TNFR Structural Operators¶

Complete Specification of the 13 Canonical Operators¶

Status: CANONICAL registry specification; generative completeness over an independently defined admissible-transformation space remains open Version source: pyproject.toml Prerequisite: FUNDAMENTAL_THEORY.md §2 (Nodal Equation), UNIFIED_GRAMMAR_RULES.md (Grammar U1–U6)

Table of Contents¶

Scope and Motivation
Operator Algebra from the Nodal Equation
The Operator Taxonomy
Generators
Integrator
Stabilizers
Destabilizers
Coupling and Propagation
Transformers
Closure and Regime Operators
Fragments and Complete Words
Operator Roles and Energy Diagnostics
Postcondition Contracts
Operator Constants Reference
Implementation Reference
Summary
Experimental Operator-Tetrad Synergies
Mechanism and Activation Audit

1. Scope and Motivation¶

Registered structural operators are the canonical semantic interface for named node and network transformations. Declared numerical solvers are a separate, explicit evolution path: they may advance EPI only through the shared nodal integrator from finite nu_f and DeltaNFR, while recording the associated provenance or residual. Ad hoc state assignment outside these two declared paths is forbidden. Both paths remain constrained by the nodal equation:

\[ \frac{\partial \text{EPI}}{\partial t} = \nu_f \cdot \Delta\text{NFR}(t) \tag{NE} \]

Each registered operator implements a specific transformation tied to (NE). The 13-operator catalog covers the engine's declared transformation categories: creation, integration, stabilization, destabilization, coupling, propagation, capacity attenuation/expansion, self-organization, phase transformation, regime transition, and multi-scale recursion. Coverage of this declared catalog does not prove that it generates every independently admissible TNFR transformation.

1.1 Why the registered catalog has 13 members¶

The registry contains 13 public semantic transformations satisfying:

Nodal equation compatibility: Every transformation must declare its effect on EPI, $\nu_f$, $\Delta\text{NFR}$, phase, or the coupling structure.
Grammar closure: The set must include generators (U1a), closures (U1b), stabilizers (U2), destabilizers (U2), coupling operators (U3), bifurcation triggers and handlers (U4), and multi-scale operators (U5).
Semantic distinction: Each member has a named executable contract. The temporal-signature owner can compare implementations on declared probes; any separation result retains those probes and observations. Global algebraic irreducibility remains open.

A proof of completeness first requires a transformation space defined without reference to this catalog. This remains the open catalog-completeness boundary (historical S10); the theory catalog routes related contract and scope questions to their owners.

1.2 Conventions¶

Throughout this document: - Glyph codes (AL, EN, IL, ...) reference the structural symbols. - Lowercase English tokens (emission, reception, ...) are executable API identifiers; title-case names are public display/class names. - Operator gains and thresholds are configured parameters. $\pi$ is the half-turn in the chosen radian phase chart, not a derivation of those parameters. - Grammar roles reference rules U1–U6 from UNIFIED_GRAMMAR_RULES.md. - Energy diagnostics and their proof boundaries are discussed in STRUCTURAL_STABILITY_AND_DYNAMICS.md.

2. Operator Algebra from the Nodal Equation¶

2.1 Structural Triad¶

Each node $i$ carries the declared structural triad. This representation is not a proof that the three attributes remain independent under every richer form representation or specified quotient:

Attribute	Symbol	Domain	Units
Form	$\text{EPI}_i$	Declared form space; scalar glyphs require a signed real chart or uniform-real BEPI embedding	Declared form unit X
Frequency	$\nu_{f,i}$	Nonnegative reals; individual operators may require strict positivity	Hz_str
Phase	$\phi_i$ (or $\theta_i$)	$[0, 2\pi)$	rad

The derived quantity $\Delta\text{NFR}_i$ (structural pressure) drives evolution. Every operator has exactly one registered primary channel among $(\text{EPI}, \nu_f, \phi, \Delta\text{NFR})$; a concrete implementation may also update declared secondary channels.

2.2 Operator as Transformation¶

An operator is an event map on its declared full input state $z$, which can include neighbors, graph data and history. Its local channel projection is $\sigma_i=(\text{EPI}_i,\nu_{f,i},\phi_i,\Delta\text{NFR}_i)$:

\[ \hat{O}: z\mapsto z',\qquad \sigma_i'=\operatorname{proj}_i(z'). \]

subject to: 1. Flow/event distinction: Continuous segments obey the declared nodal law; direct EPI-writing events contribute separate jumps. Stored pressure after an event need not equal freshly evaluated constitutive pressure. A local channel tuple alone need not determine an operator's result. 2. Grammar constraints: The operator must satisfy its declared U1--U5 role; U6 is evaluated from before/after $\Phi_s$ telemetry. 3. Contracts: Pre-conditions and post-conditions specific to each operator.

2.3 Composition¶

Operators compose left-to-right. A word's grammar acceptance does not replace live state admission, and order can change both acceptance and outcome.

The execution path determines the composition:

A public single-node operator selects its admitted action before subclass execution; its generic call is not a graph transaction (§15.2).
A supported all-target stage builds proposals from one immutable snapshot, validates them and commits graph-owned state together. A fallback or explicit execution override can instead use the declared sequential stage. A word remains ordered even when its individual stages are simultaneous.
A hybrid event schedule treats each glyph as a zero-duration jump and places elapsed time only in the declared continuous intervals. A same-time form jump is not a measured finite-time secant for Mutation.
The REMESH glyph is advisory. Delayed EPI mixing is a separate invocation whose state includes the retained history; its one-step gain cannot be multiplied across changing histories without the corresponding lifted proof.

These are execution contracts, not an autonomous invocation law. The operator-event contract owns schedules, transaction boundaries, pressure refresh, stage evidence and causal sequences. Use its atomic-stage section for snapshot/target-order guarantees and its history section for the exact and represented REMESH distinctions. Captured graph-owned state and emitted external effects have different rollback boundaries.

Finite evidence must bind the actual endpoints, ordered support, declared clock and common metric. It does not automatically prove a global binary64 runtime gain, future repetition, solver accuracy or full-channel stability. The mathematical owners are the diffusion and affine-reset theorem and REMESH history derivation; their hypotheses remain necessary when composing their results.

3. The Operator Taxonomy¶

The sections below group operators by their declared function. Groups can overlap: for example, an allowed word closure need not be an equilibrium, and a U2 stabilizer need not decrease a chosen diagnostic energy.

The grammar role registry owns the role assignments. The generated contract table owns tokens, primary channels, measurement contexts and postconditions. Neither classification is a list of every secondary write. The mechanism map in §18 distinguishes those writes from supplied selection policies and possible collective interpretations.

4. Generators¶

Generators declare creation or latent-state activation. Grammar rule U1a requires that a standalone sequence beginning from $\text{EPI}=0$ start with a registered generator.

Scope: The nodal derivative $\nu_f\Delta\text{NFR}$ is defined at $\text{EPI}=0$ whenever its factors are finite. U1a is an operator-history contract that makes the origin or reactivation of form explicit; it is not a repair for a mathematical singularity.

4.1 Emission (AL)¶

Physics: Declared activation by a positive EPI increment on an existing node. Zero EPI is a coordinate value, not absence of the nodal substrate; the map does not derive creation from a vacuum.

Primary-channel transformation:

\[ \text{EPI}' = \operatorname{clip}(\text{EPI} + b) \]

where $b>0$ is the configured AL_boost. The direct glyph handler does not also write $\nu_f$, phase, or $\Delta\text{NFR}$.

Activation threshold: $\text{EPI} < \text{EPI}_{\text{threshold}}$ where the default $\text{EPI}_{\text{threshold}}=0.8$ is configurable through AL_MAX_EPI_FOR_EMISSION (an operational gate; the AL contract fixes the channel and sign, not this threshold).

Key constants:

Constant	Value	Derivation
Emission amplitude $b$	configured positive gain	Operational parameter; channel and direction are contractual
Activation threshold	$0.8$ by default	Configurable operational gate

Properties: - Irreversible: Sets an immutable activation flag. Re-emission increments an activation counter but preserves the original timestamp. - Genealogical: Maintains structural lineage tracking (origin timestamp, parent references, derived node list). - Latency-aware: Detects and clears silence (SHA) latency state on reactivation. - All-target time basis: One accepted AL stage assigns one common timestamp to all newly emitted targets; repeated targets retain their original origin.

Grammar: Generator (U1a).

Contract: - Pre: $\text{EPI} < 0.8$ and $\nu_f$ at or above the configured basal threshold when strict preconditions are enabled. - Post: EPI does not decrease; $\nu_f$, phase and $\Delta\text{NFR}$ remain unchanged; the activation flag is set.

4.2 Transition (NAV)¶

Physics: Configured adjustments selected by latent/active/resonant state labels. These operational labels do not establish dynamical attractors or prove transitions between them.

Public/staged transformation (regime-dependent):

Regime	$\nu_f$ change	$\theta$ shift	$\Delta\text{NFR}$ reduction
Latent → Active	+20%	$+0.1$	$-30\%$
Active → Active	configurable	$+0.2$	$-20\%$
Resonant → Active	$-5\%$	$+0.15$	$-10\%$

The direct NAV glyph only performs its declared pressure action. Regime and latency effects belong to the public/staged path. NAV_STRICT and NAV_RANDOM require actual Boolean values on all paths; see the shared option admission in §15.2.

Regime detection:

\[ \text{regime} = \begin{cases} \text{latent} & \text{if } \nu_f < 0.05 \text{ or latent flag set} \\ \text{resonant} & \text{if } \text{EPI} > 0.5 \text{ and } \nu_f > 0.8 \\ \text{active} & \text{otherwise} \end{cases} \]

Properties: - Latency recovery: When transitioning from latent state, verifies EPI drift against preserved snapshot (tolerance: 1% for established nodes, $0.330$ for initial nodes). - Regime traceability: Records origin regime, before/after state, and phase shift in telemetry. Pressure telemetry distinguishes the true pre-handler value, the low-level handler result, and the final regime-retained value. - All-target stage: One immutable proposal binds each target's nu_f, phase, DeltaNFR, latency state and optional jitter draw/RNG progress before any write. A missing graph seed is resolved inside the outer transaction, copied into the detached snapshot, restored on rejection and retained on success. Stable per-node offsets and draw counts make the committed RNG progress independent of target iteration order. - Shared latency clock: Every target computes silence duration from one stage instant. Ordered warnings, histories, transition events, metrics and monitor callbacks still follow requested target order. Cache state, the opaque pressure-refresh result and relabeling equivariance remain outside the structural target-order claim.

Grammar: Generator (U1a), Closure (U1b).

Contract: - Pre: Valid regime state detectable. - Post: Configured capacity, phase and pressure proposals; applicable latency metadata is cleared. No general future-coherence bound follows.

4.3 Recursivity (REMESH)¶

Execution: The named REMESH glyph records a network advisory with a declared recursion depth. It does not execute delayed mixing or create nested structure. The separately requested network history map below mixes supplied current and past form; it does not derive fractality or preserve an arbitrary pattern identity.

Separately requested network history transformation:

\[ \text{EPI}_{\rm raw}(t) = (1-\alpha)^2\text{EPI}(t) + \alpha(1-\alpha)\text{EPI}(t-\tau_l) + \alpha\text{EPI}(t-\tau_g), \qquad \text{EPI}_{\rm new}=\operatorname{clip}(\text{EPI}_{\rm raw}) \]

where the default is alpha = 0.5, but graph configuration can override it. The runtime validates 0 < alpha <= 1 rather than clamping, so the three raw coefficients are a convex partition and sum to one. Both delays must be strict positive integers; neither booleans nor fractional values are coerced.

The operation is guarded by max(tau_l, tau_g) + 1 stored snapshots. Before that point it returns an immutable insufficient-history no-op. Empty live support is a separate side-effect-free no-op. Once the guard passes, the selected local-lag and global-lag entries must each be a node-to-EPI mapping whose support equals the current graph support. Both are temporal per-node snapshots; the global lag is not a spatial network mean. Hard or soft structural clipping acts only after the raw recurrence and can make the committed map nonlinear.

Properties: - Depth parameter: Recursion depth $\geq 1$ (validated at construction; raises error if $< 1$). - U5 assessment boundary: Declared depth records recursive structure but does not prove a parent/child coherence target. For a concrete hierarchy, assess_u5_parent_child_coherence(..., alpha=...) evaluates $C_{\rm parent}\geq\alpha\sum C_{\rm child}$ from canonical per-node $C$; $alpha$ is explicit and has no graph-independent default. - Constant-history fixed point: identical current, local-delay and global-delay EPI snapshots are preserved before clipping. - Energy scope: for one positive diagonal metric h, let V_h denote weighted disagreement from the h-weighted mean. Convexity gives

$$ V_h(x_{\rm raw})\leq \beta V_h(x_0)+\gamma V_h(x_l)+\delta V_h(x_g). $$

This bound uses all three inputs. It is not a gain bound relative to the current state alone when delayed histories are free. With the two delayed vectors fixed, write b = gamma x_l + delta x_g. The raw map of the current vector preserves the consensus subspace, and has disagreement gain at most beta squared, when b is uniform. Otherwise a consensus current vector can be sent to positive disagreement, so no finite global multiplicative gain exists. Weighted-mean preservation is reported separately from disagreement. Clipping intervention is reported separately from the raw recurrence. None of these one-step facts establishes repeated-map stability, pressure closure, a structural-charge conservation law or U2 convergence.

Execution boundary: plan_network_remesh returns the immutable proposal; apply_network_remesh commits it and returns an immutable result. Strict validation precedes writes. A commit, metadata, history or propagated callback failure restores graph-owned state, topology, caches and capturable callback state. External effects already emitted by callbacks are outside this rollback boundary. Exact evidence values that exceed the finite binary64 diagnostic range are rejected explicitly before commit. The function reads the EPI history but does not append or shift it.

Grammar: Generator (U1a), Closure (U1b).

Contract: - Pre: Declared depth $\geq 1$, configured minimum support and applicable U5 context. The network history map has its additional admission above. - Post: The node-level call records an advisory. Explicit network mixing has the stated conditional bounds, not a universal nested-identity guarantee.

Fixed-delay Cesàro surrogate (historical N15 programme):

For a finite cyclic history window, remove clipping, fix $(\tau_l,\tau_g,\alpha)$ with $0<\alpha<1$, and let $S$ be the unitary cyclic shift. The auxiliary filter

\[F=\beta I+\gamma S^{\tau_l}+\delta S^{\tau_g}\]

is a normal contraction. Its Cesàro averages converge to the orthogonal projection onto $\ker(I-F)$. If the window length is compatible with the delays, those fixed modes satisfy both $z^{\tau_l}=1$ and $z^{\tau_g}=1$; their periods therefore divide $\gcd(\tau_l,\tau_g)$; they are not determined by the historically stated $\operatorname{lcm}(\tau_l,\tau_g)$.

This finite cyclic filter is not the runtime history-update map. In particular, it does not prove a literal $\tau_g\to\infty$ limit for apply_network_remesh, whose required history, selected snapshot and clipping change with $\tau_g$. Orthogonality gives contraction only in the surrogate's declared history norm; it does not conserve the TNFR structural charge, make the structural candidate energy monotone, or provide a universal convergence rate.

The projection can be computed without registering another engine operator. That is a software-reuse observation, not a catalog-completeness theorem. A proof that the 13 operators exhaust admissible TNFR transformations remains an open inverse problem requiring an independently defined transformation space.

Full corrected analysis: REMESH_INFINITY_DERIVATION.md §§1–23. It retains the historical N15 milestones and commit anchors (a1f298fd, badac156, 48b0574a) while separating them from the current result.

5. Integrator¶

5.1 Reception (EN)¶

Physics: Captures and integrates incoming neighbour resonance from the network environment. EN does not write $\Delta\text{NFR}$; a later pressure refresh is a separate realization boundary.

On directed support, an arc $j\to i$ makes $j$ an incoming EN input for receiver $i$: the numeric snapshot reads predecessors and source discovery follows the same source-to-receiver path orientation. The current affine realization certificate remains restricted to undirected support.

Runtime transformation: let r(EPI) denote the centralized real-scalar glyph reader. It is defined for raw finite real values and uniform-real BEPI embeddings, retaining their signed value. Genuinely nonuniform or complex BEPI values have a maximum-component magnitude for generic read-only diagnostics, but Reception rejects them before proposing a blend. Let B_i=ensure_bepi(EPI_i) be the accepted target operand. Reception first forms the unweighted arithmetic mean of the runtime neighbours,

\[ \bar r_i = \frac{1}{|N_i|}\sum_{j\in N_i}r(\mathrm{EPI}_j), \]

then evaluates and structurally clips

\[ x_i' = \operatorname{clip}\!\left(\operatorname{float} \left((1-m)B_i+m\bar r_i\right)\right). \]

Edge weight does not enter this neighbour average. For every uniform-real scalar embedding, including negative EPI, float(B_i)=x_i; the runtime and pure-EPI scalar readings therefore share the same signed state. With 0<=m<=1, fixed support and values inside the declared EPI interval, the ideal real update is affine and its hard clip is inactive by convexity:

\[ \text{EPI}' = (1 - m) \cdot \text{EPI} + m \cdot \text{EPI}_{\text{in}} \]

where $m$ is the configured reception mixing fraction (the standard graph configuration uses the selected $1/(\pi+1)$ policy value).

Key constants:

Constant	Value	Derivation
Mixing fraction $m$	configured; standard policy $1/(\pi+1)$	Operational interpolation weight
Convexity term	$m(1-m)$	Algebraic mixing diagnostic, not a full-energy contraction theorem

Properties: - Source detection: Detects emission sources within configurable distance (default: 2 hops) via detect_emission_sources(). - Runtime/transport separation: Reception's unweighted neighbour mean can differ from the conductance-weighted mean defining pure-EPI diffusion. - Affine certificate boundary: certify_reception_epi_realization() keeps the ideal real map, represented binary64 coefficient matrix and actual two-stage runtime value separate. It promotes the represented map only when the declared scalar chart and clipping conditions pass and its consensus-row identity is exact. Snapshot agreement with the runtime is diagnostic, not a global floating-point linearity proof. - Pressure refresh: If a nontrivial local update changes the target by delta, retaining the pre-update pure-EPI pressure leaves the exact defect delta L_rw e_i. On connected positive conductance this is nonzero, so the pressure must be recomputed before subsequent evolution is treated as the pure-EPI channel. - Immediate coherence contract: The EN jump writes EPI, semantic kind and optional source metadata while leaving stored $\Delta\text{NFR}$ and $d\mathrm{EPI}$ unchanged. Because canonical $C(t)$ reads only those two channels, its operator-local value is unchanged at that boundary. This says nothing about $C(t)$ after pressure refresh or along a later trajectory. - Source telemetry: Records detected sources in metadata for analysis.

Grammar: Integrator (no active destabilizer/stabilizer role).

Contract: - Pre: Target EPI and signed pressure lie below their configured Reception admission ceilings. Isolation and an empty detected-source set are admitted rather than treated as hard failures; graph-backed execution with source tracking enabled emits one advisory for the latter. - Post: Neighbour EPI is integrated; stored pressure and change rate are unchanged, hence immediate operator-local $C(t)$ is unchanged.

6. Stabilizers¶

Stabilizers are assigned negative-feedback roles intended to reduce U2 debt. Grammar rule U2 requires that every destabilizer ({OZ, ZHIR, VAL}) be compensated by a stabilizer ({IL, THOL}); this finite-word condition does not by itself prove that $\int \nu_f\Delta\mathrm{NFR}\,dt$ is bounded for every runtime realization.

6.1 Coherence (IL)¶

Implemented map: Contracts stored $|\Delta\text{NFR}|$. The low-level glyph preserves EPI, capacity and phase; the public class and shared stage also apply the configured phase proposal below. Bounded future evolution still requires a specified law, gains and event schedule.

Transformation:

\[ \Delta\text{NFR}' = \Delta\text{NFR} \cdot (1 - \rho) \]

where the default retention is $1-\rho=\pi/(\pi+1)\approx0.7585$, hence $\rho=1/(\pi+1)\approx0.2415$. The gain remains configurable.

Public/staged phase locking:

\[ \theta' = \theta + \lambda \cdot \text{wrap}\!\left(\bar{\theta}_{\mathcal{N}} - \theta\right) \]

where the default is $\lambda=0.3$, with $0\leq\lambda\leq1$, and $\bar{\theta}_{\mathcal{N}}$ is the unweighted neighbor circular mean. Setting the coefficient to zero disables angular displacement. Isolates have no aligning direction; a joint-zero resultant uses the declared current-phase fallback. The same kernel owns direct public and staged proposals; an all-target stage reads an immutable snapshot.

Key constants:

Constant	Value	Derivation
ΔNFR retention factor $f$	$\pi/(\pi+1)\approx0.7585$	Default `IL_dnfr_factor`; configurable gain
Pressure-square reduction	$1-f^2\approx0.425$	Isolated $\Delta\mathrm{NFR}^2$ term at the default; not a full-energy bound
Phase locking $\lambda$	$\approx 0.3$	Configurable coupling strength

Storage and observations: The pressure-only glyph leaves $E_D+\beta V$ unchanged because that functional does not contain stored pressure. For one public IL target on simple undirected support, with all neighbor phases held fixed, write $R$ for their resultant magnitude and $a\in[-\pi,\pi]$ for the target's wrapped displacement from its direction. The ideal phase contribution is

\[\Delta V=R\bigl[\cos a-\cos((1-\lambda)a)\bigr]\leq0.\]

This single-target argument does not certify simultaneous phase changes, subsequent flow or represented transcendental rounding. Contracting stored pressure with unchanged stored EPI rates cannot decrease the configured structural $C$ read-out. An ensuing pressure refresh is a separate operation and need not preserve that comparison. Telemetry retains structural $C$, the separately named legacy pressure-dispersion statistic, pressure reduction and phase-locking observations.

Grammar: Stabilizer (U2); Bifurcation Handler (U4a).

Contract: - Pre: Active structure exists. - Post: Stored $|\Delta\text{NFR}|$ is not increased at the contraction; EPI and capacity are preserved. No future-flow or autonomous-trigger claim.

Implementation: shared IL kernel, public lifecycle; stage and signed-pressure controls.

6.2 Self-Organization (THOL)¶

Physics: An invoked pressure-reorganization map with conditional nested child creation. Its acceleration threshold, invocation and hierarchy policy are supplied inputs; executing THOL does not derive autonomous emergence.

Its primary contract channel is $\Delta\text{NFR}$ with direction reorganize:

\[ \Delta\text{NFR}'=\Delta\text{NFR}+a\,\frac{\partial^2\text{EPI}}{\partial t^2}. \]

The instantaneous magnitude can rise or fall with the signed acceleration. THOL's U2 stabilizer role comes from organizing a bifurcation while preserving global form; it is not a universal one-step pressure contraction.

Bifurcation detection and history:

\[ \widehat A=\frac{2}{h_1+h_2} \left(\frac{x_2-x_1}{h_2}-\frac{x_1-x_0}{h_1}\right), \qquad h_j=t_j-t_{j-1}>0. \]

The shared compute_d2epi_dt2 reads the latest three active samples. An available timestamped history is authoritative and must end at the current EPI. Legacy untimed histories retain the unit-step difference x2-2*x1+x0; fewer than three active samples mean unavailable acceleration, represented by zero. Cached curvature does not replace this history in graph-backed THOL. The estimate compares prior nodal rates; it does not add an independent acceleration equation.

When $|\widehat A|>\tau$, the public operator proposes sub-EPIs subject to hierarchy-depth and domain checks. The node-protocol primitive implements only the pressure channel; graphless nodes supply their own acceleration. Both paths use the shared checked pressure proposal in _thol_pressure.py.

Sub-EPI creation:

\[ \text{EPI}_{\text{sub}} = \text{EPI}_{\text{parent}} \cdot 0.3 + \text{contribution}_{\text{metabolic}} \]

where $0.3$ is the operational fractal scaling factor (a free parameter). The child is a nested coordinate with an explicit parent reference; it is not an additive mass term. THOL leaves the parent's scalar EPI coordinate and all neighbour EPI coordinates unchanged. Propagation of form belongs to Resonance (RA) and must appear explicitly in a grammar-valid word.

Key constants:

Constant	Value	Derivation
Fractal scale	$\approx 0.3$	Operational fractal nesting (free parameter)
U5 hierarchy coefficient $\alpha$	explicit per assessment	No graph-independent default
Sub-$\nu_f$ damping	$0.95$	Child inherits 95% of parent frequency
Bifurcation threshold $\tau$	configurable (default $0.1$)	From graph configuration
Pressure-reorganization gain $a$	$1/(4\pi)\approx0.0796$	Default `THOL_accel`; operational gain

THOL_MIN_COLLECTIVE_COHERENCE is a deprecated, inert compatibility alias for the general fragmentation-risk cut. It is not read by THOL and does not supply the explicit U5 coefficient.

Properties: - Nested creation: Creates child coordinates with hierarchy metadata (bifurcation level, hierarchy path, parent reference). Independent persistence and autonomous maintenance do not follow from this event. - Metabolic integration: When enabled, captures network signals and metabolizes them into sub-EPI values. - Amplitude-alignment telemetry: compute_subepi_amplitude_alignment returns $1/(1+\operatorname{var}(a_{\rm child}))$ for stored child EPI amplitudes (or zero when fewer than two exist). It is neither canonical $C(t)$ nor a U5 decision. The historical compute_subepi_collective_coherence name is only a compatibility alias. - Explicit U5 telemetry: assess_u5_parent_child_coherence evaluates $C_{\rm parent}\geq\alpha\sum C_{\rm child}$ only when the hierarchy and $\alpha$ are supplied. THOL does not enforce a hidden collective threshold. - Depth-limited: Maximum nesting depth prevents unbounded recursion (default: 5 levels). At the limit THOL still applies its signed pressure reorganization but records that no child was created. - Channel-pure: The parent and its neighbours keep their EPI coordinates; any subsequent form propagation is an explicit RA stage.

The child has no transport edge until a later admissible coupling action. Default structural pressure reads actual neighbors, not hierarchy membership. Likewise, a refresh before integration overwrites THOL's direct pressure increment; holding that increment through a physical interval instead gives the additional exact reference change h*nu_f*THOL_accel*A_hat. See THOL pressure and feedback for the execution-order budget, disconnected zero modes and finite implementation evidence.

Grammar: Stabilizer (U2); Bifurcation Handler (U4a); Transformer (U4b).

Contract: - Pre: Finite supported state and valid configuration. When enabled, the optional strict gate additionally checks configured EPI/capacity minima, positive stored pressure, connectivity and at least three active history samples. Exceeding the acceleration threshold is not required to execute the pressure action. See the shared THOL precondition owner. - Post: Parent EPI, capacity and phase are preserved; pressure follows the signed acceleration proposal. Child creation additionally requires its threshold, depth and proposal checks; a successful THOL may create no child.

7. Destabilizers¶

Destabilizers increase pressure or capacity stress under their contracts. Grammar rule U2 requires compensation by a stabilizer as a syntactic safety policy. Integral convergence additionally needs a declared pressure law, operator gains, timing, and state-space bounds.

7.1 Dissonance (OZ)¶

Implemented map: Changes stored pressure while preserving EPI, capacity, phase and support. Deterministic amplification, fixed near-zero seeding, additive noise and network propagation are distinct branches.

Transformation:

\[ \Delta\text{NFR}' = \begin{cases} f\,\Delta\text{NFR},&|\Delta\text{NFR}|>10^{-9},\\ 0.1,&|\Delta\text{NFR}|\leq10^{-9}. \end{cases} \]

where the default amplification is $f=(\pi+1)/\pi\approx1.3183$, the reciprocal of the default IL retention. It remains an operational gain with $f>1$. The fixed cutoff and nonzero seed are implementation choices, not a derived spontaneous source. With OZ_NOISE_MODE=True, positive OZ_SIGMA instead gives $p'=p+\mathrm{jitter}$; nonpositive sigma leaves the local pressure unchanged. An additive sample can decrease $|p|$, so amplification's local magnitude postcondition is not a theorem for the noise branch. OZ_NOISE_MODE uses the shared Boolean parser on direct, public and stage paths; a supported false string selects the deterministic branch. Unknown Boolean strings reject before branch effects (§15.2).

Bifurcation read-out: The implementation compares the magnitude of its observed structural acceleration with the configured threshold $\tau$. A flag records that comparison, not a bifurcation theorem for an underlying continuous law. U4a separately requires handler context for the trigger token.

Key constants:

Constant	Value	Derivation
Amplification factor $f$	$(\pi+1)/\pi\approx1.3183$	Default `OZ_dnfr_factor`; configurable gain
Pressure-square increase	$f^2-1\approx0.738$	Isolated $\Delta\mathrm{NFR}^2$ term at the default; not a full-energy bound

Properties: - Network propagation: The public class and shared stage can add positive increments to neighbors through phase-weighted, uniform or frequency-weighted policies. Positive incoming increments can cancel a negative local pressure; the local proposal and final accumulated pressure have separate contracts. - Bifurcation detection: Monitors $\partial^2\text{EPI}/\partial t^2$ against threshold $\tau$. - Telemetry: Records propagation events, affected nodes, and bifurcation flags.

Grammar: Destabilizer (U2); Bifurcation Trigger (U4a); Closure (U1b).

Contract: - Pre: Consumed finite pressure and the active branch's admitted factors. Enabled strict preconditions additionally impose configured EPI, capacity, pressure and overload policies; these do not prove absence of a dynamical bifurcation or future instability. - Post: The deterministic local branch increases pressure magnitude. Noise and incoming propagation require their separately retained observations. An acceleration-threshold flag is a configured diagnostic, not a bifurcation proof or an endogenous instruction to execute another operator.

Changing stored pressure alone leaves $E_D+\beta V$ unchanged. Recomputing pressure from unchanged constitutive inputs can erase the event's pressure write; persistent forcing or pressure memory must be supplied explicitly. Implementation: local glyph, public propagation, shared stage; signed cancellation and noise controls.

7.2 Expansion (VAL)¶

Implemented map: Multiplies capacity by the configured factor. At held pressure this multiplies the nodal product by that factor. The default EDGE_AWARE_ENABLED=True branch also scales and projects EPI; disabling it leaves EPI untouched. Neither branch directly writes pressure, phase or support.

Transformation:

\[ \nu_f' = f_{\text{VAL}}\nu_f, \qquad f_{\text{VAL}}=\text{VAL\_scale}>1 \]

With edge awareness enabled, the form factor is limited by the configured sign-specific EPI boundary, then the selected hard/soft projection is applied. Thus a positive factor does not guarantee an increase in signed EPI: negative form, saturation and projection must retain their actual endpoint. This is an event map, not an increase of graph or state-space dimension. With edge awareness disabled, $E_D+\beta V$ is unchanged at the event; with it enabled, the actual EPI reset determines the storage change.

Key constants:

Constant	Value	Derivation
Default scale factor $f_{\text{VAL}}$	$1+1/(4\pi)\approx1.0796$	Operational capacity step
Nominal square-factor change	$f^2-1\approx0.166$ at the default	Algebraic $\nu_f^2$ diagnostic; the structural candidate energy has no explicit $\nu_f$ term
Optional strict minimum EPI	$1/(2\pi) \approx 0.159$	Supplied `VAL_MIN_EPI` admission policy

Grammar: Destabilizer (U2).

Contract: - Pre: When strict preconditions are enabled, configured capacity saturation, minimum stored pressure and EPI, and optional network-size checks apply. These policies are not consequences of the nodal identity. - Post: Capacity is not decreased; no EPI, graph or state-space dimension increase follows from multiplying $\nu_f$. U2 compensation and prefix-debt limits still apply.

Implementation: shared VAL/NUL kernel; enabled/disabled boundary and signed-form controls.

Optional VAL telemetry. expansion_metrics reports signed relative form and capacity changes only when their denominators are nonzero. Missing stored acceleration and undefined growth ratios remain None; present invalid values and nonfinite or unordered thresholds reject the observation. Stored acceleration has unverified provenance and is not a temporal bifurcation certificate. coherence_above_threshold reads the current immediate-neighbor proxy, without a pre-event baseline. growth_ratio_within_policy applies a configured numeric band, not a derived fractality test. The legacy coherence_preserved and fractal_preserved keys alias these respective policies. expansion_healthy is False when a known policy fails, None when no failure is known but required evidence is unavailable, and True only when every policy passes; assessment_status, failed_policies and unavailable_policies separate those cases. This is optional post-event reporting, not a new VAL precondition, rollback promise or future-health theorem. The focused telemetry controls cover signed form, invalid authoritative aliases, policy boundaries and absence.

8. Coupling and Propagation¶

Coupling operators propose or use phase-compatible interactions between nodes. Grammar rule U3 verifies circular separation: $|\operatorname{wrap}(\phi_i-\phi_j)|\leq\Delta\phi_{\max}$.

8.1 Coupling (UM)¶

Execution: Proposes configured circular phase alignment across admitted neighbors and, when requested, functional links for subsequent interaction.

Transformation:

\[ \phi_i' \to \phi_j', \qquad |\operatorname{wrap}(\phi_i - \phi_j)| \leq \Delta\phi_{\max} \]

Two separate gates: U3 compares wrapped phase gaps with the admitted circular limit, $\pi/2$ by default. When functional links are enabled, UM_COMPAT_THRESHOLD separately gates the proposed link's configured form/Si/phase score; its default is $\pi/(\pi+1)\approx0.7585$. That score threshold is not a phase angle or an autonomous attachment law.

Key constants:

Constant	Value	Derivation
Functional-link score threshold	$\pi/(\pi+1) \approx 0.7585$	Configured `UM_COMPAT_THRESHOLD`; only the enabled link branch consumes it
Phase push	$1/(\pi + 1) \approx 0.241$	Configured circular-alignment gain; shared numeric value with EN mixing
$\Delta\text{NFR}$ reduction	$1/(2\pi)\approx0.1592$	Default phase-alignment pressure-relief gain

Properties: - Phase verification mandatory: Circular separation above the configured U3 limit rejects coupling. This gate alone supplies no wave-amplitude or destructive-interference calculation. - EPI identity preserving: Form is not modified; phase alignment, optional capacity synchronization, pressure relief and functional-link proposals have separate configured effects. - $\nu_f$ synchronization: Optional frequency alignment across coupled nodes.

UM_BIDIRECTIONAL, UM_SYNC_VF, UM_STABILIZE_DNFR and UM_FUNCTIONAL_LINKS control separate optional effects. Their shared admission is in §15.2; disabling an effect cannot bypass U3.

Grammar: Coupling (U3); requires phase verification.

Contract: - Pre: The non-disableable U3 gate requires an existing phase-compatible neighbor. Enabled strict preconditions additionally check configured EPI and capacity thresholds. Link candidates retain their separate score and post-merge phase admission. - Post: Configured circular phase proposals; EPI preserved. Optional functional proposals may add admitted links. Neither global phase-spread decrease nor the creation of at least one edge follows from a successful invocation.

8.2 Resonance (RA)¶

Physics: Propagates coherent patterns through phase-aligned nodes. It may change the target's scalar EPI through neighbour mixing while preserving the pattern's signed/kind identity, and it conditionally amplifies local structural frequency.

Let the active RA neighbourhood be the individually U3-compatible subset

\[ N_i^{\mathrm{U3}}= \{j\in N_i:|\operatorname{wrap}(\theta_i-\theta_j)|\leq\Delta\phi_{\max}\}. \]

The runtime rejects an isolate or an empty compatible subset. Its configured phase limit must satisfy $0\leq\Delta\phi_{\max}\leq\pi/2$; configuration may tighten but cannot relax the canonical gate. Only this subset contributes to the EPI mean, circular phase mean, and frequency trigger. With $r=\texttt{RA_epi_diff}$ and $c=\texttt{RA_phase_coupling}$,

\[ \bar x_i=\frac{1}{|N_i^{\mathrm{U3}}|}\sum_{j\in N_i^{\mathrm{U3}}}x_j, \qquad x_i^*=\operatorname{clip}\!\left((1-r)x_i+r\bar x_i\right), \]

\[ \theta_i^*=\operatorname{wrap}_{[0,2\pi)} \left(\theta_i+c\,\operatorname{wrap}(\bar\theta_i-\theta_i)\right). \]

Here $\bar\theta_i$ is the unweighted circular mean of compatible phases. If the runtime's exact antipodal-pair guard or exactly zero represented resultant marks that mean undefined, phase remains unchanged; this operational guard is sufficient for those cancellation classes and is not a universal symbolic zero-resultant test. When $|\bar x_i|>10^{-9}$, the selected runtime trigger,

\[ \nu_{f,i}^*=(1+a)\nu_{f,i},\qquad a=\texttt{RA_vf_amplification}. \]

The $10^{-9}$ trigger is operational binary64 policy, not a derived structural constant. Before EPI, phase, $\nu_f$, history, or RA metadata changes, the runtime requires

\[ 0\leq r\leq1,\qquad a\geq0,\qquad0\leq c\leq1, \]

finite scalar or uniform-real BEPI operands, a finite nondecreasing proposed frequency, and the identity gate below.

Key constants and domains:

Quantity	Default	Runtime domain / status
EPI mix $r$	$1/(2\pi)\approx0.1592$	`[0,1]`; convex propagation
Frequency amplification $a$	$1/(8\pi)\approx0.0398$	`[0,+infinity)`; nondecreasing capacity
Phase coupling $c$	$1/(4\pi)\approx0.0796$	`[0,1]`; shortest-arc interpolation
U3 phase limit	$\pi/2$	`[0,pi/2]`; may only be tightened
Capacity trigger	$10^{-9}$	Operational binary64 threshold on $\lvert\bar x_i\rvert$

Identity contract:

Scalar change is allowed: accepted RA replaces the target scalar by the neighbour blend; preservation does not mean “EPI without alteration.”
Sign identity: a strict negative-to-positive or positive-to-negative proposal is rejected. Exact zero is neutral: arriving at or departing from zero does not invert a pre-existing nonzero sign.
Kind identity: an established nonempty epi_kind must remain exactly the same. None and the empty string mean absent kind and may be initialized by the existing neighbour/fallback convention.
Independent gates: sign compatibility cannot compensate for a kind violation, or conversely. A failure is atomic for nodal state, glyph history, and RA telemetry.

Runtime-to-theorem boundary:

EN and RA are the first two catalog bridges. They reuse one neutral unweighted-neighbour EPI kernel even when pure-EPI diffusion uses a conductance-weighted mean.
certify_resonance_epi_realization() separates four layers: the ideal-real convex EPI blend, the affine matrix assembled from represented binary64 coefficients, the actual two-stage binary64 proposal, and the accepted identity-gated runtime snapshot. Only a represented map whose exact consensus identity and declared scalar/clipping domain pass enters the affine jump theorem.
A successful local frequency boost generally changes the post-RA metric $h_i=d_i/\nu_{f,i}$. The fixed post-RA pure-EPI flow can still be certified; arbitrary switching between pre/post generators is withheld unless their represented metrics are exactly proportional. A displayed recovery time is only an estimate; the Boolean fixed-post-flow recovery verdict comes from the exact hybrid composer.
If accepted RA changes target EPI by $\delta$, retaining the prior pure-EPI pressure creates the exact defect $\delta L_{rw}e_i$. Pressure must be refreshed before subsequent evolution is interpreted as that isolated channel. This does not identify or certify stored multichannel $\Delta NFR$.
Two-stage rounding, structural clipping, U3/identity gates, and the phase and frequency channels prevent a global binary64 runtime-affinity claim.

Global $C(t)$ telemetry: Optional before/after measurements report the realized network coherence response; propagation alone does not fix its sign.

Grammar: Propagation (U3); requires phase verification.

Contract: - Pre: Scalar or uniform-real coherent EPI; active edge; at least one U3-compatible neighbour; factor, phase-limit, identity, and finite-state gates pass. - Post: Scalar EPI may move within its preserved sign/kind identity class; $\nu_f$ is nondecreasing and is amplified when the compatible neighbour field crosses the operational trigger; any $C(t)$ change is measured rather than assumed.

9. Transformers¶

Transformers execute declared state reorganizations that require recent destabilizing context and, for ZHIR, a prior coherent base. A phase reset or threshold flag is not by itself a mathematical bifurcation of a continuous law. Grammar rule U4b uses the configured three-operation recency window; this is finite-word bookkeeping, not a measured energy threshold or universal relaxation time.

9.1 Mutation (ZHIR)¶

Physics: Controlled phase transformation. The nodal equation supplies the instantaneous prediction

\[ (\partial_t\mathrm{EPI})_{\mathrm{pred}} = \nu_f\,\Delta\mathrm{NFR}. \]

This prediction describes the current capacity-pressure product. Its strict comparison with $\xi$ is a predictive diagnostic; it is not evidence that a growth rate has been observed and does not admit ZHIR by itself.

Observed temporal evidence: the non-disableable runtime trigger instead uses the last two finite scalar EPI samples to form the signed secant

\[ \widehat{\partial_t\mathrm{EPI}}_{\mathrm{obs}} =\frac{\mathrm{EPI}_{k}-\mathrm{EPI}_{k-1}}{t_k-t_{k-1}}. \]

Timestamped epi_time_history is physical-time evidence. It requires finite samples, a finite strictly positive interval, and a final sample that represents the current EPI endpoint (exactly under the runtime default; the pure certificate can declare a finite nonnegative absolute endpoint tolerance). When this channel is supplied it is authoritative: invalid or stale physical evidence does not fall back to an untimestamped history.

For compatibility, epi_history and then _epi_history can provide legacy evidence. Their two samples are separated by one operator step, so the calculation reduces to EPI[k] - EPI[k-1]. Such evidence is explicitly marked physical_time_resolved=False and carries no physical-time or current-endpoint claim.

The observed comparison is signed and strict: $\widehat{\partial_t\mathrm{EPI}}_{\mathrm{obs}}>\xi$. Equality, contraction, sub-threshold growth, missing evidence, invalid samples, a non-increasing physical timestamp, or a stale physical endpoint reject direct execution before its phase mutation. The gate also requires finite active capacity $\nu_f>0$; an explicitly configured ZHIR_MIN_VF may tighten that condition. The default ZHIR_THRESHOLD_XI = 0.1 is an operational calibration, not a structural constant; a configured replacement must be finite and nonnegative.

The pure MutationTriggerCertificate and the SDK nodal_state() read-out keep the predicted and observed rates and crossings separate. In particular, near_bifurcation is a legacy SDK alias of the predicted crossing, while mutation_threshold_satisfied reports only the observed strict threshold. Unavailable or invalid evidence leaves observed_crossed=None instead of inventing a negative observation. rate_gap = observed - predicted is reported only for valid physical-time evidence, where those rates share a time basis; it remains unavailable for a legacy operator-step difference. Neither interface evaluates the prior-IL/recent-destabilizer requirements and neither certifies U4b execution readiness.

When a dynamic selector proposes ZHIR without valid crossing evidence or active capacity, it substitutes IL before ordinary grammar enforcement and records the requested and applied glyphs with the reason in mutation_abstentions. The SDK whole-word runner applies a stronger atomic boundary: before executing the first operator of a word containing ZHIR, it checks the temporal gate for every target node. It also rejects timestamped evidence when an earlier EPI-channel operator in that word would make the endpoint stale before ZHIR; legacy unit-operator-step evidence retains its compatibility interpretation. This preflight does not replace grammar validation or U4b. The fluent SDK's apply_evidence_gated_mutation() adds an explicitly experimental policy on top: when evidence is absent, stale, non-crossing or attached to inactive capacity, it executes a declared complete word without ZHIR and records the per-node evidence and abstention reason. It never constructs history; malformed evidence still raises, and direct sequence execution remains strict.

Phase transformation:

\[ \theta'=\operatorname{wrap}_{[0,2\pi)}\!\left( \theta+s\,\operatorname{sign}(\Delta\text{NFR})\right) \]

where the default calibrated magnitude is $s=(1/\pi)(\pi/4)=1/4$ radians unless an explicit fixed shift is supplied. The displayed sign formula concerns nonzero pressure. The represented kernel uses copysign(1.0, pressure): +0.0 selects the positive shift and -0.0 the negative shift. Zero current pressure therefore does not suppress an otherwise admitted event whose preceding observed growth crossed the threshold. The runtime preconditions and U4b history determine whether the mutation is admissible; $|\Delta\text{NFR}|$ does not scale this phase step.

Native occurrence additionally depends on selection. The heterogeneous-capacity admission study separates fresh diagnostic compatibility, declared histories, an executed default prefix and the direction of the resulting phase action.

The phase calculation and its branch-specific _zhir_* telemetry payload use one RNG-free immutable kernel. At the all-target boundary, the complete frozen proposal also binds the temporal threshold evidence, structural acceleration and U4 context to one stage snapshot. The stage commits phase and acceleration before merging histories, provenance, metrics and bifurcation events in requested target order. Its primary structural result is target-order invariant before the opaque pressure refresh; the ordered auxiliary streams and relabeling equivariance remain outside that result.

Bifurcation monitoring: compute_d2epi_dt2 estimates structural acceleration from three EPI-history samples. Timestamped histories use adjacent secant slopes over their physical intervals; legacy histories use the unit-step second difference. The result can flag bifurcation potential when it exceeds $\tau$. This acceleration diagnostic is distinct from both the instantaneous nodal prediction and the two-sample observed ZHIR gate.

ZHIR_BIFURCATION_MODE="detection" is the only supported mode. The legacy "variant_creation" value is rejected before any write. ZHIR therefore remains a phase-only transformation with bifurcation detection and telemetry; topology or sub-EPI creation belongs to THOL and must be expressed through that operator.

Key parameters:

Parameter	Default	Runtime role / status
Signed observed-growth threshold $\xi$	`ZHIR_THRESHOLD_XI = 0.1`	Strict operator-admission calibration; operational, not structural
Direct activity condition	$\nu_f>0$	Zero capacity cannot execute the phase transformation
Optional direct minimum	absent (`ZHIR_MIN_VF = 0`)	Tightens the activity condition only when configured explicitly
Branch-selection threshold	`ZHIR_BIFURCATION_VF_THRESHOLD = 0.5`	Selects whether the bifurcation router proposes ZHIR; it is not a ZHIR operator precondition
Default phase shift	$1/4$ rad	Calibrated magnitude; $\Delta\text{NFR}$ supplies direction only

Grammar: Transformer (U4b); Bifurcation Trigger (U4a).

Contract: - Pre: active finite $\nu_f>0$ (and ZHIR_MIN_VF, only when explicitly configured); valid two-sample temporal evidence satisfying the signed strict inequality $\widehat{\partial_t\mathrm{EPI}}_{\mathrm{obs}}>\xi$; prior IL for the stable base; and a recent destabilizer inside the configured U4b window. Temporal evidence is either fresh timestamped physical evidence or legacy unit-operator-step evidence under the compatibility contract. The normal grammar route or the strict precondition route enforces the two U4b context requirements. - Post: phase $\theta$ shifted; the structural identity epi_kind preserved; bifurcation potential flagged if $\partial^2\text{EPI}/\partial t^2>\tau$. After successful dispatch, source_glyph records ZHIR as provenance. That provenance field does not define or overwrite epi_kind.

10. Closure and Regime Operators¶

10.1 Silence (SHA)¶

Implemented map: Attenuates capacity without changing EPI, stored pressure, phase or support. Finite attenuation need not stop later evolution. Even along a family with $\nu_f\to0$, vanishing nodal rate requires control of the product $\nu_f\Delta\mathrm{NFR}$, not of capacity alone. A subsequent pressure refresh may depend on the changed capacity; nonzero declared forcing is separate.

Transformation:

\[ \nu_f'=f_{\text{SHA}}\nu_f,\qquad f_{\text{SHA}}=1-1/(4\pi)\approx0.9204 \]

EPI is preserved by leaving it untouched. The public class and shared stage record a latency snapshot; this metadata does not enforce future preservation.

Key constants:

Constant	Value	Derivation
$\nu_f$ suppression factor	$1-1/(4\pi)\approx0.9204$	Operational `SHA_VF_FACTOR`

Properties: - Latency state: Activates a latent flag with timestamped EPI snapshot. - Reactivation bookkeeping: Later lifecycle code can inspect and clear latency attributes. A drift tolerance or wall-clock timestamp is not a structural-time evolution law or a physical preservation theorem. - All-target time basis: One accepted SHA stage gives every target the same latency-start timestamp while preserving a target-specific EPI snapshot.

Grammar: Closure (U1b).

Contract: - Pre: Consumed capacity is finite and nonnegative. When strict preconditions are enabled, capacity must meet SHA_MIN_VF; no critical-pressure gate or nonzero-EPI requirement is imposed by that precondition. - Post: Capacity is not increased; EPI, pressure and phase remain fixed at the event, with the declared latency metadata. Finite attenuation need not produce zero capacity or freeze subsequent nodal evolution.

Since the joint storage $E_D+\beta V$ has no capacity term, SHA alone has zero instantaneous change in this functional. It changes subsequent mobility under an admitted law rather than supplying event work or deciding when UM occurs. Implementation: SHA proposal and lifecycle; shared lifecycle controls.

10.2 Contraction (NUL)¶

Physics: Attenuates capacity and rescales stored pressure by the reciprocal configured factor. This changes neither graph dimension nor the dimension of the EPI state space; “contraction” names this declared map.

Transformation:

\[ \nu_f'=f_{\text{NUL}}\nu_f,\qquad f_{\text{NUL}}=1-1/(4\pi)\approx0.9204 \]

Stored pressure is multiplied by the reciprocal capacity factor:

\[ \Delta\text{NFR}'=\frac{1}{f_{\text{NUL}}}\Delta\text{NFR} \approx1.0865\,\Delta\text{NFR} \]

The ideal product $\nu_f'p'=\nu_fp$ is preserved for this stored-pressure map. It is not a statement about pressure after constitutive refresh. As with VAL, EDGE_AWARE_ENABLED=True additionally multiplies EPI by $f_{\mathrm{NUL}}$ and applies the configured projection; the disabled branch does not consume or change EPI. Capacity-only contraction leaves $E_D+\beta V$ unchanged; the enabled form reset has its own storage budget. Local contraction toward zero need not reduce differences from neighboring forms.

Key constants:

Constant	Value	Derivation
Scale factor	$1-1/(4\pi)\approx0.9204$	Same operational $\nu_f$ step as SHA
Densification factor	$1/f_{\text{NUL}}\approx1.0865$	Reciprocal configured capacity factor

The immutable scale proposal binds the requested capacity factor, its exact binary64 reciprocal, the pre/post pressure and any EPI-boundary adaptation before commit. Each densification audit event carries its target identifier. Contraction metrics use their captured pre-operation pressure, or the latest matching target event for compatibility, and report signed pressure change separately from magnitude increase. Unchanged zero pressure is therefore not misreported as densification.

Grammar: Simplifier (no active grammar role; supports VAL reversals).

Contract: - Pre: Admitted capacity, pressure and any consumed scalar form. Enabled strict preconditions additionally impose configured minimum capacity and EPI plus a maximum density policy; these are not universal safety bounds. - Post: Capacity is not increased; stored pressure follows the checked reciprocal proposal, with zero pressure remaining zero. Subsequent pressure refresh is a separate constitutive operation.

Implementation: shared scale proposal; reciprocal coefficient and branch controls.

11. Fragments and Complete Words¶

Operators compose left-to-right. A named fragment is reusable syntax; a complete word is a concrete sequence evaluated against its execution context. Canonical U6 additionally needs before/after $\Phi_s$ snapshots and therefore cannot be certified from operator labels alone.

11.1 Named workflow fragments¶

The grammar composition section defines the reusable Bootstrap, Stabilize, Explore and Propagate fragments. They require the missing initialization, closure, history or phase context before becoming executable words. For example, [AL, IL, OZ, ZHIR, IL, SHA] supplies a generator, prior Coherence, recent destabilizer, handler and closure around the Explore body; Mutation still requires its live growth evidence.

11.2 Complete-word examples¶

Read executable recipes through list_sequences() or the fluent SDK's list_canonical_sequences() rather than copying a second name/word table. The recipe registry owns named canonical patterns; the study API exposes the registered study recipes. Names such as phase_lock, resonance_peak_hold and therapeutic_protocol are labels, not proofs of locking, peak detection, maintenance or a medical effect. Each execution retains the selected word, actual glyph trace and live admission outcome.

11.3 Validation boundary¶

The sequence validator checks U1 initiation/closure, U2 debt coverage, U3 operator awareness, U4 context, and declared U5 Recursivity depth. Actual U3 phase compatibility is checked by operator preconditions. Canonical U6 is a separate snapshot observation. Thus a sequence-level pass is not full U1--U6 trajectory certification and does not prove convergence.

12. Operator roles and energy diagnostics¶

The structural energy diagnostic $E = \frac{1}{2}\sum_i [\Phi_s^2 + |\nabla\phi|^2 + K_\phi^2 + J_\phi^2 + J_{\Delta\text{NFR}}^2]$ is nonnegative and can serve as a Lyapunov candidate for a specified dynamics. It contains no explicit EPI or $\nu_f$ term, although an operator can still change $E$ indirectly by updating phase, pressure, topology or coupled state. The grammar U2 role therefore does not determine the sign of $\Delta E$.

12.1 Grammar U2 roles¶

Use the U2 role and debt contract. It classifies operator obligations; it is not a measured energy inequality. Primary channel, grammatical role and diagnostic response are distinct.

12.2 Nominal operator parameters¶

The maps in §§4–10 state their effective factors and branches. Their configured magnitudes do not establish universal response multipliers: an IL pressure factor is not a full-state energy factor, a ZHIR phase step is not a curvature growth rate, and a VAL capacity increase has no unconditional energy sign. Read the effective factors from the actual invocation and use the configuration owners for their defaults.

12.3 Grammar U2 stability boundary¶

U2 requires destabilizers to be compensated by stabilizers, but it does not prove for every compliant sequence that

\[ \sum_{\text{ops}} \Delta E_{\text{op}} \leq 0 \]

The operator registry supplies local role and nominal parameter metadata rather than a common Lyapunov proof for all realizations. Such a proof additionally needs a specified state functional, pressure law, timing and gain bounds. The fixed symmetric pure EPI channel is one class where this has now been established, including heterogeneous positive capacities. A declared affine EPI reset can also be bounded in the same metric exactly when it preserves the consensus subspace; the resulting rational quotient-gain bound, with a weighted-Frobenius fallback, composes with elapsed diffusion time. This generic theorem does not certify any catalog operator until its runtime action is derived as that affine map on a declared domain.

Exact restricted result: see TNFR_DIFFUSION_STABILITY_THEOREM.md.

13. Postcondition Contracts¶

operator_contracts.py owns each operator's primary channel, qualitative direction, scope, measurement context and postcondition. The API contract table is generated from that registry. Introspection and the integrity monitor reuse the same records instead of maintaining another operator catalog here.

The integrity owner provides sampled audits and runtime monitoring with OFF, OBSERVE and ENFORCE modes. Report evaluated coverage, unavailable context and individual outcomes separately. A monitor's post-event exception does not by itself roll back the event; the execution path must supply any transaction guarantee. Direct glyphs, public classes and atomic stages can have different secondary effects and measurement boundaries.

The contract catalog is complete as a specification but is not instantaneously identifiable from its categorical fields. The exact certificate contract_identifiability_certificate() shows that channel, direction, scale and context leave Silence and Contraction in the same class. Their distinct semantics require a temporal or quantitative postcondition observation. This negative inverse result neither removes an operator nor proves completeness of the catalog over every admissible TNFR transformation.

14. Operator Constants Reference¶

The contract constrains a map's channel and postcondition, not a unique numeric gain. Effective factors must be admitted on their active branch before writes. Their owners are canonical constants, operator factor contracts and the individual map kernels linked in §§4–10.

14.1 The structural scale¶

In radians, $pi$ is a half-turn and sets the range of a principal phase difference. The resulting circular field bounds do not select a pressure gain, coherence threshold or physical clock.

14.2 Operator gain magnitudes (operational)¶

Mixing fractions, capacity multipliers, emission boosts, phase steps and child scales are supplied operational choices. An expression containing $pi$ is still a configured choice unless a stated theorem fixes it under explicit premises. Reproduction requires the effective configuration, not a copied table of nominal defaults. The parameter-by-parameter scope belongs to NODAL_PARAMETER_FOUNDATIONS.md.

14.3 Constant-Operator-Grammar Traceability¶

Trace an invocation through its registered contract, active factor resolution, map implementation and grammar roles. Neither the primary channel nor the sign of a gain determines a grammatical role or a full-state Lyapunov inequality. Runtime measurements and analytic proofs must use the same admitted branch and state representation.

15. Implementation Reference¶

15.1 Source Modules¶

Responsibility	Owner
Public operator classes and common call boundary	definitions.py, definitions_base.py
Registered operator contracts and active factors	operator_contracts.py, factor_contracts.py
Word roles, validation, live selection and observation	Grammar implementation map
Stage, schedule and delayed-history execution/evidence	Operator-event contract
Integrity and diagnostic energy policies	integrity.py, lyapunov.py
Declared recipes and shared CLI/SDK studies	canonical_patterns.py, study.py

Individual operator sections link their map kernels. The event contract links the specialized execution and certificate owners; this document does not maintain a second list of every event/REMESH adapter.

15.2 Base Operator Workflow¶

Operator.__call__(G, node, **kw) in definitions_base.py follows this order:

Admit the execution window, graph seed, replayable history and hard invariants. U3 cannot be disabled by turning off optional preconditions.
Validate shared arguments, consumed form, requested telemetry sinks and monitor interface, then the enabled operator-specific preconditions.
Select the grammar-admitted action before subclass writes. If fallback is selected, invoke that operator's own public workflow, metadata and metrics. Strict rejection supplies no replacement. Admit active factors before preparatory writes.
Execute the selected map and its declared secondary effects. Capture the pre-state only when metrics or the nodal comparison are requested; invoke before/after integrity hooks only when a monitor is installed.
Perform any requested held-step nodal comparison and metrics collection. The comparison uses a supplied interval after the event; it neither measures the event's duration nor identifies a hybrid jump with continuous flow.

The generic public call is not a graph transaction. Later hook or comparison failure need not undo a completed glyph. Transactional atomic stages have the separate stage contract. U6 remains a reference-relative observation rather than an implicit proof supplied by this pipeline.

Branch controls. factor_contracts.py resolves the following controls before active factors or branch proposals, on direct glyph, public-class and atomic-stage paths:

Controls	Admission
`UM_BIDIRECTIONAL`, `UM_SYNC_VF`, `UM_STABILIZE_DNFR`, `UM_FUNCTIONAL_LINKS`; `OZ_NOISE_MODE`	Shared Boolean parsing: supported false strings disable the branch; unknown strings reject. Non-string values retain the shared parser's Boolean conversion.
`NAV_STRICT`, `NAV_RANDOM`	Actual Boolean values only; strings and numeric `0`/`1` reject.

These different input contracts are intentional. Branch selection is supplied configuration, not an emergent operator choice or a relaxation of live grammar.

15.3 Executable Demonstrations¶

Use the examples catalog for runnable entry points, including operator sequences, grammar rejection and EN/RA realization studies. The event contract routes the specialized flow, REMESH and history examples to their precise hypotheses. A demonstration is an executable illustration, not an automatic execution or future guarantee.

15.4 SDK Entry Points¶

This read-only example obtains contracts and recipe definitions without executing or selecting an operator:

from tnfr.sdk import TNFR, list_sequences

emission = TNFR.operators("emission")
assert emission["token"] == "emission"
assert emission["channel"] == "EPI"
recipes = list_sequences()
assert recipes and all(recipe["operators"] for recipe in recipes)

For actual execution use the minimal admitted word or the shared CLI/SDK study workflow. Preserve its preparation, operator cycles, seeds and live rejection outcomes. A configured automatic selector is a policy, not a derived invocation law.

16. Summary¶

Use the registry for what a named operator promises, grammar for word and live admission, and the selected execution contract for what commits and what evidence is retained. The mechanism audit separates these from collective descent, autonomous occurrence and physical identification. None follows solely from a catalog label or a diagnostic score.

17. Experimental Operator-Tetrad Synergies¶

Examples 37–39 are finite diagnostic instruments. An operator request can be replaced by grammar fallback; its measured response belongs to the realized glyph and actual execution path. No numerical response table is asserted here without a retained preparation, source/configuration and response record.

17.1 Centralized Operator-Channel Classification¶

Read primary channels from the generated contract table. Capacity and pressure enter the nodal rate multiplicatively; changes to form, phase, support or history can influence their later values under a specified law. A primary-channel label does not enumerate secondary writes or determine an energy sign.

17.2 Operator-Tetrad Fingerprint Matrix¶

Example 37 measures requested/realized glyphs and before/after field changes on a supplied graph. Such a matrix is a finite response profile, not a universal fingerprint or a proof that the tetrad reconstructs the complete state. Zero sampled change does not establish neutrality under subsequent pressure refresh or evolution.

17.3 IL-OZ Tetrad Symmetry¶

Matching sampled IL/OZ read-outs would not establish equality of their maps or an exact symmetry. IL contracts stored pressure and its public/staged path also aligns phase. OZ has a different pressure map, near-zero/jitter branches and optional propagation. Interpret a response only after checking the realized glyph, auxiliary writes and pressure-refresh boundary.

17.4 Structural Potential Linear Response¶

At fixed support and distance kernel, structural potential is linear in its pressure source: $Phi_s=Bp$, hence $DeltaPhi_s=BDelta p$. This identity belongs to the field definition, not to a fitted response sweep. A topology or distance change also changes $B$. The coherence-length estimator has no corresponding general linear law; its fit/fallback provenance must remain explicit. Nonlinear sampled response alone does not establish divergence or a critical transition.

17.5 Diagnostic dependency chain¶

The read-out chain is

$$

\text{executed map} \longrightarrow \text{updated state} \longrightarrow \text{fields} \longrightarrow \text{diagnostics}.

$$

The fields observe a fully specified state; they are neither additional independent state variables nor a complete future predictor. If a controller uses a diagnostic, that feedback policy must be declared separately.

17.6 Grammar-Energy Landscape¶

Example 38 compares measured diagnostic energy with configured operator multipliers on finite sequences. U2 constrains word obligations; it does not bound that energy for every law or realization. A mismatch between a nominal multiplier and a measured change is evidence about the specified comparison, not a stronger grammar theorem.

17.7 Executable Demonstrations¶

The instruments are operator-field responses, grammar-energy comparison and nodal-row decomposition. Retain source, configuration, graph/node order, input, seed, precision, backend, pressure refresh and realized glyph trace with any evaluated response. A seed alone cannot bind an older reported value to the current implementation.

18. Mechanism and activation audit¶

This map addresses connection formation using the existing 13 operators. It does not add a second executable catalog: names, primary channels and contracts remain in operator_contracts.py. The distinction between primitive glyphs, public classes and snapshot stages is essential. A primary channel is not a complete list of writes.

For each proposed mechanism distinguish four questions: what action is implemented, when it is admissible, who selects it, and what accounts for its actual state change. Existing grammar, history gates and controllers answer parts of the first three under configured policies. They do not yet derive a unique autonomous law from an NFR's state. The storage S=E_D+beta*V below belongs to the conditional relational model, not to every operator runtime or a universal physical energy.

All thirteen actions: useful mechanisms and remaining premises¶

Operator	Actual mechanism relevant to interaction	Reusable synergy and boundary
AL — Emission	Adds/projected form on existing support; basic event holds phase, capacity and stored pressure	A local form contrast changes the following relational phase rates. Emission from uniform form costs storage; amplitude, source and occurrence remain supplied. No node or edge creation.
EN — Reception	Blends unweighted existing incoming-neighbor form; source ranking is telemetry	One unclipped target can reduce form storage and alter later phase compatibility. No U3 source filter or reception from an absent edge is implemented.
IL — Coherence	Glyph contracts stored pressure; public/staged path also aligns target phase with its neighbor resultant	A single-target phase reset can release phase storage. The pressure-only change may disappear on refresh; no global or simultaneous-stage storage theorem follows from its stabilizer label.
OZ — Dissonance	Perturbs stored pressure, with near-zero/jitter branches and optional neighbor propagation	Supplies a declared perturbation and grammar history; pressure-only change is neutral in this particular `S`. Refreshed constitutive pressure can erase it. Signed propagation can cancel another node's pressure.
UM — Coupling	Moves phase, optionally aligns capacity/contracts stored pressure and proposes links to compatible nonneighbors	A phase reset can offset a positive new-edge cost. Requires an existing compatible neighbor at the acting target. Candidate inventory, affinity/Si threshold, tie policy and time are supplied; generated conductance is generally nonunit.
RA — Resonance	Mixes form over existing U3-compatible neighbors; configured phase interpolation and triggered capacity amplification	RA then UM can reduce total storage while adding a positive-cost edge. RA itself creates no support and does not contract stored pressure. Sign/kind preservation does not imply maintained pattern identity.
SHA — Silence	Attenuates capacity and records latency information, preserving form at the event	Capacity-only reset is neutral in `S` but changes future rates. A positive retention factor does not produce zero capacity in a finite ideal event; vanishing capacity alone does not prove `nu*p` vanishes.
VAL — Expansion	Raises capacity; default edge-aware policy also scales/projects EPI	Capacity changes local speed, while the auxiliary form reset can change subsequent phase response and storage. Expansion does not derive a new state dimension, source or clock.
NUL — Contraction	Lowers capacity and rescales stored pressure; default edge-aware policy also scales/projects EPI	Ideal stored-product preservation is distinct from the freshly evaluated relational rate. Account for the actual form jump; it is not generally a storage-neutral capacity-only event.
THOL — Self-organization	Primitive signed acceleration/pressure reset; public/staged path can create an isolated nested child after history/U5 gates	Existing eligibility/dispatch and child-feedback tools can be reused. Parent metadata is not an edge. Child form/capacity and timing follow construction rules; absent edge storage does not mean free substrate creation.
ZHIR — Mutation	Configured phase reset following a live signed EPI-growth test and separate grammar context	Can alter phase geometry without adding support. A valid growth trigger is eligibility, not a derived necessity to mutate. Rich-form magnitude is not signed scalar growth; the full reset may raise storage.
NAV — Transition	Primitive pressure change; public/staged regimes can also change phase and capacity	A candidate preparation/phase-reset mechanism, with its own budget. Regime factors, optional random draw and invocation remain policies; latency metadata is not a structural event clock.
REMESH — Recursivity	Advisory glyph, delayed same-support EPI mixing, and explicitly invoked topological replacement are distinct APIs	Fixed-node rewiring can reuse joint reset accounting. Delayed-history disagreement bounds are not a theorem for `S`; community replacement may change nodes and lies outside the same-node observer.

Execution owners are linked in §15. The particularly relevant shared kernels are AL/SHA proposals, reception input, UM proposals, atomic stages, Mutation admission, and THOL dispatch.

Quantitative synergies, without inventing an activation law¶

The full-reset derivation and shared observe_relational_reset separate nodal reorganization on old support from support work at the new state. They retain the actual endpoints instead of assigning energy signs from operator names.

Form changes influence phase evolution. For held support, phase and capacity, an AL/EN jump d changes the next phase row by (w/beta)*diag(nu/H)*B*d. One unclipped EN target has nonpositive form cost; an AL jump from uniform form supplies positive cost. RA/VAL/NUL may also change form, but their auxiliary writes and clipping must be included.
Nodal relaxation can accompany attachment. Actual UM and RA-then-UM witnesses have positive bridge cost but negative complete storage change. This removes the frozen-triad budget obstruction for those declared actions. It does not derive their selection, time or subsequent maintenance.
Phase-only IL can prepare a budget. On simple undirected support, one target with fixed neighbors and nonzero resultant R*exp(i*mu) moves from theta toward mu by a fraction lambda in [0,1]. With the selected shortest displacement a=wrap(mu-theta), the ideal phase cost change is R*(cos(a)-cos((1-lambda)*a))<=0. This proves a conditional single-target result; it does not extend automatically to overlapping simultaneous resets.
Capacity controls response, not stored identity. Held form/phase/support makes a capacity-only reset neutral in S. The later dynamics still changes. VAL/NUL's default EPI effects and post-event pressure refresh prevent a blanket capacity-only interpretation. Zero capacity and nonzero Gamma must retain their separate contracts.
History can gate an event without explaining its origin. ZHIR uses a measured signed secant, THOL uses its configured history and construction, REMESH consumes declared delayed state. These are useful causal inputs; none turns a diagnostic label into an independently derived event law.

Hidden assumptions exposed by the map¶

Stored pressure versus an independently evaluated law. Pure stored-pressure changes from IL/OZ/THOL/NAV affect a held-pressure interval. If every argument of a state-based constitutive law is unchanged, its next refresh restores the same pressure. A pressure intervention needs an explicit lifetime or retained state to have a lasting role. The relational observer does not infer either.

Continuous evolution versus a finite operator action. Uniform form gives q=0 and zero instantaneous relational phase velocity, even at nonuniform admitted phase. UM can still move phase in that state. Its finite reset is not automatically a time step of the current continuous law. Any proposed internal NFR activation must justify this boundary, not merely pass a storage check.

Potential contact versus existing support. UM's search over graph nodes or a supplied sample assumes candidate access. EN/RA consume existing edges; they cannot themselves transmit input across a missing edge. THOL children start isolated. Two isolates cannot bootstrap UM because its initial U3 gate needs an existing neighbor. The potential-contact relation is therefore a separate hypothesis from the realized transport graph.

Symmetry versus a configured score. UM's absolute-form similarity changes under a common form offset, while the selected relational law does not. Si enters its affinity, and rank resolves some ties. Successful execution of that policy does not derive it from offset-covariant nodal dynamics. The existing event-action audit supplies the appropriate symmetry/clock controls.

Phase compatibility versus synchronized rhythms. A small instantaneous wrapped gap does not imply equal phase velocities or sustained locking. Equal velocities can retain a gap outside U3. A possible connection mechanism should retain both delta=theta_j-theta_i and delta_dot from the complete law, and state what makes the candidate observable before connection. Treating rhythm locking as a prerequisite is a testable hypothesis, not yet an event rule. The SDK's modal rhythm() and snapshot resonance() observations do not supply that missing temporal proof or clock. Synchronization produced by an already present link cannot circularly establish the cause of that link's birth.

Representation versus physical evidence. Native AL/EN serialized uniform-real BEPI now reaches relational execution through shared signed scalar admission. Rich or complex form is still outside that scalar model. Mutation uses that same boundary for live history matching, rather than a magnitude surrogate. These are integration corrections, not new physics.

When an internal transformation defines a collective operator¶

Let z'=f(z) be one declared complete fine law, y=R(z) a proposed NFR description, and T a supplied event map. An autonomous continuous law in y requires DR(z) f(z) to agree on every admitted state with the same R(z). An event on that same description additionally exists precisely when

R(z1)=R(z2) => R(T(z1))=R(T(z2)).

Necessity follows by evaluating the proposed collective event at the same input. Sufficiency follows by defining its output using any admitted lift; the displayed condition makes that definition independent of the lift. Admission, target/port choice, retained history and any occurrence rule must also be well defined on those classes. If a transformation changes the state space, specify the before and after observation maps separately. Covariance of an event family with a transformed target does not establish closure after discarding that target. These are conditional descent criteria, not claims that the named engine operators already satisfy them.

The unordered-pair state provides an existing exact continuous quotient under its sine-law premises. It retains internal coordinates; means alone generally fail. Whether an action at one constituent descends requires its own target and symmetry test. A continuous fine flow cannot cause a finite jump in a continuous observation; an apparent jump in a selected partition instead requires an explicit change of observation or event law. Neither interpretation creates extra fine nodes.

The full-form formation obstruction adds a complementary native-law constraint. Its monotone quantity W bounds phase acquisition during smooth flow, and its reset corollary supplies a necessary jump budget when the same graph, capacity and law are retained. W is not energy: even a storage-passive reset can increase it. Existing observe_relational_reset retains the endpoint data needed for such a comparison; its represented phase costs are not ideal trigonometric bounds. Refreshing pressure can erase a pressure-only intervention; changing capacity or support requires new theorem admission. None of these endpoint checks selects an action or proves that it follows from the fine continuous dynamics.

The AL pair comparison now supplies an exact example: the same admitted scalar form map on both interchangeable members descends, whereas a fixed singleton target generally needs a retained port mark. Its two exceptional source cases are a synchronized tip and two actual no-ops. Equal reset storage costs need not give equal collective outputs. This result uses actual AL form clipping, without claiming closure of its complete grammar, history and lifecycle runtime.

The regional-transfer identity adds a distinct realizability test: the closed normalized-sine law preserves degree/capacity-weighted form. Any actual positive AL-only increment changes that invariant, even when the event descends to a collective state. Autonomous regional gain instead has a compensating change outside the region. The synchronized-pair control also derives its subsequent phase response. This is an internal exchange mechanism under the stated law, not the autonomous execution of the registered AL event.

Thus an NFR's identity, its inherited response, a realizable event and the event's occurrence are separate proof obligations. A useful operator can be a supplied intervention while autonomous occurrence remains open. The sole research queue selects the next bounded comparison from these obligations. A justified negative result remains useful; the map provides mechanisms to test without asserting autonomous connection, substrate generation or physical identification.

Cross-References¶

Nodal equation derivation: FUNDAMENTAL_THEORY.md §2
Grammar rules U1–U6: UNIFIED_GRAMMAR_RULES.md
Energy diagnostics and scoped stability results: STRUCTURAL_STABILITY_AND_DYNAMICS.md §1
Conservation laws: STRUCTURAL_CONSERVATION_THEOREM.md
Variational formulation: TNFR_VARIATIONAL_PRINCIPLE.md
Structural field tetrad: FUNDAMENTAL_THEORY.md §3
Canonical constants: src/tnfr/constants/canonical.py
Operator-tetrad synergy experiment: examples/02_physics_regimes/37_operator_tetrad_synergy.py
Grammar-energy landscape experiment: examples/02_physics_regimes/38_grammar_energy_landscape.py
Nodal equation decomposition experiment: examples/02_physics_regimes/39_nodal_equation_decomposition.py
Glossary: GLOSSARY.md

Regime	\(\nu_f\) change	\(\theta\) shift	\(\Delta\text{NFR}\) reduction
Latent → Active	+20%	\(+0.1\)	\(-30\%\)
Active → Active	configurable	\(+0.2\)	\(-20\%\)
Resonant → Active	\(-5\%\)	\(+0.15\)	\(-10\%\)

Attribute	Symbol	Domain	Units
Form	\(\text{EPI}_i\)	Declared form space; scalar glyphs require a signed real chart or uniform-real BEPI embedding	Declared form unit X
Frequency	\(\nu_{f,i}\)	Nonnegative reals; individual operators may require strict positivity	Hz_str
Phase	\(\phi_i\) (or \(\theta_i\))	\([0, 2\pi)\)	rad

Constant	Value	Derivation
Emission amplitude \(b\)	configured positive gain	Operational parameter; channel and direction are contractual
Activation threshold	\(0.8\) by default	Configurable operational gate

Constant	Value	Derivation
Mixing fraction \(m\)	configured; standard policy \(1/(\pi+1)\)	Operational interpolation weight
Convexity term	\(m(1-m)\)	Algebraic mixing diagnostic, not a full-energy contraction theorem

Constant	Value	Derivation
ΔNFR retention factor \(f\)	\(\pi/(\pi+1)\approx0.7585\)	Default `IL_dnfr_factor`; configurable gain
Pressure-square reduction	\(1-f^2\approx0.425\)	Isolated \(\Delta\mathrm{NFR}^2\) term at the default; not a full-energy bound
Phase locking \(\lambda\)	\(\approx 0.3\)	Configurable coupling strength

Constant	Value	Derivation
Amplification factor \(f\)	\((\pi+1)/\pi\approx1.3183\)	Default `OZ_dnfr_factor`; configurable gain
Pressure-square increase	\(f^2-1\approx0.738\)	Isolated \(\Delta\mathrm{NFR}^2\) term at the default; not a full-energy bound

Constant	Value	Derivation
Default scale factor \(f_{\text{VAL}}\)	\(1+1/(4\pi)\approx1.0796\)	Operational capacity step
Nominal square-factor change	\(f^2-1\approx0.166\) at the default	Algebraic \(\nu_f^2\) diagnostic; the structural candidate energy has no explicit \(\nu_f\) term
Optional strict minimum EPI	\(1/(2\pi) \approx 0.159\)	Supplied `VAL_MIN_EPI` admission policy

Constant	Value	Derivation
Functional-link score threshold	\(\pi/(\pi+1) \approx 0.7585\)	Configured `UM_COMPAT_THRESHOLD`; only the enabled link branch consumes it
Phase push	\(1/(\pi + 1) \approx 0.241\)	Configured circular-alignment gain; shared numeric value with EN mixing
\(\Delta\text{NFR}\) reduction	\(1/(2\pi)\approx0.1592\)	Default phase-alignment pressure-relief gain

Quantity	Default	Runtime domain / status
EPI mix \(r\)	\(1/(2\pi)\approx0.1592\)	`[0,1]`; convex propagation
Frequency amplification \(a\)	\(1/(8\pi)\approx0.0398\)	`[0,+infinity)`; nondecreasing capacity
Phase coupling \(c\)	\(1/(4\pi)\approx0.0796\)	`[0,1]`; shortest-arc interpolation
U3 phase limit	\(\pi/2\)	`[0,pi/2]`; may only be tightened
Capacity trigger	\(10^{-9}\)	Operational binary64 threshold on \(\lvert\bar x_i\rvert\)

Parameter	Default	Runtime role / status
Signed observed-growth threshold \(\xi\)	`ZHIR_THRESHOLD_XI = 0.1`	Strict operator-admission calibration; operational, not structural
Direct activity condition	\(\nu_f>0\)	Zero capacity cannot execute the phase transformation
Optional direct minimum	absent (`ZHIR_MIN_VF = 0`)	Tightens the activity condition only when configured explicitly
Branch-selection threshold	`ZHIR_BIFURCATION_VF_THRESHOLD = 0.5`	Selects whether the bifurcation router proposes ZHIR; it is not a ZHIR operator precondition
Default phase shift	\(1/4\) rad	Calibrated magnitude; \(\Delta\text{NFR}\) supplies direction only

Constant	Value	Derivation
Scale factor	\(1-1/(4\pi)\approx0.9204\)	Same operational \(\nu_f\) step as SHA
Densification factor	\(1/f_{\text{NUL}}\approx1.0865\)	Reciprocal configured capacity factor