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Nodal parameter foundations and constitutive scope

Reviewed 2026-10-06. This reference owns the parameter ledger, units and cross-channel dependencies. Detailed proofs have one topical owner in the reading map below. The execution plan alone defines active research; these notes are not parallel task queues.

Form, tangent pressure and chart definitions belong to Fundamental Theory. The constitutive audit owns the auxiliary-law inventory; API contracts and the field guide own runtime and diagnostic details. Definitions, conditional derivations, configured policies and physical identification remain distinct. An observed response does not select its law.

Reading map

Read the ledger below for definitions, then the note answering the current question. Section numbers remain stable across this family so earlier mathematical references can still be located. No proof is duplicated here.

Question Detailed owner Sections
What is a connection, what vanishes when it is absent, and what must a birth law specify? Relation foundations Support, conductance, metric and causal distinctions; zero-relation admission and remaining constitutive freedom
Locality, covariance, phase-domain boundaries and the prospective pressure comparison. Pressure premises and constitutive response 4
Signed representation, pressure derivatives and finite phase/capacity compatibility. Joint form, phase and capacity response 9–11
Fine-to-coarse form, inherited pressure and metric, tetrad reconstruction and changing support. Inherited form dynamics and observation 12–14
Causal source admission, regional frames, directed exchange and the prescribed-input response. Phase-to-form source matching and exchange 15–16
Symmetry obstructions, phase-law/cutoff premises, reciprocal response, source work and actual event occurrence. Primitive phase closure and source selection 17–20; premise review
Directed diffusion, sufficient amplitude/phase observations, wave-coordinate scope and regular continuation. Derived form phase and inherited closure 21
A declared phase/form closure, finite predictions, inherited exchange scale and integration boundaries. Conditional cotangent exchange 13.21-13.26; added premise, held capacity; integration audit, unequal blocks and same-tetrad future witness
A relational storage premise compatible with native pressure, its local exchange law and an independent spectral discriminator. Relational exchange admission Conditional alternative; full mean/source balance, origin/scale admission and the arbitrary-storage P2 momentum obstruction
When do reciprocal dynamics permit oscillation, recurrence or a frequency-selective response? Resonance foundations Complete sine law; positive-loss obstruction, conservative pulse and almost-everywhere recurrence have different hypotheses
Which internal coordinates, periods and stability properties survive grouping? Nonlinear replica state, joint persistence, capacity asymmetry Exact finite-symmetry quotient with capacity-state correlations; geometric trapping survives unequal positive held capacities, while internal circulation needs its symmetry premises
Can phase identify constituent pairs without a supplied partition? Phase-only pairing Strict mutual nearest partners inside the protected tube; independent support and complete-law admission, with no claim of formation
Which equilibria does the complete law permit without choosing a target winding? Fully acute critical geometries Exhaustive doubled-C5 classification with arbitrary positive held capacities: uniform form and consensus or uniform phase winding of either sign; represented-state admission is separate

Derived information and what it adds

The purpose of retaining another quantity is to preserve information consumed by the declared dynamics. A function of the full state is not an additional independent primitive, although it can reveal information lost by a collective observation. More fitted coefficients do not by themselves improve a model. The following ledger connects existing owners without introducing another fundamental law or research queue.

Quantity or mechanism Information retained and established use Limit and owner
First/third relative phase moments Z1,Z3 Distinguish the phase information consumed by supplied sine and cubic currents and their potentials Instantaneous incident sums, not a complete future state. Classification
Rate-weighted moments M1,M3 Retain which relative velocity belongs to which phase; derive moment motion, current derivatives and incident work Generic evolution needs further correlations. The regular star chart instead recovers its existing four-dimensional relative state, with explicit singular obstructions. Motion and full-star witness
Pair phase product P, internal form square U and association W Together with mean form X and mean phasor Z, retain the full unordered pair even at cancellation Global pair state on the fixed doubled C5. Constrained derived coordinates preserve four continuous degrees of freedom per pair. Second-moment state information does not add a second-harmonic pressure law
Phase-dependent stiffness The full interaction Hessian connects local geometric protection with coupled form/phase response Law- and state-dependent; neither the bare support Laplacian nor tetrad curvature replaces it. Storage-family response
Observable dimension and retained hidden state Exact row-space growth identifies information needed for a selected linear response; hidden initialization supplies a causal source A linear observability dimension is not a nonlinear closure theorem. Shared linear observation, bridge memory
Memory kernel and reactive response matrix Derived elimination connects visible motion to its retained environment; the low-frequency matrix can encode a stationary hidden-section metric A supplied projection and complete fine law remain necessary. Neither an independent primitive capacity nor local irreversible damping follows. Memory owner
Contact phase moments and relative rates Explain exchange between a collective pulse and internal deviations while all constituents keep moving Mean amplitude or common rhythm alone does not close dynamics or establish acquisition. The pair interaction contract separates present current, retained state and hidden-environment memory. Collective feedback
Normalized contact counts and phase-offset moments Specify when grouping can inherit an exact nodal description while retaining constituent structure Exact prepared closure differs from spontaneous grouping and transverse stability. Partition owner
Winding, sector margins and own-law storage budget Separate a declared identity, its local maintenance and energetic accessibility Existing geometry need not be reachable from a given source. Formation and constitutive scope

For a proposed collective state, first test whether states with the same observation have the same observed derivative. If not, retain the missing association, derive memory under an admitted elimination, or state the closure obstruction. If they do, check finite evolution, domain boundaries and events before asserting exact closure. The tetrad remains a diagnostic observation; it neither replaces these tests nor selects the consumed constitutive premises.

1. Start from a typed law, not from its defaults

Write the unforced nodal equation as

\[ \dot x_i=\nu_i p_i,\qquad p_i=P_i(z),\qquad z=(x,\nu,\theta,G,W,\ell,h). \]

Here h denotes whatever history the declared model actually needs. It is not an assertion that all these coordinates are independent physical primitives. Some could be observations of a richer form; that requires an explicit map and a closure proof. Conversely, naming them parts of one structure does not already provide their equations of motion.

For a chosen real EPI chart, let [x]=X, [t]=T, [nu]=T^-1. Then [p]=X. The phase coordinate is an angle on the circle, expressed in radians; an explicit structural distance can have a separate unit L. All code values may be nondimensional, but the reference scales and time convention must then be declared. A finite real number does not certify dimensional consistency.

Three levels must stay distinct:

  1. Definition or identity: for example, wrapped angles have magnitude at most pi in the radian chart.
  2. Conditional consequence: diffusion dissipates its Dirichlet energy under the stated nonnegative reciprocal conductance and mobility premises.
  3. Model or numerical choice: the pressure mixture, phase/capacity update, operator word, clipping interval, threshold, seed or sampling protocol.

An identity involving a chosen coefficient proves properties of that choice; it does not derive why nature or the nodal law must choose that coefficient. The naming of constants.canonical is not an epistemic classification.

1.1 Thresholds, constants and numerical settings have different duties

The active configuration audit distinguishes the following categories. A formula containing pi, a historical CANONICAL suffix or a passing regression test does not move a value from a selected policy to a derived physical law.

Category Examples and owner Meaning and limitation
Mathematical identity / representation The radian wrap bound pi; binary64 PI in constants The geometric bound is exact; the stored floating value approximates it. sin(pi/6) is a dimensionless score, not an angular tolerance of 30 degrees.
Conditional model quantity Spectral gaps, diffusion rates, Dirichlet bounds, cycle restoring margins Derived from the specified graph, law and domain; generally state or configuration dependent. They are not universal fixed TNFR numbers.
Constitutive / control policy Pressure weights, U3 ceiling, capacity averaging, grammar windows and scalar rails They select a model, allowed event or admissible chart. Structural constraints may restrict their range without selecting a unique value.
Diagnostic / classification policy Si mixture, tetrad warnings, R-squared fit cut and comparison exponents These classify observations. Consuming them as feedback is an additional controller; a fit-quality cut is not a statistical significance test.
Numerical method / error budget RK coefficients, time step, residual tolerances and represented arithmetic Method coefficients belong to a specified numerical scheme. A tolerance needs units and a scale; it cannot establish infinite-time convergence or define a new physical law.
Implementation setting Cache capacities, worker counts and tuning scores in operational constants These organize computation, not nodal dynamics. The compatibility suffix does not give them physical meaning.

The live owners, rather than equal-looking numbers, determine what can be unified:

Configuration / observation Actual role Single owner / distinction
EPS_DNFR_STABLE=1e-3 Absolute stored-pressure admission for capacity adaptation; equality is admitted Adaptation, with the value in RemeshDefaults. Its caller must refresh pressure when fresh admission is claimed. This is not the exact equilibrium equation p=0.
VF_ADAPT_MU=0.1, VF_ADAPT_TAU=5 Snapshot-average fraction and consecutive qualifying invocation count Core defaults. Five calls are not a physical duration without a declared call cadence; successful writes do not reset the count.
SELECTOR_THRESHOLDS Cuts for Si, network-normalized absolute pressure and acceleration The shared resolver serves selection, adaptation and Si aggregation. GLYPH_THRESHOLDS is a different policy, not another source of Si defaults.
PHASE_ADAPT Global/local phase controller settings Partial mappings inherit the same core defaults as full mappings; they must not activate an obsolete private fallback table.
DELTA_PHI_MAX, UM_MAX_PHASE_DIFF Selected U3 ceiling, with an optional tighter UM limit The phase gate resolver owns execution and diagnostic interpretation. The pi/2 admission ceiling is distinct from the pi wrap bound.
Potential, gradient and curvature warnings Selected pi/4, pi/16 and 0.9*pi comparisons Field constants; U6's pi/2 reference-relative potential drift is a separate observable. Equal units or related formulas do not make the policies interchangeable.
Legacy precondition names Some retained exports no longer drive execution Precondition defaults identify active fallbacks and inert compatibility values. The old one-radian RA phase warning does not override the live U3 gate.

In particular, selector pressure ratios and the absolute adaptation threshold must not be collapsed into one constant. Their denominators, scales and purposes differ. Nor should unrelated gains be coupled merely because their current numeric values match. Centralization means one owner for one semantic quantity, with explicit compatibility aliases where needed.

Unit changes also transform admission policies

Under Section 3's positive chart/time change y=a*x+b, tau=c*t, with consistently transformed pressure p'=a*p and capacity nu'=nu/c, equivalent absolute-pressure admission requires epsilon_p'=a*epsilon_p. A signed form-rate threshold transforms by a/c, an acceleration threshold by a/c^2, capacity rails by 1/c, form rails by a*x_bound+b, and a time step by c. Dimensionless blend fractions and invocation counts do not rescale, although changing the invocation schedule changes a hybrid model. The unchanged radian chart leaves phase-angle gates unchanged.

This proves why a bare positive pressure cut cannot be a chart-independent constant selected by the nodal product. It does not choose a replacement threshold or prove complete runtime covariance: the configured normalized pressure mixture and auxiliary laws retain Section 3's separate obligations. The existing covariance controls already test that boundary; there is no second numerical campaign here.

Corrected implementation boundaries

The audit retains complete default values while removing inconsistent interpretations. Partial selector configurations now use the shared selector defaults everywhere; omitted keys no longer fall back to a different glyph policy in adaptation or Si aggregation. Nonfinite, boolean or out-of-range selector thresholds are rejected instead of clamped or silently compared. Partial phase-adaptation configurations inherit the current default policy. U3 diagnostics share the live uppercase configuration keys and validation; invalid admission inputs cannot produce a successful structural check.

Malformed consumed selector, capacity, phase and numerical policies reject under the API admission contracts; custom solvers and callbacks retain their own scope. Missing Si or pressure is unavailable capacity-gate evidence, not an observed zero; missing capacity cannot populate the consumed neighbor snapshot. The capacity-law admission separates these invocation-based events from a supplied continuous relation. Clip-aware tolerance has EPI units; the legacy unclipped rate comparison has EPI/time units. A bare tolerance cannot be carried between these modes or treated as causal execution evidence.

Field threshold diagnostics retain observation availability. Unavailable coherence length is not a successful zero, fitted length is compared in its own path geometry, and spectral fallback retains separate units/provenance. A satisfactory curvature fit cannot override observed variance violations. Precision modes do not change intended mathematical definitions, but can change rounded decisions near a cut; a mode name supplies no certified error bound. The auxiliary pulse comparison likewise enforces its requested error tolerance and requires an independent oracle before reporting success.

Importing binary64 constants no longer changes the caller's mpmath precision. Cold-import controls at 15 and 80 decimal digits preserve that context and the previous binary64 pi/log(2) values. Numerical precision is part of the experiment's provenance, not a hidden side effect of configuration import. Legacy unused tolerance names do not certify integral convergence and the unused one-degree phase tolerance does not relax U3 execution.

The grammar calibration helpers require finite positive nonboolean capacity and step with represented product 0 < nu_f*dt <= 1. Their scalar recurrence is evaluated exactly on those represented coefficients; invalid, underflowed or oscillatory inputs do not produce a spurious calibration. The 64-position U4 cap remains a policy and need not achieve the surrogate's target. U2 uses the exact reciprocal floor of the represented rate, avoiding overflow and integer-boundary rounding. Neither helper establishes graph-wide relaxation or event occurrence.

2. Parameter and dependency ledger

Family Meaning and owner Established boundary
EPI x Coherent form in a declared chart; foundation types, scalarization Signed real storage and finite BEPI storage exist. Neither storage complexity nor temporal entropy selects the necessary physical state space. Differential equivalence needs a regular chart and transformed pressure/mobility; injective storage alone is insufficient.
Capacity nu Nonnegative local mobility/rate in xdot=nu*p; adaptation Zero freezes this unforced EPI channel; it need not destroy stored form or freeze all other channels. Held capacity is not a measured oscillation frequency (section 3.4). nu<=2*pi is a configured rail, not a phase theorem. No unique autonomous capacity law follows from the product; even a fixed pressure law can have distinct capacities with the same instantaneous EPI rate (section 3.2).
Pressure p Directed tangent response, evaluated from a declared constitutive map; dnfr, support transport The EPI channel is a graph difference. The full pressure does not by itself specify a joint potential or joint dynamics. Its sign is chart-dependent and does not name an operator. Stored pressure need not be freshly evaluated pressure.
Phase theta Circle coordinate and wrapped neighbor separation; phase response A common rotation is a symmetry of the regular difference/phasor formulas. Relative phase is independent of scalar EPI in the present representation. The nodal product does not imply theta_dot=nu or any synchronization law.
Clock t, dt Declared time coordinate and numerical increment; integrator, directed structural time Physical time, operator position, invocation count and history index are different. A step size alone supplies no stability guarantee. A changing clock rate transforms the capacity law too; synchronization need not be a monotone clock (section 3.1).
Channel coefficients p=w_phi*g_phi+w_epi*g_epi+w_vf*g_vf+w_topo*g_topo; defaults Numeric sum one is a selected normalization, not physical dimensional analysis. Priority phase/EPI/capacity is configured. See section 3.
Support G Declared neighbor relation, including zero-conductance support edges; relation foundations Phase and capacity support need not equal positive EPI transport support. Isolates, self-loops, multiple edges and directionality require explicit conventions. Support availability, instantaneous response and emergent effective interaction are distinct.
Conductance W Nonnegative form-transport weight; support transport Common positive rescaling leaves row-normalized EPI transport unchanged. It is therefore not an absolute interaction rate at held capacity. The zero-relation boundary differs between a first neighbor and a new bridge between already active regions.
Distance ell, kernel exponent Structural path-distance read-out; edge semantics Explicit length wins; absent length, weight is a compatibility fallback, then unit length. Undirected distances can be a pseudometric; distinct-node zero distances are omitted from the potential and coherence fit. Parallel lengths combine by minimum; directed distances follow outgoing arcs and can be asymmetric. No equation here derives distance from conductance or selects inverse-square exponent 2 uniquely.
Tetrad Potential, phase gradient, phase curvature, coherence length; fields Required complementary diagnostics, not a proved closed state or a four-dimensional complete basis. Section 5 states their different units and domains.
Coherence C, Sense Index Si Shared diagnostic conventions; metrics common, sense index Normalized read-outs are not new dynamical laws. Their use as feedback is a separately configured controller, including where an old runtime already does so.
dEPI, acceleration Stored rate or declared event secant, then rate difference per time A stored pre-projection rate may differ from the realized clipped increment. An event secant is not automatically the continuous derivative; acceleration inherits both timestamps and rate provenance.
Currents, energies, charges Graph contractions and auxiliary models; conservation, variational principle Positivity of a sum of squares does not prove decay. A charge label does not prove conservation or quantization. Units and a symmetry/action must be supplied for a physical Noether claim.
Memory and scale Hidden-coordinate elimination; EPI memory; declared jumps in REMESH Exact projected memory is model-dependent. REMESH mixing factors and integer history delays are selected maps, not the automatically derived kernel. Nesting alone is not self-similarity or an autonomous macro-NFR.
Operators and grammar Named transformation contracts, phase gates, word admission and runtime evidence; contracts, grammar bases Channel/sign contracts do not uniquely derive magnitudes, order or occurrence. U1-U6 combine definitions, conditions and policies. Completeness of the 13 transformations remains open.
Numerical and statistical controls Clipping, tolerances, discretization, regression bins, sampling and random seeds They define an experiment or implementation. Reproducibility is not physical necessity; small residuals need an error scale and cannot replace a proof.

Variable properties and unresolved foundations

The ledger defines quantities within models; it does not settle every property needed for a generative interpretation. A variable is admitted by its type, dependencies and allowed operations, not by its name or by a floating-point storage slot. For each realization, record:

  • Independent state, derived observation, held premise, input, diagnostic or numerical policy. An independent coordinate may also be held by a stated law.
  • Domain, units, chosen frame/chart, admissible transformations and the meaning of equality. Equality of observations need not mean equality of fine states.
  • Zero and singular boundaries, positivity/realizability constraints, and which combinations of quantities can coexist in one state.
  • The prospective evolution or algebraic map, its premises, and which properties are inherited under composition or a change of scale.

The outstanding foundational questions are narrower than "all variables are undefined":

Foundation question What can already be used What remains to justify
What structure does EPI retain? A declared signed scalar chart, regular chart changes and exact collective observations Adequacy of a selected state space, dimension, metric and equivalence of forms for the proposed substrate; EPI is not defined as an amount of coherence
What fixes capacity? Nonnegative mobility relative to a common clock, with explicit zero and positivity boundaries Its independent structural dependence and evolution, or a proved effective reduction; an observed angular speed or fitted product is not automatically capacity
What determines pressure? A specified tangent response evaluated from joint state, including the complete implemented mixture Which constitutive premises select that response and its scales; an evaluated p is distinct from the map P and from a stored event write
Which phase is being used? Primitive theta on the circle and separately derived contrast orientation psi Whether a chosen model retains primitive phase, derives it or omits it; their equality requires a map and matching induced dynamics, including zero-form boundaries
Where do time and relations come from? A declared clock, support, transport conductance and separate metric length Their structural origin and evolution; synchronization alone does not supply a monotone clock, and transport weight does not fix distance
Which collective properties can coexist? Means, amplitudes and relational correlations constructed from fine state Joint realizability, retained orientation/history and observation-specific closure; independently plausible pairwise values need not define one global state

The first five rows retain the proofs in the ledger's existing owners; they are not new failed tests or laws to fill in arbitrarily. In particular, signed form and phase already show why primitive phase cannot generally be discarded at zero EPI. The inherited metric already distinguishes coordinate mobility from primitive capacity. Reusing these results prevents another investigation based on an unproved identification.

The derived-form realization has its own compact variable contract and joint-realizability result. It is a conditional mathematical realization: its fine scalar form evolves, capacity/support are held, pressure is evaluated prospectively, primitive phase is unused, and the regional phase/correlations are derived. The reciprocal sine realization instead retains primitive phase and its feedback into form; its pulse results cannot be transferred to the phase-unused realization. Neither set of choices selects a fundamental physical substrate. Resolving a variable's mathematical admission and justifying its ontological role are distinct tasks.

2.1 The implemented pressure is a specified relational map

Let N_i be the unique outgoing neighbors, k_i=|N_i|, and let W be the EPI conductance. Define the support and transport differences by

\[ (G_Uv)_i=\frac1{k_i}\sum_{j\in N_i}(v_j-v_i),\qquad (G_Wx)_i=\frac1{d_i}\sum_jW_{ij}(x_j-x_i),\quad d_i=\sum_jW_{ij}. \]

An empty support row or zero transport strength gives the corresponding zero row. On a regular phase branch, with nonzero neighbor resultant,

\[ S_i=\sum_{j\in N_i}e^{\mathrm i\theta_j},\qquad g_{\phi,i}=\frac{\operatorname{wrap}(\arg S_i-\theta_i)}{\pi},\qquad p=a g_\phi+eG_Wx+bG_U\nu+tG_Uk. \]

Here (a,e,b,t) are the effective phase/EPI/capacity/topology coefficients of the declared configuration. The default recipe is (pi/(pi+1), pi/(pi+1)^2, 1/(pi+1)^2, 0); executable coefficients retain their represented normalization. Those numbers are configured choices, not consequences of the nodal product. Section 3 gives their different units.

Only the EPI difference reads transport conductance. Phase, capacity and degree contrast use unique support neighbors, including zero-conductance edges. A self-loop is one such neighbor; parallel edges aggregate EPI conductance without repeating that neighbor. U3 restricts particular operators, not the neighborhood used by this pressure reading. The pressure owner and support transport implement these conventions. Phase-resultant cancellation, wrap boundaries and represented arithmetic retain their execution-path-specific availability and numerical scope; this formula does not assert exact parity between all backends.

The cached dense support adjacency is binary: parallel-edge multiplicity must not weight the phase or topology mean. The shared neighbor-difference reducer also evaluates coefficient-weighted linear channels before judging their representability. A finite final pressure can exist even when an unscaled difference or normalized intermediate weight overflows or underflows. Graph and dense optimization adapters reuse this realization; multiplying a rounded Laplacian is a distinct floating-point calculation, despite the same exact-real model. Matrix certificates retain the representation stated in their hypotheses.

Capacity has two separate roles: G_U nu contributes a spatial pressure contrast, while diag(nu) multiplies the complete pressure in the EPI rate. The former is not a capacity evolution equation. Likewise, t=0 removes only the explicit degree-contrast channel: the other channels still depend on the graph, and EPI transport still depends on its conductance. Degree contrast is not a general curvature or a law for evolving topology.

Writing p=eG_Wx+F, where F=a g_phi+bG_U nu+tG_U k, isolates a source relative to EPI diffusion. It need not be external forcing: all its readings can belong to the same nodal state. Computing F still does not determine the evolution of its phase, capacity or support. For fixed reciprocal positive transport strength and held F, the EPI-sector potential V=e*x^T(D-W)*x/2-x^T*D*F gives xdot=-diag(nu/d)*grad_x(V). This conditional representation does not derive a joint potential or autonomous source law; the variational closure audit owns those additional obligations.

The engine assumes a support and initial nodal state on which these differences can be read. An isolated row has no neighbor-pressure contribution; an empty graph supplies no nodal equation from which to instantiate its first node. Emission acts on an existing node, and nested birth starts from an existing parent. Their names do not derive an initial substrate. If a larger whole is modeled as one NFR, its internal regions can supply relational pressure without an assumed external environment. Identifying that whole with physical reality, deriving its initial support or placing this structure before an independently defined physical clock remains a hypothesis. A prerequisite of the current model is not evidence for an additional substance beyond coherence.

2.2 Pressure execution and representation contract

The default scalar pressure rejects nonuniform or complex BEPI rather than projecting it to a magnitude; uniform-real embeddings keep their sign. The shared neighbor-difference kernels preserve representable linear differences and validate active conductances. Pressure is computed before node pressure writes, so a nonfinite assembled vector is rejected without partial pressure assignment. This is not a transaction over every preparation cache.

The phase path uses the signed atan2(sin(delta),cos(delta)) convention, retaining tiny displacements and signed antipodal ties. Centers outside the principal chart are read through their materialized phasors before subtraction; this prevents a huge unreduced coordinate from erasing the neighbor direction. The certified acute two-neighbor midpoint remains authoritative when available. Otherwise the consumed floating neighbor sums determine the argument. Exactly zero sums have the explicit computational extension of zero phase pressure; small nonzero sums are not suppressed by an undeclared epsilon. Neither this extension nor a tiny represented resultant derives a unique physical direction at an ideal cancellation. Reduction order and transcendental rounding can still matter; there is no claim of bitwise backend equivalence or exact runtime mean conservation. JIT execution disables fast-math reassociation for this reason.

Public DNFR_WEIGHTS updates are observed at the next default refresh, including in-place edits. The engine retains a detached configuration snapshot next to its effective mix; detached forcing observers use the same resolution without writing caches. A legacy explicitly supplied _dnfr_weights mix is retained until the public configuration changes after its first preparation. This compatibility override is not the recommended configuration interface. _DNFR_META.weights_effective records the executed coefficients; weights_norm only reports normalized proportions. The default pressure and selector score policies share strict nonnegative represented-real coefficient admission. Malformed mappings, Boolean/text coefficients, negative values and nonzero inputs lost during materialization reject before use. Finite large weights normalize in scaled coordinates when their sum overflows; they do not disable a valid mixture. A zero total retains the explicit uniform-mix policy. Setting every public coefficient to zero therefore does not disable pressure. These are configuration policies, not constitutive deductions.

Optional hooks must be named as different models where their formulas differ:

Entry point Actual contract
default_compute_delta_nfr Configured four-channel map above, with weighted EPI transport.
dnfr_phase_only Neighbor-argument phase channel with the same local midpoint and represented-zero rules; not a pairwise sine oscillator law.
dnfr_epi_vf_mixed Unweighted unique-support EPI/capacity differences with fixed half coefficients.
dnfr_laplacian Unweighted unique-support EPI/capacity differences with raw public coefficients, without renormalization.
compute_delta_nfr_hamiltonian Auxiliary projector-diagonal commutator readout, identically zero; see its algebraic scope. It is not a more fundamental generating pressure law.

NumPy, optimized NumPy, Torch and JAX graph-pressure adapters delegate to the default CPU owner; an adapter name does not certify GPU pressure or a distinct constitutive law. Backend, configuration, scalar-domain and failure controls live in the pressure read tests, phase path controls and the existing linear pressure suite. The actual Numba compilation checks explicitly skip where Numba is unavailable; executing the Python kernel body is a separate validation level.

3. Joint changes of form and time units

The type audit imposes a useful positive constraint. Hold support fixed and write the configured channel readings as

\[ p=w_\phi g_\phi+w_e g_e+w_\nu g_\nu+w_Tg_T. \]

Here g_phi is a wrapped angular displacement divided by pi, g_e is the weighted neighbor EPI difference, g_nu is the unweighted neighbor capacity difference, and g_T is the declared dimensionless topology reading. Thus

\[ [w_\phi]=[w_T]=X,\quad [w_e]=1,\quad [w_\nu]=XT. \]

For a,c>0, choose a common affine form chart and a constant time-unit change:

\[ y=ax+b\mathbf1,\quad \tau=ct,\quad \nu_\tau=\nu/c. \]

Then g_e scales by a, g_nu by 1/c, and the phase/topology readings are unchanged. Coefficientwise covariance of this constitutive family uses

\[ (w_\phi',w_e',w_\nu',w_T')= (a w_\phi,w_e,ac w_\nu,a w_T). \]

This gives p'=a*p and dy/dtau=(a/c)*dx/dt, as required. Keeping all numeric coefficients fixed fails in general, even for a change of time unit alone. Renormalizing the transformed coefficients to sum one also changes the rate by 1/S, where S is their transformed sum, unless a compensating factor is explicitly introduced. The production normalized-weight interface therefore is not automatically covariant under physical unit changes.

The tests in test_nodal_parameter_covariance.py exercise the existing support/forcing owners with exact rational arithmetic on their retained finite coefficients. They include wrong-fixed-coefficient and renormalization counterexamples; they do not claim exact transcendental phase evaluation or derive new dynamics. Bounds, thresholds, external inputs and every auxiliary law would also need transformation to establish covariance of a complete runtime.

The same distinction applies to phase: theta_dot=nu treats the numeric capacity as an angular rate. If nu instead counts cycles per time, the conversion is theta_dot=2*pi*nu. Neither choice follows from xdot=nu*p; their relationship requires a stated constitutive premise. Writing angles in radians gives the exact pi bound, not a universal ceiling on a rate or a unique coefficient for other channels.

A dimensionless invariant of the selected two-channel family. With joint storage E_D+beta*V_phi, form storage has units X^2 and phase cost is dimensionless, hence [beta]=X^2. The coefficient audit derives chi=beta*(e/w)^2, invariant under the declared form/time changes, including compensated engine weight normalization. This does not reduce the full four-channel pressure or all preparation/geometry data to one number.

At uniform form and phase consensus, the selected law has real, critical or complex nonzero-mode poles separated by chi=4/pi^2. This is conditional on the phase-row and Arg/pi normalization, not a universal physical constant forced by the circle alone. Real poles do not imply monotone observations; nonconsensus phase geometry can change the boundary. Capacity separability, storage balance, clock covariance and complete synchronized replicas leave different chi values admissible.

The same owner's memory and pole identity connects chi to the hidden-phase memory kernel and two temporal poles of one resolved spatial mode. This can identify a parameter within an admitted model, not derive its universal value. Independent calibration and reserved evaluation remain distinct; no coefficient fit or physical realization is supplied here.

3.1 Capacity, clock and positivity require compatible laws

A common smooth clock change tau=T(t), alpha=T'(t)>0, preserves the unforced nodal row when nu_tau=nu/alpha and pressure is evaluated at the corresponding original state. If the pressure itself depends on capacity, its constitutive map must also be transformed; substituting nu_tau into the old formula is not that transformation. For a twice differentiable T,

d nu_tau / d tau = nu_dot/alpha^2 - nu*alpha_dot/alpha^3.

Thus a nonconstant clock change generally changes even the form of a supplied capacity law. The constant-unit result above is its special case. Phase laws, event times and memory arguments must transform as well. The existing capacity-exposure result and time-varying diffusion theorem already delimit when a common activity clock is removable. Independent local activity parameters do not eliminate the need to specify which neighbor state is read at the shared time.

Synchronization alone cannot be a universal clock: the existing positive P2 diffusion solution with equal fixed phases has phase order R=1 at every time while its EPI contrast changes. A clock reconstructed from a structural observable requires a specified domain and a strictly monotone reading there; an equilibrium or repeated reading does not supply elapsed time. This excludes that universal identification, not the possibility of restricted internal clocks.

Nonnegative capacity is a state-domain premise. A supplied locally Lipschitz continuous-time law nu_dot=F(z) must point inward, F_i>=0 at nu_i=0, to preserve the nonnegative orthant while its solution exists. Nonnegative initial data alone do not impose this condition. Multiplicative laws such as nu_dot=nu*g(z) with finite integrated g preserve positivity, but do not uniquely follow from the nodal equation. A clamp is a configured projection; positivity of a continuous flow also does not guarantee positivity of an arbitrarily large explicit Euler step. Neither a positive upper rail nor a phase-to-capacity conversion is selected by these domain requirements.

3.2 An EPI rate need not identify capacity, even with a fixed pressure law

On unit P2 with consensus phase, let delta=x_1-x_0>0, e=w_epi>0, b=w_vf>0, and f=e*delta. Both nodes have the same degree, so the actual default topology and phase channels vanish. For capacities nu=(2u,u), the declared exact pressure and rate are

p = (f-b*u, -f+b*u),
x_dot = (2u*(f-b*u), -u*(f-b*u)).

The distinct positive choices u=f/(4b) and u'=3f/(4b) give identical nonzero EPI rates with threefold capacity and one-third pressure. This is an ambiguity inside one fixed prospective pressure law, not a retrospective definition p=x_dot/nu or freedom to refit its coefficients. It concerns one instantaneous response, not equality of complete trajectories.

There is an independent prospective discriminator. Hold each supplied capacity profile and phase fixed. Since the initial EPI rates coincide, p_dot=e*G_U*x_dot coincides and is nonzero. The nodal product then gives x_ddot=diag(nu)*p_dot, so the second profile has threefold acceleration. The shared joint-response owner checks this consequence before any trajectory or calibration. Its exact controls treat the materialized e,b as rational coefficients and declare rational capacities; they do not assert exact equality after arbitrary binary64 materialization.

Support conventions introduce another, narrower ambiguity. Pure EPI diffusion on unit P2 with capacities (1/2,1/4) equals the EPI generator on the same off-diagonal conductance with capacities (1,1) and self-loop weights (1,3). In both, mobility nu_i/d_i=(1/2,1/4), the off-diagonal Dirichlet form and its energy rate coincide. Primitive capacity is therefore not identified by that transport budget if holding conductances are unrestricted. With the same consensus phases, the full default law distinguishes these states through its capacity-gradient channel; pure transport equivalence is not equivalence of the complete nodal state. Controls are centralized in constitutive capacity scope.

A supplied local relation nu=g(EPI) is a different, constrained state model. The capacity-balance owner derives its conserved coordinate, response-slope boundary and conditional restoration criterion. These results test a proposed relation; they do not select it or transfer to independently prepared capacities.

3.3 Pressure, clock and capacity charts must transform the whole law

On fixed support and a smooth admitted chart, write a declared full model as z_dot=F(z), with rows x_dot=nu*P(z), theta_dot=Omega(z) and nu_dot=A(z). The product is componentwise. A change d tau/dt=alpha(z)>0 and a capacity chart kappa_i=q_i(z)*nu_i preserve this model only when the coordinate map is regular and invertible and

\[ P_{{\rm new},i}=\frac{P_i}{\alpha q_i},\qquad \Omega_{\rm new}=\frac{\Omega}{\alpha},\qquad \frac{d\kappa_i}{d\tau} =\frac{q_i A_i+\nu_i\mathcal L_Fq_i}{\alpha}. \]

Here L_F q_i=Dq_i*F; all old-state dependencies must be evaluated through the coordinate inverse. Positive q_i alone does not ensure invertibility: q_i=1/nu_i on a positive-capacity domain collapses that coordinate to one. The pressure-preserving choice q_i=1/alpha recovers section 3.1, replacing alpha_dot by L_F alpha for a state-dependent clock. Such a choice is a valid state chart only where its coordinate map is regular; orbit time reparameterization itself does not require changing the capacity coordinate.

The product's apparent factorization freedom therefore does not supply a missing capacity or phase law. Even if A=0, transformed capacity need not be held. For example, pure P2 contrast a_dot=-2*nu*a, nu_dot=0, under q=1+a^2, has kappa=q*nu, pressure -2*a/q and necessarily kappa_dot=-4*a^2*kappa^2/q^2. At a=nu=1, the correct a_ddot is 4; incorrectly holding transformed kappa=2 gives zero. This is coordinate covariance, not a new physical capacity mechanism.

The existing joint-response owner already differentiates the implemented pressure along supplied form, phase, capacity and conductance rates and computes x_ddot=nu_dot*P+nu*P_dot. Its pressure derivatives do not select those rates. A new law, such as the cotangent correction, needs its own differentiated term; the old pressure derivative cannot silently stand for it. Support events and nonsmooth branch changes require separate event/domain handling.

Common phase rotation is also invisible to the configured pressure and tetrad: on their regular charts D_theta P * 1=0. Adding a common phase velocity is not identifiable from pressure differentiation alone. A relative phase quotient may be declared; an absolute lifted phase clock needs a reference and temporal evidence. Conversely, a genuine difference of joint orbit direction cannot be hidden in a common clock, as the exact exchange-family discriminator shows. The same-tetrad future witness then separates clock/law ambiguity from lost state under one fixed law.

3.4 Derived pulse scales and observation periods

A pulse is a property of a complete trajectory, not a fourth primitive coordinate added to the form/capacity/phase triad. The finite-amplitude doubled-C5 pulse is derived from the existing reciprocal sine rows with held common positive capacity, fixed support and zero loss. Its frequency depends on geometry, amplitude, capacity and the declared exchange/storage coefficients. Capacity scales time in that family, but neither theta_dot=nu nor frequency=nu follows. A physical frequency additionally needs an independent clock bridge.

The observation must be specified before assigning a period. The full labeled fine state has period T; its exact unordered-pair state has minimal period T/2 because a half-cycle exchanges the two constituents. A mean observation can remain constant throughout the same motion. These are different retained descriptions of one trajectory, not different microscopic clocks. A common clock change transforms all rows and periods; neither a periodic read-out nor synchronization alone gives a globally monotone time coordinate.

Existence and robustness are separate properties. The full transverse calculation retains internal and collective feedback and proves instability of the prepared waveform for sufficiently small nonzero amplitude, without a numerical amplitude radius. Changing common capacity or time units cannot remove that dimensionless instability at fixed normalized amplitude. The positive-loss obstruction and reversible alternatives also preclude treating permanent oscillation as a consequence of positive capacity alone. Zero loss remains a declared law premise, not a measured or uniquely derived microscopic property.

4. What locality and symmetry can derive

The pressure note owns the derivations, assumptions, counterexamples and finite-response comparison. Locality, covariance and regularity restrict a declared response family; they do not uniquely select the four-channel mixture or primitive phase law.

5. Geometry and the whole tetrad

5.1 Potential and local phase fields

At fixed metric kernel, Phi_s=B_G*p is linear in pressure. For inverse-square distance, [Phi_s]=X/L^2 if distances carry length units. Rescaling all distances by k>0 scales potential by k^-2; independent conductance rescaling does not change it when every edge has an explicit fixed length. With the old weight fallback those two changes are coupled by representation, not by a theorem.

|grad phi| is a mean absolute wrapped neighbor separation, not a derivative per unit metric length. K_phi=wrap(theta_i-Arg sum_j exp(i theta_j)) is a circular discrepancy; both are bounded by pi in the radian chart. Interpreting them as spatial differential operators requires a length convention and a limit. A nonzero resultant and a regular branch are needed for the displayed phase-response derivative. At very small resultants the current curvature read-out must distinguish ill-conditioning from an undefined direction. The shared observe_phase_curvature interface now sums its materialized binary64 phasor components exactly and records that numerical scope. Every nonzero represented resultant has a reported numerical direction; the former 1e-9 arithmetic-angle fallback is removed. At exact represented joint zero, the evidence marks curvature unavailable and numeric curvature/full-field adapters raise UndefinedPhaseCurvatureError. The gradient remains available; an isolated node has the explicit empty-neighborhood zero convention.

Exact summation of materialized components is not exact transcendental evaluation or a conditioning guarantee. For example, represented phases (0,0,+float(pi),-float(pi)) can have exactly cancelling represented phasors although their exact-real trigonometric resultant is nonzero. Unavailability describes the chosen numerical realization; it does not prove a physical singularity. Pressure, IL and coordination dynamics retain their separately declared kernels. This correction selects no new phase evolution law.

5.2 One coherence-fit definition across implementations

The scalar and vector implementations now share _coherence_fit.py. The fitted field is the static pressure-only read-out c_i=1/(1+|p_i|), with zero rate argument; it is not generally the full runtime coherence. The fitted statistic is

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

This is an uncentered product fit, not a connected covariance or a universal correlation-decay theorem. The shared distance channel uses explicit length, then compatibility weight, then one, taking the minimum of parallel lengths and following outgoing arcs. Undirected graphs use unordered pairs; directed graphs use ordered reachable pairs. Off-diagonal zero-distance pairs are excluded: undirected zero-length paths give a pseudometric rather than a separating metric, and directed distances need not be symmetric. Negative or nonfinite edge lengths and unrepresentable reachable distances raise rather than silently selecting the spectral fallback. A caller-supplied distance matrix must satisfy the explicit numeric domain, zero diagonal and undirected symmetry; it remains declared data, not authenticated shortest paths.

The field guide owns fit acceptance, sampling, supplied-distance admission and cache dependencies. These are estimator and implementation policies, not physical thresholds.

The previous implementations used hop distances versus conductance distances, and the vector path discarded one orientation on directed graphs. The common owner removes this disagreement. Tests use a weighted star with c_center=1, c_leaf=exp(-r_leaf/xi): every pair product is exactly the target exponential in real arithmetic. This checks a known fit, metric scaling, explicit-length independence, directionality and cache refresh without inferring a physical transition from synthetic data.

The spectral fallback remains a different diagnostic: 1/sqrt(lambda_positive) from a normalized graph Laplacian is dimensionless. It does not scale with explicit metric length and is not interchangeable with the fitted length. The legacy scalar return alone does not attest which estimator succeeded; research must retain the estimator path and conditions. Disconnected/directed graphs need their stated implementation scope rather than an automatic connected symmetric lambda_2 interpretation.

6. Telemetry, time and energy are not interchangeable

The shared coherence map C(p,r)=1/(1+|p|+|r|) is a chosen normalized numeric diagnostic. Before nondimensionalization, p and r=xdot have different units; their raw sum is not a unit-invariant physical observable. Declaring scales for the two inputs would make the definition meaningful under unit changes, but selecting those scales is still a model/measurement decision.

Network coherence applies this kernel to mean magnitudes. It generally differs from mean per-node coherence; a projected parent has another pressure again. Freshness also matters: stored pressure, stored rate and a newly recomputed pressure are not automatically simultaneous. The zero-pressure/rate predicate is not a certificate that phase, capacity and topology are at a fixed point. The shared P2 joint-response control starts with zero pressure and EPI rate, yet a supplied relative phase velocity gives nonzero pressure rate and EPI acceleration. It is an explicit counterexample to promoting the instantaneous predicate into joint equilibrium, not an autonomous evolution law. Primary coherence and dispersion now share strict authoritative-alias reads: invalid provided pressure/rate values raise instead of becoming a later alias or zero; genuinely missing values retain the public zero default.

The field guide owns authoritative source admission, detached cache reads and partial-field availability. Invalid or unrepresentable inputs cannot establish equilibrium.

Numerical realization must also be controlled before interpreting weak fields as emergence. Shared circular differences retain tiny nonzero separations; structural source reductions retain finite inverse-square responses across large and small represented distances. SDK pressure summaries use the same stable linear-reduction and dispersion owners. Backend-dependent zeros, infinities or inconsistent branch signs cannot serve as evidence for a new phase transition. These controls support the prospective phase/form comparison without selecting either candidate law.

The separate dispersion diagnostic 1-std(p)/max|p| is scale-invariant in exact arithmetic. Its duplicated implementations previously squared raw pressures and overflowed or underflowed: (m,2m) produced 0.75, 0, or 1 as the numeric scale changed. One shared kernel now normalizes before computing variance. The same fixture retains 0.75 at extreme and subnormal represented scales, with sign and backend controls. This repairs a numerical identity; it does not turn dispersion into the primary coherence or a law.

Si combines relative capacity, phase dispersion and relative pressure with configured weights and clipping. It changes when the comparison population or normalizers change. The corrected Python path refreshes the same live capacity and pressure maxima as NumPy instead of trusting stale maxima. This repairs backend-dependent values without promoting Si to an autonomous mechanism. Existing selectors/adaptation that read Si remain declared feedback policies.

The default integrator holds supplied pressure and capacity fixed within a call. Its rk4 option integrates the supported time forcing, not a newly evaluated nonlinear pressure at every RK stage. Clipping is a separate map; an unconstrained rate is not the clipped step secant. With the optional additive Gamma source the model is xdot=nu*p+Gamma: at positive capacity it can be rewritten using p_eff=p+Gamma/nu, but that rewriting is singular at zero capacity. Nonzero Gamma can move a zero-capacity node. The four forcing-scope controls verify the distinction through both integration backends. Unforced proofs require Gamma to vanish; source residuals cannot be concealed by reconstructing pressure.

The API contracts own registry-based Gamma evaluation, cache/dispatch consistency and callback/quadrature boundaries.

The solver contract owns capacity/rate admission, advancing represented clocks and restoration on numerical failure. Those safeguards do not derive a physical clock or capacity law.

Capacity-rate telemetry now uses the shared timestamped observer. For advancing recorded times, two capacity samples define the interval secant r_n=(nu_n-nu_(n-1))/(t_n-t_(n-1)). Three define B_n=2*(r_n-r_(n-1))/(t_n-t_(n-2)), using the separation of the two interval midpoints. Exact rational arithmetic on represented samples avoids inventing a duration from configured DT or rounding midpoint timestamps together. Finite public results remain approximations to those rational values. With [nu]=T^-1, these diagnostics have units T^-2 and T^-3. Under tau=c*t, nu_tau=nu/c, their values scale by c^-2 and c^-3, respectively; the irregular-sample controls check that covariance.

Each node retains at most three samples and a capacity_rate_diagnostic payload with status, availability and source times. Missing time/capacity, insufficient samples and unrepresentable results are explicit unavailable values (None). A duplicate time creates no interval; changed capacity at the same time restarts history at the right-hand value. Backward time or invalid input is rejected before capacity diagnostics are committed. The metrics callback preflights capacity/time chronology before recording its history. It records an initial sample on attachment only when runtime time is already present. Legacy untimestamped derivative fields are not evidence.

history['B'] is unavailable unless every current node supplies a representable second difference; capacity_rate_coverage reports the coverage. A genuine measured zero remains distinct from unavailable. delta_Si remains an increment, not a time derivative. The repaired secants can still contain unobserved events between endpoints, so they do not prove a smooth capacity law or exact pointwise derivatives. Event resolution and physical clock calibration remain separate obligations.

Likewise E_D=1/2*x^T(D-W)x is a conditional transport energy, while the tetrad sum of squares and auxiliary substrate Hamiltonian are other functionals. Adding quantities with different raw units requires scales/metric coefficients. Packing curvature and current into a complex number supplies neither their unit conversion nor a physical quantum state. The existing variational and conservation owners already separate trajectory residuals, auxiliary flows and restricted dissipation; reuse those distinctions for every new claim.

The joint sine storage is different from an auxiliary Hamiltonian: its balance follows from the complete form/phase rows themselves. Zero loss conserves that storage while permitting internal exchange; positive loss has its stated dissipation term. Conservation alone gives neither a particular waveform nor its stability, and does not identify this structural storage with physical energy.

7. Memory, operators and scale

Eliminating hidden coordinates in a declared linear system produces an exact memory kernel and a hidden-initial-state term. These are derived from that system and projection; no new sustaining mechanism has been discovered by renaming the kernel. The existing derived-memory analysis and epi_memory owner test continuous closure and minimal realizations. Event closure additionally requires the existing event-intertwining check.

Keeping internal state can instead give an exact nonlinear collective law. In the unordered replica description, R=cos(delta), U=u^2 and Q=u*sin(delta) retain internal phase dispersion, form contrast and their correlation. R enters the inherited phase-to-form coupling through the derived product R_i*R_j; its dynamical role follows from the fine law, not from a controller reading a coherence score. Q is not nodal pressure. This constrained state removes only pair labels, retains the continuous degrees of freedom and requires its local phase chart; R or the means alone do not close the dynamics.

REMESH instead references retained pre-jump snapshots at integer positions history[-(tau+1)]. A delay in samples becomes a fixed physical delay only under a declared uniform sampling convention. Variable cycle durations cannot be ignored. The mixing coefficient, history support, clipping and metric are part of its map; existing finite/conditional stability certificates already record those premises. They do not derive REMESH from every projected kernel.

The 13 operators specify allowed named transformations. They coexist with declared shared nodal solvers. Reciprocal scalar IL/OZ defaults do not make the full operators inverse or isometric. Reciprocal NUL capacity/pressure factors preserve their product only for the ideal two scalar rescalings; capacity is not a geometric volume. Generator/closure labels, operation-count debts and phase gates supply admission contracts, not spontaneous event timing.

Noise parameters, finite event counts and seeds likewise require a stated discrete or stochastic model. A fixed per-call perturbation is not a time-step-independent continuum noise law. A tolerance is not an exact zero, and clipping-generated persistence is not evidence for unconstrained stability.

8. Reuse and consequence for the generative objective

The review preserves a useful core: typed nodal rates, reversible transport, phase geometry, exact reduction/memory and finite event evidence. The new locality and covariance results constrain proposed completions without choosing one to manufacture a pattern. The repaired telemetry permits consistent tests of those completions; it does not supply the missing laws.

Any further proposed representation must declare its state/equivalence, explain which quantities are coordinates and which are observations, and test its directed response with the existing closure and phase-response owners. A nonuniform finite-amplitude pattern maintained indefinitely cannot be inferred from fixed positive-capacity pure diffusion on fixed connected positive-conductance support, which relaxes spatial disagreement. This does not exclude a declared finite-lived identity. Sources, finite accumulated activity, memory, changing geometry or another justified channel can alter that conclusion, but their origin and work balance must be part of the same model. This is the link to the original generative objective, not permission to add a tuned stabilizer or a desired attractor.

The execution plan alone records the current gate and next action. A universally selected multichannel closure, a unique selection of all numerical coefficients, and emergence of laboratory particles or the observable world remain unproved. The audit covers foundational parameter families and their principal owners; it is not an exhaustive proof of every implementation or historical document in this repository.

Stable destinations for links published before modularization

Current repository links point directly to the proof owner. These compact locators preserve earlier release, external and archived references.