Complete form-capacity actions on the maintained contact

This owner admits the complete primary VAL/NUL maps on the selected weighted conservative contact. Local form scaling changes relative geometry and both subsequent rate rows; it is not generally a common clock change. The retained source also gives an analytical limitation of the existing sufficient maintenance certificate. No new operator word or trajectory is evaluated. The research program alone selects further work; identity admission retains its own hypotheses.

1. State, law and execution boundary

Use the maintained contact: twenty associated nodes, forty original unit edges, one internal A0 edge and four A0--B0 edges. Retain every actual conductance W and bare support U, signed real form x, primitive circular phase with declared continuous lifts, positive capacities nu, stored pressure and execution context. The continuous law in its original structural clock is

\[ q=L_Wx,\quad S_i=\sum_{j\sim i}\sin(\theta_j-\theta_i),\quad M=\operatorname{diag}\frac{\nu_i}{\pi d_i},\quad \Phi_i=\frac{S_i}{\pi d_i},\qquad \dot x=MS,\quad\dot\theta=Mq. \tag{1} \]

Conductances enter q; unique bare neighbors enter S and d, including zero-conductance edges when present in algebraic controls. Positive weighted support and reciprocal bare support are fixed in this contact. There is no forcing or continuous loss. Capacities are held between events. The nodal pressure is specified by Phi, not reconstructed from the resulting form rate. These coefficients do not transfer to the native neighbor-resultant pressure.

The structural storage and mean weights are

\[ H=\tfrac12x^TL_Wx+V(\theta),\qquad V=\sum_{\{i,j\}\in U}(1-\cos(\theta_j-\theta_i)),\qquad \rho_i=d_i/\nu_i. \tag{2} \]

Under (1), H and both rho-weighted means are conserved. Phase means use the declared continuous lifts, not arbitrary wrapped representatives. H is the storage of this normalized model, not a measured physical energy. Event factors are dimensionless; form increments retain form units. Neither a capacity reset nor a phase rate supplies a laboratory clock or mass bridge.

The registered VAL and NUL contracts and shared proposal own the actual maps. VAL multiplies capacity by a factor greater than one and leaves stored pressure fixed. NUL multiplies capacity by a factor in (0,1) and stored pressure by its checked reciprocal. Default edge awareness also changes form: VAL limits its form factor by the sign-specific boundary before projection; NUL multiplies form before projection. The disabled branch preserves form. Actual endpoints, including projection and represented arithmetic, determine the analysis below. A requested factor is not necessarily the exact ratio of the represented pre/post capacities.

These are conditional primary stages, not complete admitted words. The direct glyph, public wrapper and atomic stage have distinct secondary effects. The immutable stage reads all targets from one source and commits together; its pointwise form certificate is not a joint form/phase identity theorem. No finite-duration pressure flow is inferred from the jump.

2. Full refreshed rows and event work

For an admitted phase/support-preserving stage, write its complete endpoint as

\[ x^+=x+h,\qquad a_i=\nu_i^+/\nu_i>0,\qquad A=\operatorname{diag}(a_i). \]

Here h includes every actual form change and is zero on untouched nodes. Fresh constitutive evaluation gives Phi^+=Phi, M^+=AM and q^+=q+L_Wh. Consequently

\[ \dot x^+-\dot x=(A-I)MS,\qquad \dot\theta^+-\dot\theta=(A-I)Mq+AM L_Wh. \tag{3} \]

An untargeted node can therefore acquire a different phase rate through (L_W h)_i, even though its own form, phase and capacity were untouched. The ideal NUL identity nu^+ p_store^+=nu p_store concerns stored pressure; after refresh the form row scales by the actual capacity ratio instead. Represented multiplication also requires its own product residual.

Since phase is fixed at this stage, the exact event work is

\[ \Delta H=q^Th+\tfrac12h^TL_Wh =\sum_{\{i,j\}\in U}W_{ij} \left[(x_i-x_j)(h_i-h_j)+\tfrac12(h_i-h_j)^2\right]. \tag{4} \]

There is no separate capacity term in H. When both endpoints of an edge change, adding their isolated work misses the cross term -W_ij h_i h_j. Use the whole simultaneous map. Local contraction toward zero need not lower relative storage: on one unit edge, x=(1/2,3/4) and NUL factor 1/2 on the first node increase form storage by 3/32. Conversely, an expansion toward a larger neighbor can reduce it. Operator names imply neither sign.

Any subsequent IL phase reset adds its own phase work and changes Phi. Thus (3)--(4) describe this stage, not an entire word containing IL. Whole-word accounting must retain every intervening map and refresh.

3. Changed mean leaves

Put rho_i^+=rho_i/a_i, R^+=sum_i rho_i^+, and let mu_x and mu_theta be the old normalized weighted means. Direct subtraction gives

\[ \mu_x^+-\mu_x= \frac{\rho^{+T}h+(\rho^+-\rho)^T(x-\mu_x\mathbf1)}{R^+}, \tag{5} \]

\[ \mu_\theta^+-\mu_\theta= \frac{(\rho^+-\rho)^T(\theta-\mu_\theta\mathbf1)}{R^+}. \tag{6} \]

The phase mean can change without a phase reset because its capacity-dependent weights change. Subsequent conservation holds on the new leaves, not the old ones. A lifted mean comparison must keep the same declared lifts.

For ideal matched, unprojected scaling x^+=Ax, the unnormalized form charge is preserved: rho^{+T}x^+=rho^T x. This does not preserve its normalized mean when R changes. For a general represented endpoint the charge defect is

\[ \rho^{+T}x^+-\rho^Tx =\sum_i\rho_i\left((x_i+h_i)/a_i-x_i\right). \tag{7} \]

Projection, boundary adaptation and rounded products must remain in (7). Discarding these terms cannot establish an invariant.

4. Common clock and coordinate-change tests

A pure global capacity change A=cI, h=0, scales both rows of (1) by c at every state. With positive constant c and otherwise identical premises, this is the exact common clock change tau=c t. A local capacity change generally is not. For one full reset, a pointwise factor c would have to satisfy both

\[ (A-cI)MS=0,\qquad (A-cI)Mq+AM L_Wh=0. \tag{8} \]

Zero rows or special states can satisfy (8) accidentally; that is not a trajectory equivalence on a neighborhood. Even uniform form/capacity scaling x^+=b x, nu^+=a nu multiplies the two evaluated rows by a and ab, respectively, and changes the preparation.

The natural stronger coordinate comparison also has a precise restriction. Let T_b(x,theta)=(b x,theta) and compare the complete vector fields through F_{a nu}(T_b y)=k D T_b F_nu(y). On a domain with both rows nontrivial, the form row requires a=kb and the phase row ab=k. For positive b these imply b=1. Thus this positive form dilation is not a flow conjugacy by one clock factor. This argument concerns T_b, not every conceivable coordinate map.

5. A continuous symmetry need not descend through a local event

The continuous law is invariant under one common form translation x -> x+c 1. Its relative form, q, phase row and storage are unchanged. But a local ideal scaling x^+=Bx on a proper subset gives

\[ P B(x+c\mathbf1)-P Bx=cPB\mathbf1, \tag{9} \]

where P removes the common form mean. For a nontrivial local scale this is nonzero. Two sources with identical relative geometry can therefore have different relative outputs. Starting with uniform form m, the output has q^+=m L_W(B-I)1, nonzero when the scaled subset has a positive weighted boundary and m is nonzero.

Thus an observation discarding the common form origin cannot implement this event by itself. Retain that origin, the full signed form, or an explicitly declared event reference. This is an event-descent obstruction, not a failure of the continuous translation symmetry. Choosing a new reference-dependent operator would be a changed map requiring its own contract; it is not an implicit correction to the current implementation.

6. Identity and the existing continuation criterion

For the contact identity, use the same reference theta_* and Euclidean centering P=P_20:

\[ Z^2=\|Px\|^2+\|P(\theta-\theta_*)\|^2,\qquad r=1/16. \]

The form-only event obeys

\[ (Z^+)^2-Z^2=2(Px)^TPh+\|Ph\|^2,\qquad Z^+\leq Z+\|Ph\|. \tag{10} \]

The existing barrier H-H_* >= Z^2/25 on Z<=r does not depend on capacity. Therefore it can still prove continuation after the reset if the complete endpoint has positive finite capacities, Z^+<r and H^+-H_*<1/6400, and the continuing law, support and storage stay as stated with no further events or inputs. New held capacities change mean leaves and mobility, not this algebraic barrier. Failing its sufficient storage condition does not prove actual departure from the identity region.

6.1 Retained source: a strict certificate limitation without a new word

Read the existing OZ_completed capture in the retained correction response. Its SHA256 is 39ddab82885e4171ed67bf61eb77a67a1e081039e48d7ca760c0ee494befb3b4. Primitive reconstruction, not its preparation label or cached verdict, supplies

\[ m=5/16,\quad x_0=m+9/8192,\quad x_1=m+7/8192,\quad x_i=m\ (i\ne0,1),\qquad Z<1/256. \tag{11} \]

In particular q is not zero. The two A0 targets each have four original unit-weight neighbors with form m. The eight distinct edges suffice for the lower bound below; the other positive-weight contributions can be discarded. The retained configuration uses hard rails [-1,1], default edge awareness and default VAL/NUL factors. No source, gain or form origin is retuned.

Consider either one hypothetical default primary stage on both A0 targets, conditional on live admission. Admit the actual represented factors through

\[ 1/13<f_{\rm VAL}-1<1/12,\qquad 1/13<1-f_{\rm NUL}<1/12. \]

The form boundary cap and hard projection are inactive for these products. With e=2^-50 as a conservative absolute multiplication-error bound, the new edge-form gaps are at least m/13-e for VAL and in absolute value at least m/13-9/8192-e for NUL. The binary64 round-to-nearest products lie in (0,1) without underflow or overflow; their rounding error is smaller than e. Hence the respective form-storage lower bounds are

\[ E_{x,\rm VAL}^+\geq4(m/13-e)^2>1/640>1/6400,\qquad E_{x,\rm NUL}^+\geq4(m/13-9/8192-e)^2>1/640>1/6400. \tag{12} \]

These lower bounds are approximately 0.00231139 and 0.00210494, compared with 0.00015625 for the old sufficient threshold. To control the sign of the phase contribution, apply the existing barrier to the auxiliary state with x=0 and this same phase. Its Z is no larger than the original Z, so V(theta)-H_*>=0. It follows that each hypothetical stage has H^+-H_*>1/6400.

At the same time only two forms change, and the centering projection is nonexpansive. With x_max=2569/8192,

\[ Z^+\leq Z+\|h\|<1/256+\tfrac32(x_{\max}/12+e) <0.043107<1/16. \tag{13} \]

Thus the endpoint remains inside the declared identity ball while lying outside the previously certified storage sublevel. This is a conditional endpoint bound, not an executed response or a prediction of future escape. It identifies a limitation of the certificate, not a failure of TNFR.

There is also a useful closure restriction. Any later map that preserves this form and support, including phase/capacity-only IL/SHA actions, cannot restore H-H_*<1/6400 while staying inside this identity ball: the same form-storage floor persists and the auxiliary-state argument again gives V-H_*>=0. This does not exclude a different maintenance criterion, a form-changing word, or separately admitted organizations. It avoids executing a phase-only closure merely to retest this already unavailable certificate.

6.2 Sharpening the same scalar storage cannot resolve the gap

The obstruction is stronger than a loose constant in the previous barrier. Let sigma equal +1 on the ten A nodes and -1 on the ten B nodes, and put s=r/sqrt(20). For any prescribed post-event rho^+-weighted means, choose

\[ x_b=\mu_x^+\mathbf1,\qquad \theta_b=\theta_*+s\sigma+c\mathbf1, \qquad c=\mu_\theta^+-\frac{\rho^{+T}(\theta_*+s\sigma)}{R^+}. \tag{14} \]

This state has both required means and Z_b=r. Common translations affect neither Z nor H. The forty original edges and the internal A0 edge retain their reference phase gaps. Only the four cross edges change, from zero to a gap of magnitude 2s. Thus

\[ H_b-H_*=4(1-\cos 2s)\leq8s^2 =\tfrac25r^2=1/640. \tag{15} \]

This is smaller than both form-storage floors in (12). The minimum energy on the identity boundary with the new means is therefore below the actual hypothetical endpoint energy. No sharper lower estimate of that same boundary energy can certify this endpoint by an H-sublevel lying inside the ball.

Restricting to a connected sublevel component does not rescue that argument. Within the ball, every phase-edge cosine is positive: its reference wrapped gap is zero or has magnitude 2*pi/5, and its perturbation is at most sqrt(2)*r, with 2*pi/5+sqrt(2)*r<pi/2. The form Hessian is positive semidefinite as well. H is consequently convex along the segment from the interior endpoint to (14). This segment remains on the same two mean leaves and within the closed ball. Its energy never exceeds the larger endpoint energy, which is H^+. The H-sublevel component containing the reset endpoint already reaches the identity boundary.

This is a configuration-space accessibility argument, not a solution of (1) or evidence of actual escape. It excludes confinement by an H-sublevel, including its connected component, restricted to the two mean leaves for these endpoints. Additional invariants, directional bounds or controlled nonlinear dynamics could still settle retention. A phase/capacity-only closure with the form and support held also retains the floor in (12); while inside the ball it cannot recover this scalar-sublevel certificate. Other form-changing words and other identity definitions have not been excluded.

7. Live admission, projection and evidence boundaries

The selected capture matters. Its target stored pressures are positive and its retained scalar values pass the default optional strict VAL/NUL policies; an earlier inherited capture has a negative target pressure and fails VAL's strict pressure policy. Neither scalar check admits a complete word. The retained OZ history still has its own U2 debt; preceding IL does not prepay that later destabilizer. NUL has no active U1/U2 role and is neither a stabilizer nor a closing token. VAL followed by NUL is not thereby a complete word. Any future word must preserve the live prefix and use the shared grammar with rejection enabled.

Default factors 1+t and 1-t, t=1/(4*pi), have ideal product 1-t^2<1; they are not reciprocals. Even supplied reciprocal factors would not make the complete actions inverses: NUL rescales pressure, projection can lose information, and histories change. A soft projection is not generally a unit-Lipschitz map; asymmetric rails can also reverse the nominal direction of a NUL form reset. Reuse actual endpoints rather than a contraction label.

Runtime admission is shared across the direct and atomic scale proposals: the edge-aware flag uses Boolean admission; active clip mode, bounds and soft steepness use the shared clip-policy resolver; active strict scalar thresholds reject nonfinite and Boolean values. Configured soft steepness must reach the proposal. Inactive clipping is not consumed by the disabled form branch. Public execution retains its existing EPI preflight domain, which is narrower than the pure disabled-branch kernel. These checks do not turn supplied thresholds into derived universal safety or identity laws.

8. Implementation and verification owners

The shared scale kernel, direct dispatcher, atomic stage and strict preconditions own execution. The event contract owns transactional and observation boundaries. No parallel operator kernel, new pressure law, event selector or field reader is introduced here.

Algebraic controls use independent edge storage, exact rational means and the existing weighted-law reader. They distinguish complete work, untouched-neighbor phase response, stored products, common clocks, charge/mean differences, form-origin dependence and projection. Read-only source controls reconstruct existing primitive state and verify the margins in (11)--(13); they execute no new maintained-source proposal or trajectory. Runtime regressions separately check invalid domains, configured clipping and rollback. Testing owns selections.

These controls are distinct from the separate IL/OZ phase-certificate correction, which resolves the numerical subclaim while preserving the frozen 8/9 report. The directional maintenance study uses additional state and nonlinear information beyond this sublevel obstruction. No physical identification, spontaneous event occurrence or universal realization of all registered operators follows from this conditional action analysis.