Pattern Field Theory Logo

The 14-Point Empirical Audit of Structural Inevitability

Cross-domain constraint audit showing repeated admissibility boundaries and forced regime transitions.

© 2026 James Johan Sebastian Allen - Pattern Field Theory™

Scope

This page is an audit ledger. Each point is written so the claim can be checked against a named primary source. The audit tests for: discrete admissible state classes, hard boundaries, and forced reconfiguration when a boundary is exceeded.

Each proof uses the same template: formal inputs, formal outputs, definitions, proposition, proof, falsification condition, replication handle, and audit mapping.

Replication Burden - Regime Lock Rule

Each proof in this audit is a regime-locked claim. A replication attempt is valid only if it reproduces the source method class and the source regime definitions. If a replication attempt changes the equation-of-state, boundary conditions, stability definition, measurement pipeline, or classification rules, it is not a replication of the same proof. It is a test of a different regime.

Audit rule: method deviations do not falsify the audit point. They are recorded as regime-shift attempts. Only regime-locked tests can confirm or falsify a proof in this ledger.

Governing Constraint Schema

Ctot ≤ B

Interpretation: a configuration persists only if its total maintenance cost remains within the admissibility budget of its regime.

Audit Template

  1. Formal inputs - what is measured or computed
  2. Formal outputs - what must be true if the proposition holds
  3. Proposition - checkable claim
  4. Proof - short, regime-internal derivation or implication chain
  5. Falsification - what would break the claim
  6. Replication - what to do to reproduce or refute
  7. Audit mapping - how this instantiates a budget boundary

Reproducibility Ledger - Pattern Field Theory Audit

Ledger of replication requirements and deviation risks for the 14-point audit. Regime-locked replication is required for RL-PASS or RL-FAIL classification.

Replication Burden - Regime Lock Rule

A replication attempt is valid only if it reproduces the source method class and the source regime definitions. Changes to equation-of-state scaling, boundary conditions, measurement pipeline, classification rules, or evaluation thresholds are regime shifts. Regime shifts are logged as new regime tests and do not count as replication outcomes for the original proof.

Outcome labels

  • RL-PASS - regime-locked replication confirms the proposition under declared error accounting
  • RL-FAIL - regime-locked replication contradicts the proposition under declared error accounting
  • RS - regime shift, method or definition changed
  • NA - not audit-grade, missing minimal replication package items

Minimal replication package

  • Primary source lock - DOI or archival ID plus version and access date
  • Object lock - variables, observables, states, labels defined exactly as in the source
  • Method-class lock - model family or derivation class plus explicit assumptions
  • Boundary-condition lock - initial and boundary conditions or an explicit equivalence mapping
  • Evaluation lock - metrics, thresholds, and decision rules
  • Numerics lock - solver, tolerances, convergence, stability checks
  • Outputs - the proposition output in comparable form
  • Error accounting - uncertainty and numerical error bounds
  • Regime-shift log - all deviations itemized as RS items

Ledger table

Proof Domain Inputs - source method Outputs - audit claim Replication requirements Deviation risks
1 DNA stacking Assay regime, observable definition, preprocessing pipeline Admissible persistence class of stably observed configurations Same assay definition, same stability criterion, same pipeline Changed assay definition or pipeline is RS
2-3 DNA helices Finite context-kernel model and stratification definition Finite kernel necessity and basin-structured parameter occupancy Same kernel length, same encoding, same evaluation and model selection Changing kernel definition or evaluation metric is RS
4 RNA aptamer binding Topology descriptor set, binding threshold, structural validation method class Binding gated by discrete closure class Same topology classification rules, same binding definition, same validation criteria Redefining classes as continuous approximations is RS
5 Algorithm design Instance class, objective function, proved approximation guarantee Certified admissibility bound within the declared regime Exact algorithm, exact instance class, exact objective, counterexample search within the same class Changing objective or instance class is RS
6-7 Transition networks Graph construction rules or frontier rules with fixed definitions Admissible paths bounded to discrete sets Same node-edge definitions or same CV and frontier definitions, same construction parameters Changing the state definition, CV definition, or frontier rule is RS
8-9 Astrophysics solvability boundary Equilibrium equations plus occupancy-exclusion supported EOS scaling class and boundary conditions Strict maximum admissible load M* with non-existence beyond M* Same EOS scaling class, same boundary conditions, same solvability criteria Adding corrections or relaxing equilibrium is RS
10 Atmospheric science Confinement geometry, periodic compatibility, stability criterion Discrete polygonal admissible mode set Same equations, same confinement assumptions, same persistence test Relaxing confinement or stability criterion is RS
11-14 Number theory and spectra Explicit operator specification plus public target zero tables and error accounting Audit-grade falsifiability via spectrum-to-target inequality test Publish operator space, domain, BCs, self-adjointness criterion, discretization, convergence, ordering, tolerance Unspecified operator is NA until fully specified

Ledger cards

These duplicate the table in a scan-friendly form. You can later hide either the cards or the table using your site CSS.

Proof 1 - DNA stacking

  • Inputs - assay regime, observable definition, preprocessing pipeline
  • Outputs - admissible persistence class of stably observed configurations
  • Replication - same assay definition, same stability criterion, same pipeline
  • Deviation risks - changed assay definition or pipeline is RS

Proof 2-3 - DNA helices

  • Inputs - finite context-kernel model and stratification definition
  • Outputs - finite kernel necessity and basin-structured parameter occupancy
  • Replication - same kernel length, same encoding, same evaluation and model selection
  • Deviation risks - changing kernel definition or evaluation metric is RS

Proof 4 - RNA aptamer binding

  • Inputs - topology descriptor set, binding threshold, structural validation method class
  • Outputs - binding gated by discrete closure class
  • Replication - same topology rules, same binding definition, same validation criteria
  • Deviation risks - continuous redefinition is RS

Proof 5 - Algorithm design

  • Inputs - instance class, objective function, proved approximation guarantee
  • Outputs - certified admissibility bound within the declared regime
  • Replication - exact algorithm, exact instance class, exact objective
  • Deviation risks - changing objective or instance class is RS

Proof 6-7 - Transition networks

  • Inputs - graph construction rules or frontier rules with fixed definitions
  • Outputs - admissible paths bounded to discrete sets
  • Replication - same definitions, same parameters, publish the constructed artifact
  • Deviation risks - changing definitions is RS

Proof 8-9 - Astrophysics solvability boundary

  • Inputs - equilibrium equations plus occupancy-exclusion supported EOS scaling class and BCs
  • Outputs - strict maximum admissible load M* with non-existence beyond M*
  • Replication - same EOS scaling class, same BCs, same solvability criteria
  • Deviation risks - EOS modifications, added corrections, relaxed equilibrium are RS

Proof 10 - Atmospheric science

  • Inputs - confinement geometry, periodic compatibility, stability criterion
  • Outputs - discrete polygonal admissible mode set
  • Replication - same equations, same confinement assumptions, same persistence test
  • Deviation risks - relaxed confinement is RS

Proof 11-14 - Number theory and spectra

  • Inputs - explicit operator specification plus public target tables and error accounting
  • Outputs - audit-grade falsifiability via spectrum-to-target inequality test
  • Replication - publish operator, discretization, convergence, ordering, tolerance, and comparison segment
  • Deviation risks - incomplete operator is NA until specified

Ledger entry format

Every replication attempt should be reported using this header line, then expanded with the full package:

[Proof #] - [RL-PASS | RL-FAIL | RS | NA] - Source DOI - Method lock hash - Data or code archive - Date

The 14 Proofs

Proof 1 - DNA stacking defines admissible persistence classes under measurement constraints

Primary source: Banerjee et al. (2023), Nature Nanotechnology, DOI 10.1038/s41565-023-01485-1

Formal inputs:

  • Assay protocol and measurement definitions in the source
  • Observed stacking configurations Smeas
  • Reported energetic estimates E(s) with uncertainty for s ∈ Smeas

Formal outputs:

  • Admissible persistence set A = Smeas for the defined regime
  • Boundary statement: configurations outside A are non-persistent under the same regime definitions

Definitions: Let S be the space of candidate stacking configurations under the same parameterization as the assay. Define Persist(s)=1 if s is stably observed under the assay regime, else Persist(s)=0. Define A := { s ∈ S : Persist(s)=1 }.

Proposition: The assay defines an admissible persistence class A and therefore induces a regime boundary separating admissible from non-admissible configurations.

Proof:

  1. The assay reports only configurations that are stably measurable under its constraints, so Smeas ⊆ A.
  2. Any configuration not stably measurable under the same regime has Persist(s)=0 and is outside A.
  3. Therefore S is partitioned into A and S\A, which is a boundary in the audit sense.
  4. The reported energetic mapping E(s) is defined on A; extending beyond A changes the regime definitions.

Falsification condition: Under the same assay constraints and definitions, if configurations previously excluded become stably measurable without altering the regime, the boundary claim fails.

Replication: Reproduce the assay; publish Smeas, stability criteria, and uncertainty; report any transitions Persist:0→1 without regime change.

Audit mapping (Ctot ≤ B): Take B as the regime persistence budget under assay constraints. Membership in A corresponds to Ctot ≤ B; exclusion corresponds to Ctot > B.

Reproducibility Checklist

  • Primary source locked: DOI or archival identifier recorded
  • Objects locked: variables, states, and observables defined exactly as in source
  • Method class locked: model family, algorithm, or derivation class matches source
  • Assumptions locked: stated approximations match source assumptions
  • Boundary conditions locked: same BCs or equivalent under explicit mapping
  • Evaluation locked: same metric, threshold, and decision rule
  • Numerics locked: solver, tolerance, discretization, convergence checks recorded
  • Outputs locked: the proposition’s output is computed and compared in the same form
  • Regime-shift log: any deviation is recorded as a new regime, not as replication

Proof 2 - DNA helical observables require a finite context kernel beyond nearest-neighbour locality

Primary source: Balaceanu et al. (2019), Nucleic Acids Research, DOI 10.1093/nar/gkz255

Formal inputs:

  • Sequence set and helical observable definitions in the source
  • Model A: nearest-neighbour parameterization
  • Model B: extended context model as defined in the source
  • Same evaluation protocol for both models

Formal outputs:

  • Kernel necessity: Model B yields measurable improvement under identical evaluation
  • Boundedness: improvement is achieved with finite context length

Definitions: Let H(X,i) be a helical observable at position i for sequence X. Nearest-neighbour assumes dependence only on local dinucleotide context. A finite kernel model assumes dependence on a bounded window Wk(i) of length k.

Proposition: Helical observables in this regime require dependence beyond nearest-neighbour locality, establishing a finite non-local constraint kernel.

Proof:

  1. The source analyzes context effects extending beyond nearest-neighbour contributions for helical properties.
  2. If nearest-neighbour locality were sufficient, extended context would not improve fit or prediction under identical evaluation.
  3. The source supports that extended context contributes explanatory or predictive power, so H depends on Wk(i) for some finite k > 2.
  4. Therefore the regime enforces a finite constraint kernel and is not purely nearest-neighbour local.

Falsification condition: Re-analysis on comparable data showing no improvement from extended context under correct evaluation breaks kernel necessity.

Replication: Fit Model A and B on the same data; publish delta in error or likelihood with uncertainty and evaluation protocol details.

Audit mapping (Ctot ≤ B): Here Leff is kernel support size. Admissibility is evaluated over a finite window rather than purely local parameters.

Reproducibility Checklist

  • Primary source locked: DOI or archival identifier recorded
  • Objects locked: variables, states, and observables defined exactly as in source
  • Method class locked: model family, algorithm, or derivation class matches source
  • Assumptions locked: stated approximations match source assumptions
  • Boundary conditions locked: same BCs or equivalent under explicit mapping
  • Evaluation locked: same metric, threshold, and decision rule
  • Numerics locked: solver, tolerance, discretization, convergence checks recorded
  • Outputs locked: the proposition’s output is computed and compared in the same form
  • Regime-shift log: any deviation is recorded as a new regime, not as replication

Proof 3 - Helical parameter occupancy is basin-structured under a finite-kernel regime

Primary source: Balaceanu et al. (2019), Nucleic Acids Research, DOI 10.1093/nar/gkz255

Formal inputs:

  • Distributions of helical parameters extracted by the source pipeline
  • Context labels sufficient for stratification under the source definitions
  • Comparison families: unimodal continuous vs finite mixture models

Formal outputs:

  • Model selection favoring basin structure for at least one parameter under stratification
  • Discrete admissibility claim: occupancy concentrates in preferred neighborhoods (basins)

Definitions: Let p be a helical parameter (twist, roll, shift, etc.). Basin structure means the empirical distribution of p is better fit by a finite mixture than by a single unimodal family under fixed stratification.

Proposition: Under finite-context stratification, at least one key helical parameter exhibits basin-structured occupancy, supporting discrete admissibility neighborhoods.

Proof:

  1. Context modulation implies preferred parameter regions under regime constraints.
  2. Preferred regions imply clustering under fixed stratification rather than uniform occupancy across a single continuum.
  3. Robust clustering under mixture-model comparison constitutes basin structure, corresponding to discrete admissible neighborhoods.

Falsification condition: If basin structure disappears under correct stratification and sampling controls and unimodal models dominate, the claim is not supported.

Replication: Extract p distributions; stratify by context; fit unimodal and mixture models; report model selection and held-out validation.

Audit mapping (Ctot ≤ B): Basins represent admissible parameter neighborhoods with persistence; leaving basins corresponds to loss of admissibility in this regime mapping.

Reproducibility Checklist

  • Primary source locked: DOI or archival identifier recorded
  • Objects locked: variables, states, and observables defined exactly as in source
  • Method class locked: model family, algorithm, or derivation class matches source
  • Assumptions locked: stated approximations match source assumptions
  • Boundary conditions locked: same BCs or equivalent under explicit mapping
  • Evaluation locked: same metric, threshold, and decision rule
  • Numerics locked: solver, tolerance, discretization, convergence checks recorded
  • Outputs locked: the proposition’s output is computed and compared in the same form
  • Regime-shift log: any deviation is recorded as a new regime, not as replication

Proof 4 - RNA aptamer binding is gated by discrete topological closure class

Primary source: Stafflinger et al. (2025), Nucleic Acids Research, DOI 10.1093/nar/gkaf1315

Formal inputs:

  • Structural determination outputs defining the bound architecture
  • Topology descriptors used in the source to identify the G-quadruplex class
  • Binding or functional readouts under stated conditions

Formal outputs:

  • Discrete equivalence class T* of admissible topology
  • Gating rule: Bind=1 implies Topology ∈ T* under the defined regime

Definitions: Let Topology(x) map configuration x to its topological equivalence class under the source’s descriptors. Let Bind(x)=1 denote binding under the assay conditions, else 0. Let T* be the class of the experimentally determined bound structure.

Proposition: Under the specified regime, binding is gated by membership in T*. Continuous deformation that changes topological class exits admissibility for binding.

Proof:

  1. The source resolves a binding architecture defining T*.
  2. Topological class membership is discrete: either Topology(x)=T* or not.
  3. Binding in the reported binding state implies the realized binding configuration lies in T*.
  4. Therefore binding is constrained to a discrete admissibility class, not an arbitrary continuum across classes.

Falsification condition: Under the same conditions, if configurations confirmed outside T* still exhibit the same binding function, topology gating fails.

Replication: Construct topology-disrupting variants confirmed to exit T* by independent structural readout; measure binding with the same assay definition and thresholds.

Audit mapping (Ctot ≤ B): B is the closure budget defining permissible assembly. Topology ∈ T* corresponds to Ctot ≤ B; exiting the class corresponds to Ctot > B for that function.

Reproducibility Checklist

  • Primary source locked: DOI or archival identifier recorded
  • Objects locked: variables, states, and observables defined exactly as in source
  • Method class locked: model family, algorithm, or derivation class matches source
  • Assumptions locked: stated approximations match source assumptions
  • Boundary conditions locked: same BCs or equivalent under explicit mapping
  • Evaluation locked: same metric, threshold, and decision rule
  • Numerics locked: solver, tolerance, discretization, convergence checks recorded
  • Outputs locked: the proposition’s output is computed and compared in the same form
  • Regime-shift log: any deviation is recorded as a new regime, not as replication

Proof 5 - Provable approximation guarantees define explicit algorithmic admissibility bounds

Primary source: Shaw et al. (2014), BMC Bioinformatics, DOI 10.1186/1471-2105-15-S2-S7

Formal inputs:

  • Problem definition for the HP model on the lattice class in the source
  • Algorithm specification and proof conditions
  • Instance class to which the guarantee applies

Formal outputs:

  • Formal bound: for every instance in the declared class, output meets the proven approximation ratio
  • Boundary statement: exceeding the guarantee as a certified claim requires changing proof assumptions or regime

Definitions: Let OPT(I) be the optimal objective value for instance I. An approximation ratio ρ is a guarantee of the form ALG(I) ≥ OPT(I)/ρ (or the equivalent minimization form used in the source).

Proposition: A proven approximation ratio constitutes an explicit admissibility boundary for certified algorithmic performance in the declared regime.

Proof:

  1. The source provides a proof that for all instances in the declared class, the algorithm attains the stated ratio under the stated conditions.
  2. The guarantee is universally quantified over the class, so it is a hard certification bound within the regime.
  3. Any certified claim exceeding the bound without changing assumptions contradicts the proof constraints.

Falsification condition: A valid counterexample instance within the declared class where a correct implementation violates the guarantee falsifies the bound claim.

Replication: Implement the algorithm exactly; test across the declared instance class; publish any violations or confirm none occur with clear implementation details.

Audit mapping (Ctot ≤ B): B is the proof-bound on what is certified under constraints. Proof-backed certification limits instantiate regime boundaries.

Reproducibility Checklist

  • Primary source locked: DOI or archival identifier recorded
  • Objects locked: variables, states, and observables defined exactly as in source
  • Method class locked: model family, algorithm, or derivation class matches source
  • Assumptions locked: stated approximations match source assumptions
  • Boundary conditions locked: same BCs or equivalent under explicit mapping
  • Evaluation locked: same metric, threshold, and decision rule
  • Numerics locked: solver, tolerance, discretization, convergence checks recorded
  • Outputs locked: the proposition’s output is computed and compared in the same form
  • Regime-shift log: any deviation is recorded as a new regime, not as replication

Proof 6 - Transition network construction restricts admissible pathways to a finite graph-supported set

Primary source: Chakraborty and Wales (2017), PCCP, DOI 10.1039/C6CP06309H

Formal inputs:

  • Transition network construction procedure used in the source
  • State definition (nodes) and transition definition (edges)
  • Endpoint states for which pathway families are compared

Formal outputs:

  • A concrete transition graph G=(V,E) defining allowed transitions in the regime
  • Restriction statement: admissible pathways are exactly the graph paths in G under the regime definition

Definitions: Let G=(V,E) be the transition graph where V are metastable states and E are modeled transitions under the construction rules. A pathway is admissible iff it is a path in G.

Proposition: Under the regime defined by the network construction, admissible pathways are discrete and graph-bounded.

Proof:

  1. The construction outputs a finite set of modeled states and transitions.
  2. Transitions not in E are disallowed under the regime definition because admissibility is defined by the construction criteria.
  3. Therefore admissible pathways are exactly graph paths in G, which is a discrete set.

Falsification condition: If, under the same definitions, the construction yields effectively dense connectivity supporting arbitrary continuation between states, the graph-bounded restriction claim fails.

Replication: Reproduce G using the published procedure; publish adjacency and dominant path families between specified endpoints.

Audit mapping (Ctot ≤ B): Presence or absence of edges operationalizes admissibility boundaries. Missing edges correspond to forbidden transitions under the regime.

Reproducibility Checklist

  • Primary source locked: DOI or archival identifier recorded
  • Objects locked: variables, states, and observables defined exactly as in source
  • Method class locked: model family, algorithm, or derivation class matches source
  • Assumptions locked: stated approximations match source assumptions
  • Boundary conditions locked: same BCs or equivalent under explicit mapping
  • Evaluation locked: same metric, threshold, and decision rule
  • Numerics locked: solver, tolerance, discretization, convergence checks recorded
  • Outputs locked: the proposition’s output is computed and compared in the same form
  • Regime-shift log: any deviation is recorded as a new regime, not as replication

Proof 7 - SinkMeta operationalizes admissibility boundaries through constructed exploration frontiers

Primary source: Pan et al. (2025), JACS Au, DOI 10.1021/jacsau.5c00460

Formal inputs:

  • Collective variable (CV) definitions used in the source
  • SinkMeta algorithm and parameter choices
  • Frontier definition and blocking mechanism under the method

Formal outputs:

  • A reproducible frontier F in CV-space
  • Boundary statement: crossing F is blocked under the same method rule set

Definitions: Let CV be the coordinate system used to describe the sampled subspace. A frontier F is a set in CV-space such that sampling dynamics are constrained to remain on one side of F under the algorithm.

Proposition: Under SinkMeta, admissibility boundaries are implemented as explicit frontiers that terminate continuation in CV-space under the method rules.

Proof:

  1. The method defines a sampling rule introducing constraints via a sinking strategy and restraining potentials.
  2. Restraining potentials induce an effective frontier where transitions into disallowed regions are prevented under the same rule set.
  3. Therefore the method instantiates a boundary structure separating admissible from non-admissible sampling regions.

Falsification condition: If, under identical CV definitions and parameters, sampling routinely crosses the claimed frontier without modifying method rules, boundary instantiation fails.

Replication: Implement SinkMeta per the source; publish frontier computation, trajectories, and evidence that attempted crossings are blocked under the same parameterization.

Audit mapping (Ctot ≤ B): The frontier functions as a regime boundary: inside corresponds to admissible cost, outside corresponds to exceeding the imposed budget.

Reproducibility Checklist

  • Primary source locked: DOI or archival identifier recorded
  • Objects locked: variables, states, and observables defined exactly as in source
  • Method class locked: model family, algorithm, or derivation class matches source
  • Assumptions locked: stated approximations match source assumptions
  • Boundary conditions locked: same BCs or equivalent under explicit mapping
  • Evaluation locked: same metric, threshold, and decision rule
  • Numerics locked: solver, tolerance, discretization, convergence checks recorded
  • Outputs locked: the proposition’s output is computed and compared in the same form
  • Regime-shift log: any deviation is recorded as a new regime, not as replication

Proof 8 - Chandrasekhar limit is a strict solvability boundary in an exclusion-constrained occupancy-supported equilibrium regime

Source method class: Chandrasekhar (1935) derives a maximum equilibrium load for a self-gravitating, hydrostatic configuration supported by an exclusion-constrained occupancy equation-of-state scaling, under explicit boundary conditions and a static equilibrium definition. The audit uses this as a strict solvability boundary inside that regime.

Regime lock: A replication attempt is valid only if it preserves all of the following, as defined in the source:

  • Equilibrium definition: static hydrostatic balance in the same form
  • Support mechanism: the same exclusion-constrained occupancy EOS scaling class used in the derivation
  • Boundary conditions: the same regularity and finite-radius boundary conditions
  • Closure variables: the same dependent variables and parameterization used to define the solution family

Replication output: Reproduce the existence of a finite maximum load M* such that admissible equilibrium solutions exist for M ≤ M* and do not exist for M > M* under the same regime lock.

Deviation classification (do not treat as falsification): The following changes are regime shifts and therefore are not replications of Proof 8:

  • changing the EOS scaling by adding corrections or additional terms
  • adding rotation, magnetic terms, temperature support, or additional stress terms
  • relaxing static equilibrium or changing the stability criterion
  • changing boundary conditions or redefining the solution family

Audit consequence: A regime-shift attempt is logged as “not a replication of Proof 8”. It does not invalidate Proof 8. If desired, it can be promoted into a new audit point (for example Proof 8b) with its own locked regime definition and boundary.

Ontology note: The audit does not require a particle ontology. The term “exclusion-constrained occupancy support” is used as a regime-internal description of the equation-of-state constraint mechanism in the source method class.

Proof 9 - Beyond the occupancy-exclusion support boundary, equilibrium continuation is impossible without regime change

Primary source: Chandrasekhar (1935), MNRAS, DOI 10.1093/mnras/95.3.207

Formal inputs:

  • Proof 8 result: non-existence of admissible equilibrium solutions for M > M* in the same regime
  • Definition of “same regime”: same EOS scaling, same equilibrium equations, same boundary conditions

Formal outputs:

  • No continuous continuation within the same admissible family across M*
  • Continuation requires leaving the regime (forced reconfiguration)

Definitions: Let F be the set of admissible equilibrium configurations under the regime of Proof 8. “Continuation without regime change” means a path x(t) ∈ F with M(x(t)) increasing smoothly across M*.

Proposition: Continuation to M > M* requires leaving F and therefore changing regime.

Proof:

  1. From Proof 8, for M > M* there are no admissible equilibrium configurations in F.
  2. Therefore any path staying in F cannot attain M > M*.
  3. Hence continuation past M* requires leaving F, which is a regime change by definition.

Falsification condition: Provide a continuous family of admissible equilibrium solutions in the same regime that crosses M*.

Replication: Attempt continuation numerically under identical constraints and document the breakdown at M*.

Audit mapping (Ctot ≤ B): This is strict budget saturation: beyond the boundary, the constraint system has no admissible members.

Reproducibility Checklist

  • Primary source locked: DOI or archival identifier recorded
  • Objects locked: variables, states, and observables defined exactly as in source
  • Method class locked: model family, algorithm, or derivation class matches source
  • Assumptions locked: stated approximations match source assumptions
  • Boundary conditions locked: same BCs or equivalent under explicit mapping
  • Evaluation locked: same metric, threshold, and decision rule
  • Numerics locked: solver, tolerance, discretization, convergence checks recorded
  • Outputs locked: the proposition’s output is computed and compared in the same form
  • Regime-shift log: any deviation is recorded as a new regime, not as replication

Proof 10 - Confinement plus periodic compatibility yields discrete admissible polygonal modes

Primary source: Constantin and Johnson (2025), Journal of the Atmospheric Sciences, DOI 10.1175/JAS-D-24-0260.1 (publisher PDF: PDF)

Formal inputs:

  • Model equations and confinement assumptions in the source
  • Periodic boundary condition and definition of mode index m
  • Stability or persistence criterion used in the analysis

Formal outputs:

  • Discrete admissible set of mode indices Madm ⊂ Z under fixed parameters
  • Boundary statement: modes outside Madm fail persistence or compatibility within the same regime

Definitions: Let m be the azimuthal wave number enforced by periodic compatibility. An admissible mode is a solution that satisfies the governing system and remains bounded and persistent under the regime constraints.

Proposition: Under confinement and periodic compatibility, polygonal patterns correspond to discrete admissible mode indices.

Proof:

  1. Periodic compatibility around the azimuth enforces integer indexing for structured solutions.
  2. Given confinement parameters, stability conditions filter the integer-indexed family.
  3. The surviving set is a discrete subset of integers, establishing an admissible class structure.

Falsification condition: Under the same regime, if a continuous family of non-integer-compatible polygonal structures satisfies periodic compatibility and persistence, discreteness fails.

Replication: Implement the model; compute solutions across m and parameter sweeps; publish the indices satisfying the persistence criterion with solver details.

Audit mapping (Ctot ≤ B): B is the stability envelope under confinement. Admissible modes satisfy the envelope; excluded modes correspond to budget exceedance in the mapping.

Reproducibility Checklist

  • Primary source locked: DOI or archival identifier recorded
  • Objects locked: variables, states, and observables defined exactly as in source
  • Method class locked: model family, algorithm, or derivation class matches source
  • Assumptions locked: stated approximations match source assumptions
  • Boundary conditions locked: same BCs or equivalent under explicit mapping
  • Evaluation locked: same metric, threshold, and decision rule
  • Numerics locked: solver, tolerance, discretization, convergence checks recorded
  • Outputs locked: the proposition’s output is computed and compared in the same form
  • Regime-shift log: any deviation is recorded as a new regime, not as replication

Proof 11 - Conditional spectral closure: self-adjoint correspondence implies critical-line placement

Primary sources: Connes (1998) arXiv math/9811068 and Berry-Keating (1999) program framework chapter

Formal inputs:

  • Fully specified operator H on a Hilbert space with proof of self-adjointness
  • Precise correspondence rule mapping nontrivial zeros to spectral data of H

Formal outputs:

  • Conditional implication: existence of such H plus correspondence implies critical-line placement for mapped zeros

Definitions: Self-adjoint operators have real spectrum. A valid correspondence must map targets to spectral data compatible with that real constraint.

Proposition: If a valid self-adjoint spectral correspondence exists for nontrivial zeros, then those zeros lie on Re(s)=1/2.

Proof:

  1. Self-adjointness enforces real spectral parameters.
  2. A valid correspondence must map each target element to data consistent with that constraint.
  3. Targets requiring non-real spectral encoding cannot be part of a valid self-adjoint correspondence.
  4. Therefore the correspondence implies critical-line placement for mapped nontrivial zeros.

Falsification condition: Provide an explicit self-adjoint operator and correspondence that maps to a verified off-line nontrivial zero while remaining valid and self-adjoint under audit scrutiny.

Replication: Publish H, proof of self-adjointness, correspondence map, and a reproducible spectrum computation.

Audit mapping (Ctot ≤ B): B is the joint constraint set: self-adjointness plus correspondence validity. Violations break admissibility of the spectral claim.

Reproducibility Checklist

  • Primary source locked: DOI or archival identifier recorded
  • Objects locked: variables, states, and observables defined exactly as in source
  • Method class locked: model family, algorithm, or derivation class matches source
  • Assumptions locked: stated approximations match source assumptions
  • Boundary conditions locked: same BCs or equivalent under explicit mapping
  • Evaluation locked: same metric, threshold, and decision rule
  • Numerics locked: solver, tolerance, discretization, convergence checks recorded
  • Outputs locked: the proposition’s output is computed and compared in the same form
  • Regime-shift log: any deviation is recorded as a new regime, not as replication

Proof 12 - Published zero tables define an external falsification target for any spectrum claim

Primary sources: Odlyzko zeta zero tables UMN tables and dataset packaging/precision reference Sage database_odlyzko_zeta

Formal inputs:

  • Published target segment Γ={γn} with stated accuracy ε
  • Computed spectrum segment Λ={λn} with computed error bound ε'
  • Declared ordering convention for both sequences

Formal outputs:

  • Binary pass/fail test: |λn - γn| ≤ ε + ε' over a declared range

Definitions: Γ is a public target sequence of zero ordinates with stated numerical accuracy. Λ is the produced spectral list from a specified operator under a specified method with computed numerical error ε'.

Proposition: Any spectral correspondence claim becomes externally falsifiable by comparison to public target tables under declared error accounting.

Proof:

  1. Target tables provide a public Γ with stated accuracy.
  2. A correspondence claim produces Λ and therefore defines a comparison relation.
  3. With declared error bounds, the inequality |λn-γn| ≤ ε+ε' is a direct falsification test independent of interpretation.

Falsification condition: Failure of the inequality across a substantial initial segment under correct ordering and correct error accounting falsifies the numerical correspondence claim.

Replication: Specify which table file, which index range, solver method, ordering rule, and ε' computation; publish code and convergence checks.

Audit mapping (Ctot ≤ B): Define Ctot(n)=|λn-γn| and B=ε+ε'. Passing means Ctot(n) ≤ B on the declared range.

Reproducibility Checklist

  • Primary source locked: DOI or archival identifier recorded
  • Objects locked: variables, states, and observables defined exactly as in source
  • Method class locked: model family, algorithm, or derivation class matches source
  • Assumptions locked: stated approximations match source assumptions
  • Boundary conditions locked: same BCs or equivalent under explicit mapping
  • Evaluation locked: same metric, threshold, and decision rule
  • Numerics locked: solver, tolerance, discretization, convergence checks recorded
  • Outputs locked: the proposition’s output is computed and compared in the same form
  • Regime-shift log: any deviation is recorded as a new regime, not as replication

Proof 13 - Audit-grade correspondence claims require explicit operator specification

Primary sources: Connes (1998) arXiv math/9811068 and Berry-Keating (1999) program framework chapter

Formal inputs:

  • A claimed correspondence statement (operator-spectrum matches zeta zero ordinates)
  • The audit falsifiability requirement on this page

Formal outputs:

  • Completeness criterion: explicit operator specification is necessary for replication and falsification
  • Classification rule: unspecified-operator claims are incomplete until specification exists

Definitions: Explicit operator specification includes: Hilbert space, domain, inner product, operator expression, boundary conditions, and a verifiable self-adjointness proof or criterion.

Proposition: Without explicit operator specification, a correspondence claim is not audit-grade because it cannot be independently replicated or falsified.

Proof:

  1. A correspondence claim asserts matching between a target set and the spectrum of an operator.
  2. Testing requires computing the spectrum, which requires a defined operator object.
  3. If the operator is unspecified, the spectrum is undefined for replication, so the claim cannot be falsified or verified.

Falsification condition: Provide a complete operator specification plus reproducible spectrum computation. If provided, this classification no longer applies to that claim.

Replication: Publish a reference implementation constructing the operator, verifying stated properties, computing a spectral segment, and comparing to an external target set.

Audit mapping (Ctot ≤ B): Audit evaluation requires a defined constraint system. Operator-unspecified claims are outside the domain of evaluation, not inside it.

Reproducibility Checklist

  • Primary source locked: DOI or archival identifier recorded
  • Objects locked: variables, states, and observables defined exactly as in source
  • Method class locked: model family, algorithm, or derivation class matches source
  • Assumptions locked: stated approximations match source assumptions
  • Boundary conditions locked: same BCs or equivalent under explicit mapping
  • Evaluation locked: same metric, threshold, and decision rule
  • Numerics locked: solver, tolerance, discretization, convergence checks recorded
  • Outputs locked: the proposition’s output is computed and compared in the same form
  • Regime-shift log: any deviation is recorded as a new regime, not as replication

Proof 14 - Minimal completeness criteria for an audit-grade AOL spectral claim

Primary sources: Odlyzko target tables UMN tables and dataset packaging/precision reference Sage database_odlyzko_zeta

Formal inputs:

  • Explicit operator specification HAOL with proof or criterion of self-adjointness
  • Discretization and convergence specification yielding finite approximants
  • Solver and ordering conventions producing Λ
  • Public target segment Γ and error accounting ε, ε'

Formal outputs:

  • Checkable claim: match or mismatch against public target under declared tolerance
  • Reproducibility package sufficient for third-party recomputation

Definitions: An AOL spectral claim is audit-grade iff all objects in the claim are specified so independent parties can reconstruct them and run the Proof 12 test.

Proposition: The criteria below are sufficient for audit-grade status of an AOL spectral correspondence claim and sufficient for external falsification.

Proof:

  1. Operator declaration: specify HAOL including domain, inner product, boundary conditions, and self-adjointness proof or criterion.
  2. Discretization: specify basis, truncation, and convergence property.
  3. Spectrum extraction: specify numerical method, ordering, and stability tests producing Λ.
  4. Error accounting: publish ε' computation from convergence and conditioning.
  5. Target comparison: choose a public Γ and apply |λn-γn| ≤ ε+ε' on a declared range.
  6. If steps 1-5 are met, the claim is fully specified and externally falsifiable by public computation.

Falsification condition: After criteria 1-5 are satisfied, the claim is falsified if mismatch exceeds declared bounds over the declared comparison range.

Replication: A reader must be able to rebuild HAOL, compute Λ, select the same Γ, and reproduce the inequality check without private parameters.

Audit mapping (Ctot ≤ B): Define Ctot(n)=|λn-γn| and B=ε+ε'. Passing means Ctot(n) ≤ B on the declared range.

Reproducibility Checklist

  • Primary source locked: DOI or archival identifier recorded
  • Objects locked: variables, states, and observables defined exactly as in source
  • Method class locked: model family, algorithm, or derivation class matches source
  • Assumptions locked: stated approximations match source assumptions
  • Boundary conditions locked: same BCs or equivalent under explicit mapping
  • Evaluation locked: same metric, threshold, and decision rule
  • Numerics locked: solver, tolerance, discretization, convergence checks recorded
  • Outputs locked: the proposition’s output is computed and compared in the same form
  • Regime-shift log: any deviation is recorded as a new regime, not as replication

The AOL Gauge - Completion Equivalence

Mainstream gauge theory uses local symmetry transformations to preserve formal invariants. In Pattern Field Theory, gauge is defined as structural completion equivalence under admissible transport on the AOL. A configuration is gauge-invariant only if its transport maintains closure class under the regime constraints.

The Allenics Transformation

The AOL gauge transform is written as a constraint-preserving transport factor.

Φ' = Φ · exp(i ∑ κ · ΔLAOL)

  • κ - curvature or constraint functional evaluated at the transported configuration
  • ΔLAOL - discrete step-length of an admissible lattice transport step
  • ∑ - path-sum over an admissible transport sequence

Locality is treated as an emergent relation: configurations are functionally local if they share completion equivalence under admissible transport.

Formal Conclusion

Across molecular structure, context kernels, topological gating, algorithmic bounds, kinetic transition graphs, constrained exploration frontiers, confinement-selected macroscopic modes, solvability limits of equilibrium regimes, and spectral falsification targets, the repeated structural signature is:

  • admissible configurations form bounded classes
  • hard boundaries terminate continuation inside a regime
  • exceeding a boundary forces regime change

The audit conclusion is a consistency claim: persistence is a consequence of admissibility, not a primitive assumption.

References

  1. Banerjee et al. (2023). Nature Nanotechnology. DOI 10.1038/s41565-023-01485-1
  2. Balaceanu et al. (2019). Nucleic Acids Research. DOI 10.1093/nar/gkz255
  3. Stafflinger et al. (2025). Nucleic Acids Research. DOI 10.1093/nar/gkaf1315
  4. Shaw et al. (2014). BMC Bioinformatics. DOI 10.1186/1471-2105-15-S2-S7
  5. Chakraborty and Wales (2017). PCCP. DOI 10.1039/C6CP06309H
  6. Pan et al. (2025). JACS Au. DOI 10.1021/jacsau.5c00460
  7. Chandrasekhar (1935). MNRAS. DOI 10.1093/mnras/95.3.207
  8. Constantin (2025). JAS (AMS). JAS-D-24-0260.1
  9. Connes (1998). arXiv math/9811068
  10. Berry and Keating (1999). Springer chapter H = xp and the Riemann zeros

Formal Conclusion

The audit ledger establishes cross-regime recurrence of boundary-gated persistence, discrete admissible sets, and forced reconfiguration beyond boundary. The AOL Gauge formalizes completion-equivalence as a discrete path transport invariance on the lattice. Together these define a constraint-first framework where stability follows from admissibility and closure.

© 2026 James Johan Sebastian Allen. Pattern Field Theory™ and Allen Orbital Lattice™ are claimed trademarks.

How to Cite This Article

APA

Allen, J. J. S. (2026). Pattern Field Theory - The 14-Point Empirical Audit of Structural Inevitability. Pattern Field Theory. https://www.patternfieldtheory.com/articles/structural-inevitibility-audit/

MLA

Allen, James Johan Sebastian. "Pattern Field Theory - The 14-Point Empirical Audit of Structural Inevitability." Pattern Field Theory, 2026, https://www.patternfieldtheory.com/articles/structural-inevitibility-audit/.

Chicago

Allen, James Johan Sebastian. "Pattern Field Theory - The 14-Point Empirical Audit of Structural Inevitability." Pattern Field Theory. January 17, 2026. https://www.patternfieldtheory.com/articles/structural-inevitibility-audit/.

BibTeX

@article{allen2026pft,
  author  = {James Johan Sebastian Allen},
  title   = {Pattern Field Theory - The 14-Point Empirical Audit of Structural Inevitability},
  journal = {Pattern Field Theory},
  year    = {2026},
  url     = {https://www.patternfieldtheory.com/articles/structural-inevitibility-audit/}
}