Geometric identity and capacity-feedback bounds¶

Local restoring response, geometric identity and prospective capacity tubes.

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

41. Geometric identity and the undefined global phase target¶

Recovering every coordinate of a stationary target is not the definition of every coherent entity. Section 32 already supplies a different conditional identity: acute cycle phases approach a uniform twist while scalar EPI approaches consensus. The limiting phase geometry remains nonuniform. This section checks that identity against the native global coordinator, without changing the earlier P3 target or assuming that its supplied sine law is the default runtime law.

Local coherent geometry can have zero global phase order¶

On a simple C5, fix the ideal phases on the circle as

theta_i = theta_0 + i*delta modulo 2*pi,  delta = 2*pi/5.

The global unweighted phasor sum is zero. Each node's two neighbors instead have the nonzero resultant

exp(i*theta_(i-1)) + exp(i*theta_(i+1))
    = 2*cos(delta)*exp(i*theta_i),  cos(delta)>0.

Thus its local phase target is its own phase, canonical phase pressure and curvature are zero, and absolute local phase gradient is delta. All edges are strictly inside the half-pi U3 limit; winding is one. With constant EPI, common positive capacity and no additional forcing, the other pressure channels also vanish on this cycle. With fresh pressure and its nodal rate, the ideal coherence read-out is one although global Kuramoto order is zero. This distinguishes two existing diagnostic meanings; it does not redefine either metric.

Positive unequal transport conductances, including section 32's retained closing-edge weight, do not change these phase facts. The local and global phase reducers use unique neighbors and unit global weights, respectively. Under section 32's supplied phase law this is a relative equilibrium, with common phase advance and unchanged relative geometry. Its conditional relaxation bounds do not transfer automatically to the native coordinator.

A symmetric phase multiset cannot select one covariant direction¶

Suppose a phase-only global target T takes values on the circle, is invariant under permutations of its inputs and covariant under a common phase rotation. Let P cyclically permute the five regular phases. Then

P*theta = theta + delta*1 modulo 2*pi,
T(P*theta) = T(theta),
T(theta+delta*1) = T(theta)+delta modulo 2*pi.

The last two conditions contradict delta!=0 modulo 2*pi. No deterministic single-valued target can satisfy both symmetries at this multiset. This is the same stabilizer logic used for parent selection in the THOL audit, applied to a circular output rather than a chosen vertex.

The argument concerns the phase multiset consumed by the global reducer. It does not assume that a cycle with a distinguished conductance has a cyclic automorphism preserving its complete weighted state. A marked node, external direction or other additional input changes the target-selection problem. None is supplied by the zero phasor resultant itself.

Any selected global target changes the uniform-twist orbit¶

The actual coordinator applies a shared global gain gamma and local gain to shortest-arc displacements. At the ideal uniform twist the local displacement is zero. If a single global target T is nevertheless supplied, its ideal phase update is

theta_i^+ = theta_i + gamma*wrap(T-theta_i) modulo 2*pi.

For 0<gamma<1, these increments cannot all agree modulo 2*pi, so the result is not merely a common rotation. Around the oriented cycle, four gaps and the one gap crossing the target's shortest-arc cut become

four gaps:       (1-gamma)*delta,
exceptional gap: delta+gamma*(2*pi-delta).

An antipodal tie can move which edge is exceptional; it does not restore equal gaps. Their unwrapped sum remains 2*pi. Every gap remains inside the open shortest-arc branch exactly when gamma<3/8; winding then remains one despite the changed geometry. Strict acuteness additionally requires gamma<1/16. At equality the exceptional gap is pi/2: the closed hard U3 gate may admit it, but the strictly acute theorem no longer applies. At gamma=3/8 that gap reaches the antipodal branch boundary.

Consequently preservation of a winding integer is weaker than invariance of the uniform-twist orbit. This obstruction needs no EPI integration or long trajectory. It also does not say that the resulting phase state has lost every possible coherent identity.

Native policy and represented arithmetic have separate scope¶

The default adaptive policy classifies sufficiently low global Kuramoto order as dissonant and retains a strictly positive global gain. That policy therefore reacts to the ideal twist's zero global order despite its regular local geometry. It is a configured preference for global alignment, not a consequence that local TNFR coherence or winding is absent.

The ideal regular polygon and its materialized binary64 angles are distinct inputs. Their transcendental sums need not agree exactly; cached trigonometry and reduction add further numerical effects. The existing retained phase audit already exposes that distinction and the conditioning of a small resultant. The shared represented-component reducer sums its supplied component pairs exactly; it does not certify the transcendental values or a rotation symmetry. The coordinator's opt-in exact_components_v1 rejects an exactly zero represented resultant when the effective global term is active. A nonzero represented sum remains a numerical direction, even if very small. The legacy path instead retains its existing numerical atan2 behavior.

The static writer controls use unit C5, represented theta_i=0.125+2*pi*i/5, EPI 0.5, capacity one, stored pressure zero, empty glyph histories, injected defaults and seed 17. One legacy invocation and a matched exact_components_v1 invocation both select the dissonant branch with kG=0.05929429797231117. Four observed gaps are approximately 1.1821256490720864 and the fifth 1.5546827108912409, matching the derived deformation. Winding remains one and the strict U3 margin is approximately 0.0161136159036559. The exact sum of cached component pairs is (-7,3)/2^55, nonzero; this does not contradict the ideal polygon's zero sum. Both calls preserve EPI, capacity, stored pressure, support and clock. They execute no glyph, nodal integration or full runtime step and establish neither future invariance nor formation of this pattern.

These results identify a model-compatibility boundary between an existing geometric identity and an existing native policy. They select no replacement target, zero-resultant threshold, phase law or autonomous event schedule.

42. A geometric-identity contract for the retained weighted C5¶

Declared state, law and identity family¶

Keep Section 32's scalar form chart, supplied clock and continuous phase/form law. The modeled state is (x,theta,nu,W), not every engine history, cache or controller variable. Retain the actual Section 31 post-UM support and its materialized nonunit closing conductance. The ordered phase support is C5; all transport conductances are fixed, positive and symmetric. Capacity is held at its captured common positive value kappa, and effective pressure weights and K>0 are fixed. No Gamma, subsequent event, active clipping or native controller is included. The phase law identifies kappa with radians per supplied time unit; neither laboratory seconds nor a clock emergent from synchronization is inferred.

The identity family is specified before evaluating a response:

I(W,kappa) = { x=c*1, nu=kappa*1,
              theta_i=alpha+2*pi*i/5 modulo 2*pi, W fixed }.

Its free labels are common rotation alpha and uniform form c in the declared chart. It excludes phase consensus and fixes winding +1 in the chosen orientation. Reversing the cycle enumeration changes the displayed winding sign, not the physical state; conjugating the physical phases is a different operation and is not quotiented out. The family preserves nontrivial phase geometry even though form is uniform. It does not require nonzero scalar-form amplitude; that would be a further identity obligation. It is not a definition of every NFR or an identification with a physical particle.

On this family the local phasor source and EPI pressure vanish, the sine corrections cancel, and theta_dot=kappa*1. Thus it is exactly invariant under the declared continuous law: c is constant and alpha rotates. The global Kuramoto order is zero while each normalized local resultant is cos(2*pi/5)>0. The global-target obstruction in Section 41 therefore remains relevant; invariance here is not invariance under that native coordinator.

A lifted shape coordinate separates rotation from deformation¶

Admit initial oriented gaps inside one strictly acute interval [m,M], contained in the fixed U3 gate, with sum 2*pi. Section 32 preserves this domain. Write delta_bar=2*pi/5, u_i=delta_i-delta_bar, and define

p_0=0,  p_i=sum_(j<i) u_j,
h_i=p_i-mean(p),
alpha=theta_lift_0+mean(p).

Then sum h_i=0, the cyclic difference of h equals u, and theta_lift_i=alpha+i*delta_bar+h_i. The last cyclic difference is valid because sum u_i=0; the phase lift itself closes after one full turn. Under the supplied equal-capacity sine law the corrections telescope in sum theta_dot_i=5*kappa. Consequently

alpha(t)=alpha(0)+kappa*t,
sum_i h_i(t)^2 <= ||u(t)||_2^2 / lambda_2(L_C5)
              <= Q^2*exp(-2*gamma*t) / lambda_2(L_C5).

The second line is the cycle Poincare inequality on centered h. This is a lifted shape norm. The circular orbit distance is inf_alpha sum_i wrap(theta_i-alpha-i*delta_bar)^2. It is no greater than the displayed lifted bound, because wrapping each residual and then minimizing over a common rotation cannot increase this chosen value. No claim is made that the lifted norm is the global circular minimum.

The offset alpha uses the ordered support and its admitted lift. It is a coordinate, not a permutation-invariant target obtained from the phase multiset, and it supplies no exception to Section 41's symmetry obstruction.

The existing CycleRelaxationEnvelope exposes this decomposition as exact affine-pi pairs in initial_phase_offset_affine and initial_phase_shape_affine, with shared rational enclosures in initial_phase_shape_enclosures. Its phase_orbit_distance_squared_upper reuses the existing sample times, gap envelopes and certified spectral lower bound. These are derived read-outs of one captured state/model, not new dynamic parameters, a second solver or another evidence record. The properties also apply to the evaluator's other admitted cycle sizes and winding sectors; the identity selected here is C5, winding one.

Form mean and the strength of the persistence claim¶

Use the actual transport strengths s, their sum S and m_s=s^T*x/S. The form residual remains D=sum_i s_i*(x_i-m_s)^2; phase and form residuals are reported separately without inventing a cross-channel normalization. Section 32 supplies D's Duhamel bound and

m_s_dot = kappa*w*s^T*g/S,
|m_s(t)-m_s(0)| <= (M_source/gamma)*(1-exp(-gamma*t)),
|m_infinity-m_s(t)| <= (M_source/gamma)*exp(-gamma*t).

Here M_source is that section's mean-rate prefactor (its symbol M there), not the upper phase-gap endpoint used above. Thus the limiting uniform form is unknown but constrained by the captured initial mean and mean_limit_offset_upper. Permitting this governed mean drift is not permission to fit an arbitrary c(t). On the invariant family itself g=0, so there is no mean drift. With the retained nonunit edge, a perturbed state can have nonzero weighted drift even though sum g_i=0.

These estimates establish conditional attraction to the identity family. They also give local orbital stability relative to that family: keep W, kappa, coefficients and a common strictly acute neighborhood fixed. As initial gap and form residuals tend to zero, their all-time upper bounds and the mean-shift bound tend to zero. The neighborhood keeps the rates bounded away from zero. Form disagreement need not decrease monotonically; the retained post-event state initially develops form contrast before it relaxes. Capacity, support, operator-policy and phase-law perturbations are outside this stability claim.

Evidence and remaining premises¶

The existing control owner reuses its retained initial capture to reconstruct the phase lift independently at high precision, check centering, affine-pi enclosures, reconstruction and the Poincare bound. Its winding is already one and its initial EPI disagreement is zero, but its phase deformation is strictly positive. Thus a winding label and uniform form do not alone identify a regular geometric shape. Reversal with relabeling retains the same nodewise shape; a consensus control has zero shape in the separate winding-zero family. No new trajectory is generated for these assertions. The earlier finite binary64 continuation remains evidence only for its stated execution horizon and numerical defects.

The derived parts are phase-domain preservation, this rotation/shape split, conditional invariance, attraction and mean bounds. Fixed support/capacity, the supplied sine phase law, its clock interpretation, pressure realization, coefficients and prepared sector remain explicit model premises. The native global coordinator is incompatible with the regular orbit; support creation still depends on the recorded UM policy. Autonomous maintenance, formation from a different sector and empirical identification remain unresolved. The single G3 queue owns the next mechanism-admission task.

43. A positive local response supplies the cycle restoring mechanism¶

Admitted class and the role of each assumption¶

Retain Section 32's fixed ordered C_n, common free phase rate kappa, actual positive symmetric transport conductances, fixed capacity and scalar form law. The phase support has two unique neighbors per node. In an admitted acute sector let

J_i = (sin(delta_i)-sin(delta_(i-1)))/2,
theta_dot_i = kappa + a_i(t,state)*J_i,
0 < a_minus <= a_i(t,state) <= a_plus < infinity.

J is the existing local sine current. The coefficient a is a response mobility in this separately specified phase row, not nodal form capacity or a new TNFR primitive. Bounds must hold throughout the declared domain and time interval; reading one positive coefficient at the initial state does not establish them. Each correction multiplies its local current; locality of the entire law also requires a_i to read only the declared local state. No derivative of a is needed for the inequalities below.

The estimates concern absolutely continuous solutions of that declared row. A sufficient existence/uniqueness contract is measurable time dependence and local Lipschitz dependence on the declared state, with locally integrable Lipschitz bounds and the displayed uniform coefficient bounds. If a also depends on form, this contract applies to the joint state; the linear form row with bounded phase source prevents finite-time escape. Positivity alone does not make an arbitrary discontinuous constitutive law well posed. Common rotation or relabeling covariance of the whole law additionally requires the corresponding covariance of a; the inequalities do not assert that symmetry for every possible coefficient function.

Acute-domain preservation and an exact dissipation identity¶

With cyclic indices, subtracting neighboring phase rates gives

delta_dot_i = [a_(i+1)*(sin(delta_(i+1))-sin(delta_i))
               -a_i*(sin(delta_i)-sin(delta_(i-1)))]/2.

On a fixed acute interval [m,M], sine is increasing. At a maximal gap both contributions are nonpositive, and at a minimal gap both are nonnegative. Therefore the interval is forward invariant. Its gap sum remains 2*pi*ell, retaining winding and the same U3 admission. Local phasor resultants remain bounded below by cos(rho)>0, where rho=max(abs(m),abs(M))<pi/2. Unequal free phase rates would add their neighbor difference to this equation and require a separate forcing analysis.

Put delta_bar=2*pi*ell/n, u=delta-delta_bar*1, Q=||u(0)||_2 and lambda_C=lambda_2(L_cycle). Summation by parts gives the exact identity

(1/2)*d||u||_2^2/dt
 = -(1/2)*sum_i a_i*(u_i-u_(i-1))
                       *(sin(delta_i)-sin(delta_(i-1)))
 <= -(a_minus*cos(rho)/2)*u^T*L_cycle*u
 <= -gamma*||u||_2^2,

gamma = a_minus*cos(rho)*lambda_C/2 > 0.

Thus ||u(t)||_2<=Q*exp(-gamma*t). This is the restoring mechanism: the local response has the dissipative sign and sufficient persistent strength on an available phase chart. Neither a winding integer nor matching fixed points supplies those properties. The same class decreases the unweighted alignment potential V=sum_edges(1-cos(delta)), since grad_theta V=-2*J and V_dot=-2*sum_i a_i*J_i^2. This potential identity does not imply monotonicity of the forced EPI disagreement or tetrad energy.

The existing phase-to-form bridge is unchanged: on this cycle g_i=(delta_i-delta_(i-1))/(2*pi), so ||g(t)||_2<=(Q/pi)*exp(-gamma*t). Section 32's weighted EPI, Duhamel, mean-displacement, mean-tail and sufficient clipping bounds apply with this gamma and their original actual transport coefficients. Section 42's lifted shape bound also applies. The regular-twist/uniform-form family is therefore conditionally attracting and locally orbitally stable with the same fixed-state and common-neighborhood qualifications. No new form source or capacity law is needed for this conditional result.

A restored shape need not keep the same phase offset¶

The centered lift still obeys

alpha_dot = kappa + (1/n)*sum_i a_i*J_i.

Since sum J_i=0, subtracting (a_minus+a_plus)/2 inside the sum and using ||J||_2<=||u||_2 gives

|alpha_dot-kappa| <= B_alpha*exp(-gamma*t),
B_alpha = (a_plus-a_minus)*Q/(2*sqrt(n)).

Consequently alpha-kappa*t has a finite limit. Its displacement from the initial offset is at most B_alpha*(1-exp(-gamma*t))/gamma, and its remaining tail is at most B_alpha*exp(-gamma*t)/gamma. This is distinct from the transport-weighted form-mean bound. A node-independent response makes the phase correction sum zero, including a common time-dependent response. The pressure specialization below also cancels its mean correction, even though its current multiplier varies from node to node.

Existing current and argument pressure belong to the class¶

For two adjacent oriented gaps a and b in the acute interval, define the positive sine divided difference by its continuous integral expression

s(a,b) = integral_0^1 cos(b+v*(a-b)) dv,
cos(rho) <= s(a,b) <= 1.

It equals (sin(a)-sin(b))/(a-b) when the gaps differ and cos(a) when they coincide. Therefore the ideal cycle pressure and current satisfy

g_i = b_i*J_i,
b_i = 1/(pi*s(delta_i,delta_(i-1))),
1/pi <= b_i <= 1/(pi*cos(rho)).

At equal gaps both readings vanish, but the continuous multiplier is 1/(pi*cos(delta_i)). No division of measured zero or near-zero current is licensed. This relation also connects to the existing curvature read-out K_phi=-pi*g on the same regular chart. It uses the common alignment potential with distinct response metrics; it does not make pressure and current identical or replace one by the other in the nodal form row.

Compare the following two separately supplied phase laws using the already declared positive coupling K:

Candidate phase correction	Current multiplier	Gap decay rate justified here
`K*J`	constant K	`Kcos(rho)lambda_C/2`
`K*g`, through the response-class bound	`K*b_i`	`Kcos(rho)lambda_C/(2*pi)`
`K*g`, using its exact cycle reduction	same `K*b_i`	`Klambda_C/(2pi)`

The last line is sharper because its exact gap equation is delta_dot=-K*L_cycle*delta/(2*pi). Its Jacobian does not contain the cosine stiffness of the sine law. Admission still requires the regular acute sector. Both candidates have alpha_dot=kappa because their respective corrections telescope. Their clock identification and positive gain remain supplied; the nodal EPI equation does not choose either phase law. In particular, reading g as pressure is implemented, while treating it as a phase velocity would be an additional constitutive choice.

Executable comparison, controls and failure boundaries¶

compare_cycle_restoring_responses in the existing cycle owner consumes its retained envelope and returns a detached CycleRestoringResponseComparison. With the shared exact pi interval [pi_lower,pi_upper], certified cosine lower bound c and spectral lower bound lambda, it encloses b_i by [1/pi_upper,1/(pi_lower*c)] and materializes the three conservative rates above. No graph is recaptured, no sample bounds are relabeled as another law, and no response, solver or controller is installed. Scalar/rate consistency checks do not authenticate caller-created reports or live state.

The existing cycle controls reuse the retained post-UM endpoint and actual pressure/current owners. Independent high-precision divided differences check the exact comparison, including the regular-twist zero-response limit. Represented current and pressure errors have separate absolute tolerances; their ratio is never treated as an exact-real certificate. Static response budgets verify the dissipation identity and maximum/minimum signs. A positive local comparison a_i=K*(1+J_i/2) has mean correction K*sum J_i^2/(2*n)>0 off the lock; this delimits the generic phase-offset claim without proposing that law.

Three negative controls delimit the sufficient mechanism:

Zero response preserves every nonregular gap configuration while the common phase advances. One actual zero-coupling production proposal retains those gaps within its stated binary64 tolerance; it supplies no restoring action. It is outside the positive-lower-bound class.
A common negative response reverses the gap-energy sign at every nonconstant acute state, despite the same regular fixed shapes and phase symmetries. The shared phase proposal rejects negative coupling; the reversed instantaneous budget is a mathematical control, not execution of an admitted engine law.
Pointwise positivity alone is insufficient. A common a(t)=exp(-t) traverses only finite effective time in the unit sine flow and need not reach its regular limit. A uniform positive lower bound is sufficient, not necessary: a nonnegative time-dependent lower envelope gives decay in accumulated action integral_0^t a_minus(v) dv. An infinite integral guarantees shape restoration under the same regularity/domain assumptions. An integrable decay bound is separately sufficient to extend the absolute source, form-mean and general phase-offset tail estimates; shape convergence alone does not provide those estimates. Exact mean cancellations can give stronger conclusions than the absolute bounds.

This completes the response-class admission at fixed support and common capacity. It identifies a shared sufficient mechanism and its failure modes; it does not derive the positive response, its clock or autonomous occurrence from the nodal product. The native global coordinator is outside this local class. Support selection, capacity evolution, formation and physical identity retain their explicit unresolved obligations in the single G3 plan.

44. Reusing phase geometry and capacity energy without adding a law¶

The preceding results connect existing state, field and transport owners. They yield the following deductions under their stated domains; they do not select the supplied phase law or establish autonomous pattern formation. The single G3 plan retains the capacity-forcing task. No additional trajectory campaign is needed to establish the identities in this section.

Curvature reconstructs the centered phase shape on the admitted cycle¶

Use Section 42's centered lift h, winding ell and oriented gap deviation u. Let L_C be the unweighted, unnormalized support cycle Laplacian. The two-neighbor midpoint identity gives, in exact arithmetic,

u_i = h_(i+1)-h_i,
g = -L_C*h/(2*pi),
K_phi = -pi*g = L_C*h/2,
h = 2*L_C^+*K_phi,
||u||_2^2 = 4*K_phi^T*L_C^+*K_phi.

Here L_C^+ is its inverse on centered vectors. For two admitted shapes on the same support and winding sector, ||h_1-h_2||_2 <= 2*||K_phi_1-K_phi_2||_2/lambda_2(L_C). Thus the existing spectral gap measures both restoring stiffness and the sensitivity of this reconstruction. This is the centered version of the phase coordinate already owned by cycle-support dynamics, not another nodal primitive.

This identity needs the full node-resolved curvature, support, winding and acute chart. It does not reconstruct common phase, capacity, EPI or transport weights. Exact regular C5 twists of winding +1 and -1 both have zero curvature and current. At uniform EPI and common capacity their ideal complete tetrads also coincide: zero pressure/potential, equal gradient magnitudes and the same correlation input. Winding therefore cannot be discarded. Nor can a lossy tetrad summary replace the full curvature vector. The common-semicircle premise of phase_quotient.py excludes a winding-one cycle; the block-constant phase premise of joint_quotient.py excludes its nontrivial regular grouping. Neither quotient automatically supplies the missing winding coordinate.

The existing cycle envelope exposes initial_phase_curvature_affine in cycle order, with mathematical pi. Controls invert L_C+11^T/n using the shared rational matrix inverse and reconstruct the retained shape independently. Production phasor/curvature rounding remains separate; exact centeredness or exact reconstruction is not asserted for uncorrected binary64 field readings.

An augmented conserved mean distinguishes the pressure-response candidate¶

Retain common fixed capacity kappa, positive EPI diffusion coefficient e, phase weight w, positive K and actual fixed symmetric conductances W. For the separately supplied candidate theta_dot=kappa*1+K*g, the centered shape satisfies h_dot=K*g. If s is the actual transport-strength vector, S=sum s_i and m_s=s^T*x/S, the form row gives

m_s_dot = kappa*w*s^T*g/S,
d/dt [m_s - kappa*w*s^T*h/(K*S)] = 0,
m_infinity = m_s(0) - kappa*w*s^T*h(0)/(K*S).

The last equality follows from exponential shape/source decay and stable EPI diffusion on the connected cycle. It predicts the uniform limiting EPI from the initial state, without fitting a late endpoint. For unit conductances, s^T*h=0 and the ordinary mean is conserved. With only the closing edge changed to beta, s^T*h=(beta-1)*(h_left+h_right); the mean shift is determined by the deformation at that edge's endpoints. This uses s^T*L_W=0, and does not replace the actual transport L_W by L_C/2.

compare_cycle_restoring_responses reports this affine-pi prediction and its outward enclosure alongside the existing candidate rates. It assumes the unclipped scalar model, with no added source, events or controllers. The original sine-law samples and clipping bounds are not relabeled as a pressure-law trajectory. For the sine law the same expression has derivative kappa*w*s^T*(g-J)/S, generally nonzero; shared fixed points do not establish a shared invariant. These are conditional model predictions, not evidence that the engine or physical observations select the pressure-rate candidate.

One capacity roughness controls two forcing channels¶

On the same cycle let f be the nonnegative configured capacity-pressure weight and define the existing Dirichlet quantity on the capacity field,

E_nu = nu^T*L_C*nu/2 = (1/2)*sum_i (nu_(i+1)-nu_i)^2,
b_i = nu_(i+1)-nu_i,
p_nu = -(f/2)*L_C*nu.

Then ||b||_2=sqrt(2*E_nu) and ||p_nu||_2 <= f*sqrt(2*E_nu), since lambda_max(L_C)<=4. The same measured capacity roughness bounds detuning in the phase-gap row and capacity forcing in the form row. The phase/capacity source also combines exactly as w*g+p_nu = -(1/2)*L_C*((w/pi)*h+f*nu) while the acute chart holds. This reuses the earlier structural-coordinate reduction and preserves signed channel cancellation. It neither adds a fundamental parameter nor identifies L_C with the weighted EPI transport operator. Topology pressure remains zero on this fixed degree-two support.

For ideal snapshot averaging with eligibility mask M, write nu_next=nu-(mu/2)*M*L_C*nu, where M is diagonal with entries zero or one. With z=M*L_C*nu and 0<=mu<=1,

E_nu_next-E_nu = -(mu/2)*||z||_2^2 + (mu^2/8)*z^T*L_C*z
               <= -mu*(1-mu)*||z||_2^2/2 <= 0.

This extends the existing compatible-capacity Dirichlet balance to partial eligibility on the cycle. It is stronger than interval invariance, but does not prove strict decay, preservation of a capacity mean, consensus or an integrable forcing budget for arbitrary masks. An actual represented update has an additional arithmetic defect; its energy change is not silently identified with the ideal expression.

For the separately supplied extension theta_dot_i=nu_i+a_i*J_i, choose a prospective fixed radius rho_star<min(pi/2,effective_U3_gate) with abs(delta_bar)<rho_star. The coefficients satisfy Section 43's positive bounds. While all gaps stay inside that chart, set gamma=a_minus*cos(rho_star)*lambda_2(L_C)/2. The differential inequality holds almost everywhere for the admitted absolutely continuous solution, with the usual norm comparison at zero; its integrated form is

d||u||_2/dt <= -gamma*||u||_2 + sqrt(2*E_nu),
||u(t)||_2 <= exp(-gamma*t)*||u(0)||_2
             + integral_0^t exp(-gamma*(t-r))*sqrt(2*E_nu(r)) dr.

A bound strictly below rho_star-abs(delta_bar) for the displayed norm envelope closes the chart bootstrap. This is a conservative sufficient condition. The radius from the unforced initial extrema cannot be reused without this check: detuning can push gaps outward. A persistent forcing bound gives a residual deformation tube, not exact restoration of the regular twist. This distinction matters for the actual averaging writer below.

For form, reuse the common energy E_x=x^T*B_W*x/2, where B_W=D_s-W, and mobility M_x=diag(nu_i/s_i). If F is the combined non-EPI source,

E_x_dot = -e*(B_W*x)^T*M_x*(B_W*x) + (B_W*x)^T*diag(nu)*F
         <= -e*m_M*lambda_2(B_W)*E_x
            + sum_i s_i*nu_i*F_i^2/(2*e),

provided nu_i/s_i>=m_M>0. The inequality is Young's inequality followed by ||B_W*x||^2>=2*lambda_2(B_W)*E_x. It reuses the time-varying diffusion owner and signed forcing ledger. Capacity-only events leave E_x unchanged, but other form/support events need their actual jump budget. No moving metric or old fixed-capacity conserved mean is assumed. Pressure realization and stale-input defects remain separate.

The runtime and clock boundaries are observable¶

The capacity synergy controls include an actual default-writer boundary: initialize unit C5 with EPI=0.5, theta_i=0.125+2*pi*i/5, capacity nextafter(1,+infinity) at node zero and capacity one elsewhere. From zero counters, five consecutive fresh-pressure, fresh-Si, adaptation calls reach the default eligibility duration at every node while leaving the capacities bitwise unchanged. The Si minimum exceeds 0.94 and the maximum pressure magnitude is below 5e-17 in this retained binary64 control. It exercises the actual writer, not invented eligibility. It is a finite writer control with phase/form held between calls, not a complete-runtime stability result. The configured positive mixing factor therefore cannot be used as a uniform realized contraction rate. The separate P2 UM lattice certificate concerns a different arithmetic map.

Physical event timing also remains essential. Even ideal full averaging can have a positive capacity profile kappa*1+epsilon*r^k*v, 0<r<1, for a nonuniform eigenvector v and sufficiently small epsilon. With held interval durations h_k, its accumulated detuning is proportional to sum_k h_k*r^k. Bounded durations make this sum finite; h_k=r^(-k) makes it diverge despite convergence by call index. This reuses Section 23's clock distinction and the response-action condition of Section 43. A combined bound must carry eligibility, physical timing and represented defects together, and distinguish exact-model recovery from a numerically resolved tube.

45. A capacity interval gives a prospective winding-retention tube¶

Scope and the bootstrap from the actual event endpoint¶

Retain the fixed weighted C5 and supplied sine phase law, but now let every capacity vary within one fixed positive interval [a,b]. The chosen execution subclass uses only existing guarded snapshot adaptation events; initialize inside the interval. Its represented interval-preservation property applies to every eligible subset, including updates rounded to zero. Interpreting these held represented values as exact coefficients gives the conditional continuous model below. The mathematical bound also covers arbitrary bounded measurable capacity schedules inside this same interval:

theta_dot = nu(t) + K*J(theta),
x_dot = diag(nu(t))*(-e*L_W*x + w*g(theta) - (f/2)*L_C*nu(t)),
0 < a <= nu_i(t) <= b,  K>0, e>0, w>=0, f>=0.

The support, actual positive reciprocal transport weights and pressure coefficients remain fixed. L_C is the unweighted support Laplacian; L_W is the actual weighted random-walk Laplacian. Topology pressure vanishes on this degree-two support. No Gamma source, clipping, form/phase jump or additional operator is admitted. A binary64 phase/EPI trajectory is separate evidence, even when its capacity values satisfy this interval premise.

The original endpoint has ||u(0)||_2 approximately 0.37161, exceeding pi/2-2*pi/5, approximately 0.31416. Thus the sufficient ball test of Section 44 cannot admit it even at zero capacity spread. This is a limitation of that norm bound, not evidence that the initial acute geometry is invalid. Its actual largest absolute gap is approximately 1.40835, below pi/2.

The centered-shape and alignment-potential owners supply a sharper bootstrap. Compare h with the constant-capacity sine reference h_0 having exactly the same initial phase. The reference needs no new simulation: Section 32 already proves that its gaps stay within their initial extrema. Let rho_0 bound those extrema and choose a prospective rho_star below both pi/2 and the effective U3 gate. On this convex fixed-winding acute lift chart,

V(h) = sum_i [1-cos(delta_bar+h_(i+1)-h_i)],
Hess V >= cos(rho_star)*L_C,
z = h-h_0,
z_dot = Pi*nu - (K/2)*(grad V(h)-grad V(h_0)),
Pi = I-11^T/n.

The interval variance bound gives ||Pi*nu||_2<=sqrt(n)*(b-a)/2. Strong monotonicity on the centered subspace therefore yields, almost everywhere and then by the usual norm comparison,

gamma = K*cos(rho_star)*lambda_2(L_C)/2,
||z(t)||_2 <= sqrt(n)*(b-a)*(1-exp(-gamma*t))/(2*gamma),
|(delta_i-delta_0_i)(t)| <= sqrt(n/2)*(b-a)/gamma.

The notation delta_0 here denotes the evolving reference gap, not a fixed initial gap. Consequently the strict prospective condition

rho_0 + sqrt(n/2)*(b-a)/gamma < rho_star

closes the chart bootstrap from the original event endpoint. Winding and the fixed U3 gate remain admitted for all times in this exact continuous model. The comparison uses the selected constant-K sine law; it does not extend automatically to every state-dependent mobility in Section 43. The radius is a proof domain, not a dynamical parameter or a controller.

Phase deformation, source and form-contrast bounds¶

Write R=b-a, B=sqrt(n)*R and Q=||u(0)||_2. Once the reference comparison admits the chart, Section 44 gives

q(t) = exp(-gamma*t)*Q + (1-exp(-gamma*t))*B/gamma,
||u(t)||_2 <= q(t),
||g(t)||_2 <= q(t)/pi,
||w*g(t)-(f/2)*L_C*nu(t)||_2 <= w*q(t)/pi + f*B.

The nonzero bound B/gamma is a residual tube, not an assertion that an actual trajectory attains it. It does not establish exact return to the regular twist. The accumulated detuning bound over a finite physical horizon T is B*T; for R>0 this estimate supplies no finite infinite-time integral. No mixing rate is inferred from call counters or the configured factor mu.

For form let E=x^T*B_W*x/2, s_max=max s_i, and use the existing exact positive lower bound lambda_W for lambda_2(B_W). With Q_star=max(Q,B/gamma), the common Dirichlet estimate gives

F_star = w*Q_star/pi + f*B,
r = e*(a/s_max)*lambda_W,
d = s_max*b*F_star^2/(2*e),
E(t) <= exp(-r*t)*E(0) + (1-exp(-r*t))*d/r.

Thus EPI contrast is bounded; for example squared distance to the ordinary constant subspace is at most 2*E(t)/lambda_W. This does not control the uniform form mode or preserve the former fixed-capacity weighted mean. The constant source bound is conservative and need not vanish even when the actual source decays. Inactivity of finite EPI clipping for all times, limiting mean, exact recovery and complete-runtime maintenance remain open.

The existing cycle owner exposes bound_cycle_capacity_forcing. It reuses the retained envelope as initial phase/EPI/support/weight data and explicitly replaces its held-capacity law by the declared interval premise. The report uses shared exact pi, square root, exponential and spectral owners. A failed margin returns nonadmission, not a negative stability conclusion. No graph write, extra trajectory or numerical solver is performed. The arithmetic controls check both increasing and decreasing bound branches against independent high-precision expressions, zero spread, failed admission and invalid inputs.

One frozen execution with the actual default gate¶

The finite writer study retains the actual post-UM weighted cycle. Before perturbation, it declares the full default pressure mixture, replacing the earlier phase/EPI-only experimental recipe so the capacity channel remains active. This change preserves the support and phase/EPI endpoint; the new mixture has its own fresh capture and prospective budget. The preparation adds 2^-12 to node zero's capacity; all other capacities remain one. This is a declared initial perturbation, not spontaneous capacity creation.

The frozen execution uses K=1/2, dt=1/8, 32 intervals and horizon T=4. At every declared endpoint, after the shared phase proposal and nodal integration, the caller refreshes pressure, computes fresh Si, then invokes the actual adaptation writer. Thresholds, duration and mixing factor remain at their defaults. The prospective interval is [1,1+2^-12] and proof radius rho_star=3/2; the reference-comparison condition admits this geometry. There is no alternate solver, gate tuning or fitted sustaining force.

The observed writer admits only node three during this horizon. Its neighbors already have capacity one, so every admitted update is an identity and the initial capacity difference persists. High Si alone does not admit the other nodes: their fresh pressure fails the stability gate. Nevertheless winding one remains observed and phase deformation decreases. This is useful partial eligibility evidence, not numerical or physical capacity consensus.

The source-bound control uses seed 17. The rational tube has displayed upper radius 1.433137, strictly below 1.5; its reference-gap allowance is approximately 0.0247903. All 33 observed endpoints satisfy their prospective comparisons with the declared 2e-13 observation tolerance. At T=4:

Quantity	Observed endpoint	Prospective continuous-model upper bound
Gap-deformation norm	0.12223481	0.35128774
Actual-conductance EPI Dirichlet energy	0.01487817	0.14148314

The actual-strength weighted form mean changes from 0.125 to approximately 0.12726777. This is a finite measured drift, not an infinite-time growth theorem. Maximum observed Euler and pressure-assembly defects are below 1.4e-17 and 7.1e-18; the independent 80-digit estimate of local phase-step arithmetic error is below 5.9e-16. These local observations do not bound global numerical error or prove a future chart margin.

The finite ledger retains the actual mask, counters and physical time, plus the represented averaging defect. With exact rationals of binary64 inputs,

nu_hat = nu - (mu/2)*M*L_C*nu,
epsilon = nu_after-nu_hat,
E_nu_after-E_nu_before
 = (E_nu_hat-E_nu_before)
   + epsilon^T*L_C*nu_hat + epsilon^T*L_C*epsilon/2.

This separates ideal masked dissipation from arithmetic effects without assuming a uniform numerical contraction. The retained one-ulp stalling control supplies the complementary rounding boundary. Fresh pressure work, capacity-channel work, Euler state defects and finite field values use the existing observation owners. Observed endpoints inside the prospective envelopes do not prove binary64 convergence or validate unobserved times.

This closes the bounded capacity-forcing admission: a conditional robust phase/winding and form-contrast tube, with actual gate limitations retained. Exact recovery does not follow. Section 46 addresses the separate absolute-form/clipping obligation.

46. Absolute form: a finite bound and an interval-law obstruction¶

Fixed observation weights separate mean drift from coordinate changes¶

Retain Section 45's fixed conductance W, strengths s and S=sum s_i. For the actual-strength mean m_D=s^T*x/S, the nodal row gives

m_D_dot = [-e*nu^T*B_W*x + (s*nu)^T*F]/S,
F = w*g - (f/2)*L_C*nu.

The first term need not vanish with heterogeneous capacity. This mean uses fixed observation weights and is well defined for measurable capacities; it needs no capacity derivative. Capacity-only events leave m_D unchanged. The read-only observe_forcing_mean_balance in the forcing owner contracts the same validated channel data as the Dirichlet observer with covector s_i*nu_i/S. It retains signed diffusion, separate channel work, fresh assembly defects and stale stored-pressure work. It does not subtract the observed mean or install a balancing force.

For one retained Euler interval of duration dt and shared state defect r_x,

m_D_after-m_D_before
 = dt*(modeled_mean_rate + kernel_defect_rate + stored_residual_rate)
   + s^T*r_x/S.

The existing 32-interval control verifies this exact rational identity and its telescope, using the same execution rather than a new producer. Its observed weighted-mean shift remains approximately 0.00226777. This is a finite source/transport budget, not an asymptotic drift prediction.

The alternative moving metric H(t)=diag(s_i/nu_i(t)) has different duties. If capacities are differentiable, h=s/nu, Z=sum h_i and m_H=h^T*x/Z give m_H_dot=s^T*F/Z+h_dot^T*(x-m_H*1)/Z. At a capacity jump its coordinate reweights even at unchanged EPI. That jump is already owned by observe_forced_support_reset, which reports mean_reweighting separately from form evolution. No derivative of a merely measurable capacity schedule is assumed by the new fixed-weight observer or interval theorem.

Prescribed capacity changes distinguish form charge from its mean¶

The moving-weight identity has a useful consequence for the complete normalized-sine law. Keep fixed finite connected simple unit support with positive degrees, fixed e>=0, w,beta>0, no Gamma or other forcing, and both prescribed sine rows. Let capacities be strictly positive C^1 time schedules, supplied independently of form and phase. Set rho_i=d_i/nu_i(t), W=sum_i rho_i, Q=rho^T*x and M=Q/W. Reciprocity cancels the instantaneous source and diffusion contributions even while these weights vary, giving

\[ \dot Q=\dot\rho^{\mathsf T}x,\qquad \dot M=\frac{\dot\rho^{\mathsf T}(x-M\mathbf1)}W. \]

For this prescribed schedule, dot M=0 at every form state is equivalent to dot rho=(dot W/W)*rho. Integrating on a connected time interval gives rho(t)=rho(0)/a(t), or

\[ \nu_i(t)=a(t)\nu_i(0),\qquad a(t)>0,\quad a(0)=1. \]

Thus a common proportional capacity schedule is necessary and sufficient for universal conservation of this normalized mean under the stated law. For raw Q, universal conservation instead requires dot rho=0. These necessities concern prescribed state-independent schedules. A state-dependent capacity law can make dot rho orthogonal to the particular centered form without being proportional; that different admission problem is not resolved here.

The common factor multiplies both consumed sine rows, so tau(t)=integral_0^t a(s) ds gives the existing fixed-capacity law in the activity clock. Along the corresponding solution, Q(t)=Q(0)/a(t) and W(t)=W(0)/a(t), while M stays constant. This is consistent with the whole-law clock contract, not a clock inferred from phase or a derived capacity mechanism. An unbounded activity horizon is additionally needed to transfer an entire fixed-capacity asymptotic limit. The held-capacity form-charge theorem and this variable-capacity accounting therefore have distinct hypotheses.

A no-reset support event must retain its changed weights¶

For two supplied snapshots on the same ordered node set, assume all old and new degrees and capacities are positive and x^+=x^-=x. Define prescribed state-independent weights rho_i^\pm=d_i^\pm/nu_i^\pm, with W^\pm=sum_i rho_i^\pm, Q^\pm=(rho^\pm)^T*x and M^\pm=Q^\pm/W^\pm. Direct subtraction gives

\[ M^+-M^-= \frac{(\rho^+-\rho^-)^{\mathsf T}(x-M^-\mathbf1)}{W^+}. \]

Mean preservation for every form state is therefore equivalent to rho^+/W^+=rho^-/W^-, or rho^+=c*rho^- for one positive scalar c. For example, a supplied update nu_i^+=(d_i^+/d_i^-)*nu_i^-/c has that property; it is one compatible event prescription, not a cause of support change or a selected capacity law. Raw charge is preserved for all forms only when rho^+=rho^-.

Common-origin covariance makes the distinction unavoidable. Replacing x by x+h*1 changes Q^+-Q^- by h*(W^+-W^-), whereas it leaves M^+-M^- unchanged. Thus requiring raw-charge conservation across an event with changing W requires a fixed form-origin convention unless additional state or event terms transform with it. This is separate from the support's Dirichlet-storage jump, which depends on form differences.

The existing observe_forced_support_reset already reports the exact same-EPI mean_reweighting and independent support-storage change; its detached snapshots do not assert that an event occurred. The static composition controls reuse that reader for a supplied P3-to-K3 comparison and check the complete sine rows under common capacity scaling. No new runtime, support event, capacity law or frozen trajectory is introduced.

The complete scalar field has a finite-prefix enclosure¶

The positive-transport maximum principle works directly with Section 45's source envelope. At a maximal form coordinate diffusion contributes a nonpositive rate; at a minimum it contributes a nonnegative rate. Consequently, with nu_max=b and A(T)=integral_0^T ||F(t)||_2 dt,

min x(0) - b*A(T) <= x_i(t) <= max x(0) + b*A(T),  0<=t<=T.

This bound controls the whole physical prefix, including times between the displayed samples. Using q_tail=B/gamma, Section 45 supplies

I_q(T) = q_tail*T + (Q-q_tail)*(1-exp(-gamma*T))/gamma,
A(T) <= (w/pi)*I_q(T) + f*B*T.

The cycle sample now exposes gap_deviation_integral_upper, non_epi_source_integral_upper and epi_interval. Shared exponential enclosures handle either sign of Q-q_tail; no fitted rate or new solver is introduced. If this entire interval lies strictly inside declared scalar rails, the exact continuous model does not require clipping on that prefix. Numerical execution still retains its own endpoint and pressure defects. For the retained full-mixture study, the computed T=4 interval lies inside the outward rational band [-0.225,0.475], strictly inside its [-1,1] rails. This certifies finite-prefix inactivity in the declared continuous model; the separately observed binary64 endpoints also lie within the band. The zero-source control preserves the initial extrema. With a persistent nonzero source bound this enclosure generally expands with T, so it alone does not imply an all-time scalar bound.

A locked phase can still drive the uniform EPI mode¶

An explicit exact-model member of the interval family rules out that stronger implication. This is a counterexample to the sufficiency of interval premises, not a proposed mechanism or a prediction of the default adaptation schedule. It retains the actual C5 transport: four unit edges and closing weight beta. Put v=(1,0,0,0,1), so s=2*1+(beta-1)*v, and choose

d = 2*pi/5,
h_* = alpha*(v-(2/5)*1),
delta_* = (d-alpha, d, d, d+alpha, d),
nu_* = kappa*1-K*J(h_*).

Here alpha is a declared small geometric displacement, not a response gain. For abs(alpha)<rho_star-d, the phase lock stays inside the same acute/U3 chart, since rho_star is below both limits. Since sine is 1-Lipschitz, abs(nu_*_i-kappa)<=K*abs(alpha)/2. Choose kappa at the capacity interval's midpoint and K*abs(alpha)<=b-a; all capacities stay inside the prescribed positive interval. The supplied phase law then gives theta_dot=kappa*1 exactly. No pressure coefficient or supporting force is fitted.

For the complete configured phase/capacity source, direct contraction gives

F_* = -(w/(2*pi))*L_C*h_* - (f/2)*L_C*nu_*,
s^T*F_* = -(beta-1)*[w*alpha/pi + f*K*cos(d)*sin(alpha)].

Choose alpha with the opposite sign to beta-1. For beta unequal to one and either positive w or positive f, this expression is strictly positive. The retained closing weight is below one, and its full default mixture has both channels positive. Thus a positive arbitrarily small alpha suffices. For unit closing weight this particular obstruction vanishes, retaining the support/transport distinction rather than discarding actual weights.

The existing compatibility and forced-support theorem in Sections 2-3 now applies with these fixed capacities and locked source:

Z = sum_i s_i/nu_*_i,
c = (s^T*F_*)/Z > 0,
m_H(t) = m_H(0)+c*t.

EPI contrast relaxes toward its bounded relative profile while every form coordinate acquires the same secular drift. For positive phase weight an explicit lower bound avoids any approximate sine evaluation:

c >= a*abs(beta-1)*w*abs(alpha)/(pi_upper*(8+2*beta)) > 0.

The lower capacity endpoint here is a, distinct from the displacement alpha. If initial EPI is uniformly x_0 and the phase starts at this lock, any finite upper rail U must be reached by the ideal unclipped solution no later than (U-x_0)/c_lower (strict crossing for larger times). Clipping changes the equation; an endpoint pinned to a rail is not a zero-pressure equilibrium.

The obstruction is not confined to initialization exactly at the lock. Choose the displacement also inside abs(alpha)<rho_star-d, freeze the same nu_ and start at the original retained phase endpoint. Section 45 admits the same interval tube. The centered phase equation is the gradient flow of the sine alignment potential plus its fixed linear tilt. Strong convexity on that admitted chart makes h(t)-h_* decay exponentially. Hence F(t)-F_* is integrable and m_H(t)=c*t+O(1). This extends the failure of a uniform all-time bound to the retained phase preparation in the exact interval class. It does not assert that the invoked default capacity writer keeps nu_ fixed forever.

The static obstruction control reuses the retained support with alpha=2^-12, kappa=1+alpha/2, K=1/2, and the full default mixture. Independent high-precision contractions verify the locked response, interval inclusion and positive source expression; separate comparisons retain errors of represented pressure and phase proposals. The sign proof above is exact, not inferred from those finite numerical tolerances. No long trajectory, new lock classification or automatic change of the constitutive law is needed.

This closes the absolute-form obligation with its proper scope: finite-prefix scalar control is available, but interval-bounded capacity and a stable phase geometry alone do not guarantee indefinite absolute EPI retention. The actual adaptation policy's first effect on this source compatibility is examined below; its repeated effect remains subject to the single execution plan.

47. Actual capacity feedback reduces the locked source without cancelling it¶

One signed balance connects capacity averaging to phase readjustment¶

Retain Section 46's ordered weighted C5, full pressure mixture and acute two-neighbor phase chart. Write L=L_C, L_U=L/2 and let M be the diagonal mask of eligible nodes. The ideal snapshot average and its represented writer defect r satisfy

delta_nu = -mu*M*L_U*nu + r,
b = s^T*F,
delta_b = -f*s^T*L_U*delta_nu
        = mu*f*(L_U*s)^T*M*L_U*nu - f*s^T*L_U*r.

This is an event identity at unchanged EPI, phase, support and weights. The phase source does not jump. Symmetry of the unweighted support gives the second equality; actual EPI strengths s still carry the unequal closing conductance. The signed cross term is different from the nonnegative capacity-energy dissipation in Section 44. Capacity smoothing alone therefore does not establish decreasing abs(b).

For the exact locked family of Section 46, a full mask and a positive blend, put t=beta-1<0, d=2*pi/5, C=cos(d)*sin(alpha)>0. Then

delta_nu = (mu*K/2)*L*J,
s^T*L^2*J = -5*t*C,
delta_b = 5*mu*f*K*t*C/4 < 0,
b_after = -t*[w*alpha/pi + f*K*C*(1-5*mu/4)].

These are exact-model expressions, before the represented writer defect. For the mathematical blend mu=1/10, the capacity contribution is multiplied by 7/8; with the exact rational represented by binary64 0.1, this factor is 7/8-2^-57. In both cases the full source remains positive. The phase contribution is unchanged by the capacity-only event.

The same event changes the supplied phase velocity from kappa*1 to kappa*1+delta_nu. Subtract its uniform component to obtain the new lock mismatch. It generally leaves the one-parameter locked family; reducing alpha is not a justified description of the new state. If the new capacity is then held while phase evolves, the acute midpoint identity gives

g_dot = -(1/pi)*L_U*theta_dot,
b_dot_phase(0+) = -(w/pi)*s^T*L_U*delta_nu
               = [w/(pi*f)]*delta_b,                 f>0,
               = 5*mu*w*K*t*C/(4*pi) < 0             (ideal full mask).

The first equality reuses the existing local phasor derivative, not a new phase law. For an imperfect represented initial lock, the displayed event formula is the change of its instantaneous phase-source derivative; the retained pre-event derivative must be added. The compatibility jump and the subsequent derivative have different units and are not two immediate source reductions. The negative tangent proves neither future monotonicity nor a finite accumulated source. It does identify a shared signed geometric contraction for the two feedback paths.

Fresh default admission gives a nontrivial first write¶

The capacity synergy controls centralize the Section 46 lock preparation in one fixture. It retains the actual post-UM weighted support (seed 17), declares the full default pressure mixture, initializes alpha=2^-12, kappa=1+alpha/2, K=1/2, and starts stability counters at zero. The bounded control executes exactly five shared phase/EPI intervals of dt=1/8. At every endpoint it refreshes full pressure and then Si before calling the unchanged capacity writer. This is the same declared composition as Section 45, on a different, explicitly prepared initial state; it is not the complete native scheduler.

All five nodes pass the fresh gates on each call: the largest absolute pressure decreases from approximately 3.005e-5 to 2.683e-5, below the default 0.001, and the smallest observed Si exceeds 0.9416, above 0.5. No capacity changes during the first four calls. All nodes become eligible at the default tau=5, physical time 5/8, with the represented default blend mu=0.1. No gate, pressure weight, state amplitude or horizon is tuned after inspecting the response.

At the resulting capacity-only event, the retained observations give:

Readout	Before	After / change
Source compatibility `s^T*F`	`1.37059588934e-5`	`1.36443423539e-5`, still positive
Frozen reversible-mean coefficient `c=b/sum(s/nu)`	`1.43509598918e-6`	`1.42864449498e-6`
Maximum absolute centered phase-rate mismatch	approximately `1.11e-16`	`2.83020e-6`
Actual fixed-strength EPI mean `m_D`	retained	exactly unchanged by the event
Reversible-metric EPI mean `m_H`	retained	reweighting of approximately `-3.40878e-11`
Maximum absolute capacity writer defect	—	approximately `1.78e-16`

The comparison uses the existing observe_forcing_capacity_difference and observe_forcing_mean_balance; it adds no production observer, integrator or controller. Exact rational contractions separate the source change from fresh pressure-assembly and stale stored-pressure differences. The capacity write does not refresh stored pressure; a detached fresh capture supplies the post-event comparison. Shared Euler residuals separately account for the preceding form evolution. EPI and phase are exactly unchanged across the capacity event, so the jump of the moving metric is not physical form motion. The frozen coefficient c=b/sum(s/nu) is a separate held-model quantity, not the instantaneous derivative of the fixed-strength mean. The signed contraction of the post-event pressure-assembly defect is about -8.87e-21; the stale stored-pressure residual is instead approximately +6.16165e-8. Ignoring freshness would hide almost the entire source change.

Applying the ideal midpoint Jacobian to the retained represented phase-rate coefficients gives a numerator pi*b_dot_phase changing from approximately -2.12e-17 to -8.01705e-7. This differentiates the declared local model, not the binary64 pressure program. Exact signed contractions and independent 110-digit evaluation check the analytic expressions above. One unapplied proposal by the shared phase owner also verifies the newly induced gap displacement; it does not extend the executed horizon or certify a new lock. These local comparisons retain finite rounding errors rather than promoting the binary64 state to an exact analytic lock.

A smaller capacity contrast is not a source-compatibility Lyapunov law¶

The existing difference controls include a complementary exact detached map. Take the same C5 support with closing weight beta=3/4, uniform phase, uniform EPI and nu=(1,1,1+R,1,1), R=2^-12. Declare a full dyadic blend mu=1/2 and pressure weights (w,e,f,topo)=(1/2,1/4,1/4,0). Its capacity range halves and its unweighted Dirichlet energy falls from R^2 to R^2/8, but

b_before = 0,
b_after = mu*f*(beta-1)*R/2 = -mu*f*R/8 < 0.

Thus abs(b) increases while both smoothing readouts improve. Unit closing weight gives zero and beta=5/4 reverses the sign with the identical capacity map. These controls are detached algebra, not observed default eligibility, locked-phase states or alternative settings for the five-step experiment. A separate masked contraction checks the same signed balance with the captured transport strengths.

This closes the first-feedback gate: the implemented default writer is admitted and weakens this specific positive source; its phase readjustment initially acts in the same direction. It does not cancel the source in one event. Recurrent admission, the accumulated source under repeated updates, binary64 residuals and indefinite absolute-form retention remain unproved. These are mechanism conditions, not evidence of physical particle emergence.