Forced-source closure and phase clocks

Phase/capacity tangency, source constraints and the limits of synchronization as a clock.

Part of Forced support balance and model boundaries. Section numbers are stable across the collection; hypotheses and model changes remain local to each result.

21. Closing the relaxed phase-capacity source

Exact model and common fixed fields

Write x=EPI, let L_W be the weighted EPI random-walk Laplacian and L_U the unweighted unique-neighbor support Laplacian. The actual canonical pressure decomposition is

p = -e L_W x + w_phi g_phi - w_vf L_U nu - w_topo L_U k,

where k is the support-degree vector. These two Laplacians generally differ. This section studies fixed physical fields of the existing phase and capacity updates, together with zero fresh pressure. It does not assume that the EPI nodal equation uniquely determines those configured update policies.

Telemetry versus dynamics. Si is a derived diagnostic of capacity, phase and pressure, not an additional nodal force or a fundamental evolution law. Computing/storing Si does not itself reorganize the triad. The implementation chooses to consume Si in a threshold-based capacity-adaptation policy. That policy has a dynamical effect because it writes capacity; its use of a diagnostic is an extra operational assumption, not a derivation from dEPI/dt=nu_f*DeltaNFR. Every result below about that gate is conditional on this implemented policy. It cannot exclude persistent differentiated forms under other structurally justified TNFR dynamics. The configured phase and averaging laws likewise remain explicit premises.

Assume finite undirected support, connected positive symmetric EPI conductance, positive capacity, e>0 and w_topo=0, as in the retained channel configuration. The phase support is connected as well. Use exact-real means and trigonometry, no scalar clipping, no named operator/reset interventions and no external changes to support or fields. Required diagnostics refer to these same fields.

The coordination owner has the ideal map

theta_i^+ = theta_i + k_G wrap(m_G-theta_i) + k_L wrap(m_i-theta_i),

with global and neighbor phasor arguments m_G and m_i. Suppose all phases admit a common real lift in an interval of width less than pi, gains are nonnegative and at least one is positive. Require a fixed point of this lifted update, excluding nonzero multiples of 2*pi hidden by normalization. Every phasor mean lies between its participating minimum and maximum, with a strict interior value unless those phases agree. At a maximum phase both increments are nonpositive. If k_G>0, a nonconstant state gives a strictly negative global increment. If k_G=0 and k_L>0, a fixed maximum forces every neighbor to share that maximum; connectedness propagates equality. Therefore the phase fixed fields in this chart are exactly consensus phases. In particular the ideal local phase-pressure channel g_phi vanishes.

For an active capacity update,

nu_i^+ = (1-mu) nu_i + mu mean_{j in N(i)} nu_j, with 0<mu<=1.

If every node is eligible, a fixed capacity satisfies L_U nu=0. The same maximum principle makes nu constant. With phase consensus and w_topo=0, the non-EPI source vanishes. Positive capacity then makes zero nodal rate equivalent to p=0, and e L_W x=0 forces uniform EPI on connected positive conductance. Conversely constant positive capacity, constant EPI and phase consensus leave these physical fields fixed. This classifies common fixed fields of the selected substeps; it does not classify cancellations between substeps of an arbitrary composite runtime cycle.

Conditional consequence of the Sense Index gate policy

Within that adaptation policy, the all-eligible premise can be weakened. At zero fresh pressure and phase consensus, the Sense Index reduces to

Si_i = clamp01(alpha*nu_i/nu_max + beta + gamma).

Assume its normalization uses the current positive nu_max and that its nonnegative weights satisfy s_max=clamp01(alpha+beta+gamma) >= si_hi. Normalized exact weights give s_max=1; represented weights need the displayed inequality, not an assumed exact sum of one. Every capacity maximum meets the Si threshold and the zero-pressure part of the stability gate. If unchanged fields are repeatedly evaluated without external counter resets, its finite stable_count reaches VF_ADAPT_TAU. At a boundary of a nonconstant maximum plateau, positive-mu averaging would strictly decrease capacity. Such a plateau cannot persist. Connectedness therefore forces uniform capacity even if some lower-capacity nodes are initially ineligible.

The existing held-capacity profile x=c*1-(w_vf/e)*nu, when L_W=L_U, remains a valid conditional balance. It is not a self-consistent heterogeneous equilibrium of this fresh-diagnostic relaxation policy. Freezing Si or disabling its refresh supplies a different closure. The native runtime refreshes pressure and optional Si before later substeps; that timing agrees at a common equilibrium but cannot be ignored along a changing trajectory. Stable counters and histories may grow even when physical fields are fixed.

What can and cannot be inferred

Channel or mechanism	Exact conclusion in this branch	Remaining sustaining condition
Attractive phase coordination	Consensus at a lifted fixed point in the common semicircle	A phase pattern outside that chart, nonstationary motion or another admitted phase feedback needs its own proof
Implemented capacity policy consuming fresh Si	Under this gate and averaging law, a persistent zero-pressure equilibrium cannot hide heterogeneous maxima behind inactive gates	This conditional obstruction does not constrain a different capacity law derived from nodal structure
Pure EPI transport after source loss	Only uniform zero-pressure form	A maintained canonical source or a different notion of identity is required for differentiation
Derived partial-observation memory	Reexpresses existing full dynamics	A memory kernel does not create a new sustaining source
Topology pressure	Absent when w_topo=0	Nonzero topology weight changes the retained configuration and still requires weighted compatibility

This is not a convergence theorem. On P2, local-only phase gain k_L=1 swaps two unequal phases within the semicircle; capacity gain mu=1 likewise permits a swap. With k_L=4 and phases (0,pi/2), increments (+2pi,-2pi) also show why wrapped stationarity is weaker than a lifted fixed point. Binary64 stalled gaps, clipping and represented phasor roundoff remain outside the exact theorem. Positive-conductance connectivity cannot be replaced by mere support connectivity with zero-weight links.

Uniform EPI is not the same as an unstructured complete triad. For local-only phase coordination on C_n, n>=5, the regular winding theta_j=2pij/n has neighbor resultant 2*cos(2*pi/n)*exp(i*theta_j) and is nonconstant but fixed. Its local phase pressure is zero: it preserves phase structure without supplying differentiated stationary EPI. Existing winding studies retain their scope; this observation does not reopen the archived C6 campaign.

A different conditional escape is already present in the pressure algebra. On a unit-conductance nonregular graph with consensus phase, constant positive capacity and symbolic w_topo>0, x=c*1-(w_topo/e)*k has zero pressure. The three-node path gives a nonuniform example without a new force term. However it holds the irregular support and changes the retained zero topology coefficient. On the actual retained weighted support, the independently read compatibility scalar is d^T g_topo = -2170365805794839/4222124650659840, which is nonzero. Activating topology pressure alone there would cause mean drift under consensus phase and uniform capacity, not a stationary pattern. No coefficient is changed or fitted in this delivery.

Disposition. The stationary branch of these configured relaxation laws is classified; stationary TNFR mechanisms in general remain open. Neither fixed-point uniqueness nor a supplied profile establishes autonomous NFR formation. The full triad, support and any genuine endogenous history must carry the identity being tested. A new sustaining closure must be explicit; configured lag schedules, REMESH echoes and target-based controllers cannot be promoted silently to a law derived from the nodal equation.

The general result is the analytic proof above, independently reviewed against the phase, adaptation and Sense Index owners. All 108 existing targeted phase, adaptation, capacity-balance and forced-support tests pass; this is regression evidence, not a numerical proof of the general theorem. Finite exact controls and the authenticated weighted-topology calculation are retained in artifacts/research/source_closure_analytic_checks_2026_09_18.json, with the script artifacts/research/validate_source_closure_2026_09_18.py. This analytic delivery changes no engine code, policy parameters or historical artifacts and executes no graph, pressure kernel or trajectory.

22. Source tangency without a telemetry controller

The diagnostic scope separates a derived observable from a justified causal law. The following result advances the primary closure question directly from the existing pressure channels; it neither uses Si nor supplies a new evolution policy.

Regular fixed-support identity

Keep fixed symmetric conductance W, finite undirected unique-neighbor support U, constant channel coefficients e>0, w_phi>=0, v=w_vf>=0 and w_topo=0. Assume positive capacity and differentiable exact-real fields on an interval without events, clipping, zero neighbor resultants or phase-wrap crossings. Write

p=-e L_W x+F, F=w_phi*g(theta)-v L_U nu, x_dot=diag(nu)*p.

These are the same pressure conventions as section 21, not a fitted source. The phase support is unweighted even when W is weighted. For S_i=sum_{j in N(i)} exp(i*theta_j), its mean-response matrix is

R_ij=1[j in N(i)] Re(exp(i*theta_j)/S_i).

Differentiating Arg(S_i) gives R. Its rows sum to one; they need not be nonnegative. On the declared regular branch,

g_dot=(R-I)*theta_dot/pi.

The exact coefficient owner already exists: derive_phase_response computes mean_response=R from admitted unit planar cosine Gram data. Use that field, not its separately composed operator-stage Jacobian. The support gradients remain owned by support_transport, and forcing_realization owns the captured channel split. Neither a Gram table nor a captured binary64 pressure is a proof that a live continuous chart is admissible.

Differentiating the declared pressure, rather than reconstructing it from measured EPI motion, now gives the identity

p_dot=-e L_W diag(nu)*p + (w_phi/pi)*(R-I)*theta_dot - v L_U nu_dot.

In particular, at p=0,

p_dot=(w_phi/pi)*(R-I)*theta_dot-v L_U nu_dot,

x_ddot=diag(nu)*p_dot.

This is the source-tangency condition: the phase-source and capacity-source changes must cancel if EPI is to remain at zero pressure. The direct capacity-rate term in x_ddot is diag(nu_dot)*p, which vanishes here; capacity still acts through its contribution to pressure. The identity does not determine theta_dot or nu_dot, nor a time for any operator to act.

At phase consensus, R=I-L_U. On connected support the condition is equivalent to spatial constancy of

(w_phi/pi)*theta_dot + v*nu_dot.

For fixed phase and v>0, this requires uniform capacity increments/rates, not uniform initial capacity. Thus a heterogeneous capacity-supported profile from section 21 is not excluded by the nodal equation itself. If v=0, capacity has no pressure-source constraint of this type; its positive mobility still sets EPI response away from p=0.

Finite events and what an instantaneous test cannot prove

Between two supplied fields on the same support with fixed x and channel weights, the exact finite counterpart is

p_plus-p=w_phi*(g_plus-g)-v L_U*(nu_plus-nu).

For a capacity-only change with fixed phase, v>0 and connected U, zero pressure is preserved exactly when nu_plus-nu is uniform. A real event that also writes EPI must include -e L_W*(x_plus-x); changing support, coefficients, clipping or stored operator pressure needs its own terms. This detached identity is not grammar admission, event activation or runtime provenance.

A concrete algebraic control uses unit P2, consensus phase, e=v=1/2, nu=(1/2,1) and x=(1,1/2). Its canonical pressure is exactly zero. A supplied capacity increment (1/4,0), with x fixed, gives pressure (-1/8,1/8) and post-change nodal rate (-3/32,1/8). The uniform increment (1/4,1/4) preserves zero pressure. These are detached states checked against the existing channel owners; no capacity law or graph evolution is introduced.

Pointwise tangency is only necessary. For a counterexample, allow the explicitly free, adversarial completion nu(t)=(1+t^2,1) on P2, phase zero, x(0)=0, e>0 and v>0; let x solve the original nodal equation. At t=0, p=p_dot=x_ddot=0, but p_ddot=x'''=(-2v,2v). This proves insufficiency of one instantaneous test, not a proposed TNFR capacity law. Conversely, if F stays constant throughout an interval and p(0)=0, the pressure equation is homogeneous and uniqueness gives p(t)=0 there. This all-time condition still supplies neither a law maintaining F nor stability after perturbation.

Consequence for the primary research question

This identifies a missing relation rather than replacing it with a controller: the nodal EPI equation and current pressure decomposition do not uniquely specify capacity, phase, support or event activation. Uniform phase rotation already lies in ker(R-I), and constant capacity shifts lie in ker(L_U). Multiple mathematical completions are therefore possible; their existence does not make them canonical TNFR mechanisms.

Geometry can preserve a phase pattern without sourcing differentiated EPI. A stationary lifted local-only phase update with nonzero local gain requires wrap(Arg(S_i)-theta_i)=0, so g_i=0 even outside a shared semicircle. Regular winding does not evade this condition. Prescribed winding or an imported particle label therefore cannot fill the missing source law.

This completes the bounded tangency calculation. Any proposed law for the missing channel rates/activation must be independently justified and checked against this identity. That requirement does not assign a new law-search task; the execution plan owns the current gate. No controller experiment, C6 restart, topology sweep or new public physics claim follows. The finite exact controls reuse existing algebra and are recorded in artifacts/research/source_tangency_checks_2026_09_18.json with artifacts/research/validate_source_tangency_2026_09_18.py. The symbolic proof above, not a finite test count, establishes the stated conditional identity.

23. Capacity exposure does not determine a phase clock

Source and implementation boundary

The original TNFR source distinguishes structural frequency from periodic oscillation. Its physical PDF pages equal the printed page numbers:

• Section 1.4.10, pages 48-49, presents a classical dynamical-system correspondence, including a Kuramoto model with omega identified with nu_f.
• Page 211 says structural frequency "no representa una oscilación periódica convencional"; pages 212-213 describe coherent-reconfiguration counts per structural-time interval.
• Page 224 qualifies the Kuramoto analogy: nu_f "mide reorganización, no ritmo". Page 218 describes structural time as an internal process coordinate.

These are compatible when the oscillator identification is a particular constitutive correspondence, not a universal deduction. The audited PDF has 378 pages and SHA256 5ba0f4a2da2d01e550e7004c3c09c6927e6620826052df84cd28babbcbd34fc3.

The shared oscillator proposal uses

theta_i_plus=wrap(theta_i+dt*nu_i+dt*K*mean_U3 sin(theta_j-theta_i)).

The nodal optimizer and FFT engine use this additional phase model beside their EPI diffusion. It numerically interprets the supplied frequency as an angular rate in radians per supplied time unit. By contrast, the ordinary runtime calls phase coordination, a per-invocation global/local mean relaxation with no dt or free nu_i advance. Neither path establishes that the other is a derived limit. The optional phase-transport formula in canonical.py also declares operational sensitivities; its name is not a uniqueness proof.

Multiplying by 2*pi would convert cycles to radians only after identifying one counted event with a complete cycle. That identification is absent from the nodal EPI law. The Hz bridge is already explicitly configured and does not supply it. No frequency factor, phase law, pressure weight or runtime behavior is changed by this result.

Exact independence, including a relative-phase witness

The EPI law and current pressure channels do not determine phase speed. On unit P2, let x=c1, nu=nu_01>0 and theta=a(t)*1. Every pressure channel vanishes for every differentiable a(t). The same initial triad therefore admits both constant phase and advancing common phase, with unchanged x and zero phase separation. This is common-phase freedom only; it does not by itself show that relative phases affect a prediction.

For that stronger distinction, take the existing zero-pressure family on P2,

nu=(nu_0,nu_1)>0, nu_0!=nu_1, x_i(0)=c-(v/e)*nu_i, theta(0)=0,

with held capacities and existing coefficients e>0, v>=0, w=w_phi>0. There are two mathematical completions of these same nodal data:

1. Keep phases fixed. The canonical source is held, p=0 and x remains fixed.
2. Let theta_i(t)=nu_it and solve x_dot=diag(nu)p with the same canonical pressure. This is an independence witness, not a proposed physical law or an executed operator sequence. Restrict time so the relative phase stays inside the chosen regular chart and U3 bound.

Set d=nu_1-nu_0 and a=e(nu_0+nu_1)>0. In the second completion, g=(dt/pi,-d*t/pi), and pressure is (q,-q), where

q_dot=-a*q+w*d/pi, q(0)=0,

q(t)=(w*d/(pi*a))*(1-exp(-a*t)).

Consequently x_0(t)=x_0(0)+nu_0*integral(q) and x_1(t)=x_1(0)-nu_1*integral(q) obey the original nodal law, with

integral_0^t(q)=(w*d/(pi*a))*(t-(1-exp(-a*t))/a).

The EPI futures differ whenever d and w are nonzero. No pressure was solved backwards from an observed derivative. These are analytic alternatives within the incompletely closed equations, not a claim that both are admitted full runtime trajectories. Thus phase closure changes structural predictions; the ambiguity is not only a global phase convention.

The section 22 tangent gives the same initial result on connected support:

p_dot(0)=-(w/pi)*L_U*nu, x_ddot(0)=-(w/pi)*diag(nu)*L_U*nu.

For nu=(1/2,1), their coefficients of w/pi are respectively (1/2,-1/2) and (1/4,-1/2). The existing sine-coupling proposal has zero interaction at initial phase consensus, so its initial free-advance consequence does not depend on selecting K. Its later coupled path is not identified with the uncoupled analytic completion above. The failed implication is universal phase speed from the EPI equation; configured oscillator models are not thereby mathematically forbidden.

What the nodal equation does derive: accumulated capacity

On a regular interval with nu_i(t)>0, define

s_i(t)=integral_t0^t nu_i(u) du.

Changing parameter along that same trajectory gives exactly

dx_i/ds_i=p_i(t_i(s_i)).

No conversion factor or additional driving term is introduced. This is accumulated capacity, not circular phase or a measured physical clock. Neighbors in p_i must still be evaluated at the shared time t_i(s_i); separate local clocks cannot be treated as simultaneous independent network clocks. A common clock removes every nodal prefactor only when the capacities agree at each time; a common factor in heterogeneous capacities leaves their relative factors. Zero capacity stalls the clock and invalidates this regular inverse. At p=0, accumulated capacity may advance while EPI stays fixed.

The pure-EPI common-clock solution and heterogeneous boundary already belong to directed transport section 6. The finite-exposure retention result in cycle support section 4 is reused, not rerun. Capacity also enters the full pressure through -v L_U nu; a time-coordinate change must not be confused with rescaling capacity while silently holding that source unchanged.

Disposition. The capacity-to-phase gate is closed with an independence result and a derived reparameterization. A fundamental law for relative phase/capacity evolution remains missing. Geometry can constrain compatible motions before a speed is assigned; the execution plan owns that next bounded question. No oscillator sweep, telemetry controller or physical-particle claim follows from this result. The finite controls and source provenance are retained in artifacts/research/capacity_phase_checks_2026_09_18.json and artifacts/research/validate_capacity_phase_2026_09_18.py.

24. Rigidity and flexibility of a held phase source

Regular rigidity from the shared mean derivative

On fixed finite support with nonempty neighborhoods, use section 22's S_i=sum_j exp(i*theta_j) and R_ij=1[j in N(i)] Re(exp(i*theta_j)/S_i). Assume nonzero resultants and a regular center-to-mean wrap chart. Then Dg=(R-I)/pi and R*1=1. This statement refers to the canonical full-support phase channel, not a new phase-update equation or a U3-filtered mean.

If R is nonnegative and its positive-entry directed graph is strongly connected, its fixed-vector space is exactly the common-rotation line. Indeed, a maximal component of a vector satisfying R*h=h is a convex mean of its neighbors. Every neighbor with a positive coefficient must share that maximum; connectivity propagates it to every node. Thus

ker(R-I)=span{1}, rank(R-I)=n-1.

A differentiable constant-g path staying in this domain satisfies (R-I)*theta_dot=0, hence theta_dot=a(t)*1 and theta(t)=theta(t0)+c(t)*1. Common rotation conversely preserves g. Its speed remains unspecified. This is a compatibility theorem, not relaxation, attraction, self-maintenance or a stability theorem for the full triad.

There is also local uniqueness modulo rotation: fix one phase coordinate. The remaining derivative has rank n-1, so n-1 independent output coordinates give a locally invertible map. Nearby states with the same full g differ only by rotation. This does not prove global reconstruction or connectedness of a level set. The already recorded consensus and regular winding on a cycle can both have g=0 and individually rigid derivatives while belonging to different local branches; no winding campaign is repeated here.

A sufficient geometric domain on connected undirected support is that every neighbor phasor has positive projection onto its row resultant:

Re(exp(i*theta_j)*conj(S_i))>0 for every support neighbor j.

This makes R positive on all support edges. A common phase lift of width strictly below pi/2 guarantees it, since every numerator is a sum of positive pairwise cosines; it also ensures regular resultants and wraps. The rowwise projection criterion is wider and must retain its separate wrap premise.

For zero phase source, a further useful corollary uses the actual U3 scale. If g_i=0 and every edge separation is strictly below pi/2, S_i points along theta_i and

R_ij=1[j in N(i)] cos(theta_j-theta_i)/|S_i|>0.

Connected support then has the same rigidity. This does not automatically extend to nonzero g: the row resultant need not point along its center. The closed pi/2 gate permits zero projections and needs separate treatment.

Connected support alone is insufficient: a cube family

Use the unit-conductance cube Q3=C4 x P2, with nodes (j,l), j modulo four, l in {0,1}, horizontal neighbors (j-1,l),(j+1,l) and partner (j,1-l). Assign the same four phases to both layers:

theta_(j,l)=(a,b,a+pi,b+pi)_j.

Each horizontal pair is antipodal and cancels exactly. The partner has the center's phase, so S_i=exp(i*theta_i) and g_i=0 for every a,b. Resultants have squared magnitude one and the center-to-mean displacement is zero. Varying b-a therefore gives a genuine finite source-preserving deformation, not merely an extra direction of a pointwise derivative. Here a,b are phase coordinates, not new force coefficients or a proposed activation law.

At quadrature b-a=pi/2, R is the vertical-partner permutation. It is nonnegative but reducible into four two-node classes despite connected graph support. Its rank(R-I)=4 and tangent dimension is four. At the exact phasor point cos(b-a)=3/5, sin(b-a)=4/5, the horizontal derivatives are signed and the rank is six, giving tangent dimension two. With c=cos(b-a), the spectra of R-I split by layer parity:

layer-symmetric: (0,0,2c,-2c), layer-antisymmetric: (-2,-2,-2+2c,-2-2c).

The displayed family supplies two finite phase parameters (one common and one relative); tangent dimension four at quadrature does not prove that all four directions integrate into a four-dimensional finite level set.

U3 boundary. The horizontal circular separations are |wrap(b-a)| and pi-|wrap(b-a)|. Requiring every edge to obey the default pi/2 bound forces quadrature. Any relative-angle departure violates one horizontal edge. This family is therefore not an all-edge U3-compatible Coupling orbit. The existing pressure channel reads all support neighbors; silently dropping U3-incompatible neighbors changes its definition and destroys this argument. No graph creation mechanism, admitted event sequence or phase-speed law is supplied by the geometric family, and the cube is not a selected emergent polyhedron or a particle identification.

The EPI consequence still follows exactly in its declared mathematical scope. The unit cube is regular, so L_W=L_U=L and topology pressure is zero. For held positive capacities, the existing family

x=c0*1-(v/e)*nu, e>0, v=w_vf>=0,

has p=-L(e*x+v*nu)=0 throughout the phase deformation. It is differentiated when v>0 and nu is nonconstant. This reuses the capacity-supported balance; it derives neither the capacity distribution nor a law maintaining it.

Reusable exact observation and claim boundary

observe_phase_source_geometry rebuilds the existing PhaseResponseReference from its primitive Gram/incidence data and observes rank(R-I) with the shared exact-rank owner. It returns the scaled source derivative, tangent dimension and mean-response sign. The merged operator-stage Jacobian is deliberately separate: at phase_factor=0 that Jacobian is the identity even when the source remains locally rigid. An exact rank n-1 also certifies conditional local rigidity for some signed R, without needing the sufficient nonnegative proof.

For example, K4 with phasors (1,0) on three nodes and (-3/5,4/5) on the fourth has negative mean entries -1/13 but rank(R-I)=3. Negative entries alone do not imply geometric freedom. Conversely the quadrature cube is nonnegative yet has more than the rotation line. Zero resultants are rejected; a live phase chart, U3 admission, causal execution and temporal stability are not established by an exact Gram or rank calculation.

The portable controls cover these independent matrices, exact cube cancellation, disconnected support, permutation covariance and rejection of tampered cached references. The general rigidity and finite-family proofs are analytic, not inferred from the number of passing controls. This closes the planned geometric gate: regular rigid regions and flexible boundary families both exist. The sole execution plan owns the continuation; no controller or new evolution law follows from this result. The subsequent grammar audit settles the strict-U3 nonzero-source rank question: a connected double-star has a two-dimensional tangent kernel strictly inside the gate, but its extra direction is obstructed at second order. Its local finite level set has only common rotation. The remaining question concerns finite geometry, not another rank calculation. The proof and portable fixture have one owner in that audit.

25. Relational time and synchronization are separate claims

The original source describes Reception in terms of shared/relational time and synchronization (PDF page 83), and internal process time on pages 218-219. This motivates a relational-clock hypothesis; it is not a derivation that global synchronization equals elapsed time. Section 23 owns the existing accumulated-capacity identity and its source/units audit.

Alignment need not advance, even while form evolves. On pure-EPI unit P2, positive equal capacities and phase consensus give R=1. Holding those phases fixed is compatible with the EPI equation while a nonuniform form relaxes: for capacity one and initial EPI (1,0), x(t)=((1+exp(-2t))/2,(1-exp(-2t))/2). Synchronization remains complete while form and its accumulated capacity change. Conversely, uniform EPI/capacity permit any differentiable common phase rotation under that same EPI identity. The alignment statistic does not identify its speed. Neither control selects a complete physical phase law.

On the retained prism with phases (-a,a,0) in both triangles, |a|<pi/4, the same statistic is R=(1+2*cos(a))/3. It is even in a and has zero derivative at a=0. Along the supplied control a=A*cos(chi), it repeats after half the full phase cycle. It therefore loses orientation and cannot serve as a globally invertible clock for that motion. This concerns phase alignment; a broader notion of coordination involving form, capacity and history must supply its own observation and evolution, not inherit this statistic's name.

A local state clock requires an already specified tangent

For a complete autonomous state law z_dot=V(z) and a scalar observation tau=T(z), a regular local time coordinate requires h=dT[V]>0. Then dz/dtau=V/h. If the relevant components of V are missing, the chain rule does not generate them. A single-valued real state function cannot increase strictly around a closed orbit: its endpoint difference is zero, whereas the integral of a strictly positive rate would be positive. An unwrapped angular clock needs a chart/history or cycle count, as well as a law for its advance; a circular phase alone supplies neither.

A regular positive reparameterization preserves the oriented path, and an onto unbounded time change preserves recurrence. Finite accumulated exposure can instead map infinite original time to a finite internal-time endpoint; that retention mechanism is already covered by the capacity results and is not a proof of continuing active oscillation. Relabeling time does not turn the pure gradient trajectories of variational section 13.10 into recurrent ones.

Curve admission can determine a speed without selecting the curve

For a declared full-state curve z(chi), let v=dx/dchi be its EPI tangent and let b=diag(nu)*p be the nodal rate evaluated independently from that state and the existing pressure law. A regular positive scalar clock must satisfy

\[ b=h v,\qquad h=d\chi/dt>0. \]

For v!=0, this is equivalent to collinearity and v^T b>0; the only possible speed is h=(v^T b)/(v^T v). The inner product merely computes the unique proportionality coefficient and introduces no physical metric or force. Every component must agree. If exactly one of b,v vanishes there is no regular positive clock; if both vanish the EPI equation leaves the clock unconstrained at that point. At an EPI turning point, phase or other coordinates may still move, so failure to identify the clock there must not be confused with complete-state stationarity.

This is an EPI admission condition for a supplied curve, not its generation mechanism or a complete phase/capacity law. The pressure must not be obtained retrospectively as h*v/nu. It provides a useful rejection test: changing a clock cannot repair a source tangent pointing in the wrong direction.

Reuse and implementation scope

The existing structural_time reader numerically accumulates supplied capacity over the supplied grid, starting at its first point. Its trapezoidal value is an estimate, not an exact integral for an arbitrary capacity function. certify_structural_time now uses that accumulated exposure as its finite structural observation window, including zero exposure; it previously used the final input timestamp instead. It remains a finite numerical diagnostic with an unassessed tail, not a derived physical clock or a general infinite-time certificate.

Controls: clock scope and existing structural-time implementation. The complementary oriented source/form work identity allows a proposed loop to be rejected before choosing its speed. The single execution plan uses these conditions within G3; no synchrony statistic is promoted to a controller or a fundamental time law.