Coarse-graining, coherence geometry and bridge results

Sections 1–6 establish restricted coarse reductions, field geometry and inverse/endpoint contracts. Each result retains its declared pressure, support, capacities and clock; none supplies closure of all 13 operators or a coupled global TNFR geometry. The nonlinear chapters distinguish the normalized-sine law from alternative reciprocal mobilities. Their shared storage balance does not make their complete dynamics interchangeable.

Chapters and scope

Chapter	Responsibility
Sufficient unordered pair state	Realizability, temporal reconstruction, capacity correlations and support symmetry, including phase cancellation.
Internal replica pulses and their perturbations	Prepared pulses, full variation, transverse splitting and persistent constituent activity.
Phase and joint form-phase grouping	Equilibrium geometry, pair observations and finite grouping windows; observation does not create support.
Pair geometry under alternative reciprocal mobility	Grouping and protected geometry under a separately declared complete law.
Pair actions, regional exchange and cancellation	Form events, autonomous currents, cancellation and the retained interaction interface.

Section numbers remain stable. The execution plan alone assigns research work.

Formation, observation and maintenance are distinct. Protection of a two-sided invariant family does not prove entry into it. The separate validated native-law transit retains its own dissipative law, preparation and capture proof.

Repository graphs keep two edge channels distinct. weight is transport conductance: it defines W, diffusion, Dirichlet energy, quotient generators and the conductance coordinate of the fixed-topology state metric; it defaults to one. length is the independent shortest-path channel used by structural potential geometry and is the second edge coordinate of that metric. An explicit length wins there; if it is absent, path geometry falls back to weight for compatibility, and then to one if neither is present. Consequently length does not change EPI diffusion, while weight also changes legacy path geometry unless length is declared. Models in which coupling and distance differ should declare both attributes explicitly. The executable policy is centralized in _edge_semantics.py.

1. Pure-EPI coarse-graining

On fixed symmetric nonnegative conductance with positive fine row strengths and positive capacities, take a partition with positive macro row strengths. These hypotheses make the inverse metric and the displayed positive macro capacities well defined. Isolates require a separate zero-row treatment and are outside this construction. Write

x' = -A x,                 A = H^-1 B,
H = diag(h_i),             h_i = d_i/nu_i.

For a node partition, let P copy one macro value to every node in a block. The reversible block average is

R = (P^T H P)^-1 P^T H,   R P = I.

The quotient conductance between two blocks is the sum of all micro conductances crossing between them. If h_bar=P^T h, its generator is

A_bar = diag(h_bar)^-1 B_bar.

This is again a TNFR pure-EPI nodal equation. Its macro capacity is nu_bar_a=d_bar_a/h_bar_a and macro EPI is R x.

Exact closure condition

The macro observable is autonomous for every micro state exactly when

R A = A_bar R.

Equivalently in this reversible construction, the block-constant subspace is invariant:

A P = P A_bar.

Thus exact coarse-graining is an intertwining structural morphism, not a fourteenth operator. The information removed by the quotient has dimension N-m, where m is the number of blocks. A nonzero intertwining defect measures the influence of those unresolved within-block modes; the effective dynamics then needs memory or additional macro variables and the nodal equation is not closed in R x alone.

certify_epi_coarse_graining constructs the quotient, reports both closure defects and a clearly named within-tolerance decision, then delegates morphism classification to the existing certificate. The exact theorem is the zero-residual identity; a caller tolerance cannot make a nonzero defect algebraically exact. An equitable reflection partition of a path has zero residual to machine precision. A path partition whose middle block mixes boundary and interior nodes is an executable counterexample.

Inherited topology source and same-law scope

Closure of EPI diffusion does not imply that every canonical pressure channel can be recomputed from the bare quotient graph. This can fail even at uniform phase and capacity, with no unresolved EPI memory. The following exact fixed-support result uses the existing unique-neighbor topology channel, not an additional physical interaction.

Take unit undirected complete bipartite support K_(a,b), with positive integers a,b, a+b>2, and common capacity nu>0. Partition into its two sides. Hold support, capacity and common phase fixed and keep the declared effective channel coefficients fixed. The fine unique degrees are b on the first side and a on the second. The existing topology response is the neighbor-mean degree minus the node's degree, hence

g_topo = (a-b) on side A, (b-a) on side B.

The reversible projection is the arithmetic mean on each side: fine weights are b/nu and a/nu, and each block has total metric weight a*b/nu. The quotient conductance is a*b, its capacity is again nu, and

A_bar = nu * [[1,-1],[-1,1]],       R A = A_bar R.

Indeed every fine neighbor mean is exactly the opposite block's average; averaging each fine derivative proves the intertwining identity for all fine EPI fields, not only block-constant ones. Uniform capacity and phase make their gradient channels zero. With y=R*x, the complete enabled pressure therefore gives the exact affine coarse law

y_dot = -w_epi*A_bar*y + nu*w_topo*(a-b, b-a).

But the simple two-node quotient has one unique neighbor at each node, regardless of the conductance a*b. Its freshly recomputed topology gradient is (0,0). For a!=b and w_topo!=0, applying the unchanged four-channel recipe to that bare quotient loses a nonzero source. Changing only its scalar coefficient cannot repair a zero gradient. In K_(2,3) at uniform EPI, nu=1 and all four effective weights 1/4, the actual projected nodal rate is (-1/4,+1/4) while the naive quotient rate is zero.

Positive inheritance result. Retaining the fine degree profile (b,a) or its calculated source restores the displayed exact affine law. This is structure inherited from the declared graph, not pressure solved from a desired trajectory. The existing forced closure observer verifies its all-state affine closure and zero hidden-to-macro coupling. Thus the result does not refute coarse autonomy; it refutes losing inherited structure while claiming the identical bare-graph pressure recipe. For this held domain the source is constant; a changing support would need its own inherited evolution and event law.

Nine executable controls compare the production pressure writer, exact represented-coefficient support/forcing records and both macro constructions. They include hidden nonuniform fine EPI, the inherited affine response, balanced bipartite graphs and a zero-topology-weight control. The default topology weight is zero, so this is not a defect in the default configuration. The all-state real identity above and the finite binary64 controls have distinct scope. Neither supplies autonomous phase/capacity/support laws, spontaneous partition selection or physical emergence. This complements the existing phase-fiber and inherited potential obstructions below rather than replacing them.

Joint constitutive reduction with inherited support counts

The next constructive bridge retains structural information rather than reapplying the fine formulas to a bare simple graph. Consider connected symmetric positive transport and a fixed partition with exactly block-constant primitive phases Theta_a and positive fine capacities kappa_a. Hold phase, capacity, support and effective channel coefficients fixed. Fine scalar EPI evolves by the declared multichannel pressure and need not be block-constant. These held rows specify a conditional fine model; the nodal product does not derive their stationarity.

Require equitable unique-support counts

N_ab = number of distinct neighbors in block b of each node in block a,
s_a = sum_b N_ab.

Internal neighbors contribute to N_aa; zero-conductance support edges still contribute to these counts. They are different from transport conductance. On the defined circular-mean branch, the fine non-EPI channels are block-constant and are reconstructed by

g_phase,a = wrap(Arg(sum_b N_ab exp(i Theta_b)) - Theta_a) / pi,
g_vf,a    = (sum_b N_ab kappa_b)/s_a - kappa_a,
g_topo,a  = (sum_b N_ab s_b)/s_a - s_a.

These follow by grouping the existing neighbor sums. They do not introduce chosen phase weights, a new pressure channel or an autonomous phase clock. The phase formula retains the production wrap orientation; a zero phasor resultant requires an explicit implementation policy and is outside an exact regular-branch interpretation.

Keep d_i for fine transport strength, so h_i=d_i/kappa_a and hbar_a=sum_(i in a) h_i. The existing cross-only quotient removes internal transport from its displayed adjacency, giving strength dbar_a and effective capacity nu_eff,a=dbar_a/hbar_a. Consequently

sigma_a = kappa_a / nu_eff,a = (sum_(i in a) d_i)/dbar_a,
f_N,a   = w_phase*g_phase,a + w_vf*g_vf,a + w_topo*g_topo,a,
p_eff   = -w_epi*L_rw,bar*y + diag(sigma)*f_N,
y_dot   = diag(nu_eff)*p_eff.

This is the projected nodal law for every fine EPI state when the existing exact EPI intertwining test passes. Indeed R diag(nu) applied to a block-constant source equals diag(kappa) applied to that source, while diag(nu_eff) diag(sigma)=diag(kappa). Thus the rescaling is fixed by the inherited metric and the selected transport normalization, independently of measured rates or desired responses. Fine capacity kappa remains the input to the capacity-pressure formula; replacing it by nu_eff generally changes that formula.

A second normalization obstruction. On unit K3, partition ((0),(1,2)), uniform fine capacity one descends to nu_eff=(1,1/2). Here N=((0,2),(1,1)), so counted capacity pressure is still zero. Using nu_eff as the capacity field on a new bare P2 instead produces gradients (-1/2,+1/2). Retaining a self-neighbor changes the second incorrect value but does not repair the first. This issue can occur with the default positive capacity-channel weight, unlike the earlier zero-default-topology counterexample. It concerns a naive reduction, not the original fine engine. The counted construction preserves the actual zero channel without a fitted cancellation.

The shared support observer owns exact multiplicities, both conductance normalizations, the inherited metric, both capacities and counted capacity/topology gradients. It rejects inequitable support or nonconstant block capacity; weighted EPI closure is a separate obligation. The joint observer then reuses the existing pressure capture and exact affine closure owners. It evaluates counted phase with the same non-JIT NumPy kernel using repeated target indices for distinct fine neighbors, rather than adding another phase implementation.

Numerical and dynamical scope. Grouped phase sums can round differently from the fine neighbor order. The observer keeps that unit-phase defect, its projected rate contribution, fresh pressure assembly error and stale stored-pressure residual separately. Its exact rational identities connect those represented-coefficient models. The inherited NumPy kernel uses an explicit zero phase-pressure extension at a represented zero resultant, where the geometric curvature reader reports an unavailable direction. The explicit phase_geometry_scope retains that distinction; successful rate identities do not certify geometric phase availability. They are not a certificate of exact transcendental evaluation, continuous phase evolution or repeated runtime execution. Held phases/capacities have zero rates in this specified model; no history or event is executed. The selected partition has not emerged. This is a derived effective description with inherited structure, not proof that the unchanged simple-graph engine represents every scale.

Controls are centralized in support tests and joint-channel tests. They distinguish the counted channels from simple-graph recomputation, including unequal phase-neighbor multiplicities and internal neighbors. Field inheritance remains a separate requirement, addressed next.

Joint evolving phase/form reduction on fixed K3

The counted construction also permits a joint evolving calculation on a restricted invariant subspace. Take fixed unit undirected K3, the selected partition ((0),(1,2)), and identical positive fine capacity kappa. Supply the existing U3-gated sine phase law with coupling K>0, identifying its free angular rate with that fine capacity. The gate is pi/2. Use fresh canonical multichannel pressure with fixed effective coefficients e=w_epi>0, w=w_phase>=0; the capacity and topology coefficients may be nonzero, but their gradients vanish because fine capacities and degrees are uniform. Hold capacities, support and the pressure recipe fixed; exclude Gamma, operator events and clipping. These are constitutive premises, not consequences of the nodal equation alone.

Assume block-constant fine EPI and primitive phase. Write their two distinct values as x0,x1 and continuous phase lifts theta0,theta1, with

q = x0-x1,                 delta = theta1-theta0,
abs(delta) < pi/2.

All fine neighbors are then U3-compatible. Node 0 has two neighbors with phase theta1; each node in block 1 has one neighbor of each phase. Their phasor directions are respectively theta1 and the midpoint (theta0+theta1)/2 in this chart. Therefore the actual counted pressure and the supplied averaged-sine phase law give

p0 = -e*q + w*delta/pi,
p1 =  e*q/2 - w*delta/(2*pi),        p2 = p1,

x0_dot = kappa*p0,
x1_dot = kappa*p1,

theta0_dot = kappa + K*sin(delta),
theta1_dot = kappa - (K/2)*sin(delta).

The two fine nodes in block 1 receive identical exact-real rows, so the block-constant subspace is invariant. This statement does not cover within-block perturbations, arbitrary partitions or unequal capacities. Eliminating the common coordinates yields a triangular relative system:

c = 3*K/2,                A = 3*kappa*e/2,
B = 3*kappa*w/(2*pi),

delta_dot = -c*sin(delta),
q_dot     = -A*q + B*delta.

The geometry supplies the factors 3/2 and the unequal block rates. It does not create a nonzero intrinsic-frequency difference. In particular,

m = (x0+2*x1)/3,          m_dot = 0,
b = (theta0+2*theta1)/3,   b_dot = kappa,

x0 = m + 2*q/3,           x1 = m - q/3,
theta0 = b - 2*delta/3,    theta1 = b + delta/3.

Here b is a lifted weighted phase coordinate, not a global circular mean or a clock derived from the tetrad. The common advancing phase still comes from the declared microscopic law. The EPI mean is precisely the inherited metric mean, since the fine transport metric is (2/kappa)*I and its block weights are (2/kappa,4/kappa).

Why effective mobility must not be reused as an intrinsic phase rate. The cross-only quotient has conductance two and effective capacities nu_eff=(kappa,kappa/2). Its counted source multipliers are (1,2), hence its effective nodal-pressure vector is

p_eff = (-e*q+w*delta/pi, e*q-w*delta/pi),
(x0_dot,x1_dot) = diag(kappa,kappa/2)*p_eff.

These unequal effective capacities derive from transport normalization; the fine phase law still reads the common microscopic kappa and the neighbor multiplicities N=((0,2),(1,1)). Substituting nu_eff for the free angular rate on a new bare P2 would instead produce delta_dot=-kappa/2-2*K*sin(delta). It already disagrees at delta=0 and can manufacture a nonzero lock from a uniform microscopic clock. That would be a changed model, not emergent detuning. Recomputing its capacity pressure from nu_eff would introduce the additional false source identified in the preceding normalization obstruction.

Exact continuous restoration ends at consensus. Choose 0<rho<pi/2 with abs(delta(0))<=rho. The phase interval [-rho,rho] is forward invariant because its vector field points inward. More explicitly,

delta(t) = 2*atan(tan(delta(0)/2)*exp(-c*t)).

Set mu=c*cos(rho)>0. Since sin(delta)/delta>=cos(rho) on this interval,

abs(delta(t)) <= abs(delta(0))*exp(-mu*t),

abs(q(t)) <= abs(q(0))*exp(-A*t)
             + B*abs(delta(0))*I(A,mu,t),

I(A,mu,t) = (exp(-mu*t)-exp(-A*t))/(A-mu)   if A != mu,
I(A,A,t)  = t*exp(-A*t).

The second inequality follows directly from variation of constants. Both relative coordinates therefore approach zero, and fine EPI tends to the constant m. The unique relative equilibrium in this chart is (delta,q)=(0,0), with linearized eigenvalues -c,-A. Common EPI level and common phase remain separate neutral coordinates. Nonzero form contrast is not maintained by the unequal effective mobilities. Phase may temporarily increase abs(q) before relaxing, so this is not a claim of monotone form contrast or monotone diagnostic coherence.

No additional pressure law is needed to express the joint response:

p0_dot = -A*p0 - (c*w/pi)*sin(delta),      p1 = -p0/2.

This identity differentiates the already declared pressure along the coupled rows. It does not add an independently adjustable force. For w=0, form is isolated diffusion with q(t)=q(0)*exp(-A*t); the phase trajectory is unchanged. A runtime control must retain the same effective e when disabling w, because setting a raw channel weight to zero can renormalize the other channels. Moving that raw weight to the identically zero topology channel on K3 supplies such a control.

Ideal Euler scope. A simultaneous exact-real explicit step is

delta_next = delta - h*c*sin(delta),
q_next     = (1-h*A)*q + h*B*delta.

For 0<h<=min(1/c,1/A), the phase keeps its sign, decreases in magnitude and remains in [-rho,rho]. Let a=1-h*mu, b_e=1-h*A; then

abs(delta_n) <= a^n*abs(delta_0),
abs(q_n) <= b_e^n*abs(q_0)
            + h*B*abs(delta_0)*sum_(j=0)^(n-1) b_e^(n-1-j)*a^j.

Both terms tend to zero; when a=b_e, the sum is n*a^(n-1) for n>=1. The EPI mean is preserved and the lifted weighted phase advances by h*kappa per step. Since q_next is a convex combination of q and w*delta/(e*pi), every ideal contrast stays in

[min(q_0,-w*rho/(e*pi)), max(q_0,w*rho/(e*pi))].

Together with x0=m+2*q/3, x1=m-q/3, this gives an explicit sufficient ideal no-clipping condition by checking the two interval endpoints against the allowed EPI range. It does not establish a future binary64 bound.

Implementation and evidence boundary. propose_joint_nodal_phase_step returns a detached JointNodalPhaseStep: its existing joint observation retains the same-snapshot EPI pressure, effective mobility and fine capture, while its new phase proposal compares actual fine endpoints with lifted counted endpoints. Both phase paths reuse the arithmetic owner in phase_evolution.py; counted neighbors retain the multiplicities supplied by the existing quotient observer. The exact signed phase_step_defect records the difference between represented endpoints, not a circular distance or an authenticated runtime certificate. This observer does not integrate EPI or write the graph. The joint phase tests separately pass the simultaneous pressure/capacity inputs to the actual shared DefaultIntegrator, compare fine and reduced one-step endpoints, and refresh pressure after the EPI and phase proposals. These finite one-step and lift controls do not establish repeated runtime closure. Binary64 trigonometry, neighbor summation, phase wrapping and Euler updates have separate represented defects; the exact equations above are not identities for every rounded row. No asymptotic binary64 convergence or future gate/clip admission follows. The theorem closes the joint calculation on the declared symmetric K3 subspace. Autonomous capacity/support evolution, partition selection, transverse stability outside the following fixed-law domain and a maintained nonuniform NFR remain open.

Transverse restoration and omission error on the same K3

The preceding invariant-subspace calculation can be tested against differences inside the selected block. Retain its fixed unit K3, common positive fine capacity kappa, supplied averaged-sine phase law K>0, and fresh pressure coefficients e>0, w>=0. Capacity and topology gradients remain zero; exclude Gamma, events, changing support and clipping. Choose continuous fine phase lifts with initial width at most rho, where 0<rho<pi/2. This is a condition on all three fine phases, not merely on their two block averages. Require the fixed effective U3 gate to admit this width, so every fine neighbor contributes to the supplied phase law throughout the invariant chart.

Define the visible and omitted coordinates by

delta = (theta1+theta2)/2-theta0,     eta = theta1-theta2,
q     = x0-(x1+x2)/2,               u   = x1-x2,

c = 3*K/2,     A = 3*kappa*e/2,     B = 3*kappa*w/(2*pi).

Each node has two neighbors whose relative phases lie in its open half-pi chart, so the neighbor-phasor direction is their midpoint. Subtracting the actual fine rows gives the exact continuous system

delta_dot = -c*sin(delta)*cos(eta/2),
eta_dot   = -K*(cos(delta)*sin(eta/2)+sin(eta)),
q_dot     = -A*q+B*delta,
u_dot     = -A*u-B*eta.

The minus sign in the hidden form forcing follows from the declared orientation u=x1-x2, eta=theta1-theta2. The mean form m=(x0+x1+x2)/3 is conserved and the lifted mean phase b=(theta0+theta1+theta2)/3 advances at the supplied rate kappa. Reconstruction is exact:

x0 = m+2*q/3,      x1 = m-q/3+u/2,          x2 = m-q/3-u/2,
theta0 = b-2*delta/3,
theta1 = b+delta/3+eta/2,    theta2 = b+delta/3-eta/2.

Thus an omitted form difference alone does not affect these macro rows. An omitted phase difference does: it changes the visible phase relaxation through cos(eta/2), which subsequently changes the visible form response. At delta=0 that macro defect vanishes even when eta is nonzero. Generic off-fiber phase states therefore lack exact two-coordinate macro autonomy; the special zero-visible-phase case must not be used to infer it.

An invariant chart and attraction of the omitted coordinates. Subtract the common rotation kappa*t. At a maximal lifted phase every sine term is nonpositive; at a minimal phase every term is nonnegative. The phase width therefore never exceeds its initial bound. In particular,

abs(eta) <= rho,       abs(delta)+abs(eta)/2 <= rho,
mu = c*cos(rho) > 0.

To obtain a quantitative rate, use sin(z)/z>=cos(z) inside this chart, continuously extended at zero, and

cos(delta)*cos(eta/2)
  = (cos(delta+eta/2)+cos(delta-eta/2))/2 >= cos(rho).

The scalar damping coefficient in each phase row is consequently at least mu. Variation of constants in the form row then yields

abs(delta(t)) <= abs(delta0)*exp(-mu*t),
abs(eta(t))   <= abs(eta0)*exp(-mu*t),
abs(u(t)) <= abs(u0)*exp(-A*t)+B*abs(eta0)*I(A,mu,t).

Here I(a,b,t)=integral_0^t exp(-a*(t-s))*exp(-b*s) ds is the nonnegative convolution already evaluated above, including I(a,a,t)=t*exp(-a*t). The selected block becomes coherent in both phase and form. The whole triangle also reaches consensus in its relative coordinates: this is attraction of a supplied grouping, not selection or persistent identity of a distinct two-node object. Every relabeled pair has the same symmetry.

A prospective absolute error for omitting the phase difference. Let delta_bar,q_bar follow the preceding eta=0 macro model with the same initial visible coordinates, common means and supplied coefficients. Its phase interval also remains inside [-rho,rho]. The omitted term in the visible phase equation is exactly

r_delta = c*sin(delta)*(1-cos(eta/2)),
abs(r_delta) <= (3*K/16)*abs(sin(delta))*eta^2.

This uses 1-cos(z)<=z^2/2, not a truncated Taylor equality. Define R0=c*abs(delta0)*eta0^2/8. The phase bounds imply abs(r_delta(t))<=R0*exp(-3*mu*t). For D=abs(delta-delta_bar) and Q=abs(q-q_bar), the mean-value theorem gives the scalar comparison D_dot<=-mu*D+abs(r_delta) in the upper-derivative sense, including at D=0. Hence

D(t) <= R0*I(mu,3*mu,t),
Q(t) <= B*R0*(I(A,mu,t)-I(A,3*mu,t))/(2*mu).

The difference of convolutions is nonnegative; equivalently the second bound is B*R0*integral_0^t exp(-A*(t-s))*I(mu,3*mu,s) ds. These expressions remain valid when A=mu or A=3*mu by the stated continuous definition of I. Both bounds tend to zero. Their all-time upper bounds are

sup D <= R0/(3*sqrt(3)*mu),
sup Q <= (B/A)*R0/(3*sqrt(3)*mu).

The first maximum follows by maximizing I(mu,3*mu,t)=(exp(-mu*t)-exp(-3*mu*t))/(2*mu); the second uses the positive form convolution with integral at most 1/A. Thus the visible omission error is quadratic in initial eta0, with no fitted coefficient or reserved-trajectory adjustment. This does not make the fine state recoverable from its block means. Against the reference lifted to the same fine graph, reconstruction gives, for example,

max_i abs(x_i-x_bar_i) <= max(2*Q/3, Q/3+abs(u)/2).

The omitted fine form and phase remain first-order information even when their effect on the macro response is second order.

Pressure and potential reuse the same error coordinates. Set P=-e*(q-q_bar)+w*(delta-delta_bar)/pi and H=e*u+w*eta/pi. The fine pressure error against that lifted reference is

p-p_bar = (P, -P/2-3*H/4, -P/2+3*H/4).

Consequently, with M=e*Q+w*D/pi and J=e*abs(u)+w*abs(eta)/pi, its maximum absolute value is at most max(M,M/2+3*J/4). Every fresh exact pressure on this unit triangle sums to zero; its fine inverse-square potential therefore satisfies Phi_s(i)=sum_(j!=i) p_j=-p_i. The same bound applies to this potential error. Both read-outs use the original fine graph and unit path lengths. Recomputing potential on a bare quotient is a different observation. This identity neither bounds every tetrad reconstruction nor removes the correlation-fit availability boundaries.

Ideal Euler scope. A simultaneous exact-real Euler step of the four displayed rows preserves the lifted phase-width bound for 0<h<=min(1/c,1/A). Indeed each fine phase update, after removing h*kappa, is a convex combination with weights (h*K/2)*sin(theta_j-theta_i)/(theta_j-theta_i) and a nonnegative remaining self-weight. Let a=1-h*mu, b_e=1-h*A. The visible and hidden phase magnitudes contract by at most a, while

abs(u_n) <= b_e^n*abs(u0)
  + h*B*abs(eta0)*sum_(j=0)^(n-1) b_e^(n-1-j)*a^j.

For the ideal eta=0 comparison initialized with the same macro state,

D_n <= h*R0*sum_(j=0)^(n-1) a^(n-1-j)*a^(3*j),
Q_n <= h*B*sum_(j=0)^(n-1) b_e^(n-1-j)*D_j.

These are exact-model discrete convolution bounds. They do not combine an exact continuous reference with an unmeasured Euler error. A condition excluding future EPI clipping must additionally use the reconstruction and form bounds; phase admission alone cannot supply it.

Implementation and arithmetic boundary. observe_k3_transverse_state reuses the fine capture to read the represented lifted coordinates, common capacity and effective pressure recipe. bound_k3_transverse_euler evaluates an exact rational upper envelope of the ideal-Euler bounds for a declared finite step count. Its computational domain uses the narrower sufficient raw phase-width condition rho<=1 radian, without wrapping the supplied chart. It replaces the transcendental constants by the justified inequalities

mu >= c*(1-rho^2/2) > 0,       B <= 3*kappa*w/(2*3),
1/pi < 1/3.

The first follows from cos(rho)>=1-rho^2/2; the others use pi>3. The evaluator propagates nonnegative rational upper bounds for the two phase magnitudes, hidden form, macro omission errors and reconstructed pressure/potential error. It does not evaluate a new trajectory or replace the existing phase/pressure/integrator owners. The one-radian restriction and finite evaluator limit are implementation scope, not an inferred structural constant or the full theorem's boundary.

The transverse controls exercise the existing averaged-sine phase proposal, refreshed canonical pressure and shared nodal integrator, keeping the reduced comparison fixed from its initial projected state. The rational envelope concerns ideal arithmetic; comparison with represented finite endpoints must retain its numerical defect scope. Binary64 trigonometry, phase normalization, midpoint rounding and integration defects are distinct from the exact-real and ideal-Euler claims. An observed finite restoration or omission error is not an asymptotic binary64 certificate. This result does not derive capacities, support, a privileged partition, the supplied phase law or a maintained nonuniform NFR.

The deterministic finite control uses node order (0,1,2), unit conductances and path lengths, kappa=1, K=1/2, effective e=w=1/2, no Gamma, hard EPI limits [-1,1] and the full half-pi U3 gate. At initialization, x=(1/4,3/16,1/16) and theta=(1/4,7/8,5/8), giving (delta,eta,q,u)=(1/2,1/4,1/8,1/8). The comparison retains the same initial delta,q,m,b and sets eta=u=0. Both invoke the shared simultaneous phase proposal and DefaultIntegrator with fresh pressure for 32 steps at h=1/8; no stochastic seed or named operator word is involved. No clipping or phase wrap occurs in these finite traces. Final macro omission errors are:

Coordinate	Observed absolute error	Precomputed ideal-Euler upper bound
Phase contrast delta	0.00010148122492026346	0.00022025172008861333
Form contrast q	0.00008750642343340054	0.00014777434227552532

The final hidden phase is approximately 0.0110270752 and hidden form -0.00620438720; form changes sign under the phase forcing and is not claimed to decrease monotonically in magnitude. Tests compare every finite endpoint with explicit 1e-12 headroom and separately record signed phase-staging, pressure-sum, field-summation and Euler defects. These observations validate the prepared finite control, not an unmeasured runtime error enclosure.

A prospective regional window on the unit barbell

This finite comparison uses two unit triangles joined by one unit edge, with node order (0,1,2,3,4,5) and bridge (2,3). Conductances and path lengths are one. This supplied geometry distinguishes two regions up to their exchange; it does not derive support or choose an autonomously maintained NFR. The model retains common fixed capacity kappa>0, fresh canonical pressure with effective weights e>0, w>=0, and the supplied averaged-sine phase law with K>0. Capacity and topology pressure weights are explicitly zero. There are no Gamma terms, operator events, capacity updates, clipping or controllers in the ideal comparison. In particular, zero topology weight is a configuration choice: the raw topology-gradient vector on this graph is

(1/2, 1/2, -2/3, -2/3, 1/2, 1/2).

It cannot be discarded by claiming the support is regular. If a topology weight t>0 were retained, its source would be -t*L*d, and after phase synchronization the corresponding fixed-support equilibrium would instead be x=constant-(t/e)*d. That is a different declared pressure recipe.

What can be grouped. Let D=diag(2,2,3,3,2,2) and L=I-D^(-1)W. The four fibers ((0,1),(2),(3),(4,5)) retain bridge endpoints and have the counted neighbor-mean matrix

P4 = ((1/2, 1/2, 0,   0),
      (2/3, 0,   1/3, 0),
      (0,   1/3, 0,   2/3),
      (0,   0,   1/2, 1/2)).

The equal-interior subspace is preserved by the ideal form and phase laws. Writing its phase coordinates as (a,b,c,d_phase), the first two phase rows are

a_dot = kappa + (K/2)*sin(b-a),
b_dot = kappa + (K/3)*(2*sin(a-b)+sin(c-b));

the other two follow by reflection. The first pressure-phase row is (b-a)/(2*pi) in the admitted chart; the second is Arg(2*exp(i*(a-b))+exp(i*(c-b)))/pi. The internal same-fiber neighbor remains in both the denominator and phasor sum. Replacing those counted rows by a simple four-node path would change the model. Two triangle averages alone are not closed: states with the same averages can have different bridge values, and hence different boundary flux and future averages.

Local joint response and its two clocks. In the common rotating chart vartheta=theta-kappa*t*1, the linearization at synchronization is

vartheta_dot = -K*L*vartheta,
x_dot = -kappa*e*L*x - (kappa*w/pi)*L*vartheta.

The sorted spectrum of L is

0, (11-sqrt(73))/12, 7/6, 3/2, 3/2, (11+sqrt(73))/12.

Thus lambda_s=(11-sqrt(73))/12 is about 0.2046663546, whereas every faster nonconstant mode has eigenvalue at least 7/6. Right eigenvectors are orthonormal in <u,v>_D=u^T*D*v. The two initialized unit modes are

r = (sqrt(73)-5)/6,
v_s = (1,1,r,-r,-1,-1)/sqrt(8+6*r^2),
v_f = (3,3,-4,-4,3,3)/sqrt(168).

Their first nonzero entries are positive. The slow profile itself has bridge to interior ratio r, about 0.5906672909; a dominant slow mode therefore does not make each triangle uniform. A regional mean and its remaining bridge profile must be retained separately.

For a modal phase coefficient c_lambda and form coefficient d_lambda, put A=kappa*e and B=kappa*w/pi. The continuous linear response is

vartheta_lambda(t) = exp(-K*lambda*t)*c_lambda,
x_lambda(t) = exp(-A*lambda*t)*d_lambda
  - B*lambda*c_lambda*integral_0^t
      exp(-A*lambda*(t-s))*exp(-K*lambda*s) ds.

The clocks K and A are distinct in general. At K=A, the integral is t*exp(-A*lambda*t); there is no singular response. The finite comparison uses ideal simultaneous Euler rather than silently treating this continuous formula as an exact numerical endpoint. With a_lambda=1-h*A*lambda and b_lambda=1-h*K*lambda, its closed prediction is

vartheta_lambda,n = b_lambda^n*c_lambda,
x_lambda,n = a_lambda^n*d_lambda
  - h*B*lambda*c_lambda*sum_(j=0)^(n-1)
      a_lambda^(n-1-j)*b_lambda^j.

For distinct factors the sum is (a_lambda^n-b_lambda^n)/(a_lambda-b_lambda); for equal factors it is n*a_lambda^(n-1). At n=0 the forcing sum is empty. The constant mode has no forcing or decay in this predictor. The reciprocal sine law conserves the ideal degree-weighted rotating phase mean by edgewise cancellation, but the nonlinear neighbor-phasor pressure need not conserve the degree-weighted form mean. Its drift is part of the source discrepancy, not removed by recentering the predicted trajectory after execution.

Prospective nonlinear envelope. Suppose the initial raw lifted phase width is at most rho<=1, the full support stays U3-admitted, and h*K<=1, h*kappa*e<=1. After subtracting the common drift, each ideal sine Euler row is a convex combination: its edge weights are h*K*sin(theta_j-theta_i)/(degree_i*(theta_j-theta_i)), interpreted continuously at zero. They are nonnegative and their sum is at most h*K. The phase width therefore remains at most rho. For the actual canonical range convention, the declared finite horizon must also stay inside the selected unwrapped chart.

The sine remainder is bounded by rho^3/6. For a neighbor arithmetic mean m, the displacements z_j=theta_j-m satisfy mean(z)=0 and abs(z_j)<=rho. Consequently abs(mean(sin(z)))<=rho^3/6 and mean(cos(z))>=1-rho^2/2>0. Using abs(atan(y))<=abs(y) gives a phasor-direction defect at most rho^3/[6*(1-rho^2/2)]. The following rational bounds thus apply to the phase-rate defect and the pressure-phase defect, respectively:

R_theta = K*rho^3/6,
R_p = rho^3/[18*(1-rho^2/2)],

where the second uses pi>3. The linear Euler phase and form matrices are row stochastic, hence contractions in the sup norm, and ||L||_infinity=2. The executable predictor restricts both step products to at most 1/2, a sufficient subdomain making every ideal modal factor nonnegative. It uses the observed initial width, which can give tighter bounds than the chosen ceiling. Starting from the same initial full state, telescoping gives

epsilon_theta(n) <= n*h*R_theta,
epsilon_x(n) <= kappa*w*(n*h*R_p
                       + h^2*n*(n-1)*R_theta/3).

These bounds compare the nonlinear ideal Euler law with its fixed linear prediction. They include all modes and possible form-mean drift. They do not bound the difference between ideal arithmetic and an arbitrary binary64 trajectory. Pressure realization, phase staging, trigonometry, spectral projection and nodal integration retain their separate numerical evidence.

Frozen finite control. The deterministic protocol fixes kappa=1, e=w=K=1/2, h=1/8, N=48, no seed or named operator word, the full half-pi U3 gate and hard EPI limits [-1,1]. Set

theta0 = (1/8)*1 + (1/32)*(v_s+v_f),
x0 = -(theta0-(1/8)*1)/pi.

The predictor reads the materialized initial modal projections, including their small represented residuals, before any trajectory is evaluated; it does not replace them with ideal fixture coefficients. The chosen width ceiling rho=1/16 holds at initialization and the full time-six phase range stays below the wrap boundary in the ideal model. This width ceiling, the amplitude and the observation times are experimental design choices, not new TNFR constants. The reference is fixed from that initial state and never refitted to an observed endpoint.

Let Pi_s be the D-orthogonal slow-mode projector, Pi_0 the constant-mode projector and Q_fast=I-Pi_0-Pi_s. At the reserved steps n=(32,40,48), corresponding to times (4,5,6), require all of the following:

• ||Q_fast*vartheta_n||_D <= 0.2*||Pi_s*vartheta_n||_D;
• ||Q_fast*x_n||_D <= 0.4*||Pi_s*x_n||_D;
• ||Pi_s*vartheta_n||_D >= 0.5*||Pi_s*vartheta_0||_D;
• full-state sup-norm prediction errors remain within the precomputed nonlinear envelopes plus the explicit finite numeric allowance 1e-12.

These tests mean that faster disturbances have subsided while a predictable interregional contrast remains. Projector errors are at most sqrt(14)*epsilon when a full-state sup bound is epsilon, since sum(d)=14. Before execution, the displayed ideal prediction and rational envelopes give phase-ratio upper bounds approximately (0.151,0.104,0.079), form-ratio bounds (0.377,0.284,0.229), and slow-phase retention lower bounds (0.652,0.585,0.524) at the three selected times. These are prospective bounds, not measured responses. The 1e-12 allowance is an explicit finite numerical comparison tolerance, not a proof of future runtime accuracy.

The phase-response owner supplies the fixed linear predictor and envelopes; the reserved barbell controls exercise the existing shared phase proposal, canonical pressure refresh and nodal integrator. No second nonlinear solver is introduced. The finite control must retain source and execution defects, chart/clipping checks, weighted mean and bridge-profile observations. Its acceptance or failure concerns this configured transient regional window only, not autonomous phase-law selection, permanent differentiated maintenance, emergent topology or physical NFR identity.

Finite outcome under the frozen protocol. The shared engine passed all three reserved acceptance checks. Rounded observations were:

Supplied time	Fast/slow phase norm	Fast/slow form norm	Slow phase retained
4	0.133885	0.332843	0.662348
5	0.0809865	0.221152	0.597526
6	0.0489883	0.144229	0.539048

The observed initial width is approximately 0.03252325. At time six, full-field phase and form sup errors were 5.92e-8 and 3.05e-8, versus respective ideal nonlinear bounds 1.72e-5 and 2.26e-5, with the separately declared 1e-12 finite numerical allowance. The weighted form mean changed by 1.59e-8: the nonlinear source was retained, not replaced by the linear Laplacian. The maximum observed nonlinear pressure-phase discrepancy was 1.35e-7. Independent 70-digit source evaluations separate this discrepancy from binary64 source materialization (below 2.83e-16); phase staging was below 6.94e-16, and signed nodal Euler defects below 4.21e-19. These are measured finite residuals, not interval certificates. Exact regional endpoint budgets retain internal dissipation, bridge exchange, forcing and numerical defects. The result admits this temporary regional window; it does not establish permanent maintenance or an autonomous origin of the supplied phase law.

Geometric observation can require an additional state coordinate

For the same held fine law, write p=-e L x+f, x_dot=-A x+b, and let K be the actual inverse-square fine distance kernel. Observing the averaged potential gives

O = -e R K L,             o0 = R K f,
R Phi_s = O P y + O Q x + o0,       Q=I-PR.

EPI autonomy requires R A Q=0. Potential determination by the current macro EPI additionally requires O Q=0. This is weaker than the full arbitrary-pressure factorization R K=Kbar R; the latter must not replace the actual observation test when only this constitutive family is claimed.

A minimal exact countercontrol uses unit-conductance P3, common capacity one, pure EPI pressure and blocks ((0,2),(1)). Set the existing explicit edge lengths to 1,2, so the endpoint distance is 3. The transport still closes exactly, but for the hidden vector delta_x=(1,0,-1):

R delta_x=0,       R A delta_x=0,       -R K L delta_x=(0,-3/4).

The missing geometric observation is retained by one derived coordinate, h=(x0-x2)/2. With u=(x0+x2)/2, v=x1,

u_dot=v-u,       v_dot=u-v,       h_dot=-h,
R Phi_s=(-37*(v-u)/72, 5*(v-u)/4 - 3*h/4).

The stacked linear observation (R;O) has rank three. Thus three scalar coordinates are necessary and sufficient for this exact instantaneous linear state/field description; the explicit equations also close its evolution. No new force was introduced: the extra coordinate was already part of fine EPI. Equal edge lengths supply the two-coordinate positive control, with inherited kernel ((1/4,1),(2,0)). The asymmetric lengths are declared input geometry, not a claimed spontaneously selected metric.

Independent rational and production-field controls verify the two domains and the decoder. This is one potential observation, not full-tetrad sufficiency. In particular, primitive phase fields and the nonlinear coherence-length fit have their own dependencies.

Joint EPI/potential realization and field provenance

The joint geometry observer now derives the joint observation directly from a captured held fine law. With A=e diag(nu) L, it constructs

O_joint = (R; -R K diag(1/nu) A),
o_joint = (0; R K f).

The inverse-capacity conversion is essential: the potential aggregates pressure, whereas A generates the EPI rate. Using -R K A would silently change the field at heterogeneous capacity. The fixed source f comes from the existing non-EPI channel capture, independently of a desired response or stored pressure. Its potential offset stays in the decoder, without adding a constant state coordinate.

The observer delegates these rows to the single affine realization owner. That owner derives a minimal closed linear state s=Cx with s_dot=-G s+C b and outputs D s+o_joint. Both EPI means and averaged model potential, including their rates, are reproduced for every initial scalar EPI in the held model. EPI partitions that fail instantaneous closure are allowed: the same invariant-row construction retains the missing directions. Redundant field rows do not inflate the state dimension. The two P3 controls give dimensions two and three as derived above; dimension three is a change of coordinates of the complete P3 form, not compression.

Distance provenance. The exact kernel uses the shared edge reader's materialized binary64 lengths: explicit length, legacy weight, then unit default, with the minimum over parallel edges. This differs from the summed parallel conductance used in transport. Detached rational shortest paths and inverse squares give K; self, zero-distance and unreachable pairs contribute zero, following the canonical field policy. The original raw edge attribute need not equal its binary64 materialization. Runtime path sums, inverse powers and field accumulation are separate numerical operations: for example, exact represented edge lengths 1 and 2^-54 have rational path sum 1+2^-54, while binary64 addition gives 1.

For fresh pressure-kernel defect delta_p_kernel, stored-pressure residual delta_p_stored and the actually materialized fine field Phi_runtime, the observer verifies the exact diagnostic telescope

R Phi_runtime = (O x + R K f)
              + R K delta_p_kernel + R K delta_p_stored + delta_field,
delta_field  = R Phi_runtime - R K p_stored,
O            = -R K diag(1/nu) A.

Thus numerical field evaluation is not silently identified with the exact model observation. These potential corrections have no capacity multiplier. They are detached measurements, not certified future error bounds. Geometry controls independently check capacities, source offsets, nonclosed partitions, length/transport conventions, the separate defects and graph immutability. Phase, capacity, support, materialized lengths and channel coefficients remain held data. This result introduces no graph evolution, partition selection, complete-tetrad reconstruction or physical-emergence claim.

Remaining tetrad dependencies and the limit of linear compression

observe_forced_support_tetrad_dependencies composes the same geometry realization with the existing phase and coherence readers. It does not create another field implementation or reduction method. The retained observation is s=Cx, with CT=I. Set

M = -diag(1/nu) A,       Dp = M T,       Ep = M - Dp C,
p = Dp s + f + Ep x.

Thus all fine model pressure is determined by s exactly when Ep=0. If column j of Ep is nonzero, delta=(I-TC)e_j satisfies C delta=0 and M delta=Ep e_j != 0: it is a same-state/different-pressure witness. The observer tests the whole residual matrix, not just its action on the current state, which can vanish accidentally.

There is a useful limit. On the admitted connected positive-capacity model, ker(M)=span(1). Because C retains R and R1=1, ker(C) intersect ker(M)={0}. Consequently Ep=0 implies ker(C)={0}; the converse follows from TC=I at full rank. Retaining all fine pressure and the partition means therefore requires the full fine linear dimension. Pressure completeness is sufficient input for a pressure-based model diagnostic, but it is not necessary for every nonlinear scalar diagnostic. Failure of this test must not be labeled failure of every possible xi decoder.

The dependency contract is:

Read-out	Retained input and result	Boundary
Averaged model potential	The shared affine realization supplies the exact linear output and its rate	Actual stored pressure and numerical field residuals remain separate
Averaged phase gradient and curvature	Fine primitive phases and unique-neighbor support are held; the existing phase observer supplies these constant read-outs	They are not derived form angles; represented cancellation leaves curvature unavailable in any average containing it
Coherence-length fit	Nonlinear function of fine pressure products, runtime floating paths and the estimator's sample/bin policy	A current value from full stored pressure is not a forecast from s; model pressure completeness only supplies a sufficient input criterion
Coherence-length fallback	A selected normalized-Laplacian mode scale with explicit method metadata	It is dimensionless, not fitted path-length correlation; the cutoff can skip the true smallest positive mode

The observer evaluates xi on a detached graph copy so that the canonical spectral cache does not mutate the caller's graph. Its observed_coherence_length uses actual stored pressure. It preserves the autocorrelation_fit, historical spectral_gap, or unavailable method and associated provenance. The fallback selects the smallest numerical eigenvalue greater than 1e-9; it equals the usual gap scale only when that gap passes the selection. An independent P4 with conductances (1,2^-40,1) demonstrates a skipped small mode. The fit bins equality of floating shortest-path distances, not the rational edge-path model used by the potential realization. These are estimation conventions, not new TNFR equations.

An exact nonlinear obstruction on the existing P5 reduction. Use the unit path, common capacity one, pure EPI pressure, equal primitive phases and reflection blocks ((0,4),(1,3),(2)). Both its transport and inverse-square kernel respect reflection. The existing three-coordinate orbit projection therefore retains EPI means and averaged potential and is already invariant. For the hidden form

x(a)=a*(-2,-1,0,1,2),      Cx(a)=0,
p(a)=(a,0,0,0,-a),        R Phi_s(a)=0,

the retained state and all its modeled outputs are the same for every a. Nevertheless the canonical coherence products differ. At a=1, the three accepted distance bins 1,2,3 have exact means (3/4,2/3,1/2); at a=2 they have (2/3,5/9,1/3). Their pair counts are (4,3,2); the tenth pair at distance four is a singleton and is excluded from regression. The exact real log-linear fits give

xi(a=1)=2/log(3/2),       xi(a=2)=2/log(2).

The production reader agrees numerically on prepared inputs whose stored and model pressures coincide. These are successful fits, not two values of the spectral fallback. Hence no single-valued xi output of this three-state realization exists on the full admitted fine-form domain. No trajectory or new force is needed for the counterexample. Completing its pressure inputs with the shared affine algorithm gives dimension five, as the theorem predicts.

The converse boundary matters: every finite-pressure P3 has fewer than the required ten pairs, so its fitted branch is unavailable independently of pressure. Its fixed selected spectral value can be constant despite missing pressure coordinates. Also a and -a on P5 have opposite pressure but identical xi by reflection. These facts motivate a symmetry-based nonlinear observation test; they do not establish a minimal nonlinear state or an autonomously selected geometry.

Controls reuse the existing P5 reduction owner in tetrad dependency tests, with separate spectral selection controls. Held phase, capacity, source and support are still premises. This completes the explicit observation/dependency contract, not full-tetrad minimality or a complete evolving law for the fine substrate.

A complete reflection-invariant form state on the retained P5

The P5 observation obstruction can be resolved without declaring the missing shape irrelevant. Write the existing fine form as

x=(a+r, p+s, c, p-s, a-r).

Reflection fixes (a,p,c) and sends (r,s) to (-r,-s). Reuse the same unit-support, pure-EPI P5 generator with common positive capacity nu. Its equations give

a_dot=nu*(p-a),       p_dot=nu*((a+c)/2-p),       c_dot=nu*(p-c),
r_dot=nu*(s-r),       s_dot=nu*(r/2-s).

The three quadratic observations

J00=r^2,       J01=r*s,       J11=s^2

are unchanged by reflection. Their valid image requires both J00>=0,J11>=0 and J00*J11=J01^2; the determinant equality alone is insufficient. They determine the hidden pair up to its simultaneous sign: when J00>0, choose r=sqrt(J00) and s=J01/r; when J00=0, necessarily J01=0, so choose r=0,s=sqrt(J11). The zero triple gives zero hidden form. Thus (a,p,c,J00,J01,J11) distinguishes exactly the two-element reflection orbits, with the symmetric form as the one-element orbit.

The product rule derives a closed law, rather than selecting one:

J00_dot=2*nu*(J01-J00),
J01_dot=nu*(J00/2+J11-2*J01),
J11_dot=nu*(J01-2*J11).

For Delta=J00*J11-J01^2, Delta_dot=-4*nu*Delta. More fully, the matrix J=h h^T evolves by J_dot=B J+J B^T, where B=nu*((-1,1),(1/2,-1)). Congruence with the exact linear hidden flow preserves positive semidefiniteness and rank; the valid invariant image is forward invariant in the exact continuous model. Keeping only the diagonal squares fails: (r,s)=(1,1) and (1,-1) have the same squares, but J00_dot is respectively 0 and -4*nu. The cross term carries necessary shape information.

What is preserved. Reflection commutes with this transport generator and with the unit-path inverse-square kernel. Pressure and fine potential are therefore recovered up to reflection; their labels are not recovered uniquely. Orbit-averaged potential and EPI, uniform held-phase read-outs and the scalar coherence diagnostic are invariant. Reflection bijects the full unordered node pairs while preserving their distances and coherence products, so it preserves the fit input and its branch, as well as the selected spectral fallback. This exact statement concerns the declared model and estimator; arbitrary stored-pressure defects and numerical order effects need their own provenance. The invariant state retains the previously missing magnitude information, so it distinguishes the P5 a=1 and a=2 hidden-line examples.

Dimension and boundary. Six displayed coordinates obey one independent constraint away from zero hidden form: the quotient still has intrinsic dimension five. It removes the duplication between reflected descriptions, not two continuous physical degrees of freedom. The observation Jacobian has rank five off r=s=0 and rank three on that symmetric stratum; the latter does not make its surrounding state space three-dimensional. A chosen representative can change sign branch at r=0 even while the invariants remain continuous. That change of representative is not a physical jump. The exact symmetric stratum stays symmetric under this unforced law; no spontaneous symmetry breaking or geometry selection has been derived.

The implementation lives beside the existing reduction in p5_reduction.py. observe_p5_reflection_invariants derives the observation and its rates from the shared fine generator. decode_p5_reflection_invariants supplies an exact rational representative on the rationally liftable image, including both axes and the zero state. A valid real image with irrational lift is outside that exact rational decoder and is explicitly rejected; no floating square root silently replaces it. Generated rational/represented fine snapshots lie in the supported image. The mathematical real quotient is broader.

No integrator is added. In particular, advancing quadratic invariants with ordinary Euler is not the pushforward of a fine Euler step: squaring h+dt*h_dot also gives dt^2*h_dot*h_dot^T. An independently supplied Euler step on the displayed J law need not stay on its rank-one cone. Independent controls verify the quotient, mixed-term obstruction, derivative and singular domains, decoder, and field invariance using the existing readers.

Continuity boundary for a later approximate reduction

Removing a discrete reflection redundancy differs from neglecting a small hidden amplitude. For the preceding hidden line, write its amplitude as alpha>0 to distinguish it from the even coordinate a. Endpoint coherence is z=1/(1+alpha). The retained bin means are ((1+z)/2,(1+2*z)/3,z), and their exact-real fit is

xi(alpha)=2/log(1+alpha/2).

This expression applies while all three bins pass the current floor, 0<alpha<10^9-1. Hence it diverges as alpha -> 0+, whereas the exactly uniform pressure case has no admissible fit and returns the distinct finite P5 spectral scale sqrt(2+sqrt(2)). This is a discontinuity between tagged estimator branches, not a divergence of the nodal state or a phase transition proved by the dynamics. Binary64 coherence rounding can switch branches earlier; the expression is not a binary64 asymptotic theorem. It follows that small pressure error alone cannot justify a uniform error bound for the unqualified displayed xi value near this boundary. Any later dynamical reduction must retain that distinction instead of imposing continuity or altering a nodal force to improve the diagnostic.

Absolute error from omitting the decaying odd form on P5

Keep the preceding fixed unit-conductance P5, unit structural edge lengths, common constant capacity nu>0, and pure-EPI law x_dot=-nu*L_rw*x. Primitive phases are equal and held; no additional source, capacity/support motion or hybrid event is admitted. Replacing the full form by its lifted reflection means gives

x_hat=(a,p,c,p,a),       h=x-x_hat=(r,s,0,-s,-r),
z=(r,s),               z_dot=-nu*D*z,
D=((1,-1),(-1/2,1)).

Both full and lifted forms follow the same declared fine law. Their difference therefore evolves in the invariant odd subspace. This five-to-three approximation discards actual form information; it is not the preceding reflection quotient, which retains five intrinsic dimensions. It also retains the even contrast u=c-p: the separate three-to-two memory approximation below discards a different coordinate.

Decay in the inherited metric. Use the degree metric H=diag(1,2,2,2,1), equal to nu times the existing reversible metric. The squared norm, without a factor of one half, satisfies

N=||h||_H^2=2*r^2+4*s^2,
N_dot=-4*nu*(r^2-2*r*s+2*s^2).

In coordinates (sqrt(2)*r,2*s), the hidden generator is symmetric with diagonal entries one and off-diagonal entries -1/sqrt(2). Its dimensionless rates are lambda_minus=1-1/sqrt(2) and lambda_plus=1+1/sqrt(2). Hence, for every t>=0,

N(t) <= N(0)*exp(-2*nu*lambda_minus*t).

The rate is sharp over the real state space, attained by a nonzero slow mode with s=r/sqrt(2). The rational evaluator does not replace this irrational eigenvector by an approximate vector and call it exact.

Pressure and potential errors. Write signed errors as full model minus lifted model. Fresh model pressure is p_model=-L_rw*x, independently of nu. Thus

delta_p=(s-r, r/2-s, 0, s-r/2, r-s).

The same unit-path kernel B_ij=1/|i-j|^2 off the diagonal, B_ii=0, gives delta_Phi=B*delta_p. On odd pressure (q0,q1,0,-q1,-q0), its first two components are K*(q0,q1), where

K=((-1/16,8/9),(8/9,-1/4)),
delta_Phi=(73*r/144-137*s/144, -73*r/72+41*s/36, 0,
           73*r/72-41*s/36, -73*r/144+137*s/144).

Cauchy-Schwarz in the hidden metric diag(2,4) bounds a row (u,v) by (u*r+v*s)^2 <= (u^2/2+v^2/4)*N. Applying it to these rows proves

(delta_p_i(t))^2   <= C_p[i]*N(0)*exp(-2*nu*lambda_minus*t),
(delta_Phi_i(t))^2 <= C_Phi[i]*N(0)*exp(-2*nu*lambda_minus*t),
C_p   =(3/4, 3/8, 0, 3/8, 3/4),
C_Phi =(9809/27648, 2897/3456, 0, 2897/3456, 9809/27648).

The row constants are sharp instantaneous bounds in this norm; joint attainment with the slow-rate envelope is not asserted. All three orbit means of h, delta_p and delta_Phi are exactly zero. Thus exact orbit means and orbit-averaged potential can coexist with nonzero fine-field error. No commutation of the potential kernel with the diffusion generator is required: reflection invariance of each suffices.

Admitting an absolute approximation. For positive requested tolerances eps_p,eps_Phi, both infinity-norm errors are within budget for all t>=T, where the sufficient analytical time is

M=max(1, (3/4)*N(0)/eps_p^2, (2897/3456)*N(0)/eps_Phi^2),
T=log(M)/(2*nu*lambda_minus).

Zero initial hidden form gives exact agreement immediately. Nonzero hidden form cannot give identically zero pressure or potential error at finite time; the hidden flow and both restricted output maps are invertible. Tolerances are approximation budgets, not new parameters in the nodal dynamics.

bound_p5_hidden_form derives its hidden generator and metric from the shared P5 reduction, and the potential from the existing exact geometry kernel. It returns rational squared bounds, using a positive rational lower rate g<=nu*lambda_minus certified by the shared semidefinite test and an upper rational enclosure of exp(-2*g*t). sample.within_tolerances(pressure=..., potential=...) compares those bounds directly with squared budgets. A false result means the bounds are insufficient, not that the actual error exceeds the budget. Conservative rate/enclosure slack is distinct from model omission error. The evaluator supports nu*t<=2048; this resource limit does not restrict the all-time analytical theorem. Binary64 execution defects are outside both claims. Independent controls cover the shared matrices, factors, rational bounds and admission boundaries.

Identity and tetrad boundary. Absolute decay alone does not justify forgetting the remaining shape. For nonzero pure odd form the lifted approximation is zero, and the relative H-norm error is one at every finite time. More generally, the retained nonuniform even modes decay at nu and 2*nu, both faster than the slow omitted odd mode. With that slow component present, its fraction of the remaining nonuniform form can approach one. The result therefore admits an absolute pressure/potential approximation, not preservation of relative pattern identity. Held equal phases give the same phase read-outs, but the preceding xi fit/fallback discontinuity still precludes an unqualified uniform xi error guarantee. This is a conditional reduction of a supplied passive law, not autonomous NFR formation, maintenance, or a law for the full evolving substrate.

Derived memory when closure fails

For fixed reversible pure-EPI diffusion, the unresolved coordinate z=(I-PR)x can now be eliminated exactly. With Q=I-PR, the projected equation has instantaneous generator RAP, kernel K(t)=RA exp(-QAQ t) QAP, and initial hidden source f(t)=-RA exp(-QAQ t) Qx_0. The convolution enters with a positive sign. The weighted identity H_bar K(0)=(QAP)^T H(QAP) proves that its kernel vanishes exactly when the quotient closes for every state.

The proof and independent P4/P5 controls are centralized in Derived EPI memory. observe_epi_memory reuses the same partition geometry and records finite numerical samples of the full equation and the two omission controls. Numerical generator/propagator checks reject loss of the necessary diffusion invariants; they are not accuracy enclosures. This is an offline observation, without a REMESH or complete-runtime claim.

For the explicit P5 partition, the same note's section 8 derives uniform and causal error bounds for a shortened history window. Its separate rational reference preserves the initial hidden source, encloses the actual truncated solution and keeps evaluation width separate from model error. This restricted approximation has no fitted memory rate or asserted REMESH correspondence.

The P5 reflection partition ((0,4),(1,3),(2,)) supplies a complementary exact closure on a genuine three-node quotient with conductances 2,2. Observing only its endpoint mean and combined inner/center mean is precisely the second reduction that generates memory. Uniform unclipped REMESH commutes with this reflection projection and its lift; the fine stationary-history energy splits into quotient and discarded energies. These identities reuse the existing companion theorem in the quotient's metric. They do not identify a REMESH echo with a forward diffusion step, as an exact causal-history counterexample shows. See section 9 of Derived EPI memory.

Consequence for S9

The pure-EPI diffusion channel is a fixed family under every exact quotient: its generator remains diag(nu_bar)L_rw,bar. This does not show that Emission, Coherence, REMESH or the other nonlinear operators close under the same map. Operator RG flow therefore remains open beyond this one channel.

Field inheritance is another independent obligation. On the unit triangular prism, the triangle-mean quotient closes exactly and has zero memory kernel, yet a self-excluded scalar macro potential reverses the sign of the averaged fine potential. The inherited kernel retains within-block sources. Keeping four internal modes and the mean contrast instead reconstructs full exact model pressure and potential; primitive phase and field provenance remain separate. See the single macro-state and tetrad derivation.

Generic fixed-vector operator test

For any declared micro map F, macro map F_bar, projection R and right inverse lift P, two logically independent identities are relevant:

R F = F_bar R,        F P = P F_bar.

The first makes the projected dynamics autonomous for every micro state; the second keeps lifted macro states inside the lifted subspace. Exact strong closure requires both. A deterministic counterexample shows that projected closure need not imply lift invariance.

certify_operator_quotient checks both identities globally for matrices and names every Boolean decision as within the declared numerical tolerance. For nonlinear callables it reports only sampled residuals and repeated-evaluation consistency on declared macro and micro probes, including the defect between two micro states with the same projection. A cubic componentwise map closes on block-constant probes yet depends on unresolved within-block fibers, providing an explicit nonlinear obstruction. Graph mutation, operator history and nested-EPI changes are rejected as outside a fixed-dimensional vector-map test. Thus the certificate advances S9 without claiming closure or completeness of the 13-operator catalog.

Circular phase extension and obstruction

Choose an open semicircle chart q so every relevant wrapped pairwise difference stays on one branch. The pairwise phase-pressure realization is then

r_pair(q) = -(1/pi) diag(nu_f) L_rw q.

It is linear and inherits the reversible pure-EPI projection and lift identities. This exact fixed-branch statement concerns pairwise edge differences. It is distinct from the engine's canonical phase channel, which uses the argument of the unweighted sum of neighboring phasors:

g_i(q) = -(1/pi) wrap(q_i - Arg sum_{j in N(i)} exp(i q_j)),
r_i(q) = nu_f_i g_i(q).

For a lifted block-constant phase field, this nonlinear channel closes on the macro support under a sufficient fixed-support domain: neighbor-count profiles are equitable inside each fiber, fibers have no internal edges, every active macro neighbor has the same multiplicity within its source block, capacity is exactly block-constant, the chart stays on one wrap branch, and all phasor resultants are nonzero. The canonical support is the unweighted NetworkX adjacency, including edges whose transport weight is zero; the reversible pairwise projection continues to use the conductance-degree-over-capacity metric.

That lifted-subspace result does not make the projected canonical dynamics autonomous for arbitrary micro phases. On K3,3, a nonconstant micro state and its lifted representative have the same macro chart coordinate but different projected nodal rates. This is a constructive counterexample to global canonical phase closure: unresolved within-fiber phase affects the macro derivative. Branch crossing, zero resultants, evolving capacity, changing support and finite-time phase evolution remain outside the certificate.

certify_phase_nodal_coarse_graining reports the pairwise matrix quotient, the restricted canonical lift test and the same-macro-state counterexample without reading or changing EPI. The capacity hypothesis uses exact represented equality within each fiber; the separately rounded macro-capacity match has its own numerical residual. In a K3,3 fiber, capacities 1 and 1+2^-35 can pass a 1e-10 tolerance while unequal lifted nodal rates still exclude exact closure. Numerical closeness must not satisfy that algebraic premise. A small nonzero phasor resultant can likewise fail the implementation's numerical margin without being an exact circular-mean singularity. The regression owner is the phase quotient tests.

2. Geometry forced by canonical coherence

For the signed local chart p=DeltaNFR, v=dEPI, the constitutive kernel is

C(p,v) = 1 / (1+|p|+|v|).

For 0<c<1, its exact level set is

|p|+|v| = 1/c-1.

It is an L1 diamond. It is smooth on each open edge and nondifferentiable at the four axis vertices. Its Euclidean distance from equilibrium is not fixed: it ranges from r/sqrt(2) to r, whereas its L1 distance is exactly r=1/c-1. At c=1 the level collapses to (0,0).

coherence_level_set_geometry exposes this exact geometry. The result rejects the assumption that canonical coherence alone supplies a smooth Riemannian manifold. It naturally supplies an L1 gauge in the local constitutive chart. A smooth information metric on the full graph state would require an additional modeling choice.

For a fixed nonempty network with N nodes, canonical mean aggregation gives

C_N = 1 / (1 + (sum_i |p_i| + sum_i |v_i|) / N).

Thus C_N=c is the boundary of a 2N-dimensional cross-polytope with total L1 radius R=N(1/c-1), intrinsic dimension 2N-1, 4N vertices and

f_k = 2^(k+1) binom(2N,k+1)

k-faces. Its 2^(2N) open facets form the regular locus; every lower face is a nonsmooth absolute-value stratum. The Euclidean radius ranges from R/sqrt(2N) to R, the regular gradient norm is c^2 sqrt(2/N), and the superlevel set C_N>=c is closed and convex.

The fixed-capacity nodal equation v_i=nu_f_i p_i cuts this ambient level down to

sum_i (1+nu_f_i)|p_i| = N(1/c-1).

This is a weighted N-dimensional cross-polytope. The executable certificate reports its pressure-axis vertices, Euclidean radii in the induced (p_i,nu_f_i p_i) metric, face stratification and regular gradient norm. network_coherence_level_set_geometry and fixed_capacity_coherence_level_set_geometry make both scopes explicit. They describe one instantaneous chart; capacity evolution, changing node count, basin boundaries and temporal attraction remain outside the result.

Fixed-topology structural-state metric

One such explicit choice is now available within a declared topology and label class. For each node, form the five-nodal-channel state

q_i = (EPI_i, nu_f_i, phase_i, DeltaNFR_i, dEPI_i),

and assign each edge both its weight conductance coordinate (unit when weight is absent) and its effective structural-length coordinate (length, then the compatibility fallback weight, then one). Declare one finite positive reference scale per channel, use circular geodesic distance for phase, and minimize the scaled Euclidean product over node and edge coordinates across all topology- and declared label-preserving graph isomorphisms. The compatibility API lets an omitted edge_length scale reuse edge_conductance; dimensional studies should declare both. Because a finite graph has finitely many isomorphisms and they act by isometries, the minimum is an exact metric on the resulting state-isomorphism classes. The certificate also reports the two L_infinity residuals of dEPI=nu_f*DeltaNFR; it does not project inconsistent inputs onto the nodal equation.

fixed_topology_structural_state_distance implements this quotient metric for finite simple graphs. It preserves phase wrapping and node relabeling, accepts explicit node and edge labels, validates finite nonnegative conductance and structural length, and rejects nonisomorphic supports, mixed direction and multigraphs. It reports every distance minimizer, uses an explicit lexicographic component refinement, and withholds a node mapping when that refinement remains ambiguous. The exhaustive isomorphism search can be factorial. Cross-topology edit costs, nested EPI identity and operator-history geometry remain open, so this is a structural state metric rather than a completed global information geometry.

The structural-length coordinate is a numeric cost. A caller that requires identical path geometry may additionally include "length" in edge_label_attributes, which makes exact attribute equality part of the admissible-isomorphism test rather than merely charging a nonzero distance.

3. Dissipative-symplectic direct product

Let z be the 4N-dimensional harmonic substrate coordinate and x the EPI field. Define

X = (z,x),
H = (1/2)||z||^2,
V = (1/2)x^T Bx,
J = diag(J_sub,0),
G = diag(0,diag(nu_i/d_i)).

Then

X' = J grad(H) - G grad(V)

simultaneously reproduces the specified auxiliary harmonic-substrate flow and fixed symmetric EPI diffusion. The degeneracy identities

J grad(V) = 0,             G grad(H) = 0

give H'=0 and V'<=0. This is an exact metriplectic-style direct product. verify_metriplectic_product checks the antisymmetry, positive semidefiniteness, degeneracies and both vector-field residuals.

The graph's stored DeltaNFR initializes part of the auxiliary coordinate z through Phi_s and J_DeltaNFR. The dissipative EPI block nevertheless constructs its pressure independently as p_epi=-D^-1 Bx from the weight conductance and supplied EPI. It does not replace the stored value. The certificate returns stored_pressure_consistency_residual=||DeltaNFR_stored-p_epi||_2 and the separate relative-tolerance decision stored_pressure_matches_epi_channel. Neither is included in is_decoupled_metriplectic_bridge: the block-product identity can pass while stored pressure contains other channels or is otherwise inconsistent with pure-EPI diffusion. A caller asserting a pure-EPI graph state must therefore require both the bridge Boolean and the pressure-consistency Boolean. This separation is exercised by test_metriplectic_product.py.

The cross blocks are zero, so the result does not derive how a pulse changes EPI relaxation or how dissipation feeds back into the substrate. A coupled bridge requires a TNFR derivation and preservation of the realizable graph-field image, as well as its stated balance laws. Nonzero cross tensors alone are insufficient. Fixed mutual-singleton P2 already excludes nonzero continuous harmonic evolution of the same geometric read-outs; its fixed-phase pure-EPI potential/flux sector instead has a closed dissipative law with rate nu_0+nu_1. The exact proof and production regressions are centralized in the variational note, section 3.7. Broader TNFR-derived realizability and coupling remain open under S12.

4. Inverse identifiability

Contract snapshot

contract_identifiability_certificate partitions the catalog using only declared executable contract features. With the richest non-tautological tuple (channel, direction, scale, context), eleven operators are singletons and one class contains both Silence and Contraction. Both are node-scale decreases of nu_f observed at network context.

This is an exact negative result for instantaneous contract identification. It does not say their trajectories are always identical: Silence targets latency, whereas Contraction also densifies pressure. Resolving them requires temporal state changes or a quantitative postcondition observer. Catalog coverage is therefore distinct from catalog identifiability and from universal catalog completeness.

Quantitative one-step signatures

probe_canonical_operator_identifiability enumerates the 13 operators from the current contract catalog and applies each one to a fresh deterministic heterogeneous path graph at the known target node n_nodes//2. Its features contain the target-node change, the across-node mean and RMS change in the four raw channels EPI, nu_f, DeltaNFR and wrapped phase; the corresponding target, mean and RMS change in nodal velocity nu_f*DeltaNFR; and node/edge-count changes. Operator names, glyphs and contract categories are evaluation labels and never enter the feature matrix. Executed history is checked separately to reject a fallback mislabeled as the requested operator.

With the declared default three probes, all 13 rows are distinct after the declared decimal quantization. Matrix rank and affine rank are 12. Row distinctness exactly partitions this finite generated hypothesis matrix; rank is only a numerical diagnostic. Silence and Contraction separate because both reduce nu_f while only Contraction changes EPI on these probes. The result is measured on this finite family.

This is closed-set classification against the fixed current catalog and a known target, not target localization or operator discovery. Although mean and RMS summaries of the raw changes are included, the protocol does not invert from aggregate TNFR telemetry alone: C(t), Si, phase synchronization and the tetrad are absent, and target-indexed raw coordinates are present. Identification from those aggregate read-outs, localization of an unknown target, unseen states, compositions and arbitrary grammar words remain open.

For any declared positive feature scales, let delta be the minimum pairwise distance between these finite prototypes in the scaled L-infinity or L2 norm. The triangle inequality gives an exact robustness statement: an additive error strictly smaller than delta/2 cannot cross a nearest-prototype boundary. The noise-margin certificate compares the supplied binary64 values through exact rational arithmetic, rounds the minimum distance downward and halves that lower bound. It therefore reports a conservative radius and marks midpoint ties as uncertified. Stable L2 accumulation avoids false underflow, large declared feature scales avoid false subtraction overflow, and unrepresentable scaled separations are rejected. The default operator probe attaches a unit-scaled L-infinity margin after applying the same decimal quantization as its row partition. Those unit scales define a coordinate convention, not a calibrated sensor or process-noise law. This handles bounded perturbations around the fixed prototypes only. It does not provide a stochastic noise law or extend the prototypes beyond the declared states.

5. Executable S16 endpoint certificate

certify_core_research_integration turns the restricted S16 intersection into one inspectable endpoint certificate. It accepts two independently frozen states, one partition and explicit StructuralChannelScales. The inputs must be finite undirected simple graphs with exactly the same node identifiers and bare edge support. At each endpoint the effective positive-conductance graph must also be connected, every capacity must be positive and frozen, and all raw state channels required by the constituent certificates must exist. Identical bare support alone is therefore insufficient when zero weight disconnects effective conductance. Conductance, structural length and state values may differ between endpoints.

At each endpoint it runs the fixed heterogeneous pure-EPI stability certificate, the graph-specific full-potential EPI reconstruction certificate and the reversible pure-EPI partition certificate. It also computes the declared fixed-topology structural-state distance between the endpoints. Finally, it checks two state-consistency conditions independently at each endpoint: stored DeltaNFR must match -L_rw EPI, and stored dEPI must match nu_f*DeltaNFR, after scaling by the declared pressure and EPI-rate scales.

The result exposes all constituent certificates, raw and scaled consistency residuals, numerical_conditions with fifteen named entries, and failed_conditions. joint_numerical_conditions_pass is true only when every entry passes. Numerical rank, closure, balance and consistency decisions use the single declared relative tolerance; the exact metric-on-isomorphism-classes entry remains a structural Boolean. A nonclosing partition, a pure-EPI pressure mismatch or an independent nodal-equation mismatch blocks joint promotion while leaving the other evidence inspectable; changed support is rejected before composition. These positive and negative paths are executable in test_core_research_integration.py.

A passing Boolean demonstrates simultaneous endpoint membership in this shared restricted numerical hypothesis class under the declared tolerance. It does not certify a trajectory or persistence between the endpoints. Phase dynamics, nonlinear or multichannel pressure, operator histories and words, S15 inverse identification, REMESH/nesting, changing support/topology and nonzero dissipative-symplectic coupling remain open. The exact L1 coherence level-set result also remains a separate theorem; the current S16 Boolean does not consume it. Explicit length values contribute to the numeric state distance; they may still differ unless the caller includes "length" among the exact edge-label constraints.

6. Time-resolved S16 boundary

certify_core_research_trajectory extends the endpoint intersection to a finite ordered sample path without claiming unobserved interpolation. It retains persistent node identifiers and fixed bare edge support, applies the endpoint certificate to every adjacent pair, and checks the left-explicit update

EPI^(k+1) - EPI^k = dt_k diag(nu_f^k) DeltaNFR^k

in scaled L-infinity norm. Stored dEPI remains an independent endpoint channel; it is not substituted for the vector field in this step test.

Two extra stability conditions prevent a merely self-consistent data sequence from being called stable. First, each timestep must satisfy the modal explicit-Euler condition for its left-hand transport regime. Its stationary-mode resolution uses a separate dimensionless tolerance relative to the fastest decay rate; the EPI residual tolerance is not reused as an absolute frequency cutoff. Second, all sampled regimes must share the exact projective d_i/nu_i metric required by the common switching theorem. The implementation evaluates the resulting common quadratic at every supplied state. At each snapshot it recomputes that metric's instantaneous weighted projection onto the consensus subspace before evaluating disagreement energy; it does not freeze the first snapshot's consensus coordinate. This is necessary because the represented binary64 generator can contract disagreement in the displayed metric without preserving that metric's weighted mean as an exact rational identity. The certificate limits both each positive increment and the cumulative positive variation over the whole path by one declared EPI-squared scale budget. This prevents individually small increases from accumulating without bound as samples are added.

The Euler residual and both local and cumulative Lyapunov-budget decisions compare exact rational values of the represented binary64 inputs with the exact rationalization of the caller tolerance. Their exposed float magnitudes are diagnostics only and do not decide a boundary case.

The analytic switching theorem covers arbitrary piecewise-constant continuous solutions among this finite regime family. The snapshot certificate only checks its supplied Euler steps: it does not prove that the samples lie on such a continuous solution, identify a switching law between them, or cover an unseen regime.

The companion refinement comparison certifies both paths before comparing them. It separates the strict path tolerance from the coarser agreement tolerance, requires every coarse time to match exactly one fine time, and requires a smaller fine-grid maximum step. Direct EPI differences on persistent ids determine agreement. Quotient structural distance is exposed only as a diagnostic because a different minimizing isomorphism at each time is not a node trajectory. The joint result also requires the caller to set same_dynamics_declared=True. That recorded assertion closes the previous semantic gap in which identical equilibrium samples from different generators could be promoted as a refinement pair; snapshots cannot independently verify the declaration.

The deterministic 161_core_research_trajectory.py uses two nested stable Euler meshes for one non-equilibrium fixed generator and also evaluates the analytic fixed-generator semigroup numerically through the symmetric similarity H^(-1/2) B H^(-1/2). The fine solution is closer at every noninitial common time in this experiment. This finite result does not prove numerical convergence or its order. Phase evolution, nonlinear/multichannel pressure, operator histories, REMESH/nesting, adaptive/event timesteps, changing support and nonzero metriplectic cross-coupling remain open.

Section link directory

These aliases route existing citations to their substantive owner.

• Exact scale, coherence-geometry and bridge results

• sine-replica-inheritance

• 7. Nonlinear sine inheritance with retained internal coordinates

• 7.1. Supplied support and an invertible description

• 7.2. Internal phase geometry modulates inherited coupling

• 7.3. Exact synchronized inheritance and storage scaling

• 7.4. A nearby exact obstruction to autonomous block means

• 7.5. Trapping permits persistent internal organization

• 7.6. Meaning and limits of the scale result

• sine-replica-unordered-state

• 8. An exact collective state for unordered pairs

• 8.1. Realizable invariants and their fibers

• 8.2. Closed dynamics with internal state retained

• 8.3. Evolution, boundaries and lift independence

• 8.4. Storage and identity pass to the exact quotient

• 8.5. What has and has not been reduced

• sine-replica-inherited-poisson

• 8.6. The collective state inherits the same Poisson structure

• sine-replica-internal-observability

• 8.7. Temporal resultant information recovers the unordered internal state

• sine-replica-internal-pulse

• 9. A finite-amplitude internal pulse with persistent collective identity

• 9.1. The exact invariant preparation

• 9.2. Conserved amplitude and exact periods

• 9.3. Collective identity and the stronger fine-edge condition

• 9.4. Boundaries and scientific scope

• sine-replica-pulse-variation

• 10. Complete variation around the finite internal pulse

• 10.1. Direct differentiation of the full retained law

• 10.2. Exact spatial decomposition without discarding directions

• 10.3. Zero-amplitude control and dimensionless parameters

• 10.4. Symplectic blocks and the correct half-period return

• 10.5. Amplitude-dependent phase shear is a neutral-family effect

• 10.6. A discriminating stability question, not a stability verdict

• sine-moving-pulse-work-response

• 10.7. Internal motion produces a finite same-type work-response asymmetry

• Matched form inputs and storage-work outputs

• Time ordering supplies the first nonzero antisymmetric term

• A finite positive bound at the declared horizon

• Reversal controls and scope

• sine-moving-pulse-finite-work-response

• 10.8. A controlled finite probe of the complete fine law

• Uniform nonlinear error, before evaluating a finite response

• Domain, preparation and interpretation

• sine-moving-pulse-parametric-representation

• 10.9. Parametric response and complete-history reversal

• sine-pulse-stiffness-discriminator

• 10.10. A clock-independent stiffness-family discriminator

• Exact obstruction for one affine stiffness control

• sine-replica-pulse-splitting

• 11. Small-amplitude transverse splitting with collective response retained

• 11.1. Fix the period before expanding the coefficients

• 11.2. The induced collective motion changes the splitting

• 11.3. Exact algebraic signs for the two spatial modes

• 11.4. Analytic remainder and the actual sufficiently-small conclusion

• 11.5. Interpretation and what remains unmeasured

• sine-replica-joint-persistence

• 12. Collective identity with independently active constituents

• 12.1. An admitted open full-state family

• 12.2. Every nontip pair remains internally active

• 12.3. Bounded internal circulation without a common period

• 12.4. Almost-everywhere recurrence in the same family

• 12.5. Overlap with the prepared pulse and its instability

• 12.6. Captured-source admission and evidence

• sine-replica-capacity-asymmetry

• 13. Unequal constituent capacities: persistence and its symmetry boundary

• 13.1. Exact asymmetric mean and internal rows

• 13.2. Collective trapping and ambient recurrence survive

• 13.3. Exact counterexamples to tip invariance and pointwise activity

• 13.4. The unordered state must retain capacity correlations

• 13.5. Weighted centers remove direct cross terms, not internal information

• 13.6. Shared engine scope

• sine-phase-pairing

• 14. State-derived phase partners and independent law admission

• 14.1. A branch-independent comparison of circular distances

• 14.2. The protected tube separates true partners

• 14.3. A prospective nearest-partner certificate and its abstentions

• 14.4. Phase pairing does not certify the paired support

• 14.5. Observation and admission interfaces

• sine-replica-acute-critical

• 15. Exhaustive fully acute equilibria on the doubled cycle

• 15.1. Complete equations and the sector being classified

• 15.2. Zero phase rates force uniform form

• 15.3. Member-phase equality follows without a pair chart

• 15.4. Acute sine balance and circular closure give exactly three windings

• 15.5. Exact families, captured phases and observer abstention are distinct

• 15.6. The strict sector boundary and what has not been selected

• 15.7. Shared classification and captured-state evidence

• sine-pairing-transition

• 16. Local onset and loss of a support-compatible observed grouping

• 16.1. Differentiate the observation using the complete law

• 16.2. One frozen represented preparation

• 16.3. All initial competitors and the two transverse ties

• 16.4. A complete local transition follows from smoothness

• 16.5. Form reversal controls the direction with the same phase geometry

• 16.6. What has appeared and what has not

• 16.7. Implementation and evidence

• sine-pairing-window

• 17. Whole-box prediction throughout a fixed observation window

• 17.1. The frozen question and complete state

• 17.2. A global full-field remainder, without a reduced trajectory

• 17.3. Exact rational margins for the whole window

• 17.4. The two reserved predictions and their limits

• 17.5. Shared consumer and numerical evidence

• sine-pair-support-symmetry

• 18. Which attachments permit an unordered pair-state law?

• 18.1. Equivalence concerns the entire member state on a fixed graph

• 18.2. Necessary and sufficient fixed-support condition

• 18.3. Internal edges preserve symmetry but change the inherited rows

• 18.4. Frozen one-edge control: identical retained state, different future

• 18.5. What a larger collective description must retain

• 18.6. Shared support reader and state-only evidence

• sine-mixed-pair-state

• 19. Sufficient collective state on asymmetric support

• 19.1. Remove only the independently redundant member labels

• 19.2. Inherited rates use the actual fine attachments

• 19.3. Realizability, synchronized tips and the chart boundary

• 19.4. The retained receiver control now discriminates the two states

• 19.5. Shared observer and scope

• sine-star-moment-chart

• 19.6. A regular phase/rate chart of the existing unordered state

• sine-pairing-constitutive-scope

• 20. Constitutive scope of the local grouping mechanism

• 20.1. The common work identity leaves a mobility law to specify

• 20.2. Positive reciprocal response preserves the local direction of grouping

• 20.3. A frozen relative-rate discriminator

• 20.4. Shared comparison and evidence boundary

• sine-replica-constitutive-nonselection

• 20.5. Recursive inheritance and the equilibrium tangent do not select one law

• collective-mean-closure-obstruction

• 20.6. Universal collective-mean closure forces a constant circular current

• sine-mobility-relative-geometry

• 21. Protected relative geometry without an assumed invariant volume

• 21.1. Quotient only the common origins of the actual complete law

• 21.2. The existing storage barrier protects the relative pattern

• 21.3. Mean motion and divergence change although storage does not

• 21.4. What is still required for an almost-everywhere return claim

• 21.5. Shared balance and admission evidence

• sine-pair-emission-descent

• 22. Local and whole-pair form actions on an exact unordered NFR

• 22.1. Keep the continuous quotient and the supplied event separate

• 22.2. Exact obstruction and the two exceptional cases

• 22.3. A sufficient marked port and a symmetric control

• 22.4. Storage and occurrence remain separate obligations

• sine-autonomous-regional-transfer

• 23. Autonomous regional transfer and supplied form injection

• 23.1. The regional balance of the complete sine law

• Relative inventory and a physical-property boundary

• sine-two-contact-orientation-response

• A finite orientation-sensitive response with retained backreaction

• 23.2. A positive control inherited from the existing pair pulse

• 23.3. Why a positive AL-only endpoint is not that closed flow

• 23.4. Shared current and endpoint observations

• sine-joint-identity-window

• 24. Joint form-phase identification during autonomous exchange

• 24.1. A storage-defined observation, not an extra dynamical rule

• 24.2. Exact reference identity survives a phase-only collision

• 24.3. A full-field perturbation bound over a declared window

• 24.4. A fixed full-exchange control and genuine degeneracies

• 24.5. Shared observation and window admission

• sine-joint-grouping-comparison

• 25. The frozen phase transition does not change the joint pairing

• 25.1. Compare two observations of the same complete preparation

• 25.2. A different joint partition is already strictly present

• 25.3. The same joint pairs persist on the complete frozen boxes

• 25.4. Observation change, collective state and formation are distinct

• 25.5. A projection of the existing window, not another forecast

• sine-joint-boundary-acquisition

• 26. Acquiring a joint pairing at its exact observation boundary

• 26.1. A separately declared critical preparation

• 26.2. Both rows determine the crossing direction

• 26.3. One-sided acquisition and qualitative full-state robustness

• 26.4. What is acquired, and what remains supplied

• sine-joint-recurrent-episodes

• 27. Repeated finite episodes of joint identification

• 27.1. Identity, loss and a common acquisition interval

• 27.2. One finite invariant ambient family

• 27.3. Infinitely many distinct finite matching episodes

• 27.4. Measure, interpretation and integration boundaries

• sine-zero-resultant-restoration

• 28. Zero pair resultant and autonomous restoration of form current

• 28.1. A global current observation without a midpoint angle

• 28.2. The retained phase channel and internal form information

• 28.3. One frozen antipodal preparation and its complete response

• 28.4. Immediate restoration, a magnitude cusp and exact silence

• 28.5. Existing source owners and exact-phase scope

• sine-moving-pattern-interface

• 28.6. Retained interaction contract for a moving constituent

• sine-global-pair-state

• 29. Global unordered pair state through phase cancellation

• 29.1. Derived coordinates and exact realizability

• 29.2. The inherited polynomial field

• 29.3. Storage, current and global continuation

• 29.4. Why cancellation needs the retained phase product

• 29.5. Implementation and numerical scope

• sine-pair-cancellation-observability

• 29.6. Observable internal state at cancellation