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Physical phenomena atlas and regime correspondences

Status: Technical reference Version source: pyproject.toml Reviewed: 2026-10-06

1. Scope

This document records implemented graph-transport and auxiliary-wave results, and separately declared classical adapters. Each comparison retains its assumptions and evidence. The nodal equation \(\partial\mathrm{EPI}/\partial t = \nu_f\,\Delta\mathrm{NFR}(t)\) directly gives a first-order structural drift law. It does not, by itself, derive Newtonian, quantum or thermodynamic dynamics. Those labels apply only to the adapters or auxiliary models named below. Retired supplied-target demonstrations and unimplemented thermal proxies are not research routes or validation evidence.

This is also the single owner of the physical-emergence atlas: correspondence cards, reuse and evidence boundaries. The execution plan alone selects work and its order. The atlas does not replace the generative objective or establish that physical entities emerge from TNFR.

Vacuum, ether and a relational substrate

An underlying structure is a broad hypothesis; the historical luminiferous ether was a more specific proposed medium for light. Einstein's 1905 formulation explicitly dispenses with that medium and an absolutely stationary space. This is not a mathematical exclusion of every possible relational ontology, nor evidence in favor of a TNFR substrate. Calling the latter "ether" would not supply light propagation, a physical clock, a preferred-frame prediction or compatibility with observed electrodynamics.

TNFR currently identifies neither its fine nodes nor zero structural pressure with a physical vacuum. Its substrate admission distinguishes an empty domain, a quiescent but responsive background, channel cancellation and an unobserved environment. The native pressure-state counterexample tests informational sufficiency within one supplied nodal law, not vacuum physics. A proposed physical identification must first specify the background state, inherited observables, clock and response; any comparison then follows the same independent terrestrial P1-P5 protocol. No new empirical campaign or imported vacuum dynamics is admitted by this terminology.

Verification Status

Comparison Implementation External reference Test coverage Status
Classical adapter classical_mechanics.py Harmonic and circular-orbit closure test_classical_mechanics.py Finite regression
Kinematic adapter classical_mechanics.py Two-train analytical Embedded in example Demonstrated
Finite graph spectra and auxiliary waves structural_diffusion.py Declared graph generator and wave equation test_structural_diffusion.py Conditional algebra and finite numerical controls
Sampled ring modes and winding winding_certificates.py, structural_diffusion.py Declared cycle and phase field test_emergent_wave_particle_scope.py Aliasing, branch and spectral controls; no particle identification

Physical emergence atlas

The useful question is which known mechanisms admit a justified reduction, controlled approximation or falsifiable comparison with a specified nodal law. Writing a known vector field as nu_f*DeltaNFR by assignment establishes a representation, not an independently derived mechanism. Replace intuitive ratings such as "very high correspondence" with the evidence states below. No entry currently has complete physical admission under the P2 protocol.

Six fields for each correspondence card

  1. Phenomenon: precise regime and evidence status, rather than a whole physical theory claimed from one resemblance.
  2. Reference equation: source, full state, constitutive assumptions, units, boundaries and scale/size limit. External equations are comparisons, not undeclared TNFR laws.
  3. Observables: instruments or simulated observations, preparation, clock, uncertainty and accessible inputs. Identify hidden state explicitly.
  4. TNFR dictionary: the plan's definition/identification/evolution contract, with each quantity marked primitive, derived, supplied or diagnostic.
  5. Reduction: state/observation map and matching of the full vector field on the stated domain, or a controlled approximation with an error bound. In one declared clock, a closed observation H requires DH(z) F(z)=F_ref(H(z)); matching one trajectory or one sine term is weaker.
  6. Prediction and decision: predeclared response, fair comparator, reserved evidence and rejection/abstention criterion. If the models are equivalent in scope, record "no discriminating prediction in this regime"; identify an additional justified premise before claiming predictive novelty.

Proofs and raw evidence stay in their existing owners. The table is an index of cards and prerequisites, not independently active campaigns.

Current evidence map

Phenomenon Present TNFR evidence and reusable owner Missing bridge or limiting condition
Diffusion / relaxation Exact conditional reduction: isolated EPI transport and its stability theorem; separate TCLab exploratory response Measurement/clock and constitutive admission remain open; driven thermal input is not unforced diffusion.
Synchronization / Kuramoto-type dynamics Exact conditional observed-phase reduction and mode selection in derived form phase; supplied primitive-phase models are separate Physical observation/clock map remains open; the admitted passive capacity family locks generically while fading, with no positive-capacity-ratio threshold.
Josephson / superconductivity Constitutive comparison candidate: sine interaction and phase-current readers are reusable algebra Electrical current/voltage, gauge-invariant junction phase, storage/dissipation and a superconducting bridge are absent.
Dynamical phase transitions Open model-specific test: existing restoration/response-slope criteria Select a complete law, control parameter and order parameter; distinguish bifurcation, finite crossover and thermodynamic transition.
Kibble–Zurek Prerequisites missing: phase, memory and winding readers may support a future admitted model Establish critical relaxation/correlation scaling, quench protocol, order-parameter manifold and defects first.
Turing / reaction–diffusion Conditional mechanisms and obstructions reusable: pure diffusion versus the same multichannel capacity closure A justified local reaction/feedback law and a spatial-mode instability test; nonuniform appearance alone is insufficient.
Critical phenomena No identified physical critical observable: field scope and spectral relaxation remain useful controls Physical connected correlations, ensemble/dynamics, finite-size protocol and uncertainties; no critical exponent follows from the name xi_C.
Vortices / topological defects Conditional discrete winding evidence: retention and branch loss Spatial/cell realization, interpolation, physical order parameter and defect core; graph winding alone is not a physical vortex.
Solitons / localized coherent structures Deferred candidate: conditional identity and recovery A justified supporting law and the claimed propagation/stability or collision property; persistence alone does not identify a soliton.
BEC / superfluidity Deferred physical bridge: classical phase observations are available Quantum state/statistics, observables and constitutive bridge; classical phase order does not establish condensation or superfluidity.
Renormalization / universality Exact restricted coarse reductions and memory: scale bridge, elimination Declared scale/time/field transformations and effective-model flow/fixed points; REMESH and nesting are not automatically RG.
Emergent geometry Conditional inherited geometry: the same scale and observation owners Demonstrate realizability and retained dynamics; a tetrad readout does not derive a fundamental physical geometry.
Time-dependent directional response Exact tangent representation and controlled finite kicks: moving-pulse oscillator/reversal identities, physical comparisons Transient, steady-state, work and power responses differ. Physical preparation, observation, clock and uncertainty remain unadmitted.
Fundamental particle properties / quantum theory Collective-property admission: compare the mechanisms and missing observations below No particle or quantum identification; a bounded symmetry/interaction comparison is distinct from deriving a complete physical theory. Gravity, cosmology and substrate-origin campaigns remain deferred.

Collective properties and fundamental phenomena

The target is a generative explanation, not a primitive-variable dictionary: joint nodal dynamics -> organized pattern -> interaction signature -> physical observation. A combination of form, phase, capacity, support and internal motion can have properties absent from any individual coordinate. Whether those properties explain charge, spin or another observed phenomenon is a further question. The ontology contract and the sole execution plan govern that distinction. The following are reusable mechanisms and tests, not established correspondences or independent research queues.

The oscillator preparation/readout admission illustrates this requirement: documented support and repeated scalar recordings still need a justified state reduction, complete law and measurement model before they can discriminate the TNFR candidates.

Physical question Actual TNFR mechanism available for reuse Discriminating obligation
Can one substrate generate distinct material families? Collective organization and scale hypotheses, with conditional pattern and reduction results Derive distinguishable dynamics, composition and measured responses. A shared substrate or scale change alone does not identify elements or derive the substrate's origin.
Can an organized excitation carry a conserved, transferable property? Weighted-form balance and boundary currents, under fixed support, positive held capacities and the specified unforced sine law Apply the charge and magnetic-response boundary: conservation, sign and additivity do not establish electric charge or electromagnetic coupling.
Can topology distinguish persistent identities and their responses? Conditional winding sectors, formation and retention, and interface reflection controls A cycle integer is not generally conserved across phase slips. Determine whether a retained probe distinguishes the proposed identity, rather than merely reading a supplied oriented cycle.
Can internal motion produce spin-like observable behavior? Global unordered pair state, exact internal phase action and retained constituent motion Derive physical spatial rotation, its observable action and outcome statistics. Circular phase symmetry and exchanging constituent labels do not supply these.
Can pulse, storage and response account for energy or inertia? Same-law joint storage, regional work and the nonlinear storage representation; conditional maintained internal pulses Derive the observation of motion and force, clock and energy units. A storage invariant, oscillation frequency or graph kinetic metric alone supplies neither mass nor an energy-frequency quantum relation.
Can collective interaction produce measurable nonadditive work or energy? Distinct-neighbor response and full-minus-tangent storage/loss excess under the same supplied law Compare an independently observed interaction residual with matched lower-order alternatives; the classical measurement below illustrates an observable, not an admitted TNFR correspondence or selected research target.
Can mediated interactions acquire an effective field description? Causal hidden-state elimination retaining memory, initial state and forcing Establish the effective field's dynamics and response, including its observation geometry. Directional response and gauge diagnostics retain the magnetic-identification boundary.
Can collective patterns exhibit quantum properties? Phase, internal motion, interference and memory are possible ingredients under their declared laws Meet the quantum-emergence obligations, including transformations and detector statistics. These ingredients do not by themselves derive quantum mechanics.

Nonadditivity has a measurable classical counterpart. Brunner et al. measured a three-body contribution among optically trapped charged colloids, subtracting separately measured pair potentials from the total interaction. This motivates a physical test of inherited collective interaction. The present TNFR excess is a transient storage/loss contrast against its complete tangent model; it is not the experiment's equilibrium effective potential or an identified potential of mean force. A comparison needs an independent observation map, physical units, environmental and thermal dynamics, preparation, input and clock models, and propagated errors. Calibration must be separate from reserved predictions and equally informed alternatives. The storage-observer admission prepares that mathematical observable; it does not establish experimental attainability, quantum behavior or a new fundamental force. The generative objective remains explaining formation, identity and interaction from justified nodal dynamics, not reproducing this apparatus. This illustrative comparison remains governed by the sole execution plan and activates no experimental campaign.

Geometry can guide this comparison. In fundamental physics the relevant geometry often concerns states and their transformations, not a spatial polyhedron. TNFR already represents circular phase and its common-rotation symmetry. Superconducting phase differences and closed-loop constraints are a useful physical comparator, but their electromagnetic coupling, units and quantum state are additional obligations; see Feynman's superconductivity discussion. The geometry of spin-one-half transformations instead concerns two complex amplitudes and their response to spatial rotations; see Feynman's spin derivation. The current TNFR phase action does not supply that rotation/measurement model.

The balanced-neighborhood admission makes the internal issue concrete: a zero sum of phase vectors does not by itself determine every admissible phase current. Requiring cancellation for all such geometries adds the information restriction that selects sine within the stated kernel class. A geometric resemblance is therefore a way to pose a discriminating question, not a substitute for its complete dynamics.

Three exact boundaries already sharpen this comparison:

  1. The conserved weighted form changes under a common form-origin shift, while the declared relative dynamics do not. If the physical observation discards that origin, the uncentered total cannot alone define its intrinsic charge. The common-origin owner supplies the distinction; it does not exclude all future charge mechanisms.
  2. In the global pair coordinates, common internal phase rotation acts as (X,Z,P,U,W) -> (X,exp(i alpha)Z,exp(2i alpha)P,U,exp(i alpha)W). Every coordinate returns at alpha=2*pi. This is not an action of physical spatial rotations; the integer weights are not spin quantum numbers. Square-root branch exchange reconstructs the same unordered pair, not an antisymmetric quantum amplitude. The auxiliary U(2) construction has its own premises and does not repair that missing derivation.
  3. Reflection can reverse a donor's winding while fixing its sole contact and the entire environment. Equivariance and uniqueness then give identical external responses for the matched preparations. A second labeled contact can break this indistinguishability, but its geometry is an explicit input. The finite two-contact sine response makes that distinction constructive: opposite acute C5 twists with equal storage give separated ordered probe responses while retaining their winding through the specified readout. The contacts and backreaction are retained. This is an orientation-sensitive interaction, not intrinsic charge.

Charge and magnetic response remain physical identifications to establish. The relative-inventory construction removes the common-form-origin dependence above and inherits the same boundary current under its fixed-support, positive-held-capacity, unforced sine law. It remains a continuous excess relative to a fixed whole support. Its zero total follows from centering; it is not evidence of physical neutrality or charge quantization. Equal inventory also need not give equal subsequent interaction. The balance is a reusable mechanism, not an electromagnetic law.

The two-contact response already establishes orientation-sensitive interaction in a specified finite experiment with evolving source and probes. The moving-pulse work response supplies a different finite-delay contrast about an internally moving pattern. Neither overrides the equilibrium same-type reciprocity result. Their preparations, probes and observables differ; an ordered transient or finite-delay work contrast is not a winding-odd Hall susceptibility. The magnetic-texture admission records the missing spatial-motion, drive and measurement bridge.

The auxiliary gauge readout is a vertex-derived connection with identity U(1) cycle holonomy on its admitted domain. Individual wrapped differences may sum to a multiple of 2*pi; this does not supply independent magnetic flux or curvature. Likewise, the chirality contraction is a diagnostic, not a derived magnetic moment. Names assigned to diagnostic charges do not establish physical sources or conserved electric charge.

A proposed electromagnetic realization must connect a pattern's property, its transport and its interaction under compatible complete laws, with independently justified spatial orientation/motion, observation, clock and units. It must predict the force or field behavior of the claimed regime; see the electromagnetic force and field relations. Supplying those physical laws as extra equations is a constitutive model, not their emergence from TNFR. These obligations refine the common research target without opening a separate charge or magnetism campaign.

Quantum emergence remains a hypothesis. A common relational substrate could be proposed as an explanation of quantum properties, but no such derivation is established here. Phase, interference, persistent patterns or hidden memory alone do not supply quantum detector statistics. A claimed realization must derive the relevant preparation, interaction with a detector and outcome probabilities, including the Born rule where claimed; see Feynman's comparison of wave and quantum interference. Spin requires the rotation and measurement obligations above. Correlations between separated measurements must also respect the restrictions exposed by Bell experiments: a locally causal hidden-state model with measurement-setting independence obeys Bell bounds. A shared history or rhythm does not evade those premises. Graph adjacency is not an established physical causal structure. A proposed TNFR realization must state which assumptions it satisfies; adding a desired quantum probability rule does not derive it. These are future acceptance obligations, not a quantum implementation or an additional research queue.

Independent physical comparisons. These primary sources identify concrete phenomena and measurement obligations; they do not validate TNFR:

  • Saminadayar et al., fractional quasiparticle charge measure charge through tunneling noise in a fractional quantum Hall system. This demonstrates a measurable collective excitation property in an already material, quantum system. It motivates testing transport and measurement together, without deriving electrons or quantum dynamics from TNFR.
  • Jiang et al., skyrmion Hall response observe transverse motion of magnetic textures under applied current. This provides an example of geometric organization connected to a measured response. Its magnetic substrate and driving remain physical inputs; a graph winding or persistent TNFR region alone does not reproduce that law.
  • Fan et al., single-electron magnetic moment supplies a stringent, dimensionless response benchmark. A proposed electron would need mutually compatible charge, spin and magnetic-response predictions, not a tuned value in one channel. The spin-one-half rotation reference explains why a relative spinor phase in an interference/measurement model is different from a classical coordinate branch or an unobservable global sign.

The useful near-term comparison is pattern identity plus interaction response. The first two examples concern collective excitations of existing matter, whereas the third concerns an elementary particle; this difference in explanatory depth must survive any future successful comparison. Published results are known constraints, not unseen validation data. No raw response archive from these studies has been admitted here, no electron constants have been fitted, and no additional force or quantum measurement rule is installed.

Selected comparison: directional motion of a magnetic texture

Admission result: physical_bridge=not_admitted, evaluation=not_tested. The Jiang experiment and methods provide an ordinary-laboratory comparison of collective identity and response: optical MOKE imaging measures planar texture motion under electrical pulses. Its topological sign belongs to a two-dimensional magnetization texture, an S2-valued field. Its current-driven motion and magnetic preparation are independent experimental inputs. Pinning and boundary effects limit a simple rigid-texture description. These observations are published prior knowledge.

For a declared interval and nonzero longitudinal displacement, a candidate dimensionless observation is R=Delta_y/Delta_x. A matched, approximately linear drift regime motivates three distinct controls:

Operation Comparison obligation
Reverse texture polarity at fixed drive and axes Test the change of transverse response relative to longitudinal response
Reverse drive at fixed texture and axes Test whether both displacement components reverse, preserving their ratio in the claimed regime
Reverse a reporting axis only Track the conventional sign change without claiming a different physical state

This is an admission design, not an assumption that arbitrary experimental regimes satisfy those parities. The publisher's supplementary listing contains methods and movies, but the published movie descriptions concern different drive regimes. They are not a verified matched reversal holdout. No independently indexed raw trajectory archive or calibration/evaluation split was located in this bounded review; movies and response arrays were not opened. No claim of exhaustive dataset absence is made.

The missing TNFR bridge is specific. The present two-contact result observes x_A-x_B on supplied fixed support. It has no admitted observation of a translating spatial center, magnetization texture or physical drive. S1 cycle winding does not identify the S2 texture degree. Joint TNFR variables could in principle support a more complete collective observation, but that map must follow their dynamics and preparation rather than fitting the already published Hall angle. Importing a magnetic/Thiele equation would supply the missing dynamics instead of deriving it.

There is also an exact dynamical screen, before any data fitting. The same-type port reciprocity result shows that equilibrium linear form-input/work-output response is symmetric and unchanged by reversing a critical twist. The signed two-contact transient therefore cannot be reinterpreted as a winding-odd linear Hall susceptibility. An antisymmetric mixed form/phase response is insufficient as well: an ordinary canonical oscillator already supplies it.

Retained internal motion supplies a scoped mathematical distinction. The moving doubled-C5 pulse has a certified nonzero finite-delay response contrast between two distributed form probes with matched work outputs. Its varying symmetric stiffnesses need not commute over time. Static reciprocity, winding and motion reversal, and member/port relabeling remain separate controls. That result concerns infinitesimal probes about a moving preparation. The finite-kick continuation provides a sufficient positive probe radius, but does not resolve that physical admission. It is not identified magnetism. The execution plan determines whether this comparison is reopened after its information, complete-law and measurement dependencies are admitted. No large data download or magnetic-law fit is justified by the present comparison.

Moving-pattern response: precise physical synergies

The closest admitted mathematical comparison is with time-dependent oscillator response, a mechanism used in physics of wave transport. This does not identify a particle; it makes the collective response amenable to established symmetry, energy and observation tests. The exact representation and reversal proof own the TNFR derivation. The following cards identify reuse and missing premises; the execution plan remains the sole queue.

Time-varying oscillator card.

  1. Phenomenon. Source/receiver interchange can change vibration response in a modulated system. Wu and Yousefzadeh, Eqs. (1)-(7) study two oscillators with prescribed stiffness modulation, damping and harmonic forcing. Their response comparison includes phase; unequal transmitted signals need not mean unequal time-averaged output norms.
  2. Reference equation. Their model has the form r''+2*zeta*r'+K(t)*r=f(t), with symmetric K and phase-shifted modulation of its diagonal entries. It is a driven model, not a proposed TNFR law.
  3. Observables. Their exchanged source/receiver responses are steady quasiperiodic displacements, including sidebands, phases and norms. TNFR's present observation is instead a finite-delay centered work difference after initial form kicks. These are distinct experiments.
  4. TNFR dictionary. The collective tangent coordinates satisfy y''+S(tau)*y=0, with y=D^(-1/2)q and symmetric S=D^(1/2)*C(tau)*D^(1/2). Its variation is inherited from the autonomous internal pulse. No primitive variable is identified with a sensor or mass.
  5. Reduction. This is an exact invertible transformation of the admitted tangent block. The actual work kernel is H=20*D^(1/2)*G'(tau)*D^(1/2), where G(0)=0,G'(0)=I. It therefore compares initial-velocity to velocity, not force to displacement or a steady scattering matrix. Finite kicks retain the full fine nonlinear law and have a separate error theorem.
  6. Prediction and decision. The fixed pulse produces a finite centered contrast with a positive bound; the frozen-stiffness linear control does not. This is a mathematical compatibility result, with no physical discriminating prediction established. A physical comparison must admit the collective preparation, matched readout, clock and noise, and retain the source of modulation energy. Do not import the external pump and then describe the resulting dynamics as autonomous TNFR emergence.

Reversibility card. A forward-window directional contrast is compatible with a reversible complete law. Brandner, Saito and Seifert, Eq. (42) relate thermodynamic response coefficients after reversing the driving protocol and magnetic field; their stochastic, cycle-averaged setting permits unequal off-diagonal coefficients for a fixed asymmetric protocol. Its Onsager-Casimir coefficients are not the deterministic TNFR work kernel. Similarly, Wapenaar, sections 7, 10 and 11 derives endpoint-swapped reciprocity and propagator relations for time-dependent wave equations. Their transferable method is to compare the complete two-time propagator and its bilinear invariant, retaining endpoint and field parities. TNFR supplies its own exact M_R=R*M^-1*R identity; the reversed history starts at the reversed original endpoint. There is no thermal ensemble, temperature or measured magnetic field identified here. The empirical obligation is a declared reversal operation and measured response under both preparations; an asymmetric matrix entry cannot itself be scored as microscopic time-reversal violation.

Effective geometric-force card, conditional and deferred. Berry and Robbins derive geometric reaction forces from an explicitly supplied classical slow-fast Hamiltonian, including a magnetic-like antisymmetric contribution. Their example contains a classical spin rather than deriving one. TNFR's retained hidden-state/memory results suggest examining such reduction methods only when a justified time-scale separation and collective observation demand them. A candidate must specify slow and fast state, preparation, clock, controlled elimination error and the resulting measured force response. Neither that reduction nor a Berry curvature is supplied by the present finite-delay commutator. In fact the existing stiffness depends on one regular coordinate delta and retraces its path over a libration; its adiabatic stiffness-mode holonomy is trivial. Other retained-state or nonadiabatic constructions have separate obligations. The term "geometric magnetism" therefore labels the external comparison, not a TNFR discovery of magnetic charge or spin.

The immediate empirical limitation is quantitative as well as conceptual: the sufficient finite kick has a raw four-reading lower margin of about 1.15e-15 in the declared structural units. There is no admitted conversion to a laboratory signal or noise floor. A positive mathematical margin does not select a practical experimental regime. Published responses above are known constraints; no raw dataset, holdout prediction or physical fit has been admitted in this comparison.

Mechanical temporal-interface candidate: data exist, direct bridge excluded

Decision: physical_bridge=not_admitted, evaluation=not_tested. Wang et al., temporal refraction and reflection in elastic beams provide the strongest mechanical candidate inspected in this bounded intake. Their Eq. (1) uses prescribed scalar bending stiffness D(t) and fixed mass density. Thirty controlled piezoelectric patches modulate the medium; a separate actuator launches three-cycle bursts, and laser Doppler vibrometry measures transverse velocity. The mechanical subsystem exchanges energy with the shunting circuit. Its externally controlled preparation is not the autonomous pulse plus matched initial kicks of the current TNFR protocol.

Data status. The public record lists a 23.8 MB Source Data workbook. The successfully inspected article XML identifies 41467_2025_64530_MOESM3_ESM.xlsx. Retrieval did not complete; no workbook sheets or response cells were read, no local archive/hash was created, and no calibration/evaluation split was admitted. This is verified published availability, not absence of public data.

The model obstruction precedes fitting. Under the ideal fixed-mass, fixed two-mode reduction of uniform bending stiffness, K(t)=D(t)*K_0. The clock/basis-independent discriminator excludes its equality with the current varying TNFR pulse stiffness: the determinant-versus-trace quadratic coefficient is at most one quarter for any symmetric affine one-control family, whereas the selected TNFR geometry requires (1160+480*sqrt(5))/1089. Changing the time scale or fitting an arbitrary temporal D cannot remove that distinction. This is an exact restriction on the declared families, not a fitted discrepancy in the data.

The source also describes frequency-dependent effective stiffness and coupled electromechanical numerical equations; its cited supplementary approximations were not inspected. Therefore the obstruction is not a claim about every circuit-coupled beam model, projection or additional mode. Prescribed stiffness switching and the published input/output roles also do not supply the current TNFR preparation or work kernel. The unverified noise/readout bridge remains open independently of the structural exclusion.

A related microwave time-interface experiment announces data and code at PURR. Its optically switched circuit is another externally controlled comparison; no repository files or response arrays were inspected, and it was not selected as a fallback physical fit.

Composed mechanisms and the constituent hypothesis

The negative direct comparison does not exclude properties emerging from a combination of TNFR mechanisms. A scoped composition can retain geometric identity, internal motion, storage/boundary work and interface response under one complete law. The present stiffness discriminator specifies a property of that combined motion which is absent from a simpler scalar control. It does not add a primitive variable or identify a matter constituent.

For a proposed physical property, declare the joint state and interactions, derive its symmetry and composition rules, and test an independently observable response. A conserved quantity, a winding integer and a directional response cannot be combined into electric charge or spin by naming them so; the physical transformation and measurement laws must agree as well. Retained circuit or material degrees of freedom may supply relevant comparisons, but their external laws remain comparators until a justified TNFR reduction has been established. The plan determines whether a specific comparison is needed by an admitted information/law claim; this paragraph does not start multiple constituent campaigns.

Interaction and musical analogies: questions, not identifications

Particle collisions motivate an input-pattern to output-pattern question: which identities survive, transform, split or appear, with what measured response and balances? Collider experiments actually reconstruct particle products, momenta and energies through detectors; see the CMS detector description. TNFR currently has no physical collision energy, particle species, cross sections or quantum outcome rule identified by its contact study. Its prepared contact and formation boundary can inform the mathematical interaction question without being called a particle-collision explanation. This analogy opens no collider-data campaign and does not change the workstation/ordinary-laboratory empirical scope.

Music suggests complementary observations: frequency relationships, beating, phase locking, rhythm and the response to a perturbation. Acoustic structure and perceived consonance require separate explanations. Experimental work distinguishes harmonicity from beating in consonance perception, while cross-cultural measurements show that consonance preferences are not uniform across populations. An appealing frequency ratio or musical pattern therefore does not establish one universal structural preference or a shared particle-generating law.

For TNFR, reuse the existing modal, derived-phase and synchronization owners before making an acoustic correspondence. A mode of a supplied Laplacian, a damped form oscillation and a maintained audible tone have different dynamics and measurement requirements. A future comparison needs a declared output signal, clock and prediction; sonifying telemetry does not validate the dynamics. These are supporting prompts for the sole research queue, not two new experimental programmes or a reason to select pleasing results.

Binding: a stable relation is not the origin of interaction

Three laboratory examples distinguish formation of a bound configuration from creation of a previously nonexistent fundamental interaction. They supply mechanism comparisons, not admitted TNFR reductions or additional task queues.

Observed phenomenon Mechanism and supplied conditions Useful TNFR question
Single NaCs molecule assembled from two trapped atoms Two separately trapped atoms are brought together; optical excitation produces an excited molecular state. Atomic structure, electromagnetic interaction and laser preparation are inputs. Can the declared dynamics form a joint state, and what interaction and preparation make that transition possible? Phase agreement alone is not a molecular binding law.
Optical binding of dielectric microspheres An applied optical field induces separation-dependent forces with alternating sign and experimentally observed bound configurations. The illuminating field and particles already exist. Can a retained mediator produce a restoring relative geometry, including configurations that do not bind? A stable relative configuration need not be a newly created primitive edge.
Orbiting pairs of walking droplets A vertically driven liquid bath mediates wave interactions. Spatial wave damping and adaptation of impact phase affect orbital stability. The bath, droplets and driving are supplied. Can phase, internal deformation and retained history predict capture or loss of a collective state, with the driving and dissipation accounted for?

The droplet mechanism is particularly close to the question about rhythm and geometry: controlled changes of impact phase also rearrange bound droplet lattices. This is evidence for the reported fluid system, not for a primitive TNFR phase or the emergence of matter. Its external vibration must not be silently identified with an internally derived pulse.

For these comparisons, keep four obligations separate: the channel that lets components influence each other; an admitted joint state; a dynamical route into it; and its response to perturbations or separation. A favorable final state alone supplies neither a capture trajectory nor its timing. Quantum molecular binding, driven optical organization and dissipative fluid orbits also need not share the same energy function or constitute one mechanism.

TNFR's positive form/phase storage is not an identified chemical binding energy. Adding a negative edge reward to make attraction occur would be a new constitutive premise, not a derivation from these experiments. The connection-mechanism audit instead reuses native memory, support work and joint reset accounting to separate effective interaction, reinforcement and primitive support birth. Explaining known binding as a later consequence of a more primitive theory is a legitimate research objective; these analogies do not establish that such a theory exists or that its microscopic mechanisms must be identical.

Magnetic binding: orientation is a test, not an identification

Magnetic interactions can organize matter before permanent contact. In field-induced colloidal assembly, dipolar attraction organizes coated cobalt particles into chains; a separate process fixes those chains permanently. Experiments on magnetic Janus rods also show how particle shape and permanent dipoles affect the assembled geometry. These results concern existing matter and magnetic interactions, not their origin.

A magnetic dipole comparison requires an orientation-sensitive observable and response. The dipole energy and torque depend on the dipole moment and the local magnetic field; a scalar attractive response or phase agreement alone does not identify that mechanism. A physical bridge must independently specify the spatial orientation, field/response, units and applicable regime. Magnetic assembly is also not a general account of chemical bonding. No magnetic force or dipole law is imported into TNFR by this comparison.

The charge and magnetic-response boundary centralizes the available TNFR mechanisms and their limits. Two-contact orientation responses and moving-pulse work contrasts are established under their respective hypotheses; the single-contact reflection obstruction retains its own scope. The remaining physical question is whether an independently admitted preparation and observation connect a derived interaction to a specified magnetic regime. Neither these responses nor the assembly examples establish magnetic binding or a pre-material origin from TNFR.

Vibrational organization and effective binding

A dynamically maintained relation can be an effective bond; it need not be a permanent material connector. The relevant distinction is between response to a shared drive and interaction mediated by the driven environment.

In ordinary Chladni experiments, grains reveal a vibrating plate's or membrane's mode geometry by accumulating near vibration nodes. A measured space-dependent diffusion model explains accumulation through lower grain mobility in those regions. This is a tested mechanism in the reported regime, not a universal law for all particle sizes, media and excitation amplitudes. The resulting pattern alone does not establish mutual binding between grains.

There are also actual wave-mediated bonds. In acoustically levitated lock-and-key grains, secondary sound scattering creates shape-dependent attractive interactions and selective assembly. A separate experiment measures interparticle scattering forces and collective deformation. Thus common vibrational organization and binding can coexist in one physical system, while requiring distinct causal evidence. These experiments supply matter, an acoustic medium and external driving; they do not derive them.

For TNFR, reuse the modal scope below and the derived mediator memory. The useful control holds the external preparation fixed and perturbs one region: does the other's response depend on that perturbation through the mediator, and is a declared relative configuration restored? Shared motion alone does not establish that dependence; dependence alone does not prove capture or binding. Current supplied-support mediation establishes an interaction channel, not spatial acoustic forces or spontaneous support. This comparison informs the existing work package without importing a sound wave equation or reopening a separate empirical campaign.

Flocking: maintained organization with changing neighbours

Starling flocks provide a comparison for collective coherence maintained by active responses. A three-dimensional field study inferred an interaction neighbourhood of roughly six to seven nearest birds, across the observed flock densities. This is a domain-specific empirical finding, not a universal neighbour count or a TNFR support law. Measurements of collective turns found propagating direction changes with little attenuation and motivated a model including behavioural inertia, rather than purely diffusive information transport.

The reusable question is how local interaction, response time and changing neighbourhoods maintain a collective organization through deformation. Coherence need not require a rigid outline or permanently identical edges. The birds supply propulsion and sensory responses; a shared pattern does not identify magnetic or acoustic binding, nor does it equate flight heading or wingbeat with the primitive TNFR phase. A TNFR reduction would have to justify its state, support updates and propagation law and then predict a reserved collective response. The structural analogy is a supporting comparison, not evidence that these systems share one microscopic law.

First card: derived and supplied phase dynamics

The reference comparison is a declared Kuramoto-type law theta_dot_i=omega_i+sum_j K_ij*sin(theta_j-theta_i). Coupling normalization, frequency distribution, support and finite/continuum limit are part of that model. Transition type is not universal: the cited frequency-dependent Kuramoto study derives and numerically studies different transitions for specified choices. It is not experimental validation of TNFR.

The existing coupled directed triangles supply a stronger starting point than assigning a sine law. With equally oriented unit triangles, reciprocal unit links between corresponding vertices, common fixed capacity nu>0 and pure EPI diffusion, the exact reduction gives, for positive contrast amplitudes,

\[ \dot\psi_a=-\frac{\sqrt3\nu}{4} +\frac\nu2\frac{r_b}{r_a}\sin(\psi_b-\psi_a). \]

Equal amplitudes form a restricted invariant family with delta_dot=-nu*sin(delta). This is a Kuramoto-type two-phase reduction in a rotating frame, with coefficients inherited from the supplied fine graph. General angular prediction needs the amplitude ratio; absolute form needs amplitudes and means, and zero contrast needs Cartesian continuation. Total contrast strictly decays. This provides an emergent observed interaction, not sustained oscillators, a many-body threshold or primitive-phase identity.

Mechanism Actual role and reusable owner Correspondence verdict
Derived form phase Regional contrasts observed from the fine diffusion generator; existing controls Exact conditional amplitude/phase reduction; the two observed phases close without amplitude information only on restricted invariant preparations.
Supplied sine evolution propose_u3_gated_phase_step; P2 locking, K3 reduction Capacity-as-angular-rate, gain, admitted-neighbor averaging and U3 gate are supplied premises; not unrestricted Kuramoto.
Native phase coordination Per-call circular-mean relaxation Different execution law; it has no supplied physical dt or free angular advance.
Conditional joint form/phase exchange Native pulse scope and precontact locking Geometry-dependent damped modes and a reversible periodic boundary are properties of the stated law. Matching rhythms does not derive an edge, and a modal graph-wave spectrum is not an observed nodal pulse.
Phase contribution to EPI pressure Arg of a neighbor resultant, through configured pressure Not the sine phase row or an electrical current. The existing Arg/current discriminator must not be repeated as a new discovery.
Named UM/RA events and cycle readouts Operator contracts and winding certificates U3 compatibility is admission; RA primarily blends EPI. A name or gate does not select an oscillator law.

The observed quantities are phase separation, amplitude ratio and full contrast budget, not automatically C or the tetrad. Physical instrument mapping is open. The existing phase-only, radial-response and branch-loss witnesses are negative controls; the source-matching owner defines the full-state obligation before feeding observed phase into pressure. Exact agreement on the restricted reduction is an equivalence result, not an exclusive physical prediction. Further contrasts require a stated new premise and prospective response; this card does not authorize a large sweep.

The completed unequal-capacity extension now provides that bounded contrast. For fixed positive capacity within each triangle, generic contrast directions approach a predicted capacity-dependent lag bounded by atan(sqrt(3)/5); an exceptional fast eigenline is retained. All amplitudes still decay. Starting with equal amplitudes does not justify holding them equal: at relative phase pi/2, the full relative-phase acceleration is -5*(nu_0-nu_1)^2/8, while the frozen-ratio approximation predicts zero despite matching the initial relative angular velocity. The retained finite control separates these predictions without fitting a phase gain. This closes the first mathematical card. The conditional measurement contract now specifies the spatial observation and uncertainty test, but admits none of the existing public sources for this directed law. The subsequent six-study review also admits none; the exact-model physical test is parked. MOSTR is a deferred three-tank transport lead, not evidence of this phase mechanism. The primary collective-emergence objective does not require one-to-one sensor mappings for primitive EPI. Reciprocal scalar RC/thermal dynamics cannot be identified with its complex contrast spectrum. The independent physical bridge remains open. A synchronization threshold or KZ interpretation would be unsupported for this particular family.

Corrections that govern later cards

Josephson: periodicity and exchange antisymmetry allow more than one harmonic; they do not single out sine. The cited junction review specifically discusses departures from sinusoidal current–phase behavior; a primary calculation retains first and second harmonics. A physical card must address voltage–phase evolution as well as current–phase behavior and independent electrical units. Even the measured Josephson voltage/frequency relation is additional physics, not implied by a graph sine term. Josephson is a separate comparison, not a prerequisite for Kibble–Zurek.

Correlation and relaxation: current xi_C is an uncentered static coherence-product fit with a separately tagged spectral fallback. Neither branch is automatically a connected physical correlation length. Keep that diagnostic's definition unchanged; a critical observable needs its own admitted observation map. Likewise configured C is not automatically an order parameter. Even common-capacity diffusion has modal relaxation times tau_k=1/(nu_f*lambda_k) for positive modes, not simply 1/nu_f. The relevant full generator/Jacobian and observation determine relaxation in other models. A finite plot or fitted power law does not establish universality; the critical-phenomena reference provides physical context, not a TNFR bridge.

Coupled form-phase relaxation and underdamping: The relational spectral owner already derives the consensus tangent of the opt-in two-channel law. For common positive capacity its mode block is nu_f*lambda_k*[[-e,-w/pi],[w/(beta*pi),0]]; P2 has lambda_k=2. For e>0, its nonzero-mode poles are negative real, repeated, or a damped complex pair according to the sign of e^2-4*w^2/(beta*pi^2). At e=0 the nonzero poles are purely imaginary, not damped. These statements concern the local derivative of the conditional phase law, not the full operator runtime.

The coefficient audit connects this classification to units, capacity composition and chi=beta*(e/w)^2. The numeric consensus boundary 4/pi^2 uses the chosen phase-source normalization; it is not a universal physical constant. A uniform cycle twist instead has boundary chi*cos(kappa)=4/pi^2. General phase geometries require their actual metric and Hessian. The same default coefficients give real poles at consensus and complex poles on winding-one C5.

Pole class is not a monotonicity test on a measured signal. Even the overdamped P2 tangent can overshoot, and mixtures of pure-diffusion modes can be nonmonotone at one node. A reversible scalar-diffusion model does have only real poles, but a physical comparison needs the actual observation and an equally informed alternative. Damped modes alone neither select TNFR nor establish a physical phase transition.

Realizability and calibration (P1-P3). No independent physical bridge for this candidate is admitted here. Similarity to a two-dimensional oscillator does not establish the same nonlinear law, spatial structure or instrument dictionary; not every amplitude-phase system has this spectrum. Conversely, no absence theorem for physical realizations or quantitative tests has been proved. The atlas permits either exact reduction or a controlled approximation with a declared error budget.

The auxiliary digital-PLL admission examines one published terrestrial realization with explicit filtering and delay. It is not admitted to the fixed sine inference: the current-state history obstruction has a stated preparation/mapping scope, and a different observation or controlled reduction still needs its own error evidence. This comparison neither excludes all oscillator realizations nor makes a direct circuit realization a prerequisite for generative pattern research.

The same-mode pole identity can identify chi when both rates are observable and excited; a single decay or unresolved modal mixture need not suffice. The calibration contract permits fitting identifiable combinations on separate data. Freeze that model, observation, clock and uncertainties before evaluating reserved responses. Estimating a coefficient is neither deriving its universal value nor evidence against the possibility of later quantitative testing. The missing bridge remains an admission obligation, not a declaration that the paradigm is exhausted.

Discrete winding: on a declared cycle, the wrapped edge sum divided by 2*pi is invariant along continuous trajectories that avoid antipodal edges. A branch crossing can change it with fixed support and well-defined node phases. The existing C8 counterexample already demonstrates this. Physical vortex charge needs an admitted spatial order parameter and interpolation; net winding is not a count of all defects. Retained winding, autonomous defect formation and NFR identity are separate.

Turing: in pure EPI diffusion xdot=-diag(nu(x))*L_rw*x, positive smooth state-dependent mobility on fixed reciprocal nonnegative support does not destabilize uniform form. At x=a*1, its derivative multiplies L_rw*x=0, so the Jacobian is -diag(nu(a*1))*L_rw. Reuse the diffusion obstruction. The existing multichannel nu=g(x) model additionally places capacity in pressure and can have a destabilizing combined slope; it is a different constitutive class, already classified in capacity closure. A Turing claim requires a stable local homogeneous system and a nonuniform mode destabilized by transport, not merely contrast growth or threshold crossing. Turing's original model supplies coupled reaction and diffusion laws. Any new TNFR closure must justify its counterpart before searching for patterns.

Kibble–Zurek: first establish the model's transition, order-parameter manifold, spatial correlation and relaxation laws. Under the conventional algebraic scaling assumptions and a linear quench, the freeze-out argument gives xi_hat proportional to tau_Q^(nu_crit/(1+z*nu_crit)); defect density additionally depends on codimension and the measurement/coarsening protocol. Here nu_crit is a critical exponent, not capacity nu_f. These are conditional reference relations from the KZ review, not TNFR predictions. Size, noise/preparation, sampling time and defect definition must be frozen; a different transition class need not obey this power law. The 2023 Ising-domain experiment studies binary structural domains, not automatically circular-phase vortices. Its public raw-data suitability remains unassessed. The cited 2026 holographic disk study is a numerical gravitational-dual model, not laboratory observations; it is excluded from this project's experimental shortlist.

Scale and stronger interpretations: an RG card needs a coarse map, space/time/field rescaling, controlled closure and a demonstrated effective-law flow or fixed point. Existing memory and quotient results supply tools, not that conclusion. Soliton, BEC and superfluid cards likewise need their defining state and dynamical observables; the BEC reference does not make a classical phase reader a quantum condensate. Retain these as deferred entries under the same evidence contract.

2. Classical Adapter and Overdamped Limit

2.1 Regime Conditions

A high-coherence structural snapshot requires small aggregate pressure and EPI rate:

\[ \operatorname{mean}|\Delta\mathrm{NFR}| \ll 1, \qquad \operatorname{mean}|d\mathrm{EPI}/dt| \ll 1. \]

Under these measured assumptions the canonical read-out \(C(t)=1/(1+\operatorname{mean}|\Delta\mathrm{NFR}|+ \operatorname{mean}|d\mathrm{EPI}/dt|)\) is close to one. Phase spread and a constant capacity may be assumptions of a selected mechanical comparison, but neither quantity enters the constitutive kernel directly. A small phase gradient alone does not establish high \(C(t)\).

2.2 Observable Mapping

Model quantity TNFR comparison Scope
Position \(q\) Selected spatial component of EPI Adapter convention
Velocity \(\dot q\) Selected flow component Adapter convention
Mobility \(\nu_f\) Exact coefficient in the first-order nodal law
Inertial mass \(m\) Explicit adapter parameter No identity \(m=1/\nu_f\) follows from TNFR
Force slot \(F\) Optional \(\Delta\mathrm{NFR}\) bridge Must be declared and dimensionally specified
Potential comparison \(\Phi_s\) Structural source aggregation, not mechanical potential energy
Action comparison Optional phase accumulation Mapper does not construct this bridge

2.3 First-order law and second-order adapter

Reading an EPI coordinate as \(q\) and pressure as \(F\) gives the exact algebraic form

\[ \dot q = \nu_f F, \]

so \(\nu_f\) is a mobility. Newtonian motion instead requires an additional state and a separately chosen law,

\[ \dot q=v,\qquad \dot v=F/m. \]

The classical examples implement this second-order adapter. Their inertial mass is an adapter parameter; substituting \(\nu_f=1/m\) is an optional dimensional identification, not a consequence of the nodal equation. A damped graph wave has a restricted pure-EPI diffusion limit under its own stated hypotheses. Neither construction derives the full inertial regime from the isotropic auxiliary substrate.

2.4 Force Interpretation

Adapter mechanism Directly available telemetry Optional TNFR comparison
Externally supplied pair force \(q\), \(p\), force/acceleration and classical energy \(\Delta\mathrm{NFR}\) and \(\Phi_s\) only after an explicit scalar pressure bridge
Configured damping Adapter state and dissipation rate \(C(t)\) only after pressure and EPI-rate channels are materialized
Configured restoring force Adapter state and force Phase-gradient and curvature only from a separately constructed graph state

These rows specify possible comparisons. The mapper and N-body solver do not materialize canonical pressure, \(C(t)\) or the tetrad, and the tetrad does not generate the adapter's force law.

2.5 Integration Scheme

The Verlet/Yoshida integrators in src/tnfr/dynamics/symplectic.py preserve the symplectic structure of the declared Hamiltonian map up to numerical error. This property does not certify all TNFR structural invariants or any arbitrary operator schedule.

Workflow:

  1. Select and record the adapter, force law, units and integrator order.
  2. Export \(q\), \(p\), force/acceleration and adapter energy with numerical tolerances.
  3. If a graph comparison is required, declare how adapter state maps to scalar EPI and \(\Delta\mathrm{NFR}\), materialize \(d\mathrm{EPI}/dt\), and record its provenance before computing \(C(t)\) or the tetrad.
  4. Compare the adapter trajectory against its stated reference.

2.6 Validation

The focused tests check one-period harmonic and circular-orbit closure under explicitly supplied force evaluators. The Kepler demonstration (examples/02_physics_regimes/12_classical_mechanics_demo.py) plots a selected inverse-square central-force trajectory. These finite checks validate the declared force/integration paths at their tolerances; they do not derive gravity from the nodal equation.

3. Zero-pressure and Kinematic Comparisons

3.1 Canonical zero-pressure statement

\[ \Delta\mathrm{NFR}=0 \quad\Rightarrow\quad \partial\mathrm{EPI}/\partial t=0. \]

This gives zero instantaneous unforced EPI rate. A fixed EPI trajectory needs pressure to remain zero; changing phase, capacity or support can generate later pressure. Constant translation does not follow unless a separate kinematic adapter stores velocity and advances an external position coordinate.

A reproducible zero-pressure experiment should record the pressure and EPI-rate residuals, fixed/moving coordinate convention, operator schedule and structural telemetry. Excluding named destabilizers is insufficient: any custom pressure law or initial state can still carry nonzero pressure.

3.2 Two-train adapter

examples/02_physics_regimes/15_train_crossing_demo.py uses prescribed constant velocities:

Parameter Train A Train B
Initial position \(x=0\) km \(x=600\) km
Velocity \(+300\) km/h \(-250\) km/h

Its analytical crossing is

\[ t_c=\frac{600}{300+250}\approx1.0909\ \mathrm{h},\qquad x_c=300t_c\approx327.27\ \mathrm{km}. \]

Numerical agreement checks the kinematic adapter. It does not turn the zero-pressure fixed EPI chart into a derivation of Newton's first law.

4. Finite Spectral and Wave Correspondence

4.1 Scope

Finite symmetric graph operators possess a discrete orthonormal eigenbasis. The choice of graph and operator remains an input. Large wrapped phase gradients can be useful stress diagnostics, but they do not by themselves select a quantum model or cause quantization.

4.2 Observable Mapping

Auxiliary-model quantity TNFR comparison Boundary
Complex wave field \(\psi\) Separate from \(\Psi=K_\phi+iJ_\phi\) A state map and dynamical bridge would be required
Modal energy/frequency Eigenvalue-derived model value Not identical to nodal \(\nu_f\)
Potential \(V(x)\) Declared function compared with \(\Phi_s(x)\) Matching must be specified
Mode index \(n\) Eigenmode or winding label Depends on boundary operator
Damping/selection Explicit dissipative adapter IL/SHA labels alone do not define measurement collapse

4.3 Retained graph-mode calculation

For fixed reciprocal nonnegative conductance and common positive capacity, the pure-EPI generator has modes exp(-nu_f*lambda_k*t). The separately supplied graph wave has frequencies sqrt(lambda_k); this wave equation is not the first-order nodal law. Reuse structural_diffusion.py and its stability scope, rather than imposing an array of desired levels. Boundary conditions, initial state, clock and damping remain declared inputs.

On a unit cycle, the sampled phasor exp(2*pi*i*k*j/n) is a normalized Laplacian eigenvector with eigenvalue 1-cos(2*pi*k/n). Sampling aliases k modulo n; the winding reader separately checks the wrap branch and U3 domain. This exact graph identity does not turn the phasor into the geometric field Psi: at uniform phase the phasor is one while Psi is zero. The retained ring-mode controls exercise this distinction without claiming particle creation or quantum duality.

4.4 Validation boundary

The retired example 13 adjusted an energy array toward an imposed phase reticulum using a handwritten noisy update. It did not construct a cavity operator or derive its spectrum; it is not retained as spectral evidence.

The supplied-target quantum mapper has also been retired: its level generator ignored the declared length and its two state mappings were not inverses. There is no replacement physical quantum API. Graph spectra, winding and auxiliary wave calculations retain their own mathematical contracts; they do not establish probability amplitudes, detector statistics or physical energy levels.

5. Fourier Width and Interference Correspondence

5.1 Fourier-width product

An analytical Fourier-width bound requires specified normalization, integrable signals and a compatible second-moment definition. A finite sampled window does not automatically inherit that bound. Every reported product must name its width estimator, Fourier convention and sampling window; no fixed TNFR constant has been derived by this comparison.

5.2 Two-path interference

A two-path auxiliary wave model can compare detector intensity with wrapped phase separation. Constructive and destructive bands change phase-order and phase-gradient diagnostics. They change structural \(C(t)\) only when the model also specifies how phase affects \(\Delta\mathrm{NFR}\) or \(d\mathrm{EPI}/dt\) and the recorded channels confirm the resulting change. Phase separation is not a direct argument of the constitutive \(C(t)\) kernel.

5.3 Validation boundary

The retired example 14 combined a raw-array Fourier-width calculation with a separately forced wave-grid model. Its magnitude-weighted, truncated frequency read-out did not validate a stated analytical uncertainty bound; its wave intensity was not canonical structural coherence. No quantitative validation or canonical pressure-to-phase bridge is retained for this proposed comparison.

6. Phase and balance boundaries

Phase dispersion, structural coherence and the field energies are distinct read-outs. None is a physical temperature or thermodynamic entropy without an independently justified measurement and evolution model. The former thermal-proxy proposal had no retained validated driver and is not maintained as a research route.

6.1 Pairwise U3 compatibility is not transitive

U3 checks each active pair independently. If A and B are each within \(\Delta\phi_{\max}\) of C, their mutual separation may reach \(2\Delta\phi_{\max}\) and fail the gate. Therefore a synchronized cluster needs explicit pairwise or edgewise evidence; compatibility through a shared neighbor is insufficient.

6.2 Balance boundary

Conservation of a structural current requires a stated symmetry and model. The auxiliary symplectic substrate conserves its declared charges under its exact flow. A general engine operator sequence, Kuramoto experiment or sum of \(J_\phi\) values has no automatic conservation law; its residual must be measured along the actual trajectory.

6.3 Stochastic desynchronization boundary

Random perturbations can increase phase spread in a specified stochastic schedule, while sufficiently strong coupling can also resynchronize the same network without invoking IL or THOL. The direction of \(C(t)\) depends on the observed pressure and EPI-rate channels. No universal time arrow follows from omitting named stabilizers or closure operators.

7. Regime Transition Summary

The comparisons can be organized by their declared pressure and phase conditions. This table is a model index, not a phase diagram derived from TNFR:

Comparison \(\Delta\mathrm{NFR}\) \(\lvert\nabla\phi\rvert\) Primary telemetry Declared model
Zero-pressure chart \(=0\) Measured separately EPI rate, \(C(t)\) Fixed EPI
Classical adapter External force value; graph bridge optional Optional low-spread regime Adapter \(q,p,F\); tetrad only through a declared bridge Explicit \(F=ma\) adapter
Finite spectral model Model-dependent Model-dependent Eigenpairs and declared wave coordinates Declared boundary operator

These regimes organize several implemented TNFR read-outs and declared auxiliary models. Only reductions with explicit hypotheses, such as fixed-graph EPI diffusion under its pressure realization, follow from the stated nodal model. Classical adapters and auxiliary waves supply extra laws. Their presence in one package does not derive the corresponding physical theories.

8. Implementation Reference

Component Location
Classical mechanics mapper src/tnfr/physics/classical_mechanics.py
Graph transport and auxiliary waves src/tnfr/physics/structural_diffusion.py
Declared-cycle winding src/tnfr/physics/winding_certificates.py
Symplectic integrators src/tnfr/dynamics/symplectic.py
Structural field computation src/tnfr/physics/fields.py
Central-force demonstration examples/02_physics_regimes/12_classical_mechanics_demo.py
Two-train kinematics examples/02_physics_regimes/15_train_crossing_demo.py

Implementation & Examples

TNFR ↔ Classical Mechanics Dictionary

Originally docs/TNFR_CLASSICAL_MAPPING.md. Consolidated here as the canonical mapping reference.

Scope: This is an adapter dictionary. dynamics/nbody.py assumes the Newtonian potential, while dynamics/nbody_tnfr.py assumes a different regularized phase-coupled pair law. Neither model derives Newtonian mechanics from coherence or from the nodal equation.

TNFR Quantity Definition (TNFR) Classical Analog Notes
EPI Canonical coherent form Adapter payload for \(q\) and \(\dot q\) The payload convention does not redefine EPI generally
νf Reorganization rate (Hz_str) Mobility in the overdamped projection High νf → faster drift for fixed pressure
ΔNFR Scalar structural pressure in the engine Optional force/acceleration bridge The Newtonian solver returns acceleration but does not materialize canonical graph pressure
Φ_s Inverse-square ΔNFR accumulation Potential-like readout U6 monitors drift between declared snapshots; a well analogy adds no bound
|∇φ| Local desynchronization Optional stress comparison No mechanical stress or force identity is implied
K_φ Phase torsion read-out Curvature comparison Does not generate the adapter force
ξ_C Static product-fit length or separate spectral fallback Correlation-range comparison only on the fit branch Does not set either N-body pair law; estimator provenance is required
Ψ = K_φ + i·J_φ Complex auxiliary geometric field Phase-space comparison No Hamilton-Jacobi identity is established
Operator sequences Canonical engine transformations Work/impulse comparison No equivalence between grammar validity and mechanical admissibility is established

Structural Triad ↔ Phase Space comparison: - Form (EPI) → configuration coordinate in the overdamped projection - Frequency (νf) → mobility; inverse inertial mass is only an assignment in a separately specified second-order adapter or model - Phase (φ/θ) → oscillator phase; action-angle status requires the auxiliary Hamiltonian construction

Field Tetrad ↔ Energetics: - Φ_s → source-aggregation diagnostic; mean \(|\Delta\Phi_s| < \pi/2\) is a selected before/after policy, not a potential-energy bound - |∇φ| → local phase-stress comparison - K_φ → phase-curvature comparison; no centripetal/Coriolis force follows from it - ξ_C → fitted correlation-range comparison, with the dimensionless spectral fallback reported separately; the N-body force laws do not read it

The tetrad guide owns the fit, fallback, distance and sampling conventions. A comparison table does not override those observational dependencies or provide a force law.

Executable Demonstrations

Example Concept from this document
11_classical_limit_comparison.py Finite comparison of Newtonian and declared phase-coupled N-body adapters
12_classical_mechanics_demo.py Declared inverse-square central-force trajectory
15_train_crossing_demo.py Prescribed constant-velocity kinematic adapter

Key Source Modules

  • src/tnfr/physics/classical_mechanics.py — Explicit classical adapter and diagnostics
  • src/tnfr/physics/structural_diffusion.py — Declared diffusion generators, spectra and auxiliary waves
  • src/tnfr/physics/winding_certificates.py — Cycle, branch and winding evidence

9. References