Mathematical Scope of Structural Diagnostics and Grammar¶

Status: Exact identities under stated hypotheses, canonical engine policies, and open sufficiency questions. This note fixes the distinction between these three categories without changing the operator catalog or numerical policies.

The nodal equation is

x'(t) = ν_f(t) p(t),    x = EPI,    p = ΔNFR.

It specifies the relation between structural change, capacity, and pressure. A theorem about its solutions also needs a pressure law, graph assumptions, initial conditions, and, for a discrete evolution, an integrator and step size. Operator contracts supply additional constraints; they do not follow from the scalar identity without those premises.

1. Existence, boundedness, and convergence¶

If f(t) = ν_f(t)p(t) is locally integrable on a finite-dimensional structural coordinate space, then

x(t) = x(t₀) + ∫[t₀,t] f(s) ds

defines an absolutely continuous trajectory on each finite interval, with the nodal equation holding almost everywhere. Continuity of f gives the usual pointwise derivative. In particular, x = 0 is not a singularity of this equation: ν_f = 1 and p = 1 give x' = 1 at zero. U1's generator requirement is an initialization contract for the operator language, including its supported pre-existing-structure context, rather than a claim that this derivative is undefined.

On an infinite horizon the following properties are different:

Property	Exact statement
Bounded trajectory	The partial integrals of f remain bounded.
Convergence to a finite limit	The vector improper integral of f converges.
Absolute integrability	The integral of the norm of f is finite. This is sufficient for convergence and gives finite total variation.

Neither boundedness nor p(t) tending to zero establishes the last two properties:

x(t) = sin(t), ν_f = 1, p(t) = cos(t) is bounded and satisfies the nodal equation, but x(t) has no limit.
p(t) = 1/(1+t), ν_f = 1 gives x(t) = x(0) + log(1+t). Its pressure is bounded and tends to zero, while its accumulated displacement diverges.
On a finite connected undirected graph, fixed homogeneous ν_f > 0 and the pure EPI law p = −L_rw x give autonomous negative feedback. Relaxation does not mathematically require an additional named operator during that flow.

An example of a sufficient analytic assumption is

||ν_f(t)p(t)|| ≤ M exp(−a(t−t₀)),    M < ∞, a > 0.

Then the tail displacement is at most (M/a) exp(−a(t−t₀)). For repeated operator applications, proving such a bound requires control of their gains, elapsed times, and intervening feedback. U2's prescribed stabilizers and debt accounting remain mandatory engine policy; grammar acceptance by itself is not an infinite-horizon convergence certificate.

On a fixed undirected graph with positive edge weights, homogeneous frequency, and pure EPI diffusion, each nonstationary mode of an explicit Euler step has multiplier

q_k = 1 − ν_f dt λ_k.

Its amplitude after n steps is q_kⁿ times the initial amplitude. Decay requires |q_k| < 1; a negative multiplier alternates sign and need not decay. The continuous solution instead has multiplier exp(−ν_f λ_k t). These are different evolution laws at finite dt.

For a loopless graph without isolated vertices, trace(L_rw)/N = 1. This is the mean of all eigenvalues, including stationary modes, not the relaxation rate of each mode. Isolates and self-loops require different trace accounting.

The functions derive_bifurcation_window_from_physics and derive_u2_debt_capacity_from_physics retain their public names and formulas. They use the fixed surrogate rate ρ = 1 and q = 1 − ν_f dt ρ. At the canonical ν_f = 1 and dt = 0.5:

The first n with qⁿ < 1/(π+1) is 3, the U4b recency policy.
The floor of 1/(1−q) is 2, the U2 debt-capacity policy.

The geometric sum describes a scalar recurrence with unit forcing and 0 ≤ q < 1. Its finite steady state is not a maximum physical pressure or a criterion for integrability of a sustained forcing. Mapping its floor to a count of destabilizing operators is the engine's calibration choice.

The helpers admit finite positive nonboolean inputs only when their represented product satisfies 0 < nu_f*dt <= 1. Rational evaluation then uses that binary64 product and the represented target band. The exact q=0 surrogate crosses in one step; negative or invalid multipliers are rejected. The U4 search remains capped at 64 positions even when the target is not reached. U2 evaluates the reciprocal floor exactly, including subnormal rates. These are scalar grammar calibrations, not stability results for graph Euler modes.

Finite-graph witness. On the 21-node unweighted path,

λ₂ = 1 − cos(π/20) = 0.012311659404862...
q₂³ = (1 − 0.5 λ₂)³ = 0.981645960340198...
1/(π+1) = 0.241453007005224...

The Fiedler amplitude needs 231 steps to cross that target. The same graph still has mean eigenvalue 1. Thus the three-operation grammar window is not a mode-uniform relaxation theorem. A trajectory bound must use the relevant spectrum, norm, Euler stability region, stationary-mode treatment, and operator-induced gains. Directed non-normal and heterogeneous-frequency generators require their own analysis.

The read-only diagnose_euler_relaxation_window now performs that analysis for frozen symmetric pure-EPI diffusion with heterogeneous frequency. It returns the actual modal multipliers, explicit-Euler stability limit and first solver step below a declared target. It also reports the canonical U4 window for comparison while keeping the units distinct: solver steps are not operator positions. Directed non-normal transport and any policy mapping remain open.

3. Assumption-explicit model thresholds¶

physics.life is a supplied-series diagnostic model, not a consequence of the nodal equation and not a biological classifier. It rejects Boolean, non-finite, multidimensional and implicitly broadcast inputs; the combined detector additionally requires nonnegative EPI, matched lengths and strictly increasing times. Its selected event \(A(t)>1\) means only that the declared time-local ratio \(G(t)\dot x(t)/(|\Delta\mathrm{NFR}_{\rm ext}(t)|^2+\epsilon_{\rm num})\) exceeds one on the supplied samples. The reported time is the first sample already above one or the linear interpolation of the first upward crossing. It does not establish persistence, autonomy or biological life. Formulas and parameter domains are centralized in Structural Stability and Dynamics.

4. Structural potential and topology-dependent bounds¶

For a fixed distance convention, define

B_ij = d(i,j)⁻² for reachable j ≠ i with 0 < d(i,j) < infinity; otherwise 0,
Φ_s = B p.

The field is linear in pressure. The triangle inequality gives the exact bound

|Φ_s(i)| ≤ Σ_j B_ij |p_j|,
||Φ_s||∞ ≤ b_G ||p||∞,    b_G = max_i Σ_j B_ij.

For two pressure fields on the same graph,

||Φ_s(t₁) − Φ_s(t₀)||∞ ≤ b_G ||p(t₁) − p(t₀)||∞.

If the graph changes, an additional term is necessary. Writing B₀ and B₁ for the two kernels gives

Φ₁ − Φ₀ = B₁(p₁−p₀) + (B₁−B₀)p₀.

These bounds explicitly depend on pressure, graph geometry, and normalization. There is no phase angle in Bp. Setting every phase to zero and p_j = 1 on K₄ gives Φ_s(i) = 3 at every vertex. On K_n it gives n−1, and multiplying p by a multiplies Φ_s by a. A transition from p = 0 to p = 1 has the same n−1 drift.

Consequently the canonical π/4 per-node and π/2 drift thresholds are selected safety policies. A state can exceed them and trigger telemetry; phase wrapping does not enforce them. Establishing them as trajectory bounds would require explicit pressure and topology hypotheses implying the inequalities above. U6 remains a read-only monitor, separate from the word grammar.

Its implemented statistic is the mean absolute nodewise drift from a declared reference, not the maximum drift or an intermediate-time measurement. Equal endpoint fields can hide an excursion between them. For a fixed reference and smooth aligned kernels, y=Bp-Phi_reference satisfies y'=B p'+B' p. A forward-confinement theorem for ||y||_1/N would need a well-posed law and inward boundary conditions, plus compatible event jumps. The nodal EPI identity and word labels do not supply those derivatives. Rescaling every distance by a>0 gives B_new=a^-2 B; a fixed numerical pi/2 policy therefore depends on the declared distance convention.

The inverse-square exponent α = 2 is the canonical kernel choice. Chain summability does not uniquely select it: Σ d⁻α converges for every α > 1, and independent unit-variance pressure gives variance Σ d⁻²α, convergent for every α > 1/2. On a graph family with shell counts bounded by C r^(d−1), α > d is sufficient for absolute accumulation and 2α > d for independent unit-variance accumulation. These are sufficient shell-growth conditions; they need not hold on every graph family. Particular ζ(2) and ζ(4) values do not establish universal π-fraction confinement.

In contrast, the phase definitions give exact kinematic bounds:

|∇φ| ≤ π,    |K_φ| ≤ π.

The curvature warning level 0.9π is a selected margin inside that bound. The phase-gradient warning level π/16 is also a policy value.

5. Diagnostic channels and state reconstruction¶

The tetrad selects four useful readouts: source accumulation Φ_s, local phase mismatch |∇φ|, local circular curvature K_φ, and correlation length ξ_C. These are canonical telemetry channels. Canonical status does not establish that their values reconstruct all state variables or every independent observable.

On a graph, edge derivatives and Laplacian powers can be generated by composing a gradient and a Laplacian. Algebraic generation does not imply that successive operators, or lossy scalar summaries of their outputs, are linearly dependent. If L has at least four distinct eigenvalues, I, L, L², and L³ are linearly independent: a dependence would give a polynomial of degree at most three vanishing at four distinct points. More generally, independence is controlled by the minimal polynomial, not a universal second-order cutoff.

An independent obstruction comes from the nodal equation. At fixed graph, phase, and pure-EPI pressure, rescaling homogeneous ν_f leaves the tetrad unchanged but rescales x' = ν_f p. Therefore the tetrad does not determine the full dynamical state. Canonical coherence also depends on pressure amplitude: with dEPI = 0 and uniform |p| = 1 it equals 1/2; doubling pressure gives 1/3. Only the separate normalized dispersion statistic is scale invariant.

A reconstruction or minimality theorem must specify the state space, gauge identifications, full fields versus node/global summaries, allowed observables, and the desired equivalence relation. A diagnostic removal study instead needs explicit fixtures and detection thresholds. Neither a test count nor an operator-composition identity supplies those missing hypotheses. Completeness and minimality in this stronger sense remain open.

For fixed connected undirected pure EPI diffusion, this specification yields a conditional reconstruction result. The full nodal potential has observation operator O=-K L_rw; if rank(O)=N-1, it recovers EPI modulo a uniform shift. The conserved d_i/nu_i mean supplies that last scalar. Capacity is still required to determine the generator. The repository certificate evaluates the rank per graph; the condition has been measured on the 142 connected simple graph-atlas cases through six nodes and is not asserted universally.

6. Authoritative entry points¶

Unified grammar: current U1–U6 requirements.
Tetrad scope: interpretation and limitations.
Field definitions and API.
Canonical calibration functions.
Threshold values.
The deterministic finite-graph witnesses used by this scope statement are recorded directly in the sections above and covered by the linked tests.

7. Derived observables and dynamical closure¶

A diagnostic can be a function of nodal structure without being a primitive force or a dynamically sufficient emergent variable. Write the full declared state as z and an observable as y=f(z). If a closed dynamics z'=F(z) has already been justified, then y'=Df(z)F(z) on differentiable charts is a derived law. An autonomous reduced law y'=g(y) additionally requires Df(z)F(z) to agree for all states with the same y. Defining f alone proves neither statement. Maxima, clamps, phase branches and discrete events require their own nonsmooth rules.

Conversely, using an observable in F(z)=H(z,f(z)) is not intrinsically invalid: substitution gives a state-dependent law. The dependence H must be derived from the declared structural dynamics or identified as an additional constitutive/control assumption. A diagnostic label neither forbids causal use nor justifies it. This is the distinction relevant to generative TNFR research; calling any computed quantity "emergent" does not close the dynamics.

Sense Index: definition, information loss and consumers¶

The shared Si owner computes

Si_i=clip(alpha*|nu_i|/nu_max + beta*(1-disp_i) + gamma*(1-|p_i|/p_max)),

with clamped normalized components, wrapped phase dispersion, configured normalized weights and graph-wide absolute maxima. A zero maximum uses denominator one; both current Si backends refresh maxima from the effective stored aliases before normalization. Thus Si is a defined structural read-out. Its formula discards pressure sign, and common positive capacity rescaling preserves the normalized capacity term. It is not a complete dynamical coordinate or a proved stability certificate. Writing default coefficients in terms of pi supplies reproducible values and a normalization identity, not a unique derivation of their causal role.

Three finite controls make the distinction explicit. They use exact dyadic values and a chosen threshold to expose an existing policy's dependency; they are not proposed research parameters or new physical laws.

On a mutual-neighbor pair with capacity (1/2,1), phase consensus and stored pressure zero, weights (1,0,0) give Si=(1/2,1), while (0,1,0) give Si=(1,1). At si_hi=3/4, identical nodal fields can therefore produce different capacity-gate decisions when only diagnostic configuration changes. If the counter condition is met, the existing mu=1/2 averaging proposal changes the left capacity to 3/4; this is a conditional proposal, not an executed experiment or a new coefficient choice for the primary study.
Add a disconnected isolated node. Raising its capacity from 1/2 to 2, with fresh maxima and weights (1,0,0), changes the unchanged pair's Si from (1/2,1) to (1/4,1/2). The right gate now fails at that same threshold. This is a graph-global control dependency, not transport through an edge. A global law could be postulated explicitly; local emergence has not thereby been derived. These stored-pressure examples test the consumer contract, rather than asserting fresh full-channel equilibrium.
With pure EPI pressure on the same unit pair, unit capacities and equal phases, x=(0,1) and x=(1,0) have opposite pressures and opposite nodal rates. Their nodewise Si values agree for any common diagnostic weights. Thus the full Si vector alone cannot determine the direction of EPI change.

Computing Si writes only its diagnostic/cache state, not the nodal triad or support. Its consumers are separate: the capacity gate, glyph candidate selectors, and the enabled functional-link branch of Coupling. That branch includes Si similarity in a configured link compatibility score; U3 and final admission remain separate. No Si-dependent phase evolution is inferred from the mere existence of Si sensitivities.

step(..., use_Si=False) skips refresh, not the consumers of stored Si. It is therefore not an implementation of telemetry-independent dynamics. The present scope clarification changes no runtime behavior. For the primary research line, retain Si as a read-out; do not use its weights, thresholds or counter policy to fill an unexplained causal step. Its existing controllers remain available as explicitly conditional engineering models.

The built-in selectors validate the stored decision inputs and use one shared snapshot/kernel across execution backends. Each new batch renews its pressure and acceleration normalizers and score weights; engine-owned snapshots are released even after failure. This prevents past maxima from silently changing a later decision, but does not recompute the underlying pressure or Si. Missing Si retains the explicit baseline 0.5; missing pressure/acceleration retains zero. Finite Si is clipped to [0,1]. These defaults, thresholds, hysteresis, repetition and lag rules remain controller choices, not deductions from the nodal equation. The decision controls test this boundary and scalar/worker agreement without treating a candidate as a live operator-admission certificate.

The corresponding reusable constraint is the source-tangency identity: it derives what phase/capacity evolution must satisfy to preserve zero pressure, without selecting that evolution through Si. The finite witnesses are reproducible with artifacts/research/validate_source_tangency_2026_09_18.py; their successful checks do not prove a general autonomous Si reduction or an emergent mechanism.

The separate capacity/phase result applies this same distinction to the word frequency: accumulated capacity follows from the nodal equation, while identifying it with circular phase requires an additional relation. The configured oscillator and phase-relaxation models are not uniquely selected by that equation.

Uniform capacity and default selector reachability¶

The configured diagnostic and selector have a concrete interaction on the retained prism. More generally, consider a nonempty graph with valid finite readings, equal positive capacities, fresh Si normalization and the default nonnegative normalized SI_WEIGHTS. Every node has nu_norm=1, so

\[ \mathrm{Si}_i=\operatorname{clip}_{[0,1]} \{\alpha+\beta(1-\mathrm{disp}_i)+\gamma(1-|p_i|_{\mathrm{norm}})\} \ \ge\alpha. \]

The configured primary weight represents pi/(pi+1)>3/4, while the default selector's upper cut is si_hi=1/2. Hence every node's default base choice is IL, independently of its pressure sign/magnitude and phase alignment. The represented normalized coefficients also satisfy this strict separation; the claim is not based on rounding them to displayed decimal values. Both current Si backends use the actual positive maximum, including positive subnormal capacities; their denominator-one fallback concerns only an exactly zero maximum. This theorem excludes that all-zero inactive class.

The complete decision path must still be distinguished from its base choice. The native selector can force AL or EN through its configured lag counters. The shared grammar fallback tries IL,THOL,EN,SHA,RA,NAV,AL, never ZHIR. Thus this ordinary default path cannot introduce ZHIR while the stated diagnostic condition holds. When IL itself is admitted and neither lag forces a replacement, the applied glyph is IL. The parametric selector, custom callables, stale or skipped Si refresh, heterogeneous capacities and changed diagnostic weights/cuts have different scope. No conclusion about all future steps follows if those conditions change.

A valid signed growth secant and U4b context can therefore make Mutation eligible without making it selected. A manually requested Mutation or a custom selector returning it would test a different occurrence rule. The configured Si formula does not prove that the physical nodal structure forbids a phase change. Conversely, altering Si weights or disabling its refresh to obtain the desired event would not derive a missing TNFR mechanism. This is a control-policy reachability result, not a newly prescribed policy or a defect warranting an arbitrary threshold change.

Controls: diagnostic lower bound and selection. The single finite causal check and source-work interpretation belong to the execution plan.

8. Grammar derivation: premises, language and trajectories¶

The grammatical audit distinguishes four claims: an identity following from the nodal equation, a conditional theorem for a specified realization, an operator/invariant contract, and a selected engine policy. A contract can be mandatory without being a theorem of the scalar equation. A classification predicate that returns membership in a declared set is executable consistency, not an independent mathematical derivation of that membership.

Rule	Structural content that can be established	Additional premise or policy	What acceptance does not establish
U1a	The rate is zero if capacity or pressure is zero; EPI zero is not singular.	A new word uses a generator unless existing form is declared.	Origin of basal capacity, unique birth mechanism or spontaneous selection.
U1b	Suppressing capacity suppresses the rate when pressure is bounded.	The supported endpoint set is SHA/NAV/REMESH/OZ.	A stationary endpoint, accumulated-change convergence or equivalence with the older PDF closure set.
U2	Accumulated change obeys the flow/jump balance below.	Named coverage, causal unit debt and capacity two.	A bound on physical pressure, an energy gain or stability under arbitrary repetition.
U3	Circular separation and the pairwise phasor cross term are computable.	A nonnegative cross-term domain motivates the default pi/2 gate; the coupling contract requires it.	All-support compatibility, phase-source rigidity or forward invariance.
U4	On smooth segments, acceleration is the derivative of capacity times pressure plus capacity times pressure derivative.	Trigger/handler roles, prior IL and three-position context.	A mathematical bifurcation, threshold crossing, unique next operator or a physical waiting time.
U5	A declared differentiable hierarchy obeys the chain rule; exact reduction requires closure.	Depth-local stabilization and a separately specified coherence target.	Autonomous macro dynamics, identity preservation or a universal coherence inequality.
U6	Potential is Bp; changes contain pressure and distance-kernel terms.	Mean absolute reference drift below pi/2.	Pointwise bounds, intermediate-time confinement, future safety or metric-scale invariance.

The unified specification owns the supported rules. This section owns their logical scope. The operator-role registry and operator effects must remain separate: sharing a primary channel does not determine a grammatical role. For example, IL and OZ both act on pressure but have different roles. THOL nesting and network-scale REMESH also show why target scale is not the number of state entries an operator may modify.

Necessary and sufficient for what? A grammar predicate can be necessary for admission to a declared engine API. It need not be mathematically necessary for every solution of the nodal equation, and it is not thereby sufficient for stable evolution. Pure diffusion without named stabilizers is the existing counterexample to the first promotion. Gain, timing, source and history obligations prevent the second. A grammar-valid word specifies possibilities; it does not select targets, invocation times or one autonomous successor.

9. U2 and U4 in the actual flow/jump model¶

For locally finite event times and piecewise absolutely continuous EPI, using right-continuous (post-event) endpoint values, the engine's hybrid interpretation requires both terms:

x(t)-x(t0) = integral D_nu(s) p(s) ds
            + sum_{events in (t0,t]} [x(event+)-x(event-)].

This assumes a common coordinate support; births, removals and remeshing need explicit coordinate/history maps. An EPI-writing zero-duration operator is a jump, not an ordinary finite pressure integrated over zero time. Between jumps the nodal equation holds; the shared event executor records the separate maps. An infinite-horizon sufficient condition is absolute integrability of the flow rate plus absolute summability of jump magnitudes, with events locally finite. U2 role counts do not prove either bound. The earlier flow-only integral statements apply to continuous segments, not silently to all jumps.

A useful reusable restricted energy balance comes from fixed symmetric nonnegative conductance W with positive degrees d and positive held capacity. Let B=D-W, H=diag(d_i/nu_i), z=x-1*(1^T Hx)/(1^T H1), and let the declared pressure be p=-e L_rw x+F with constant e>0. Then

V = (1/2) z^T H z,
V' = -e z^T B z + z^T D F.

Thus diffusion dissipates disagreement while the actual canonical source can add or remove it. At a same-support, same-metric jump j=x_plus-x_minus, with P the H-centering projection,

V_plus-V_minus = z_minus^T H Pj + (1/2)(Pj)^T H(Pj).

These identities make a grammar-to-stability proof obligation concrete: bound the source work and each actual jump, then combine them over the declared times and history. Changing metric/support and delayed REMESH add their own terms. Reuse regional balances, represented event schedules, and the delayed-history results, rather than fitting an energy meaning to the unit debt counter. Existing certificates remain restricted to their declared maps, norms and realization defects.

For U4, on a smooth segment,

x'' = D_nu_dot p + D_nu p_dot.

The EPI equation does not by itself specify nu_dot or p_dot, which can depend on phase, support and other state. A large acceleration or a configured threshold crossing is an event diagnostic; a bifurcation theorem requires a declared family of dynamics and a change in its equilibria, stability or other invariant structure. Event secants, continuous derivatives and operator-index recency are different quantities. Existing Mutation evidence and event clocks preserve this separation. No new threshold is needed here.

10. U3: exact geometric content and a strict-gate counterexample¶

For two nonnegative phasor amplitudes a,b,

|a exp(i theta_i)+b exp(i theta_j)|^2
  = a^2+b^2+2ab cos(theta_i-theta_j).

Demanding a nonnegative pairwise cross term yields circular separation at most pi/2 when both amplitudes are positive. This gives a geometric interpretation of the selected gate. It does not derive that design requirement or the full coupling map from x'=D_nu p. The actual UM/RA gates apply to their participating relations. Existing support and the full canonical phase-pressure neighborhood are not globally filtered through U3. Zero-weight support can still participate in phase pressure. Tightening an operator's gate does not change that owner.

On fixed undirected support with no isolated nodes, strict gaps below pi/2 on every edge imply Re(exp(-i theta_i) S_i)>0, where S_i=sum_neighbors exp(i theta_j). Hence all S_i are nonzero and the displacement from each center is inside its regular open half-pi chart. But the source derivative

g_i = wrap(Arg(S_i)-theta_i)/pi,
Dg = (R-I)/pi,
R_ij = 1[j in N(i)] Re(exp(i theta_j)/S_i)

depends on neighbor-to-resultant angles, not only neighbor-to-center angles. The previously proved nonnegative irreducible criterion is an additional hypothesis not implied by strict U3.

Exact six-node witness. Use a unit double-star: centers u,v joined by an edge; leaves a,b attached to u; leaves c,d attached to v. Set

(theta_u,theta_v,theta_a,theta_b,theta_c,theta_d)
  = (0, pi/3, -pi/3, -pi/3, 2pi/3, 2pi/3).

Every edge has separation pi/3, strictly within U3. The resultants are S_u=3/2-isqrt(3)/2, S_v=isqrt(3), and each leaf sees its center's phasor. Their squared magnitudes are (3,3,1,1,1,1), and

g=(-1/6,1/6,1/3,1/3,-1/3,-1/3).

The two center-to-center entries of R vanish exactly. Each center assigns 1/2 to each of its leaves; each leaf assigns one to its center. Thus R is nonnegative but reducible into two three-node classes despite the connected support. The exact shared observer gives rank(R-I)=4 and tangent dimension two. Independent infinitesimal rotation of either star preserves g to first order. This refutes the pending implication that strict U3 and connected support force rank n-1 for nonzero source.

The apparent freedom is obstructed at second order. Locally fixing leaf sources forces theta_a=theta_b=theta_u-pi/3 and theta_c=theta_d=theta_v+pi/3. Write delta=theta_v-theta_u. Fixing g_u=-1/6 then requires

Im[exp(i*pi/6)*(exp(i*delta)+2exp(-i*pi/3))]
  = sin(delta+pi/6)-1 = 0.

Near delta=pi/3 this forces delta=pi/3 exactly; the real part remains positive. The local finite fixed-source set therefore has only common rotation. Along the extra tangent which rotates v,c,d by eta,

g_u''(0)=-1/(pi*sqrt(3)),   g_v''(0)=1/(pi*sqrt(3)).

Leaf source derivatives stay zero. A rank-deficient linearization is therefore not a finite relative-motion mechanism, even strictly inside the gate. The exact regression fixture uses rational cosine Gram data and the existing derivative/rank owners; it does not simulate a trajectory or assert runtime operator admission.

Finally, an instantaneous phase gate is not a forward-invariance theorem. For a declared differentiable relative phase delta(t), the upper active boundary requires delta_dot<=0 and the lower boundary delta_dot>=0. An invariant-set proof additionally needs a well-posed phase law and the appropriate boundary conditions throughout that set, plus checks on jumps. The EPI equation alone supplies no such phase law. A merged UM stage may verify its used relations without certifying all later flows.

11. U5: exact scale dynamics and coherence ordering are separate¶

For a differentiable representation y=f(x), y'=Df(x)x' is an identity along the declared micro trajectory. An autonomous macro law additionally requires this value to be identical on every fiber f(x)=y. For a fixed affine EPI system x'=Tx+b and linear read-out y=Qx, the exact condition is

QT=A_bar Q,

equivalently invariance of ker(Q) under T; Qb supplies the macro source. Failure produces hidden-state dependence. Reuse the repository's quotient, observability and memory derivations; adding a scale stabilizer does not establish this algebraic condition.

Even exact closure does not order nonlinear coherence. On unit K4 with unit capacity and pure EPI pressure, partition the vertices into {0,1} and {2,3}. Take x=(0,-3/4,3/8,3/8)+c*1; then

p=x'=(0,1,-1/2,-1/2).

This equitable partition closes exactly. Block means y=(-3/8,3/8)+c*1 obey a two-node diffusion with effective capacity 2/3 and pressure p_bar=(3/4,-3/4). In the first block, y'=1/2. Using the same canonical C=1/(1+|p|+|x'|) at both scales gives

C_parent=4/9 < (C_child_0+C_child_1)/2 = 2/3.

This is a failure of the proposed average-coherence inequality with alpha=1/2 despite exact dynamical closure. The effective capacity is fixed by the two external neighbors among three, not fitted to the inequality. An arbitrary uniform offset can make every EPI positive without changing it. More generally, at zero pressure/rate, m equilibrium children have C=1 and their sum is m, so an unnormalized universal target needs alpha<=1/m even in that simple case. The supplied U5 assessment correctly requires an explicit hierarchy and coefficient. No new physical coefficient is derived.

12. Audit integration and remaining obligations¶

The grammar owner specifies shared word admission, profiles, live rejection and observation. Signed nonzero form, causal debt, prior Coherence, actual recursion metadata and live phase checks are separate inputs. A diagnostic waiver is not a normal word pass; missing phase/reference observations are unavailable rather than evidence of safety. Tests establish those implemented contracts, not their physical necessity.

There is no complete derivation of the thirteen maps, their invocation law, role catalog or numerical policies from the scalar nodal identity alone. Nor is there a proof that six grammatical rules are minimal or sufficient for all admitted TNFR dynamics. These limitations do not justify discarding an existing contract without a replacement argument.

The strict-U3 witness in §10 disproves the proposed rank implication while its finite fixed-source set is locally rigid. The U5 witness in §11 separates exact macro closure from coherence ordering. Both remain reusable counterexamples; neither selects a phase/capacity/support law or establishes NFR maintenance. The execution plan alone assigns current work.

13. Structural grammar refactor and the full nodal system¶

Physical conclusions require explicit structural and dynamical premises. Renaming a calibrated constant, differentiating an underdetermined identity or removing a contract cannot supply the missing law. The grammar basis registry records which parts are identities, conditional theorems, contracts or policies; these are provenance declarations, not proof-carrying permission to execute.

The word and execution interface owns profiles, rejection modes and read-only observations. Its finite typed execution evidence can bound the represented EPI composition in a common metric. Gain at most one proves nonincrease for those maps; strict inequality proves contraction. Gain above one does not prove that the actual observed trajectory expanded. Missing evidence remains unavailable.

Such a result does not authenticate another graph, supply a next operator, or extend to a global binary64 runtime, future schedule, full-channel or tetrad theorem. The following equations identify the additional closure obligations.

Differentiation exposes missing laws; it does not invent them¶

Let z=(x,nu,theta,W,h) denote a declared complete state, including relevant support/weights and history h. On a regular continuous chart with pressure p=P(z), the EPI component is

x'=D_nu P(z),
x''=D_nu_dot p + D_nu p'.

The derivative p' includes the derivatives of every channel on which P depends. For the fixed-support, fixed-coefficient realization p=-e L_W x+w_phi g(theta)-v L_U nu with no changing topology source,

p'=-e L_W D_nu p + (w_phi/pi)(R-I)theta' - v L_U nu'.

If conductance changes smoothly, add -e L_W' x; if any other source or coefficient changes, its derivative must also be retained. Actual edge birth/deletion and operator writes are hybrid events, not silently smooth terms. These identities reuse sections 9-10 and the existing source-tangency derivation. They constrain proposed laws for nu, theta and support; they do not select those laws. The same-initial-state completions already prove nonuniqueness of the EPI future without an additional phase relation. Differentiating the original identity adds no independent relation capable of removing that ambiguity.

For fixed symmetric conductance, positive capacity and the pure EPI channel, the closed diffusion and its Dirichlet dissipation are already derived. They are a reusable positive example of structural dynamics without operator-count thresholds. Extending that result requires accounting for source work, changing geometry, auxiliary channels and jumps, not assuming the extension. The auxiliary symplectic model and its conserved quantities have their own premises; the variational bridge cannot be promoted into a derivation of the complete nodal dynamics without proving the correspondence of their states and trajectories.

The tetrad participates in the same closure problem¶

All four canonical fields remain required read-outs, computed by the existing field owner:

Field	Joint structural dependency	Obligation for a dynamical conclusion
Phi_s	Pressure and the distance kernel B of the current geometry.	Phi_s'=B p'+B' p on smooth aligned charts; event and reference changes separately.
Phase gradient magnitude	Circular phase differences on the declared neighborhood.	Actual phase/support law; wrapping and absolute-value boundaries cannot be treated as globally smooth.
K_phi	Circular curvature on that same support.	A regular-chart derivative or an explicit branch/event treatment; no global substitution by the EPI Laplacian.
xi_C	The declared correlation field, graph distances, fit and fallback.	State/scale and estimator regime must be specified; the spectral fallback is not an identity for all correlation fits.

For a differentiable tetrad observation T(z), T'=DT(z)F(z) follows only after a full law z'=F(z) is specified. An autonomous tetrad law additionally needs that derivative to agree on every state sharing T(z). The existing capacity rescaling counterexample in section 5 excludes an unconditional claim that the tetrad determines the evolution. Singular charts and estimator switches need separate treatment. Geometry may legitimately enter a derived law, but its diagnostic definition alone is not that derivation.

A candidate autonomous NFR mechanism must therefore address the triad, pressure, support, memory and tetrad together. An EPI energy bound, an instantaneous U3 gate or a U6 threshold alone cannot stand in for this joint claim. Reuse the existing exact quotient/memory tests to ask whether a proposed macro observation closes; use the tetrad to report its geometric and correlation consequences without imposing a diagnostic target as a new force. These are constitutive admission requirements. The single execution plan owns their current priority; this scope note does not open another trajectory campaign or promote a configured selector to emergent physics.

14. Constitutive closure audit from the nodal law¶

The constitutive question starts with structural form and x'=diag(nu)*p, not with the claim that each implemented formula is a necessary TNFR law. The foundation audit owns EPI types, coordinate changes, units and the original source's conflicting definitions. Source fidelity preserves the underlying hypothesis; it does not require keeping an inconsistent dimensional equation or a lossy chart claim.

Existing laws and independently supplied premises¶

Component	Implemented or derived content	Missing implication
EPI flow	Shared integrator implements capacity times declared pressure; isolated EPI diffusion has its exact Dirichlet balance.	The nodal identity alone does not choose the state space or pressure functional.
Detached smooth-sine research	Reciprocal form/phase exchange derives joint balances and response under a complete, globally defined comparison law.	The sine kernel, cosine storage, exchange/loss scales, held capacities/support, clock, preparation and events remain premises. The storage counterfamily retains balances and organized equilibria without uniquely selecting that law; it is separate from native argument-pressure execution.
Phase relaxation	coordination.py blends global/local circular means per invocation with configured, possibly adaptive gains.	Invocation time, gains and their adjustment policy are not derived by the EPI identity.
Oscillator proposal	phase_evolution.py adds `dt*nu` and a configured sine coupling over admitted neighbors.	Capacity as angular speed and the coupling coefficient are additional constitutive premises. This is distinct from mean relaxation.
Capacity adaptation	adaptation.py applies clipped neighbor averaging after stored Si/pressure gates and a consecutive-call count.	Averaging gain, thresholds, waiting count and use of that diagnostic as a controller are supplied policies; there is no independently derived capacity rate.
Optional extended system	canonical.py supplies explicit nonlinear phase/pressure responses; the integrator retains the live additive Gamma source in the EPI row.	Its coefficients, source and flux correspondence need justification; it provides no capacity law and does not follow from the EPI product.
Edge creation	_coupling_stage_kernel.py combines candidate sampling, phase checks and configured link scores.	U3 constrains which proposed links may be accepted; it does not select their occurrence, weight or persistence.
Remeshed topology	remesh.py provides declared MST/neighbor/random rewiring policies.	EPI-distance construction and seeded reproducibility are not a derived support evolution law.
Variational completion	Variational section 13 derives the joint potential family compatible with a specified EPI mobility.	Its free non-EPI function, auxiliary mobilities and invocation law remain undetermined.

The ordinary runtime integrates EPI before phase coordination and capacity adaptation. Disabling glyph selection or fresh Si computation does not by itself disable those later steps. Their retained gain/history state belongs in any declared complete model.

The optional extended pressure response is exactly -(1+0.135)*div(J_p); it has no restoring dependence on p. Constant nonzero supplied divergence therefore gives a constant pressure slope, not automatic relaxation. With capacity one, p=1/2 and zero flux, the configured phase rate is 0.575, not zero: zero flux removes transport terms, not all auxiliary dynamics. These documentation errors are corrected without changing the arithmetic.

Optional pressure composition and source consistency¶

Composing the actual shared readers sharpens the preceding scalar-input audit. On fixed unique-neighbor support let L_U=I-N^-1 U, with zero rows at isolates. The pressure-contrast reader gives J_p=-L_U p; the integrator's scalar "divergence" gives diag(sqrt(k))*L_U J_p, where k counts unique neighbors. The optional pressure response therefore implements

\[ \dot p=(1+0.135)\operatorname{diag}(\sqrt{k})L_U^2p. \]

This is not an incidence divergence of oriented edge fluxes. On a regular undirected graph the nonconstant eigenmodes have positive squared-Laplacian rates, rather than diffusion decay. The formula is a continuous vector-field interpretation of the supplied update; finite Euler steps, rounding and clipping retain separate effects.

The retained unit prism provides an in-band, zero-phase-gap control. Set x=1/2*1-P/4, P=(1,-1,0)^2, unit capacity, and pure-EPI pressure. The shared canonical pressure is p=P/4 and L_U P=P. Differentiating that same constitutive law using the shared joint-response owner requires p_dot=-L_U*x_dot=-p. The optional block instead gives p_dot=(1+0.135)*sqrt(3)*p, with the opposite sign. Both expressions refer to the same prepared state; no trajectory, fitted parameter or numerical step size is needed to detect the incompatibility.

The optional path remains an explicitly configured independent-pressure model, disabled by default in update_epi_via_nodal_equation. Ordinary runtime dispatches its resolved integrator directly and then coordinates phase and adapts capacity; this wrapper's flag alone does not replace that path. Reversing one sign would not derive agreement with the full multichannel chain rule. Its arithmetic is retained while unwarranted conservation/invariant claims are removed. Three unused private synthetic flux helpers are removed; execution already used the shared real field readers and divergence owner. The fresh-pressure phase comparison has a separate exact dissipation identity on a regular prism chart and is not silently substituted for the optional model. Controls: extended-pressure scope.

Capacity remains undetermined even under dissipation¶

On unit P2 with consensus phase and homogeneous positive capacity a(t), both the capacity-gradient and topology sources vanish. With EPI-channel coefficient e>0, x(0)=c1+d(1,-1), c>d>0, compare two mathematical completions:

a_A(t)=1,     x_A(t)=c*1+d*exp(-2*e*t)*(1,-1),
a_B(t)=1+t,   x_B(t)=c*1+d*exp(-2*e*(t+t^2/2))*(1,-1).

Time t>=0 is expressed in one fixed normalized structural unit for this witness. Both solve the same nodal equation and the same canonical pressure realization, share the complete initial triad/support/history, initial tetrad and EPI rate, keep positive EPI and capacity, and strictly dissipate Dirichlet energy. Yet their initial accelerations differ by -2*e*d*(1,-1). Thus adding positivity, phase compatibility and dissipation does not determine capacity evolution. These are logical countermodels, not proposed TNFR mechanisms or two executions of a single configured deterministic runtime. Their exact controls reuse the actual pressure and field owners in test_constitutive_capacity_scope.py.

Moving conductance: an exact balance and an unobservable scale¶

On fixed active symmetric support, write B=D-W, g=-D^-1 Bx and r=nu*p. For supplied differentiable positive conductances and positive row strengths,

g_i' = mean_W(r_j-r_i)
       + [sum_j W_ij'*(x_j-x_i)-d_i'*g_i]/d_i,
E_D' = (Bx)^T r + (1/4) sum_ij W_ij'*(x_i-x_j)^2.

The first derivative separates form evolution from changing geometry; the energy balance separates nodal work from the change of its conductance weights. There is no explicit capacity derivative in this energy because E_D depends on x and W. The shared support derivative observer now evaluates both identities exactly on declared rational/represented input, including loops and isolates. It reuses the snapshot, Laplacian and energy owners, validates symmetric edge rates and does not select W', refresh pressure or execute a graph change. Active-set changes need the reset owner.

For any positive c(t), W(t)=c(t)*W_0 gives the identical normalized walk D(t)^-1 W(t)=D_0^-1 W_0. At fixed unweighted support all other canonical gradient channels are likewise unchanged by this weight scaling. Functions with c(0)=1 but different c'(0) give distinct conductance histories with the same initial state and the same instantaneous EPI generator. Nevertheless E_D scales by c. In the derivative balance the geometry term in g' vanishes while conductance work equals (c'/c)*E_D. This is nonidentifiability of the weight law from EPI transport, not evidence that weights physically change.

The tetrad must retain its metric conventions. With explicit fixed edge length, structural potential is unchanged at fixed pressure; with weight as the compatibility length, distances scale by c and Phi_s by c^-2. Phase gradient and curvature use their declared support/phase. The coherence-length fit has estimator-specific distance conventions; a small-graph spectral fallback must not certify every fit or backend. Portable controls in test_constitutive_support_scope.py state which estimator they use. No complete tetrad-autonomy theorem follows.

What is now derived and what must come next¶

This subsection records mathematical dependencies, not a chronological task queue. The execution plan determines the current next step.

The variational calculation supplies useful positive constraints: the EPI law fixes the x-dependent part of a compatible potential, requiring reciprocal couplings when auxiliary gradient-flow premises are chosen. It also gives a source-compatibility criterion for boundedness below. The strict-U3 star counterexample rules out only a constant positive diagonal gradient metric for the actual nonlinear phase source near consensus; it leaves other metrics and non-gradient dynamics open. These are consequences to reuse, not reasons to impose a preferred energy minimizer.

Before proposing missing laws, define the intended EPI state/equivalence, directed pressure action and clock independently of a retrospective fit. Test a proposed representation with the existing closure and minimal-realization owners, observability and operator quotient. The continuous fibre condition and event intertwining are in the foundation note. A scalar observable may close without reconstructing the full state; a rich storage object may still lack any justified evolution. Retain all four tetrad channels as consequences to account for, and introduce neither a physical clock nor a selection functional solely to obtain stability.

When an additional law is justified¶

A predictive model needs additional laws or declared input histories whenever a consumed channel has no specified evolution or justified elimination. Holding capacity, support or another channel fixed is already an explicit completion; every such channel need not immediately acquire autonomous evolution. Missing closure does not by itself invalidate the nodal row or establish a need for a more general equation. Mathematical closure, well-posedness and physical justification are separate obligations. A proposed law may be an independently stated constitutive hypothesis, a consequence of stronger premises, or an inherited reduction with retained memory and error. Its status must survive integration into the engine.

Evaluate each proposed law or hypothesis against its existing owner: state its domain and units, classify its logical status, expose its independent premises, and derive the consequence that needs it. Test an alternative or counterexample before claiming necessity or uniqueness. Record whether the claim is derived under those premises, remains a supplied hypothesis, is underdetermined, or is rejected in the declared domain. A revised foundational premise needs the same scrutiny and a dependency review of earlier results; a failure in one completion does not invalidate every model of the nodal row.

NFR and operator properties are part of this admission, with their logical status retained. Use the NFR property requirements and participation in form to specify identity, form/capacity domains, retained internal and boundary state, circular phase and any claimed nested description. A law must respect the admitted domains and symmetries; a reduction must preserve the information consumed by its claimed response or account for its loss. Name which requirements are definitions, independent constitutive premises or consequences of the chosen model.

For claimed operator actions, select the actual execution path and apply its event-map and composition contracts, including live admission, target/history dependence, primary and secondary state changes, ordering and event costs. A collective description also needs the operator descent criterion. The grammar scope remains distinct from continuous-law selection and autonomous event occurrence. A detached continuous comparison has its own laws and declared interventions; matching a named operator's apparent effect does not establish that operator's contract. The canonical catalog does not by itself prove generative completeness or uniquely determine the continuous pressure/phase law.

The existing parameter ledger identifies the missing premises. The phase/storage alternatives and replication counterfamily show why balance, symmetry, organized equilibria or a shared local response cannot alone select the complete nonlinear law. An added information or selection principle needs independent justification; reproducing a desired pattern does not provide it. Distinguish genuinely different completions from coordinate or clock changes that transform every row of the same model.

Use the derived organizations as test cases for those premises. Specify which generative claim requires more than the present preparation, identity and interaction results, then revisit the premise that blocks it. For example, uniqueness at full equilibrium does not supply spontaneous initial activity, and finite positive-loss dynamics restricts sustained nonstationary motion. Neither restriction excludes static organized identities or every other admitted completion. Autonomous source preparation, contact occurrence, scale selection and a physical observation require their own arguments; they are not consequences of a successful prepared response. The bound-organization target also needs a stated composite identity and binding criterion; a signed storage excess alone supplies neither.

Integrate an admitted completion through its declared-law and shared execution owners, with matched domains, sources, events, balances and observations. Diagnostics and grammar do not silently provide its missing controller. Preserve valid conditional objects and counterexamples while testing a prospective discriminator, equivalence or obstruction between alternatives. Reusing a proof method or numerical kernel does not transfer the old law's derivative, regularity, source or trapping bounds. Match every consumed hypothesis before inheriting a certificate; otherwise derive it for the new complete law. A local response coefficient alone is not a finite-response separation with an admitted error budget. The execution plan selects the investigation; this admission criterion creates no parallel queue.

Cross-channel foundation audit¶

The follow-through is centralized in NODAL_PARAMETER_FOUNDATIONS.md, not another constitutive-law inventory. It covers every foundational parameter family, derives coefficientwise form/time covariance and the conditional local diffusive generator, and records telemetry, metric, energy and memory units. The shared coherence-length estimator retains distinct fit and spectral fallback provenance. Configured gains and phase-period interpretations require their stated premises; a numeric value alone does not close autonomous phase/capacity/support dynamics.