does not require an explicit microscopic model of the Allen Orbital Lattice (\(\mathrm{AOL}\)). It uses \(\mathrm{AOL}\)as a structural hypothesis layer: if a discrete substrate imposes directional selection rules, then multi-regime collider measurements should exhibit consistent ordering behaviour at the level of angular observables. The present work provides the comparative metric framework required for later \(\mathrm{AOL}\)-specific modelling.

Directional selection in a discrete substrate implies quantized radial transport layers, as illustrated in Fig. 2.

Discrete radial transport structure implies quantized propagation distances. Such transport constraints produce measurable scale-dependent power distributions in spatial or momentum representations.

Consistent directional ordering across cascade, medium, and intrinsic alignment regimes implies that angular organization is not an isolated phenomenon of specific interaction dynamics, but reflects constraint structure that persists across physical scales.

Within Pattern Field Theory, this persistence is interpreted as evidence that directional selection is governed by structural properties of the underlying transport substrate. The Allen Orbital Lattice provides a candidate discrete geometric framework capable of imposing such scale-independent directional constraints.

Falsifiability and Controls

Null expectations

Three null outcomes falsify the cross-domain common-ordering hypothesis:

1. Jets: ordering metrics fluctuate randomly with scale proxy and show no consistent trend.

2. Flow: harmonic power does not correlate with system constraint proxies.

3. Spin: polarization strengths are consistent with zero across kinematics (within uncertainty) and show no stable structure.

Metric robustness

Robustness checks:

• replace \(O_H\) with \(O_{\text{tail}}\) for jets and require consistent conclusions,

• vary \(x_0\) within a reasonable range and check ordering stability,

• vary \(N\) in \(O_F(N)\) for flow and check stability of monotonic behaviour.

Implications

If consistent ordering scaling is present across cascade, medium, and intrinsic channels, then directional organization should be treated as a structural invariant class rather than an isolated feature of one interaction regime. In PFT terms, this supports the view that constraint-mediated selection is a unifying description for directional transport signatures across physical domains.

Directional transport constraints generate characteristic scale-dependent spectral structure.

The resulting spectral organization is shown in Fig. 4.

Discussion

Directional organization in high-energy collisions has traditionally been interpreted within domain-specific dynamical models. Jet collimation is treated as perturbative cascade evolution. Collective flow is treated within hydrodynamic or transport frameworks. Polarization is analyzed within spin-transfer or hadronization models.

The present work does not replace these descriptions. Instead, it introduces a structural comparison layer across regimes. By converting experimental observables into scalar ordering strengths, a common quantitative language is established. This permits testing whether directional organization follows shared scaling behaviour independent of microscopic interpretation.

If cross-domain scaling is confirmed, directional ordering should be regarded as a structural invariant class rather than an isolated dynamical feature. Within Pattern Field Theory, such invariance supports the view that directional selection reflects transport constraints of an underlying substrate architecture. The Allen Orbital Lattice is proposed as a candidate discrete geometry capable of imposing such scale-independent directional constraints.

If scaling fails, the hypothesis of common constraint structure is rejected, and ordering remains domain-specific. Either outcome provides informative structure about the hierarchy of angular organization in high-energy interactions.

Scope and Interpretation Boundaries

The present work does not claim that existing experimental results establish substrate structure. Instead, it defines a quantitative framework that permits testing whether directional organization exhibits shared scaling behaviour across independent physical regimes.

The structural interpretation proposed within Pattern Field Theory represents one possible explanatory model for such scaling. Empirical confirmation or refutation depends entirely on the numerical analysis defined here. The framework therefore functions as a measurement-based test of structural invariance, independent of any specific microscopic model.

Conclusion

This paper defines a unified angular ordering framework that permits structural comparison of directional organization across CERN high-energy collision measurements. Event-shape distributions provide orientation-quotiented cascade ordering magnitudes. Flow harmonics provide orientation-preserving medium ordering. Polarization provides intrinsic dipole alignment ordering. A set of scalar ordering metrics and statistical tests is specified to evaluate monotonic scaling under domain-appropriate constraint proxies.

Replication Protocol (Minimum)

1. Acquire published unfolded event-shape histograms and compute \((O_H, O_{\text{tail}})\) per scale bin.

2. Acquire published \(v_n\) and compute \(O_F(N)\) per centrality or system bin.

3. Acquire published \(P\) values and compute \((O_S, Z_S, \Delta P)\) per kinematic bin.

4. Compute normalized ordering \(\tilde{O}\) per domain and evaluate monotonicity.

5. Report robustness under metric variants and threshold changes.

Data Sources, Provenance, and Exact Datasets Used

This work does not rely on private detector files. All quantitative inputs are taken from public, versioned analysis records (HEPData where available) and the corresponding collaboration publications.

Primary dataset used in this paper

• CMS event-shape distributions (pp, \(\sqrt{s}=13\) TeV, Run 2). Unfolded distributions for five event-shape variables computed from charged particles inside jets, with analysis binning in \(H_{T,2}\) and jet kinematics as defined by the collaboration.

  • Publication: CMS-SMP-22-004, CERN-EP-2025-192, arXiv:2602.17509.

  • Tabulated data (exact dataset): HEPData record for this analysis (2026), DOI: https://doi.org/10.17182/hepdata.167810.

Secondary reference datasets (used for cross-context comparison, not for fitting)

If you include cross-system comparisons (collective flow and charm-baryon polarization), cite them explicitly as reference datasets:

• CMS OO and NeNe collective flow (if used). CMS-HIN-25-009, arXiv:2510.02580. HEPData record for this analysis (2025), DOI: https://doi.org/10.17182/hepdata.165513.

• ATLAS OO and NeNe azimuthal anisotropy (if used). arXiv:2509.05171 (ATLAS Collaboration). If no HEPData record is available yet, cite arXiv plus the CERN Document Server plots/record associated with the paper.

• LHCb charm-baryon polarization in fixed-target pNe (if used). LHCb-PAPER-2025-060, CERN-EP-2026-016, arXiv:2602.17184. If no HEPData record is available yet, cite the paper and the CERN Document Server record.

ATLAS light-ion flow dataset (primary collective-flow reference)

The collective-flow comparison in this work uses the ATLAS measurement of azimuthal anisotropy coefficients in light-ion collisions:

ATLAS Collaboration, Measurement of the azimuthal anisotropy of charged particles in \(\sqrt{s_{\mathrm{NN}}}=5.36\) TeV \(^{16}\)O\(+^{16}\)O and \(^{20}\)Ne\(+^{20}\)Ne collisions with the ATLAS detector, arXiv:2509.05171 (2025).

This dataset provides the first measurements of flow harmonics \(v_n\) (\(n=2\)–4) in oxygen–oxygen and neon–neon collisions at the LHC. The coefficients are reported as functions of transverse momentum, collision centrality, and event multiplicity. They are extracted using two independent methods:

• two-particle correlations with template-fit subtraction of short-range non-flow,

• four-particle subevent cumulants suppressing non-flow and probing fluctuations.

The measurements exhibit the characteristic hierarchy \(v_2 > v_3 > v_4\) and non-monotonic \(p_T\) dependence with a maximum near 2–4 GeV. Enhanced elliptic flow in central Ne+Ne collisions reflects the deformed nuclear geometry of neon relative to oxygen.

The tabulated harmonic coefficients \(v_n\) from this measurement are used as the collective anisotropy inputs in the cross-domain ordering comparison.

Interpretation of CMS Jet Structure in Pattern Field Theory

In Pattern Field Theory, jet event–shape observables measured by CMS are interpreted as indicators of directional ordering in fragmentation dynamics. Rather than treating jet structure solely as a consequence of perturbative shower evolution and hadronization, PFT interprets collimation and energy distribution within jets as signatures of constrained directional transport during cascade development.

Unfolded multijet event–shape distributions, including transverse thrust complement, total jet broadening, and hemisphere mass variables, are used to quantify the degree of angular dispersion and directional concentration of energy flow. Suppression of high-dispersion tails and reduced entropy in these distributions are interpreted as evidence of directional selection within the fragmentation cascade.

These observables are mapped to scalar ordering metrics that quantify the strength of directional organization in jet formation. The resulting measures can be directly compared to medium-level ordering in heavy-ion flow and intrinsic ordering in spin polarization.

Interpretation of LHCb Polarization Measurements in Pattern Field Theory

In Pattern Field Theory, polarization measurements reported by LHCb are interpreted as indicators of intrinsic directional alignment at the particle formation level. Rather than describing polarization solely as a consequence of spin–orbit coupling or hadronization dynamics, PFT treats polarization as a manifestation of orientation selection during the formation of composite states.

Measurements of charm–baryon polarization magnitude, transverse momentum dependence, and charge asymmetry provide quantitative measures of preferred orientation relative to production geometry. The absolute polarization magnitude and its statistical significance are interpreted as measures of intrinsic alignment strength.

These polarization observables define a third ordering regime that operates at the level of individual particle formation. When expressed in normalized form, intrinsic alignment strength can be directly compared with directional organization in jets and collective flow.

Unified Cross-Domain Ordering Framework

Pattern Field Theory interprets high-energy collision observables across multiple experimental domains as manifestations of directional organization operating at different physical scales.

Three regimes of ordering are identified:

• Cascade Ordering — directional organization in fragmentation processes, measured through jet structure observables (CMS).

• Medium Ordering — collective directional organization in expanding collision matter, measured through harmonic flow coefficients (ATLAS).

• Intrinsic Alignment — directional organization in particle formation, measured through spin and polarization observables (LHCb).

These three regimes represent directional organization at progressively different physical scales. Cascade ordering governs transport during fragmentation. Medium ordering governs collective response of interacting particle ensembles. Intrinsic alignment governs orientation selection during particle formation.

The scalar ordering metrics defined in this work provide a unified measurement language that permits quantitative comparison of directional organization across these structurally distinct regimes.

Each regime is quantified using scalar ordering metrics derived from the corresponding experimental observables. These metrics allow direct cross-domain comparison of directional organization across cascade dynamics, collective medium behaviour, and intrinsic particle alignment.

This unified framework enables tests of structural scaling across physical regimes and provides a basis for evaluating whether directional organization follows common quantitative behaviour across distinct dynamical processes.

Cross-Domain Scaling Hypothesis

If directional organization is governed by common structural constraints, ordering strength measured in different physical regimes should exhibit systematic scaling behaviour when expressed in normalized form.

The central testable hypothesis is that ordering magnitude evolves with constraint scale according to a shared functional dependence across:

• energy scale in jet fragmentation,

• system size and centrality in heavy-ion collisions,

• kinematic scale in particle polarization.

Quantitative evaluation of these scaling relations constitutes the primary objective of the numerical analysis stage.

Likelihood-Based Scaling Fit

To test whether a single structural function describes ordering across domains, a parametric scaling model \(\tilde{O}(\chi;\theta)\) is fitted simultaneously to all normalized datasets.

For measurements \(\tilde{O}_{i,k}\) at constraint values \(\chi_{i,k}\) with uncertainties \(\sigma_{i,k}\), the likelihood is

\[\begin{equation} \mathcal{L}(\theta) = \prod_{i} \prod_{k} \frac{1}{\sqrt{2\pi}\sigma_{i,k}} \exp\!\left[ -\frac{ \left(\tilde{O}_{i,k} - \tilde{O}(\chi_{i,k};\theta)\right)^2 }{2\sigma_{i,k}^2} \right]. \end{equation}\]

Parameter estimates are obtained by maximizing \(\mathcal{L}(\theta)\) or equivalently minimizing

\[\begin{equation} \chi^2(\theta) = \sum_{i,k} \frac{ \left(\tilde{O}_{i,k} - \tilde{O}(\chi_{i,k};\theta)\right)^2 }{\sigma_{i,k}^2}. \end{equation}\]

Goodness-of-fit is evaluated using reduced \(\chi^2\) and likelihood ratio tests comparing alternative functional forms.

Candidate Universal Scaling Function

Transport-constrained directional organization is expected to exhibit bounded growth with increasing constraint strength. A minimal structural form is a saturating exponential:

\[\begin{equation} \tilde{O}(\chi) = 1 - \exp(-\lambda \chi), \label{eq:universal_saturating} \end{equation}\]

where \(\lambda\) is a structural rate parameter.

For small \(\chi\), cascade amplification may produce a power-law regime:

\[\begin{equation} \tilde{O}(\chi) \propto \chi^{\alpha}, \qquad \chi \ll 1. \end{equation}\]

A combined piecewise model is therefore

\[\begin{equation} \tilde{O}(\chi)= \begin{cases} \chi^{\alpha}, & \chi \le \chi_c,\\ 1-\exp[-\lambda(\chi-\chi_c)], & \chi>\chi_c. \end{cases} \end{equation}\]

Compatibility of jets, flow, and polarization with a shared parameter set \((\alpha,\lambda,\chi_c)\) constitutes the primary empirical test.

Reproducibility note

All derived quantities in this manuscript must be reproducible from the cited HEPData tables (or directly from the cited publications where HEPData is not available). Any intermediate processing steps (bin merging, normalization, reweighting, angular reparameterizations) must be described and versioned in the analysis section of this paper.

Glossary

Angular anisotropy - deviation from uniform angular distribution

Cascade ordering - directional structure in branching momentum transport

Flow harmonic - Fourier coefficient describing anisotropic emission

Polarization - alignment of intrinsic angular momentum

Ordering metric - scalar measure of directional structure

References

Data Provenance and Reproducibility

All quantitative inputs used in this work are derived from publicly available, versioned CERN collaboration datasets and corresponding peer-reviewed publications. No private detector-level data are used.

Each dataset is referenced by collaboration identifier, arXiv record, and when available, HEPData DOI. All ordering metrics are computed exclusively from published unfolded distributions or tabulated harmonic coefficients.

The analysis pipeline is deterministic and reproducible:

1. obtain tabulated experimental observables,

2. compute ordering metrics defined in Section X,

3. evaluate monotonic behaviour across system constraints,

4. compare normalized ordering across physical domains.

No parameter fitting or model-dependent reconstruction is performed. All transformations are explicitly defined and invertible where applicable.

Exact Table and Figure Extraction Map

This section specifies the precise experimental observables used as inputs for directional ordering metrics and identifies their source within each collaboration analysis.

CMS multijet event-shape dataset

Source: CMS-SMP-22-004, arXiv:2602.17509.

Extract the following unfolded event-shape distributions:

• transverse thrust complement,

• total jet broadening,

• hemisphere mass variables.

These observables are binned in leading-jet scale or \(H_{T,2}\) as defined in the CMS analysis.

Data sources:

• unfolded distribution figures,

• HEPData tables for each observable,

• associated statistical and systematic uncertainties.

Derived ordering metrics:

• entropy ordering,

• tail fraction ordering,

• scale-dependent directional organization.

ATLAS light-ion flow dataset

Source: ATLAS Collaboration, arXiv:2509.05171.

Extract harmonic coefficients:

• \(v_2(p_T)\),

• \(v_3(p_T)\),

• \(v_4(p_T)\),

• centrality dependence.

Use both analysis methods when available:

• two-particle correlations,

• four-particle subevent cumulants.

Data sources:

• harmonic coefficient figures,

• supplementary material tables if available.

Derived ordering metrics:

• harmonic power ordering,

• harmonic hierarchy ratios,

• system-size scaling of anisotropy.

LHCb charm-baryon polarization dataset

Source: LHCb Collaboration, arXiv:2602.17184.

Extract:

• polarization magnitude,

• transverse momentum dependence,

• particle–antiparticle asymmetry,

• statistical uncertainty.

Derived ordering metrics:

• absolute alignment strength,

• significance ordering,

• alignment asymmetry.

Observable to Ordering Metric Mapping

Replication Workflow

Step 1: Data acquisition

Obtain tabulated data from HEPData or digitize published collaboration figures.

Step 2: Standardization

Convert all inputs to a uniform format:

• observable,

• bin center,

• value,

• statistical uncertainty,

• systematic uncertainty.

Step 3: Metric calculation

Jets:

• probability normalization,

• entropy computation,

• tail integral evaluation.

Flow:

• quadratic harmonic sum,

• harmonic ratio computation.

Polarization:

• absolute magnitude,

• uncertainty normalization.

Step 4: Scaling analysis

Within each domain evaluate:

• ordering versus constraint proxy,

• Spearman rank correlation,

• bootstrap uncertainty.

Step 5: Cross-domain normalization

Apply min–max normalization and compare ordering trends.

Step 6: Structural interpretation

Compare cascade, collective, and intrinsic ordering strengths.

Statistical Uncertainty Propagation

Histogram-derived metrics

Apply Monte Carlo resampling:

• sample bin values within uncertainties,

• recompute entropy,

• obtain ordering distribution.

Flow harmonic power

Analytic propagation:

\[\sigma_{O_F} = \frac{1}{O_F} \sqrt{\sum (v_n \sigma_{v_n})^2}\]

Polarization

Direct significance:

\[Z = \frac{|P|}{\sigma}\]

Monotonic trend significance

Evaluate using:

• Spearman rank p-value,

• bootstrap confidence intervals.

Status of Numerical Evaluation

The ordering framework and metric definitions are fully specified in the present manuscript.

Application of these metrics to the cited CERN datasets constitutes the next stage of analysis. The datasets have been identified, the extraction mapping is defined, and the statistical pipeline is complete.

Numerical evaluation and cross-domain scaling results will be presented in a subsequent analysis document or supplementary results paper.

Interpretation of ATLAS Flow Measurements in Pattern Field Theory

In Pattern Field Theory, harmonic flow coefficients measured by ATLAS are interpreted as indicators of medium-level directional ordering rather than purely hydrodynamic response. Measurements from oxygen–oxygen and neon–neon collisions are used to quantify anisotropic emission patterns, with enhanced elliptic flow in neon attributed to geometry-induced directional selection.

The harmonic coefficients are combined into a scalar ordering strength to enable quantitative comparison with jet collimation and spin polarization observables from CMS and LHCb.

The present work establishes the metric framework. The next stage is numerical evaluation of ordering as a function of system size and testing for cross-domain scaling behaviour across jets, collective flow, and polarization.

Medium-Level Directional Ordering

Within Pattern Field Theory, harmonic flow coefficients measured by ATLAS are interpreted as indicators of medium-level directional ordering rather than purely hydrodynamic response.

Measurements from oxygen–oxygen and neon–neon collisions quantify anisotropic particle emission patterns. The observed enhancement of elliptic flow in neon collisions is attributed to geometry-induced directional selection arising from the intrinsic spatial deformation of the colliding nuclei.

The harmonic coefficients are combined into a scalar ordering strength, allowing quantitative comparison with cascade ordering in jets and intrinsic alignment in polarization measurements.

The present work establishes the metric framework required for this comparison. Numerical evaluation of ordering as a function of system size and tests of cross-domain scaling constitute the next stage of analysis.

Planned Numerical Evaluation

The next stage of this work applies the defined ordering metrics to the identified CERN datasets to obtain quantitative cross-domain ordering curves.

Planned results include:

• ordering magnitude versus energy scale in jets,

• harmonic power versus system size in light-ion collisions,

• polarization ordering versus kinematic scale,

• normalized cross-domain comparison.

Citation

James Johan Sebastian Allen (2026). Emergent Directional Organization in CERN High Energy Collisions. Pattern Field Theory.
Available at: https://patternfieldtheory.com/
ORCID: https://orcid.org/0009-0009-9594-6803

Document Timestamp and Provenance

This document is part of Pattern Field Theory (PFT) and the Allen Orbital Lattice (AOL). It defines the Phase Alignment Lock (PAL) constraint and specifies methods and replication procedures used by subsequent papers in the series. Any research, derivative work, or commercial use requires an explicit license from the author.