Skip to content

TNFR Fundamental Theory

Status: Canonical reference Version source: pyproject.toml Origin: March 2026 Scope review: September 19, 2026

1. Scope

This document describes the nodal equation, canonical structural diagnostics, operator policies, and explicitly specified model correspondences. The tetrad is a canonical selection of diagnostic channels; complete state reconstruction and universal minimality are not established. Mathematical hypotheses and counterexamples are centralized in DIAGNOSTIC_AND_GRAMMAR_SCOPE.md, which governs the interpretation of grammar calibration and field thresholds here.

2. Governing Dynamics

2.1 Nodal Equation

On a continuous segment in a declared structural chart, nodal evolution satisfies

\[ \frac{\partial \mathrm{EPI}}{\partial t} = \nu_f(t) \, \Delta \mathrm{NFR}(t) \tag{1} \]

where:

Symbol Definition Units
EPI Primary Information Structure — structural configuration in a declared chart Declared form unit X
\(\nu_f\) Reorganization capacity, conventionally named structural frequency Hz_str: inverse declared structural time
\(\Delta\mathrm{NFR}\) Nodal field response — directed structural pressure X when capacity has inverse-time units

Equation (1) is a nodal identity within a complete model. A prediction also requires a pressure map and laws for every consumed phase, capacity, support, input and event, including an explicit premise when a quantity is held fixed. A derived oscillation is a trajectory of those joint laws; the identity alone does not supply an oscillator or select its frequency.

2.2 Structural Triad

The engine's structural triad distinguishes three attributes:

  1. Form (EPI): coherent structural configuration in a declared state space. A Banach-space formulation and the engine's signed scalar chart are representations whose scope must be stated; the nodal equation alone does not choose their dimension. Named operators use their jump contracts; declared solvers use the shared nodal integrator.
  2. Capacity (\(\nu_f\)): nonnegative reorganization rate relative to the declared clock; \(\nu_f=0\) suppresses continuous EPI flow, without necessarily erasing stored form. A held capacity is not automatically the frequency of a periodic trajectory.
  3. Phase (\(\phi\) or \(\theta\)): circular synchronization coordinate; coupling uses \(|\mathrm{wrap}(\phi_i-\phi_j)|\leq\Delta\phi_{\max}\). The EPI equation alone supplies no phase clock.

2.3 Integrated Form and Stability Criterion

On one fixed vector chart without jumps, integrating Eq. (1) over \([t_0,t_f]\) gives:

\[ \mathrm{EPI}(t_f) = \mathrm{EPI}(t_0) + \int_{t_0}^{t_f} \nu_f(\tau) \, \Delta\mathrm{NFR}(\tau) \, d\tau \tag{2} \]

On an infinite horizon, a sufficient condition for convergence is absolute integrability of the structural velocity:

\[ \int_{t_0}^{\infty} \|\nu_f(\tau) \, \Delta\mathrm{NFR}(\tau)\| \, d\tau < \infty \tag{3} \]

Boundedness only requires bounded partial integrals and does not imply (3) or convergence to a limit. Local integrability of a supplied time-dependent velocity defines its finite-horizon accumulated change; it does not prove existence or uniqueness for an unspecified state-dependent law. See the integrability distinctions. U2 prescribes stabilization and a two-unit prefix-debt policy; it motivates control of accumulated change but does not independently prove an infinite-horizon estimate. Its numerical calibration and the three-operation U4b window use a mean-rate surrogate, not every graph mode's decay time.

For locally finite operator events, add their actual EPI jumps to (2). Changing node support or structural chart requires an explicit identification map. The integral is a Banach-valued integral when EPI is Banach-valued; neither subtraction nor integration is defined by a bare set of forms alone.

2.4 Physical concepts, mathematical types and implementation

The research starting point is coherent form and its reorganization through the nodal law. A historical formula or a Python type is evidence of a proposed formalization, not an additional axiom that makes that formalization necessary. The joint closure review uses the following distinctions before selecting any new dynamics.

Quantity Minimum mathematical meaning What remains to be established
EPI, denoted x A structural configuration with specified equivalence of forms and a chart or vector-space realization. Which independent coordinates retain the intended structure and its dynamics; what observations identify them.
DeltaNFR, denoted p A directed structural response in the tangent space of x, or its vector-chart representative. A constitutive map from the joint nodal/network state to that response. A scalar distance alone gives no direction.
Capacity nu A nonnegative multiplier converting structural response into a rate relative to the declared time. Its independent state dependence, evolution and physical calibration. It is not automatically angular frequency.
Time t A common ordered evolution parameter with explicit units and event conventions. Its relation to laboratory time or a derived internal clock. A simulation step index is not such a derivation.
Phase theta A point on the circle with an explicitly defined relation to structural configuration. Whether it is independent, a derived coordinate or a history observable; and its evolution in that realization.
Support, conductance W and metric length Distinct structural data specifying interaction, normalized transport and distance. Which data emerge from form and which are separate state; admissible changes and their law.

A scalar value EPI=0 is not absence of the graph node. Its capacity, phase and incident support still exist in the current engine and can affect other nodes. Consequently a representation that discards phase at zero EPI needs a projectability proof. Calling zero form a "vacuum" or invoking Emission does not derive creation of the substrate or of its phase/capacity law.

If x lies in a differentiable manifold M, the type-correct statement is x_dot=nu*p, with p in T_x M. In a Banach vector chart this tangent space can be identified with the ambient Banach space. A metric space supplies distances, but not by itself the tangent vector appearing in this equation. If p denotes an operator instead of its evaluated vector, its argument and action must be specified, for example p=P(z), where z retains the relevant joint state and history. Neither interpretation determines P uniquely.

The current graph engine supports a signed real scalar EPI coordinate and its exact uniform-real BEPIElement embedding. The richer implementation stores finite complex continuous-field samples and a finite coefficient array; it is not an exact representation of every element of C^0([0,1]) direct-sum ell^2. The supplied grid need not be [0,1], and no interpolation or infinite tail is inferred. Its maximum-component scalar read-out loses information outside the uniform-real embedding. Its historical direct_sum method adds aligned components; it does not enlarge the state-space dimension. Its regularity functional is not the canonical coherence or a Banach norm. See epi.py, spaces.py and the shared chart boundary.

Scalar numerical solvers use that same finite signed reader: uniform-real BEPI is supported, while nonuniform or complex live/serialized forms are rejected before solver-owned writes instead of being projected to a magnitude. This is a representation contract, not a derivation of rich-EPI dynamics. See solver boundary controls.

Consequently neither the existence of BEPI storage nor the use of floats proves the physically necessary EPI type. Temporal spectral entropy of a scalar series likewise proves neither a vector-valued state nor equivalence to a complete spatial/modal representation. Adequacy requires a specified observation/reduction map and closed dynamics or explicit missing-state memory. The existing quotient and memory derivations provide that test; storage labels alone do not.

A faithful chart of a full structural state and a closed scalar observable are different claims. For an observation h and a specified full law z'=F(z), autonomous observable evolution requires Dh(z)F(z) to agree for all states with the same h(z). On a smooth domain where h is a submersion onto its image, this defines a smooth reduced vector field. Events additionally require h(J(z))=J_bar(h(z)); event guards and admission must also descend to the observation. Matching jump values alone does not fix when a jump occurs. An injective chart is sufficient to retain state but is not necessary for such a closed observable. Capacity, phase, support and history remain inputs unless their projected evolution also closes. These conditions test a proposed representation; they do not supply the missing full law by assuming it.

If a declared group action g satisfies h composed with g = h, full equivariance F(gz)=Dg(z)F(z) is sufficient but stronger than necessary: projection only requires Dh(gz)*(F(gz)-Dg(z)F(z))=0. A component along discarded directions can be invisible. With a declared positive clock change d tau/dt=alpha(z), apply the criterion to F/alpha. Equal tangent directions without equal rates establish at most the same unparameterized orbits.

2.5 Dimensional consistency and structural activity

Write [x]=X, [t]=T, [nu]=T^-1. The nodal product then requires [p]=X. For normalized dimensionless x, p is dimensionless and x_dot has units T^-1. This does not identify x with energy, probability or the diagnostic C(t).

The preserved original source, printed/physical pages 212-219, contains incompatible alternatives that must not be silently combined:

  • Pages 212 and 216 define DeltaNFR as a time derivative of internal structure. Page 213 instead uses a mean metric displacement between consecutive forms. The latter is nonnegative and cannot alone be a signed/tangent pressure: positive capacity would then prohibit decreasing scalar EPI, contrary to the explicit P2 diffusion control.
  • Page 218 assigns both [nu]=T^-1 and [DeltaNFR]=X/T. Their product is X/T^2, not X/T. The stated substitution lambda=DeltaNFR/nu leaves nu^2*lambda with the same wrong time dimension. It renames the expression without repairing the balance. An explicit normalization or a different constitutive convention is necessary; the source supplies no unique one.
  • Pages 212-213 define capacity by counts of reorganizations per interval. For literal locally finite discrete events, the interval count divided by interval length tends to zero away from events as the interval shrinks to the observation time, and may be singular at an event. A smooth nonzero capacity requires a declared averaged intensity, continuum limit or independent continuous definition; the count formula alone is insufficient. Potentially possible events and realized events also differ.

The engine's directed pressure convention is dimensionally usable in its normalized chart; it does not resolve the source's interpretation by fiat. Adding phase, EPI, capacity and topology gradients requires compatible units or explicitly normalized component scales. If capacities retain units T^-1, the coefficient multiplying a capacity difference must supply the missing X*T scale. Normalizing four numeric weights to sum to one does not establish that physical normalization. This remains a constitutive/metrology obligation.

One useful consequence does follow without a new force. On a continuous interval with positive capacity, define each node's accumulated activity

s_i(t) = integral nu_i(tau) d tau,     dx_i/ds_i = p_i.

Thus pressure is change of structural form per unit accumulated activity in this representation, not a second time derivative or an unsigned distance. This is a reparameterization of a supplied trajectory, not a definition of the unknown capacity law. Zero capacity makes the local clock singular; heterogeneous nodes generally do not share one removable clock. See the existing derivation. The Hz bridge is an explicit conversion convention; its default value does not independently calibrate laboratory time.

2.6 A structural coordinate must transform its pressure

For a nodewise differentiable chart y=f(x) with differentiable inverse, with the same time and capacity, the chain rule requires p_y=Df(x)*p_x. Keeping only the written form y_dot=nu*p_y does not justify reusing an unchanged coordinate formula for p. For a chart mixing several nodes with unequal positive capacities, the network formula instead is p_y=D_nu^-1 Df D_nu p_x if those same capacities are retained. The simpler pushforward requires commutation with D_nu.

On unit P2 with x=(1,2), capacity one and pure EPI diffusion, p=(1,-1). The positive-chart transformation y=x^2 requires p_y=(2,-4), whereas recomputing the same Laplacian formula at y=(1,4) gives (3,-3). More generally, preserving the pure scalar neighbor-difference formula on every pair requires f(b)-f(a)=f'(a)*(b-a), hence a common affine f. This conditional result does not make the chosen affine chart physically unique. In the full pressure p=-e L_W x+F, an affine rescaling y=ax+b also needs F_y=aF; unchanged phase/capacity source coefficients do not automatically satisfy that.

Changing capacity cannot generally repair a missing pressure transformation. On unit P3 with x=(1,2,3) and unit capacity, pure EPI pressure is (1,0,-1). The valid positive-domain chart y=x^2 requires rate (2,0,-6), whereas recomputed neighbor differences are (3,1,-5). Every positive center capacity then gives a nonzero center rate. This discrepancy is not a clock calibration. For a convex differentiable nodewise f and nonnegative difference weights, the discrepancy is the sum of Taylor remainders sum_j a_ij*(f(x_j)-f(x_i)-f'(x_i)*(x_j-x_i)) >= 0.

A valid coordinate change preserves the existing gradient-flow mechanism if its metric is transformed too. For x_dot=-M*grad_x E, a diffeomorphism with Jacobian J gives E_tilde=E composed with f^-1 and y_dot=-(J M J^T)*grad_y E_tilde. Positive semidefinite mobility and the energy dissipation identity survive. Reusing the old Euclidean mobility and Laplacian in y generally defines a different model. A finite Euler step also need not commute with a nonlinear chart; the implemented integrator is a solver in its declared scalar chart, not a coordinate-independent manifold solver.

Injectivity alone is insufficient for differential equivalence. On pure unit-capacity P2 diffusion from x(0)=(0,1), the unique solution is x=((1-exp(-2t))/2,(1+exp(-2t))/2). The injective encoding y_i=x_i^3 has a singular inverse at zero. Its pushed equation is y_i_dot=3*abs(y_i)^(2/3)*(cbrt(y_j)-cbrt(y_i)). Both the cubed true solution and y=(0,exp(-3t)) solve this equation with the same initial value. The latter lifts to (0,exp(-t)), whose first coordinate violates the original nodal row. No information was discarded by the encoding, but its vanishing Jacobian introduced a spurious absorbing branch. A differentiable inverse supplies the general solution-equivalence guarantee; faithful storage alone does not. A singular encoding can still preserve a particular law, so this is not a claim that every singular representation necessarily fails.

The portable foundation controls exercise the actual pressure owner, signed scalar embedding, lossy rich-EPI projection, transformed mobility and the singular-chart counterexample. They do not select a new state space or evolve a new production model.

2.7 All-channel parameter foundations

The parameter foundation audit extends these types to capacity, phase, time, pressure coefficients, support, conductance, distance, all four tetrad fields, diagnostics, energy, history and operators. It owns the joint form/time covariance law and the conditional derivation of a local diffusive generator. That derivation requires explicit locality, linearity, shift, equilibrium and maximum-principle premises; reciprocity and graph symmetry supply further restrictions rather than following from the nodal product alone. Numeric defaults and telemetry remain distinct from autonomous constitutive laws.

Its pressure map specifies the four channel signs, weights and distinct neighborhoods. The low-degree reduction and nonlinear boundary explain why phase/capacity compensation on a path does not imply a general conservation law for the mixed pressure. Capacity acts both as mobility and as a spatial source; a zero explicit topology coefficient does not remove the graph from the dynamics.

2.8 Representation dimension does not select a maintaining mechanism

Scalar-EPI exclusion results must retain their scalar-chart hypotheses. They do not rule out every manifold-valued TNFR realization. However, merely adding components to the same reciprocal diffusive pressure does not remove its dissipation. On a common finite-dimensional real inner-product fiber, let the rows of X be nodal forms, B=D-W for fixed symmetric nonnegative conductance, and M=diag(nu_i/d_i) with zero mobility at isolates. The componentwise extension of the declared pure-EPI model gives

\[ \dot X=-MBX,\qquad E_D=\tfrac12\operatorname{tr}(X^TBX) =\tfrac14\sum_{ij}W_{ij}\|X_i-X_j\|^2, \]
\[ \dot E_D=-\operatorname{tr}((BX)^TM(BX))\le0. \]

This follows by differentiating the quadratic and using symmetry of B; it needs no assumption about the number of components. It is a conditional mathematical extension, not an implemented rich-EPI engine law. A faithful smooth chart change transports an existing Lyapunov function by composition with its inverse. Consequently a richer chart alone cannot convert those same dissipative trajectories into recurrent ones. A new connection, nonlinear response, internal generator or coupled channel would be new dynamics whose origin must be justified separately. Rich storage and auxiliary Hilbert operators do not provide that justification by themselves.

2.9 Assumed substrate and emergence between scales

A primitive of one description is not thereby external to TNFR or fundamental at every scale. The present engine takes a nodal state and its admitted laws as inputs. A coarser nodal state could instead be an observation of finer TNFR structure. The restricted EPI quotient already derives macro form, conductance and capacity from that structure. This does not derive the existence of the fine substrate or the selection of the observed partition. Conversely, taking a fine substrate as a premise does not assert a substance outside the framework. Structural scale is not an assertion of an earlier physical time.

A larger NFR would be an organization of its internal constituents, not their replacement or disappearance. Changing the scale of description must preserve their influence through retained state or justified memory. The conservative replica calculation demonstrates why this matters: internal phase geometry changes collective response even when the observed mean form and phase are unchanged. A supplied partition still needs its own formation and identity evidence.

In particular, a non-EPI pressure contribution from neighboring phases or capacities can be an internal relational effect, even when a decomposition calls it a "source". It need not be a force from an ambient external reality. Conversely, reading those relations on an initialized graph does not derive the first network or establish that an observable physical world emerges from it. That claim needs a closed formation model and an independently testable observation bridge.

For a declared fine state Z with a complete law Z_dot=F(Z), let Y=Pi(Z) be a proposed effective nodal description. On a smooth fixed-support domain, an autonomous effective law Y_dot=G(Y) requires

\[ D\Pi(Z)F(Z)=G(\Pi(Z)). \]

Thus two admitted fine states with the same Y must give the same observed derivative. A time-dependent observation additionally contributes partial_t Pi; an admitted event map J requires the separate relation Pi_after(J(Z))=J_bar(Pi_before(Z)). Neither event occurrence nor a moving partition follows from the continuous identity. Memory-dependent models must retain the required history, not silently claim instantaneous closure.

Writing Y_dot=nu_bar*p_bar by defining pressure from the observed derivative would not pass this test. The effective pressure, capacity, phase, support and their laws must be constructed independently from the fine structure. The scale bridge owns the restricted positive result and the phase, topology and field-inheritance obstructions; derived memory supplies exact lost-state accounting in its declared linear domain. A failed reduction can require an inherited coordinate or memory rather than a new force. It does not, by itself, prove that TNFR needs a different underlying theory. A claimed emergent NFR must also establish formation and identity: closure of an analyst-selected partition alone is insufficient.

The hypothesis that the present nodal description is an effective part of a more complete TNFR formulation is therefore open and mathematically testable in scoped models. It is not a demonstrated identification of all physical entities with one substrate. Repeating a nodal description at a finer scale also leaves that finer state/law as premises; it is not an explanation of existence from no assumptions.

Environment, substrate and a proposed vacuum. In the current relational model, an environment consists of other nodal degrees of freedom and their relations, not an independently defined surrounding substance. A fine node is not thereby a physical particle or a point in a pre-existing physical space. A hypothetical relational substrate could underlie both patterns and their apparent environment, but this identification is an open generative claim. Connected support is a premise of the relational executor; it does not prove that all physical reality belongs to one connected network.

Keep the following cases separate:

Preparation or description What remains in the current model
Zero or uniform EPI Phase, capacity and support remain; choosing a form origin does not remove nodes
Zero total structural pressure Channels can cancel while phase evolves; this need not be full equilibrium
Full nodal equilibrium Present rates vanish, but response to a perturbation can be nonzero, including mediated transmission
Unobserved environment Its state can enter the retained dynamics through memory and a hidden initial-state contribution
No nodes or relations The present nodal pressure has no such substrate-free domain; this is a model boundary, not proof that physical emptiness is impossible

For an exact full equilibrium of the admitted smooth unforced law, uniqueness keeps the state there. Being able to respond does not generate an initial perturbation or spontaneous fluctuations. A region can instead receive a disturbance from another part of the same fine structure; an environmental source in its reduced description need not be external to the whole model.

The same-law pressure-state counterexample rules out using the current pressure vector alone as an autonomous state in general. Its constructive P2 chart retains pressure plus one relative form coordinate, or the corresponding exact memory and initial datum. That is a change of description of the same dynamics, not a new substance or a uniquely fundamental pressure law. A stored pressure, an independently supplied source Gamma, and a freshly calculated P(z) have different contracts. The mediator chart extends this distinction to interacting regions: retaining the environment also requires its initial information and the movement of its boundary.

Calling a background a physical vacuum therefore requires an independently specified state class, its response and a measurement bridge. No present zero condition makes that identification. Likewise, a relational substrate is not automatically the historical light-carrying ether; the physical comparison keeps those hypotheses distinct. The effective-link study can investigate which environmental information carries a relation without first asserting either physical identification.

A pulse can be derived without becoming a new primitive. The selected zero-loss sine law has an exact finite-amplitude internal pulse on a supplied doubled-C5 geometry. Reciprocal form/phase evolution sustains the prepared internal motion while the collective mean form and phase remain fixed. No external periodic drive or additional pulse coordinate is inserted. Its existence depends on the complete law, held capacities, support and initial state; it does not derive those premises or the substrate's origin.

Initial activity is not a universal sustained pulse. Positive capacity with zero pressure gives zero EPI rate. Nonzero nu_i*p_i at an initial state gives activity, but need not give oscillation: pure-EPI diffusion provides active relaxation. An auxiliary wave spectrum or a configured phase clock supplies a different claim. The permanent-pulse boundary proves approach to the equilibrium set for the finite connected, unforced sine law with positive loss and held nonnegative capacities. Equilibria also remain stationary in the reversible law. A universal primordial oscillation or an independently selected microscopic zero-loss law does not follow.

Identity, waveform and clock are distinct. The internal pulse is periodic in the full labeled state, while its exact unordered-pair description has half that period. Capacity is a factor in the derived timescale; geometry, amplitude and the other law coefficients also matter. A repeated observation does not itself define a monotone clock. Moreover, the small-amplitude transverse result proves orbital instability of sufficiently small nonzero members of this prepared pulse family, without storage loss. Maintaining a geometric identity and maintaining one precise waveform require different proofs; an unstable waveform does not by itself imply loss of all organization. The parameter foundation owns the resulting capacity, clock and observation distinctions.

Resonant susceptibility is a separate derived property. The resonance foundation proves a bounded positive-frequency work-port gain maximum near stable critical patterns of the complete sine law. It follows from the joint form/phase dynamics and positive stiffness/loss, including cases with no oscillatory free poles. The declared probe is not an inferred primordial drive; pure diffusion and the nodal identity alone do not inherit this theorem.

Resonance and fractality are precise obligations, not a complete law. Reciprocal exchange, a response peak, a permanent pulse and formation of an identity require different evidence. The complete native law already contains form-to-phase and phase-to-form feedback; a negative formation result does not by itself reveal a missing resonance term. Its loss coefficient is nevertheless a constitutive premise. The full consensus derivative shows that the current half-weight law has overdamped relative modes on any admitted connected support with positive held capacities. That classification does not forbid frequency-selective driven response or finite nonlinear formation. It identifies a restriction of that law near consensus, rather than a restriction of every possible TNFR completion.

The nonlinear full-form bound separately excludes the aligned unit-winding target from every phase-consensus preparation in the fixed two-ring form budget. It accounts for full native exchange and loss, without a linear approximation or a symmetry restriction. Global continuation remains unproved for that whole class. Its event corollary also shows why an energy-passive reset need not preserve the continuous-flow obstruction. The collective-operator criteria require closed state, action and target information before an internal event can be interpreted as an operator of an effective NFR.

For dynamical scale inheritance, specify a fine law, a scale map and the family of states on which the inherited law has the same declared form after any stated unit changes. Repeated inheritance must compose consistently. A nested container, repeated graph or fitted dimension alone does not meet this condition. The replica construction establishes it on a supplied synchronized submanifold. Outside that submanifold, the existing sufficient collective state retains internal form, phase and capacity information; those constituents have not disappeared. Autonomous selection and formation of such a hierarchy remain separate obligations.

These structural requirements also leave constitutive freedom. The replication counterfamily preserves the same storage, synchronized inheritance and equilibrium derivative while changing nonlinear response. This consolidates existing nonselection results: replication consistency and small-perturbation behavior do not uniquely select a microscopic pressure/mobility law. Stronger claims about a complete nonlinear response, pulse or recurrence still require their own proof.

Finally, determine what loss means at the retained scale. In the declared positive-loss laws it is a negative total-storage rate. In a conservative fine model, a visible region may instead exchange storage with retained hidden nodes. The finite-memory calculation retains that hidden state and possible return; it does not derive an exact memoryless damping term. Interpreting microscopic loss as unresolved internal exchange therefore needs a reduction and approximation bound, not relabeling. These distinctions locate the unfinished foundations without discarding valid conditional results or adding feedback merely to obtain a preferred pattern.

The joint reuse map connects these foundations to actual collective feedback, phase-offset closure and finite full-state retention. It keeps driven resonance, conservative pulse, scale inheritance and formation as separate obligations. In particular, the derived mean-contact balance exposes information lost by a scalar pulse description; compatible motion does not generally close without the internal contact deviations. Correlated storage and ordered geometry then constrain which preparations could reach a maintained identity. These are consequences of declared complete models, not a new universal term in the nodal equation.

2.10 Foundation decision ledger: state, equivalence and origin

This ledger records the scoped F1 reference admission of the revised research plan. It is not a new axiom list or a declaration that every entry is uniquely necessary. The preceding sections own the definitions and counterexamples.

Decision Admitted working statement Unresolved identification or rejected shortcut
Form type The engine has a signed real scalar chart and exact uniform-real BEPI embedding; richer BEPI is finite storage The nodal identity does not fix ontological dimension; spectral entropy cannot decide it
Form zero Zero is a value in the selected chart; the node, phase, capacity and support can remain present Zero EPI is not absence of substrate; dropping its phase can change a neighbor's response
Form origin/scale A regular chart change transforms the full law, its storage, inputs and policies Offset symmetry of the pressure row does not prove offset symmetry of the joint law or a physically preferred origin
Primitive phase In scalar reference S, theta is an independent circle coordinate, including at x=0 Combining signed form and phase as xexp(itheta) is not generally a sufficient observation
Derived phase In reference D, regional orientation and amplitude are computed from fine form; Cartesian coordinates can remain regular at zero amplitude Orientation is unavailable there; it is not automatically reference S's independent theta
Capacity and clock Capacity is nonnegative relative to a declared clock; a held value is an explicit model premise A measured form rate, fitted product or phase speed need not identify it; time rescaling transforms auxiliary rows
Relational data Support, conductance, neighbor multiplicity and path length have distinct roles A smaller bare graph need not preserve inherited dynamics or observations
Mathematical state versus execution state A fresh algebraic law can use p=P(z); the hybrid runtime additionally consumes retained pressure/history/context A triad snapshot or serialized recipe is not automatically a complete runtime checkpoint
Equivalence Define the relation and prove its compatibility with the selected full law or target observation Python comparison, numerical closeness and equal diagnostic values cannot establish the relation

Reference S and reference D have compatible but different scope. S retains the fine scalar state with independent primitive phase. Under a specified transport law, D is an observation of that same fine form, not a competing microscopic foundation. Identifying D's regional orientation with S's primitive phase is a separate constitutive hypothesis. The cards below make the admitted sector, target observations and lost information explicit. Neither establishes minimality for every observable or physical ontology.

An implementation comparison is not structural equivalence. The legacy BEPIElement.__eq__ compares arrays approximately and numeric operands through a scalar projection. For a=BEPI((1,-1),(0,0)) and b=BEPI((-1,1),(0,0)), on the same grid, it gives a==1, 1==b, but a!=b. The mixed comparison is not transitive. Its compatibility behavior must not define a quotient of forms. Likewise, allclose does not define structural equivalence, and magnitude projection is not an injective state chart. Scalar solver admission uses the separate represented-real reader, not this equality. No equality API or physical state space is changed by recording this boundary.

Origin decision. The cotangent candidate assigns dynamical meaning to form relative to the origin in its stipulated momentum/storage. Its same-tetrad witness and positive chart control separate changing that state from changing coordinates. A passive translation can preserve the entire law; resetting its formulas unchanged need not. Thus the audit supplies no evidence of a physically absolute EPI origin. Whether a reference is independently meaningful, inherited from a finer state or removable by a valid reduction remains a constitutive question.

The source audit records which current and historical calculations support these decisions.

Reference S: fine scalar state and a complete held baseline

Fix a finite nonempty ordered node set, support U and nonnegative conductance W; directed support is admitted. Support neighbors and positive-conductance neighbors need not coincide. The mathematical state is (x,theta,nu) in R^n x (S^1)^n x [0,infinity)^n. Metric lengths are supplied only for observations that consume them. State and metadata must all transform under node relabeling; within frozen ordered data only their automorphisms are graph symmetries. No bitwise runtime equivariance follows from this algebra.

Use a declared common clock, form units X and capacity units T^-1, so pressure has units X. Effective channel coefficients and their units follow the parameter owner; a normalized numerical recipe is not independent metrology. Zero form retains the other coordinates, and zero capacity is admitted. Where phase is consumed, require nonzero neighbor resultants and regular wrapped displacements; isolates use the specified zero row. Primitive phase has a circle value even at x=0.

A complete existing baseline S0 holds phase, capacity, support, conductances and coefficients, uses fresh configured pressure, and excludes inputs, events and clipping. With N=diag(nu), G_W=-L_rw and constant non-form source F, its equations are

\[ p=eG_Wx+F,\qquad F=a g_\phi(\theta,U)+bG_U\nu+cG_Uk,\qquad \dot x=Np,\quad \dot\theta=\dot\nu=0. \]

The pressure owner defines the distinct neighborhoods and zero rows. Holding phase preserves its admitted chart. S0 is a constant-coefficient affine ODE, so it has a unique solution for every finite time, including at zero capacity. Its forcing may cause indefinite mean drift; this is not a boundedness or formation theorem. The exact-real domain is real form; configured runtime rails and finite represented arithmetic are separate execution restrictions.

For targets consisting of relative form, relative primitive phase and pressure, let C=I-11^T/n, y=Cx and u_i=exp(i*(theta_i-theta_ref)). At fixed metadata/capacity, declare two S0 states equivalent precisely when y and u agree. This discards a common form offset and common phase rotation. Both are symmetries of this held baseline, and the quotient closes:

\[ \dot y=CN[eG_Wy+F(u,\nu,U)],\qquad \dot u=0. \]

Pressure and nodal rates follow from y, u and retained metadata. To recover absolute form, retain the initial mean and integrate m_dot=1^T*N*(e*G_W*y+F)/n. A form scale change is covariance with transformed units/coefficients, not equality of these observations. These are sufficient declared equivalences, not a classification of every possible symmetry.

There is also a useful baseline consequence of the existing joint chain rule: p_dot=e*G_W*N*p. The initial pressure must be realized by this fixed source, p(0)=e*G_W*x(0)+F; the pressure flow preserves F+image(e*G_W). Pressure alone has closed evolution at fixed N, W and e under these holds, although it is not complete state. Opening phase, capacity, support or events requires additional terms and projectability checks. This does not conflict with the same-tetrad future witness under the different cotangent law. S0 is a declared continuous reference, not the default runtime's sequence of pressure writes, held intervals, glyphs and adaptation.

Reference D: a derived collective observation of the transport sector

Restrict S to fixed pure-EPI transport x_dot=-diag(nu)*L_rw*x with held positive capacities. Primitive phase is held and unused. Choose the existing two equally oriented directed unit triangles with matched reciprocal unit links; outgoing degree is two. The partition and the orthonormal frame U=[(1,-1,0)/sqrt(2), (1,1,-2)/sqrt(6)] are supplied relational data.

For each region, x_a=mu_a*1+U*(Re z_a,Im z_a)^T. Thus (mu,z) is a complete real coordinate change; it adds no complex microscopic form. The derived angle psi_a=arg z_a exists only at nonzero amplitude. Cartesian coordinates remain defined at zero. The selected observation is (mu,Q) with Q=zz^dagger; Q is the Gram H of its proof owner, not a cotangent metric. Its units are X^2; mu and z have units X.

Admission item Reference D contract
Target Regional contrast squares, retained means, relative orientations where defined, and cross-region contrast-budget terms
Exact equivalence Same means and Gram; for nonzero z, representatives differ by one common contrast rotation at fixed frames
Realizable domain Q is Hermitian positive semidefinite of rank at most one; a positive diagonal entry reconstructs a representative by z_i=Q_iq/sqrt(Q_qq)
Zero stratum Q=0 means zero internal contrasts, not zero means or absence of support; each zero contrast has no angle and a zero Gram row/column
Excluded targets Fine EPI, vertex pressure/current and general tetrad values can depend on the discarded common orientation
Capacity/support scope For the six-node graph, capacities are constant within each triangle for all-state Gram closure; the regional constants may differ. General interfaces need the existing matrix criterion
Numerical restrictions PSD/rank realizes real fine form, not necessarily configured EPI rails; constrained domains require checking admitted lifts

For regional capacities nu_0,nu_1>0, the existing complete reduction is

\[ \dot\mu_a=\frac{\nu_a}{2}(\mu_b-\mu_a),\qquad \dot z=Az,\quad A=\begin{pmatrix} -\nu_0(5+i\sqrt3)/4&\nu_0/2\\ \nu_1/2&-\nu_1(5+i\sqrt3)/4 \end{pmatrix},\qquad \dot Q=AQ+QA^\dagger. \]

This is a derived law once the fine transport has been admitted, including the zero stratum through the Cartesian flow. The exact capacity-domain theorem and interface criterion own necessity, sufficiency and counterexamples; arbitrary capacities or ports do not inherit it. Cross blocks of the joint Gram are required when regions interact, and pairwise realizability alone is insufficient. The real-fine-state lift must realize one joint matrix.

The held affine-source extension now admits the same observation beyond pure transport precisely when the existing generator conditions and U^T*N*F=0 both hold on the full real-form domain. It permits a regional mean source, but a fixed contrast source can retain orientation information lost by the Gram observation. This conditional extension leaves the transport reference above intact and supplies no new pressure law or complete-runtime closure.

A reporting-frame change rotates z and conjugates both Q and A consistently. That is different from an active common modal rotation at fixed frames, which changes fine EPI but leaves this particular observation unchanged. Rescaling form changes Q quadratically and is not equality in this quotient.

F1 verdict and the specific obligations passed to F2

S0 and D now have explicit domains, complete baseline laws, target observations and admitted equivalences. D is an exact lossy description of a restricted S transport sector. Neither their equivalence nor the existence of circle angles identifies D's regional psi with S's independently stored nodal theta. That proposed identification needs an explicit lift and tangency of the full law, including mean, amplitude, capacity and any source it activates.

The unequal-capacity D theorem uses pure-EPI pressure. In the full mixture, capacity differences can activate an additional source. Common capacity, equal primitive phases and equal outgoing degree make the other channels vanish and give the existing embedding with rate nu*w_epi when w_epi>0. At w_epi=0 the motion is the frozen zero-rate limit, not the stated positive-rate D family. This is not a general permission to discard channels. A phase-derived source fed back into pressure must be counted as a model change unless its full inherited response matches. Exact quotient equations do not create an additional physical force.

F1's reference admission is complete in this scope. F2 must justify the fine pressure and support choices, the held capacity/phase premises or their replacements, and any identification between levels. Physical state minimality, a universal clock and autonomous selection remain open. Existing proofs and controls are reused rather than presented as newly discovered dynamics. The first F2 identification control now separates preserved contrast from changed means, necessary phase tangency from a supplied phase row, and a regular Cartesian zero from an undefined angle-fed pressure. Its restricted positive bridge does not select a general law or sustain contrast.

3. Structural Field Tetrad

TNFR exposes four canonical diagnostic channels. They complement the structural triad, frequency, and graph state; they do not determine the full dynamics. Their computation and storage depend on the selected telemetry path.

3.1 Structural Potential (\(\Phi_s\))

\[ \Phi_s(i) = \sum_{j \neq i} \frac{\Delta\mathrm{NFR}_j}{d(i,j)^2} \tag{4} \]

Aggregates surrounding pressure using the selected inverse-square kernel. Explicit edge length defines distance; absent length, weight is the compatibility fallback, then unit length. The sum includes only reachable distinct nodes at strictly positive finite distance; zero-distance pairs are omitted under the shared read-out convention. U6 monitors its change under a declared policy; the aggregation alone does not prove stability.

3.2 Phase Gradient (\(|\nabla\phi|\))

\[ \lvert\nabla\phi\rvert(i) = \frac{1}{|\mathcal{N}(i)|}\sum_{j\in\mathcal{N}(i)} \big|\mathrm{wrap}(\theta_j - \theta_i)\big| \tag{5} \]

Quantifies local desynchronization between a node and its neighborhood; isolates return zero. A large value is a diagnostic observation, not a law selecting a Coherence event.

3.3 Phase Curvature (\(K_\phi\))

\[ K_\phi(i) = \mathrm{wrap\_angle}\big(\theta_i - \mathrm{circular\_mean}(\theta_{\mathcal{N}(i)})\big) \tag{6} \]

Measures circular phase curvature, with \(|K_\phi| \leq \pi\) by construction. The displayed circular mean requires a nonzero resultant. The shared read-out now distinguishes nonzero represented direction from exact represented cancellation, with explicit unavailable evidence or a numeric API error; it does not invent an arithmetic phase direction. Exact sums of retained phasor components do not certify exact trigonometry. Curvature alone does not certify a bifurcation. See the API domain.

3.4 Coherence Length (\(\xi_C\))

Estimated from uncentered products of static pressure-only coherence \(c_i=1/(1+|\Delta\mathrm{NFR}_i|)\), grouped by structural path distance:

\[ q(r)=\operatorname{mean}_{d(i,j)=r}(c_i c_j)\approx A \exp(-r / \xi_C) \tag{7} \]

Both backends now share the distance/pair/fit definition. This is not connected covariance and has no goodness-of-fit acceptance test. If unsuitable, fitting yields to a spectral fallback: the inverse square root of the smallest normalized-Laplacian eigenvalue above the implemented 1e-9 cutoff. On an admitted connected undirected graph this equals \(1/\sqrt{\lambda_2}\) only when that gap clears the cutoff. It is a dimensionless mode scale, whereas a successful fit has path-distance units. A large length is not a proof of criticality. See estimator scope.

3.5 Complex Geometric Field (\(\Psi\))

Phase curvature and phase current unify into a single complex field:

\[ \Psi = K_\phi + i \cdot J_\phi \tag{8} \]

This complex packaging retains the two real coordinates. Correlation measured on particular trajectories does not reduce their algebraic degrees of freedom or establish completeness of the field representation.

3.6 Emergent Invariants

The following algebraic read-outs combine the fields. Their conventional names do not prove conservation, quantization or physical dimensional compatibility:

Conventional diagnostic name Definition Supported interpretation
Energy density \(\mathcal{E}\) \(\Phi_s^2 + \lvert\nabla\phi\rvert^2 + K_\phi^2 + J_\phi^2 + J_{\Delta\mathrm{NFR}}^2\) Nonnegative diagnostic functional; decay needs evidence
Topological charge \(\mathcal{Q}\) \(\lvert\nabla\phi\rvert \cdot J_\phi - K_\phi \cdot J_{\Delta\mathrm{NFR}}\) Bilinear diagnostic; no general integer or conserved-charge theorem
Chirality \(\chi\) \(\lvert\nabla\phi\rvert \cdot K_\phi - J_\phi \cdot J_{\Delta\mathrm{NFR}}\) Signed channel contrast; geometric handedness needs an explicit symmetry action
Symmetry breaking \(\mathcal{S}\) \((\lvert\nabla\phi\rvert^2 - K_\phi^2) + (J_\phi^2 - J_{\Delta\mathrm{NFR}}^2)\) Channel contrast; transition interpretation needs a protocol
Coherence coupling \(\mathcal{C}\) \(\Phi_s \cdot \lvert\Psi\rvert\) Product of potential and geometric-field magnitude; no cross-scale closure theorem

4. The Structural-Field Tetrad

4.1 Statement

The four fields organize source aggregation, local phase mismatch, circular curvature, and correlation. Higher derivative operators can be formed by composition, but this does not prove that their outputs can be recovered from four lossy diagnostics. Universal minimality and complete reconstruction remain open under a specified state space and equivalence relation. π gives the exact phase-wrap maximum; writing other values as π-fractions does not prove their physical necessity.

Field Symbol Operational limit Structural scale
Structural potential \(\Phi_s\) Drift policy \(\pi/2\); per-node policy \(\pi/4\) Pressure and graph-kernel dependent
Phase gradient \(\lvert\nabla\phi\rvert\) Warning \(\pi/16\) Exact maximum \(\pi\)
Phase curvature \(K_\phi\) Warning \(0.9\pi\) Exact absolute maximum \(\pi\); local Laplacian comparison, with exact low-degree/chart cases
Coherence length \(\xi_C\) Configured length comparisons Correlation fit or spectral reference using the stated positive-mode cutoff

The phase bounds are kinematic identities. The warning margins and potential thresholds are unchanged engine policies. Fitted correlation lengths need not equal the graph-spectral reference for every state. The exact curvature exception and its support/chart premises belong to the low-degree phase result.

4.2 The four fields

The tetrad groups four selected diagnostic channels by their construction:

        Φ_s (0th — global aggregation)
             /|\
            / | \
           /  |  \
  |∇φ| ------+------ K_φ
  (1st)      |  (2nd; π-bounded)
          \   |   /
           \  |  /
            \|/
          ξ_C (non-local — product fit / spectral fallback)

4.3 Derivation Outline

The detailed arguments have one owner in Minimal Structural Degrees, section 4:

Field Established fact Remaining distinction
Potential Fixed-kernel linearity and norm bound Neither a universal pi-fraction bound nor unique selection of exponent 2
Phase gradient Mean absolute wrapped mismatch is at most pi Warning margin and measured synchronization onset are separate
Phase curvature Wrapped bound where the circular mean is available Represented cancellation is an availability condition, not invented zero curvature
Coherence length A static product fit with a separately identified spectral fallback No universal decay law or criticality follows from either estimate

This summary does not replace the linked graph, distance, representation and estimator hypotheses.

4.4 Grammar Integration

Grammar obligations and field readouts have distinct enforcement paths:

Rule Primary fields Enforcement
U1 (Initiation/Closure) Context and endpoint roles Supported generator and closure contracts
U2 (Convergence policy) Operator roles and debt Stabilizer presence and prefix debt at most 2
U3 (Resonant Coupling) Actual wrapped phase mismatch Phase alignment verified before UM/RA
U4 (Bifurcation Control) Operator context Handlers, recent destabilizer, prior IL for Mutation
U5 (Multi-scale Coherence) Declared hierarchy Deep Recursivity requires nearby scale stabilization
U6 (Structural Confinement) \(\Phi_s\) Read-only drift policy \(\pi/2\)

5. Core Structural Metrics

5.1 Total Coherence \(C(t)\)

The shared numeric kernel is \(C(p,r)=1/(1+|p|+|r|)\); the network read-out applies it to mean pressure and rate magnitudes. It differs from mean local coherence and assumes declared input scales. Stored inputs need not describe a fresh simultaneous nodal rate. The selected cuts \(\pi/(\pi+1)\) and \(1/(\pi+1)\) label diagnostic bands; neither cut is a stability theorem.

5.2 Sense Index \(Si\)

Configured, clipped combination of relative capacity, phase dispersion and relative pressure. Both numerical backends normalize against live maxima. The score is telemetry; thresholds and its use in a selector or adaptation loop are additional policies. It is not an independent nodal primitive or proof of future stability. See the telemetry and time audit.

6. Multiscale Domain Mapping

Domain-correspondence studies use a specified pressure law and reduction procedure. A derivation must state the assumptions at each step:

6.1 Reduction Procedure

  1. Declare the fine law: Specify form coordinates, pressure dependencies, capacity, phase, support and time. A diffusive or solenoidal decomposition requires its own operator, metric and domain assumptions.
  2. Specify the observation: Choose the coarse variables and prove that equal observations have equal projected rates, or retain the required hidden coordinates and memory. A graph quotient is not automatically a PDE.
  3. Separate flows and events: Derive the actual effect of each supplied operator map. An EPI jump or stored-pressure edit is not automatically a continuous source or damping term.
  4. Define the read-outs: Compute the tetrad where available and state which information it discards. Use the scale and geometry contracts and derived memory rather than assuming tetrad closure.

6.2 Regime Summary

Model comparisons and research applications (each retains its own assumptions):

Domain Regime condition Telemetry priorities Governing reduction Verification
Overdamped drift Specified restoring pressure and frequency Structural velocity and pressure First-order mobility law; frequency is not inverse mass Requires the stated pressure law
Instantaneous EPI stationarity \(\Delta\mathrm{NFR}=0\) Structural velocity Zero unforced EPI derivative; full equilibrium also needs the other state laws Direct nodal identity
Discrete-mode analogy Bounded graph with specified boundary conditions Graph spectrum Standing graph modes; no quantum-state identification Spectral calculations
Spectral factorization Specified Paley/residue graph construction Graph-spectral and available field diagnostics Partitioned periodicity detection under that construction Number-theory scope

6.3 Tetrad Requirements per Domain

Every domain study must report the four structural-field channels, including explicit unavailability when its state or estimator does not support one:

  • \(\Phi_s\): Report distributions and gradients; compare against the selected \(\pi/2 \approx 1.571\) drift policy with its baseline and aggregation.
  • \(|\nabla\phi|\): Monitor the heuristic early-warning level (\(\approx \pi/16 \approx 0.196\); not derived — the kinematic bound is \(\pi\)).
  • \(K_\phi\): Report available curvature and the selected \(0.9\pi\) warning; a crossing neither selects Mutation nor proves a transition.
  • \(\xi_C\): Report fit quality, path units and estimator/fallback provenance; a critical-scaling interpretation requires a separate finite-size protocol.

7. Emergent Geometry from the Nodal Equation

The following constructions have different coordinates and evolution laws. Their certificates apply to those specified models; identifying them with the full four-channel nodal dynamics requires an explicit mathematical bridge.

7.1 Transport Layer (Structural Diffusion)

For the pure EPI channel on a graph with the stated neighbor-weight convention,

ΔNFR_epi = −L_rw EPI,
EPI' = −diag(ν_f) L_rw EPI.

On a fixed undirected graph with positive homogeneous frequency, Laplacian modes decay as exp(−ν_f λ_k t), and equilibrium is constant on each connected component. Degree-weighted total EPI is conserved. For fixed positive heterogeneous frequencies the invariant weights are degree_i/ν_f_i and modal rates come from diag(ν_f)L_rw. Isolated nodes have no diffusive coupling.

The Dirichlet energy ½ EPIᵀ(D−W)EPI has degree-metric gradient L_rw EPI. This supplies an exact gradient-flow identity for the isolated EPI channel. It is a different functional from the sum of squared tetrad fields and does not prove a variational identity for the full four-channel pressure.

Implementation: structural_diffusion.py and variational.py. The connected homogeneous formulas require their stated assumptions; the spectral stability of nonstationary modes does not alone settle stationary sources, nonlinear operator gains, or general U2 compliance.

7.2 Auxiliary Symplectic Substrate

The substrate implementation specifies an ambient phase space with pairs (K_φ, J_φ) and (Φ_s, J_ΔNFR). Its isotropic Hamiltonian is

H_sub = ½ Σ_i (K_φ² + J_φ² + Φ_s² + J_ΔNFR²),

with the phase-gradient term treated as a fixed background in the corresponding readout. The canonical symplectic form and harmonic flow are well-defined on these independent ambient coordinates. Hamiltonian flow preserves symplectic form and phase volume; that theorem does not certify all 13 implemented operator maps.

The isotropic model has its specified U(1)/U(2) invariances and oscillator charges. Action-angle coordinates apply away from zero actions; singular levels and global quotient topology need separate treatment. Extracting coordinates from a graph can impose dependencies that an ambient-coordinate certificate does not remove.

Implementation: symplectic_substrate.py. See its current certificate assumptions and the variational note.

7.3 Orthogonal Structure and the Overdamped Projection

A separately specified damped graph wave with stiffness L_rw has an overdamped diffusion limit under its damping, time-scale, and coordinate assumptions. The isotropic substrate evolves with identity stiffness, so the graph-wave calculation is not a derivation of the full nodal equation from that substrate.

Likewise an orthogonal decomposition of selected graph currents is a result in its specified graph metric. It does not establish a universal orthogonal decomposition of every four-channel nonlinear evolution. The exact EPI Dirichlet gradient flow, the graph-wave approximation, and the independent harmonic substrate should be reported separately.

8. Empirical Validation

Field and operator experiments test defined protocols, with topology, weights, initial state, gains, time steps, and seeds recorded. A correlation between potential drift and coherence loss is evidence for that protocol, not a universal upper bound or a proof of state reconstruction.

The exact definition-level facts are linearity of structural potential in pressure and the π phase-wrap bounds. Potential policies π/4 and π/2, curvature margin 0.9π, and phase-gradient warning π/16 retain their current values. The finite-graph witnesses and the distinctions they require are recorded in DIAGNOSTIC_AND_GRAMMAR_SCOPE.md. A test count measures tested behavior; it does not establish an open theorem.

9. Practical Guidance

  1. Monitoring: Export available tetrad fields and explicit failures after each operator batch. Threshold crossings are policy flags; any resulting action needs its own declared controller and grammar admission.
  2. Operator design: Specify a proposed map's state changes, domain and postconditions, then verify its field response. A field interpretation does not establish a new canonical operator or its autonomous selection.
  3. Model calibration: Compare like units and declared observation scales. Phase ratios such as \(|\nabla\phi|/\pi\) are dimensionless; dividing dimensional \(\Phi_s\) by the number \(\pi/2\) alone is not. A potential ratio requires a reference with the same pressure/path units. See the parameter foundations.
  4. Correlation diagnostics: A large \(\xi_C\) warrants checking fit quality, spectral fallback, and finite-size effects before interpreting a critical regime. Any subsequent operators must satisfy their grammar and contracts.

10. Implementation Reference

Component Location
Structural field computation src/tnfr/physics/fields.py
Grammar validation (U1–U5 context; U6 is a separate observer) src/tnfr/operators/grammar.py
Structural balance diagnostics src/tnfr/physics/conservation.py
Integrity monitor src/tnfr/physics/integrity.py
Canonical constants src/tnfr/constants/canonical.py
SDK access (tetrad, conservation) src/tnfr/sdk/simple.py
Auxiliary symplectic substrate src/tnfr/physics/symplectic_substrate.py
Structural diffusion (transport) src/tnfr/physics/structural_diffusion.py
Test suite tests/ (current executable verification; counts are obtained from pytest)

11. Implementation & Examples

SDK Entry Points

from tnfr.sdk import TNFR

net = TNFR.create(20).ring().evolve(5)    # Configured engine evolution
tetrad = net.tetrad()                      # Structural Field Tetrad
telem = net.telemetry()                    # C(t), Si, phase, νf
analysis = TNFR.analyze(net)               # Comprehensive analysis

Executable Demonstrations

Example Concept from this document
01_hello_world.py SDK initialization, one requested operator word and canonical C/tetrad observations
04_operator_sequences.py Flat grammar admission versus an independent illustrative pressure proxy
07_phase_transitions.py Prepared state ensembles and descriptive finite-size diagnostics
10_simplified_sdk_showcase.py SDK topology builders, configured evolution and comparison read-outs
99_structural_diffusion.py Declared diffusion and auxiliary graph-model correspondences
179_phase_form_driven_response.py Prescribed phase contrast and derived EPI response; no autonomous maintenance claim

Key Source Modules

  • src/tnfr/physics/fields.py — Structural Field Tetrad computation
  • src/tnfr/operators/definitions.py — 13 canonical operator implementations
  • src/tnfr/operators/nodal_equation.py — Nodal equation ∂EPI/∂t = νf·ΔNFR(t)
  • src/tnfr/sdk/simple.py — Simplified SDK with TetradSnapshot

12. References