Conservative source preparation and finite retention

Energy-speed retention, reverse-source certification, sign/reflection saddle reduction, nonlinear corridors, source-to-target construction and its finite operational and constitutive limitations.

Part of Sine pattern geometry and dissipative capture. Section numbers are stable across the collection; hypotheses and model changes remain local to each result.

46. A conserved budget bounds finite relative-phase travel

The full storage need not lie below the acute-face barrier to give a useful finite retention window. A trajectory has limited phase speed while it remains in an acute winding sector. This supplies a target certificate that does not require proximity to an exactly rigid family or a small contact pulse.

Keep the complete conservative unit-capacity law with e=0, w=beta=1, fixed connected simple undirected unit support, no forcing or events, and tau=t/pi. Select a cycle on five distinct nodes and retain every other node and edge in the full law. Extra cycle chords are permitted: they add nonnegative storage and contribute to the actual degrees. Let L_full be the full Laplacian and K=diag(1/d_i). The full form storage and the minimum acute unit-winding phase storage on the five selected edges are

\[ F=\frac12x^{\mathsf T}L_{\rm full}x,\qquad V_5=5\left(1-\cos\frac{2\pi}{5}\right) =\frac{25-5\sqrt5}{4}. \]

An edge-speed bound from the actual full form storage

For a selected edge oriented i->j, put b_e=e_j-e_i and a_e=K b_e. Its lifted phase gap satisfies

\[ \delta_e'=b_e^{\mathsf T}KL_{\rm full}x =a_e^{\mathsf T}L_{\rm full}x. \]

Cauchy–Schwarz for the positive semidefinite Laplacian gives

\[ |\delta_e'|^2\le (a_e^{\mathsf T}L_{\rm full}a_e) (x^{\mathsf T}L_{\rm full}x) =2\kappa_e F,\qquad \kappa_e=\frac1{d_i}+\frac1{d_j}+\frac2{d_i d_j}. \]

The coefficient is a_e^T L_full a_e, computed from the full degrees and the selected connecting edge. On the C5/private-leaf support of Sections 35–45, receiver degrees are three, so every kappa_e=8/9 and the bound is |delta_e'|^2<=16F/9. Environmental form is retained in F; replacing it by receiver form storage would invalidate this estimate.

While every receiver gap lies in (-pi/2,pi/2) and their winding is sigma=+1 or -1, convexity of 1-cos(delta) on that interval and sum(delta)=2pi*sigma give V_R>=V_5. Every other edge's phase storage is nonnegative. For a conserved full-storage upper bound H_bar, therefore,

\[ F\le H_{\rm bar}-V_5,\qquad \boxed{|\delta_e'|\le c_e :=\sqrt{2\kappa_e(H_{\rm bar}-V_5)}.} \]

A nonempty admitted acute target necessarily has H_bar>=V_5. A smaller bound cannot be repaired by clipping the square-root argument to zero. Write c_H=max_e c_e; for the private-leaf support this is 4sqrt(H_bar-V_5)/3.

First-exit certificate, including full-state uncertainty

Suppose every state in a declared full-state target has the same acute unit winding, initial acute margin at least m>0, and full storage at most H_bar. For any supplied finite T>0, the strict inequality

\[ \boxed{m-c_HT>0} \]

certifies that every such trajectory remains acute with its original winding throughout [-T,T], with margin at least m-c_H T. Indeed, up to a proposed first boundary exit, the speed estimate bounds each gap's displacement by c_H times elapsed time. Reaching a face by time T would require a displacement at least m, contradicting the strict bound. The same argument applies backwards. This is a first-exit bootstrap, not an assumption that the trajectory remains acute after the initial observation.

At a uniform unit twist on the private-leaf support, m=pi/10. For T=1, the admissible storage ceiling is consequently

\[ H_{\rm bar}<V_5+\frac{9\pi^2}{1600} =3.51043155\ldots\;>\;\frac72. \]

Thus the finite retention condition has a nonempty overlap with the storage budgets necessary for acquisition from zero winding. For example, at H_bar=7/2 the strict duration ceiling is 3pi/(40sqrt(7/2-V_5))=1.10967355.... This does not establish acquisition at the barrier or select any trajectory reaching the target.

For a general independent box around a supplied full state (x*,theta*), with form radii r_x and lifted-phase radii r_theta, one valid upper bound is

\[ \begin{aligned} H_{\rm bar}={}&H(x^*,\theta^*) +\sum_i|(L_{\rm full}x^*)_i|r_{x,i} +\sum_i|S_i(\theta^*)|r_{\theta,i}\\ &+\frac12\sum_{\{i,j\}\in E} \left[(r_{x,i}+r_{x,j})^2 +(r_{\theta,i}+r_{\theta,j})^2\right]. \end{aligned} \]

This follows from the exact form quadratic and the global phase remainder bound cos(delta)<=1. It retains nominal torques at represented phase centers; approximating a critical twist numerically does not permit declaring those torques zero. The initial acute margin must likewise account for the sum of both endpoint phase radii on every selected edge.

An explicit target with uncertainty in all twenty coordinates

On the C5/private-leaf support, let L_R be the receiver cycle Laplacian and take the nonzero centered form perturbation

\[ q=\kappa(1,-1,0,0,0),\qquad \kappa=\frac1{128}. \]

For receiver indices i=0,...,4 and their matched leaves, choose centers

\[ x_R^*=\frac u4\mathbf1+q,\qquad x_Q^*=-\frac{3u}{4}\mathbf1+(I_5+L_R)q,\qquad \theta_{R,i}^*=\theta_{Q,i}^*=\frac{2\pi(i-2)}5, \qquad u=\frac7{50}. \]

Allow independent errors of absolute size at most epsilon=1/4096 in every fine form and lifted phase coordinate. The capacities and support remain fixed premises. The initial winding is one throughout this box and the initial acute margin satisfies

\[ m\ge\frac\pi{10}-2\epsilon. \]

The center has contact form contrasts r=u*1-L_R*q, which are all positive, and form storage 5u^2/2+13kappa^2. Its cycle form differences have total absolute magnitude 4kappa, while its contact contrasts sum to 5u. Every edge form difference acquires an error of at most 2epsilon. The receiver phase errors telescope around the cycle: their edge increments sum to zero. Taylor's bound 1-cos(alpha+eta)<=1-cos(alpha)+sin(alpha)eta+eta^2/2 therefore cancels the cycle's linear term. Contact phase gaps are at most 2epsilon. Together these give the full, correlated bounds

\[ V_5+\frac52(u-2\epsilon)^2\le H\le H_{\rm bar}:=V_5+\frac52u^2+13\kappa^2 +10u\epsilon+8\kappa\epsilon+40\epsilon^2. \]

The lower bound also uses the already verified acute unit-winding geometry, which gives V_R>=V_5, and Jensen's inequality for the five contact form contrasts, whose mean lies within 2epsilon of u. It does not discard a possibly negative phase term. For these declared rational values,

\[ \frac52u^2+13\kappa^2+10u\epsilon+8\kappa\epsilon+40\epsilon^2 =\frac{13147281}{262144000},\qquad \frac52(u-2\epsilon)^2=\frac{51022449}{1048576000}. \]

All target states have H>7/2; the enclosing full-storage interval is approximately [3.50357382,3.50506793]. At T=1, the retained acute margin is greater than 1/100 radians. This strict rational conclusion follows already from pi>157/50 and c_H<3/10; the sharper evaluated lower margin is about 0.0150730 radians. The box has positive width in every form and phase coordinate. Its center is already nonrigid: Section 45 gives zero first and second relative phase derivatives but a nonzero third derivative for this compensated form preparation. The retained identity therefore includes actual changing geometry, not only errors around an exactly moving collective family.

What this closes, and what it leaves open

This provides a full-state retention target for scaled duration one (original-clock duration pi) whose entire storage range is above the zero-winding acquisition barrier. It avoids requiring entry into a particular moving-family window. It does not supply a zero-winding preparation whose actual trajectory enters this target, and overlapping source and target budgets do not prove that connection.

Because the certificate also protects the preceding duration one, an identity-absent preparation must precede that backward window. An entry checkpoint is not necessarily the first instant at which acute identity appears. Longer or indefinite retention, an attractive basin, autonomous preparation and physical identification remain outside this certificate.

The shared assess_sine_cycle_retention reader applies this argument to actual admitted centers, full-coordinate error budgets and graph degrees. Its contract distinguishes the exact source set from displayed interval bounds. The rational example retains the actual small phase torques of a near-twist center; it does not replace that center with the symbolic geometry used in the proof above.

A directed preparation for a possible reverse-time check

One directed checkpoint center selects phase travel toward the sector saddle in reversed time. On the same private-leaf support, set

\[ \theta_{R,i}=\theta_{Q,i}=\frac{2\pi(i-2)}5,\qquad x_R=\frac1{32}(-4,-4,0,4,4),\qquad x_Q=\frac1{32}(-6,-5,0,5,6). \]

Its receiver and contact form storages are 48/1024 and 5/1024. Consequently H=V_5+53/1024=3.50667284..., with strict duration-one acute reserve pi/10-sqrt(53)/24>0. The actual full rows give

\[ \theta_R'=\theta_Q'=\frac1{32}(-2,-1,0,1,2),\qquad \phi'=0. \]

Under the exact reversing involution R(x,theta)=(-x,theta), the vector field obeys

\[ \mathcal F(Rz)=-R\mathcal F(z). \]

Reversing this checkpoint therefore gives initial oriented cycle-gap rates (-1,-1,-1,-1,4)/32: the four consecutive gaps decrease from 2pi/5 toward pi/3, while the closing principal gap increases toward 2pi/3. These rates select the direction of the sector saddle geometrically. They do not establish its crossing, later zero winding, or constancy of that initial direction. The factor 1/32 is a supplied experimental amplitude, not a coefficient derived for the constitutive law.

The strict retention margin permits a nonzero-width full-state checkpoint box, admitted with the general box-storage estimate above. Any finite reverse check must declare that box, its horizon and numerical budget before evaluation. If a later validated check proves that the whole backward image of an open checkpoint box B has zero winding at time -T_back, the corresponding source is the exact flow image Phi_{-T_back}(B). Its openness follows from the smooth invertible flow, and its forward trajectories return to B. An outer rectangular enclosure of that image is not interchangeable with the source: an independently supplied rectangular source requires its own forward-containment proof. Neither construction derives spontaneous preparation. Section 47 specifies the represented-point evaluation and its evidence obligations; this symbolic recipe alone leaves formation open. The execution plan owns any prospective check.

47. A reversible enclosure can certify an independent formation source

An open backward image in Section 46 need not be a rectangular preparation that can be specified independently. The complete law also supplies a global error bound that can bridge this distinction. The resulting theorem requires actual validated endpoint evidence and tests the whole proposed source; it does not infer formation from reversibility or storage overlap.

Complete state, clock and global error bound

Keep the same conservative unit-capacity law and fixed full support. In this section Phi_A denotes the flow for original-clock duration A. Only the twenty form and lifted-phase coordinates vary on the private-leaf support. Capacity is held at one and is not a perturbation coordinate. For two full states z=(x,theta) and z_tilde, use the common sup norm of all form and phase differences.

In the scaled clock, the form row is a normalized sum of neighbor sines, and the phase row is K L_full x. Since sine is globally Lipschitz with constant one and ||K L_full||_infinity<=2,

\[ \|\mathcal F_\tau(z)-\mathcal F_\tau(\widetilde z)\|_\infty \le2\|z-\widetilde z\|_\infty. \]

Therefore the complete original-clock flow obeys

\[ \boxed{\|\Phi_A(z)-\Phi_A(\widetilde z)\|_\infty \le e^{2|A|/\pi}\|z-\widetilde z\|_\infty.} \]

This comparison uses continuous phase lifts. It does not replace a lifted distance by independently wrapped endpoint differences. The global bound retains all environmental coordinates and is independent of the trajectory's intermediate winding sector.

From a reverse endpoint to a forward source ball

Declare an exact target center z_* and an independent box B_epsilon(z_*) already certified by Section 46 to retain acute unit winding for scaled duration T. The reversal R(x,theta)=(-x,theta) is an isometry in the same norm and satisfies

\[ \Phi_A(Rz_*)=R\Phi_{-A}(z_*). \]

Suppose a validated positive-time forecast from Rz_* supplies an outward endpoint enclosure I_A containing Phi_A(Rz_*). Let m_A be its exact componentwise midpoint and eta_A the maximum of its form/phase half-widths. Put

\[ c_A=R m_A,\qquad s_A=\Phi_{-A}(z_*). \]

Then ||c_A-s_A||_infinity<=eta_A. For every state in an independently declared source ball B_delta(c_A), the forward error bound gives

\[ \|\Phi_A(z)-z_*\|_\infty \le e^{2A/\pi}(\eta_A+\delta). \]

Consequently the following conditions suffice for formation followed by the declared retention interval:

1. Every point of the full source ball has receiver winding zero, certified with the same node order and unique circular branches on the complete source phase intervals.
2. The target box has the stated full-state retention certificate, including its actual law, support, capacity, phase margins and conserved budget.
3. An outward exponential upper bound E_A>=exp(2A/pi) satisfies the strict inclusion E_A*(eta_A+delta)<epsilon.

Every source member then enters B_epsilon(z_*) at the common original-clock time A and retains acute unit winding for at least the following original-clock duration pi*T. Unlike a mere outer enclosure of a backward flow image, the ball in this theorem has an independent forward-containment proof. It has nonzero width in every form and phase coordinate and does not constrain their conserved origins to one lower-dimensional surface.

assess_sine_reversible_preparation owns this conditional composition and returns SineReversiblePreparation. It rebuilds the target through assess_sine_cycle_retention and retains every grid endpoint's source, error bound, conditions and availability.

The target certificate also holds backwards. A zero-winding preparation must therefore precede that protected interval: require A>pi*T, checked against an outward upper bound for pi*T. Reversal constructs a comparison initial condition; it is not a support event, a sign-flipping operation inserted into the physical evolution, or a source of forcing.

Numerical admission is part of the composition

The forecast must enclose the declared exact initial point Rz_* and retain a valid chain from time zero to the selected endpoint. Read and re-admit primitive state, full support, law, clock, node order and interval evidence before consuming that endpoint. A source identifier, cached success flag or midpoint-only phase observation cannot replace those premises. Structural admission alone does not authenticate a claimed enclosure or replace the numerical method's proof obligations and retained provenance.

Compute midpoint, half-width and source endpoints without float conversion. The source may be declared by exact rational endpoints c_A +/- delta; outward interval arithmetic can enclose these endpoints for admission. If instead the exported source is the enlarged materialized box, its actual maximum radius about c_A must replace delta in the forward error bound. The same rule applies to any enlargement of the endpoint enclosure. The exponential uses an upper exponent, including the lower bound of pi in the denominator; strict inequalities cannot rely on rounded midpoint values or equality accepted as tolerance.

Conservation supplies an additional consistency check: each source member's full storage equals that of its reached target member. Independently computed source and target storage intervals must not contradict this identity. Their overlap alone is not a proof of entry. The forward inclusion, rather than that overlap, establishes the required connection.

A valid prefix is enough for a certificate at its endpoint. A later failure to extend a forecast does not undo an already validated prefix, but the declared full-horizon status must remain separately visible. No failed or unavailable step may be skipped to construct a later endpoint certificate.

Frozen bounded evaluation and its scope

The declared preparation uses the form vectors at the end of Section 46 and the exact represented phase centers

\[ \theta_{R,i}^*=\theta_{Q,i}^* =(i-2)\frac{1256637}{10^6}. \]

These are the actual rational phases consumed by the forecast; they are not declared equal to (i-2)*2pi/5. The target box has epsilon=1/4096 in every form and phase coordinate, with scaled retention duration T=1. Its retention preflight does not establish any backward crossing.

One bounded evaluation is specified in original time by horizon 128/5, step 1/10, Taylor order 16 and at most 256 steps. The independent source radius is fixed at delta=2^-100. Evaluate the predeclared grid endpoints in increasing time, after the backward-window guard, and select the first endpoint satisfying every source and target condition above. This is a deterministic constructive rule fixed before the response, not permission to retune the checkpoint, radius, horizon or solver after observing failure. The selected source is a derived mathematical preparation, not an independently observed natural occurrence or a reserved physical prediction. The nonzero radius 2^-100 gives the source box a nonempty interior; it is not a practically demonstrated laboratory tolerance or a broad basin.

These numerical choices are an evaluation budget, not constants selected by the nodal law. Incomplete coverage without a passing prefix is unavailable; complete coverage without a passing endpoint is no_certificate_on_declared_grid. Neither rules out formation through other preparations. The execution plan remains the sole task queue.

Retained response: the directed checkpoint does not supply a zero-winding source

The single evaluation retains its declaration, frozen protocol, source archive and response. The response has no evaluation error and covers all 256 steps through original time 128/5. Its verdict is no_certificate_on_declared_grid.

Declared obligation	Retained result
Target retention	Certified for T=1; full-box storage upper bound about 3.5068583304, strict retention margin greater than 0.00978
Whole source winding zero	Fails at all 256 endpoints: each exact source ball has winding +1
Strict source-to-target radius inclusion	Passes at all 256 endpoints
Source/target storage compatibility	Passes at all 256 endpoints
Endpoint after the protected backward interval	Passes at the final 225 endpoints
Complete numerical coverage	All 256 steps; no unavailable suffix

At the last endpoint, the enclosure halfwidth is about 4.06837e-30, the amplification upper bound is about 1.19647e7, and the propagated source-to-target error is less than 5.82e-23, well below 1/4096. These decimal values summarize exact retained bounds. Numerical precision does not explain this candidate's failure: the missing property is initial zero winding. The successful inclusion tests connect already wound source balls to the retention target and therefore do not establish formation.

A separate retrospective reading, without solver replay, checks the unique branches of every whole-time tube. All 256 tubes retain winding +1, so the enclosed reverse trajectory never reaches zero winding anywhere on this declared horizon, including between endpoints. For example, the closing principal-gap hull is contained in (1.2561, 1.8298), below the sector saddle's closing gap 2*pi/3. Only 98 tubes certify acuteness throughout their own interval; sector preservation is a different assertion from acute identity retention. This reading does not alter the original grid verdict or establish behavior beyond 128/5.

The same retained tubes and the full linear phase row also bound the closing gap's scaled-time derivative strictly inside (0.0287, 0.1254) throughout the horizon. Its directed motion therefore persists in this finite response; the record does not demonstrate a turn-back, permanent trapping or impossibility of later crossing. A prospective continuation would need its own justified bound and evidence, rather than rewriting the completed horizon.

The local initial gap direction in Section 46 is consequently insufficient to establish travel to the saddle under full contact feedback. This result does not prove a new trapping invariant, nor refute the reversible-source theorem. Sections 48-50 establish nonlinear connections under a separate exact preparation, using coupled phase/contact dynamics. They do not extend this completed response or declare its checkpoint successful.

48. An exact sign/reflection reduction retains the sector saddle

This section uses the same complete zero-loss, unit-capacity sine law on the five-node receiver cycle and its five matched private leaves. Receiver indices are 0,...,4, leaf indices are 5,...,9, and the cycle orientation is 0 -> 1 -> 2 -> 3 -> 4 -> 0. Every contact remains live. With tau=t/pi, the complete rows are

\[ x'=K S(\theta),\qquad \theta'=K Lx, \qquad K=\operatorname{diag}(1/d_i). \]

The reduction below is an exact representation of a declared invariant family of those rows. It does not replace the nonlinear law with its tangent, choose a preparation, or discard a coupled environmental coordinate.

Exact invariant family and reconstruction

Let P reflect receiver indices by i -> 4-i and reflect the matched leaves in the same way. It is an automorphism of the full support and preserves the capacities and degrees. The vector field F obeys both

\[ F(Pz)=P F(z),\qquad F(-z)=-F(z),\qquad z=(x,\theta). \]

The second identity follows from sine oddness and the linear phase row. Thus Tz=-Pz is an involutive equivariance. Uniqueness of the globally smooth full flow preserves Fix(T) at every time. With real phase lifts, its exact coordinates are

\[ \begin{aligned} x_R&=(a,b,0,-b,-a), &x_Q&=(c,d,0,-d,-c),\\ \theta_R&=(u,v,0,-v,-u), &\theta_Q&=(p,q,0,-q,-p). \end{aligned} \]

The fixed receiver and leaf have zero form and phase in this declared origin convention. Their full derivatives vanish by paired cancellation; they remain nodes of the system. No coordinate is an inferred sensor or an automatically wrapped phase. Common form and phase origins can be restored by their exact shift symmetries, but such shifts are a separate declared reconstruction.

Writing X=(a,b,c,d) and Theta=(u,v,p,q), the eight nonlinear rows are

\[ \begin{aligned} 3a'&=\sin(v-u)-\sin(2u)+\sin(p-u),\\ 3b'&=\sin(u-v)-\sin v+\sin(q-v),\\ c'&=\sin(u-p), &d'&=\sin(v-q),\\ 3u'&=4a-b-c, &3v'&=3b-a-d,\\ p'&=c-a, &q'&=d-b. \end{aligned} \]

Substitution reconstructs every full nodal row, including both fixed nodes. Conversely, every full state fixed by T has this representation. The result holds globally in real phase lifts, including nonacute gaps; it is not a small-oscillation or near-equilibrium approximation.

The general shared owner assess_sine_involution_reduction checks an admitted signed involution and supplies SineInvolutionReduction. Its .evaluate(form_coordinates, phase_coordinates) reconstructs the family's full state and rows. For this reflection the representatives are (0,1,5,6), giving the coordinate order above. That API reports rates in original structural t; divide the displayed scaled-time rows by pi when comparing them. The source anchors the law and support and need not already belong to the declared family.

This T differs from the time reverser of Section 47: R(x,theta)=(-x,theta) satisfies F(Rz)=-R F(z). The map T commutes with forward evolution; R exchanges forward and backward evolution. Neither is an event applied during a trajectory.

Inherited storage and oriented winding

The complete storage restricted to this family is exactly

\[ \begin{aligned} H={}&(a-b)^2+b^2+2a^2+(c-a)^2+(d-b)^2\\ &+2[1-\cos(v-u)]+2[1-\cos v]+[1-\cos(2u)]\\ &+2[1-\cos(p-u)]+2[1-\cos(q-v)]. \end{aligned} \]

No environmental storage is omitted. With D=diag(1/6,1/6,1/2,1/2), the reduced equations satisfy

\[ X'=-D\nabla_\Theta H,\qquad \Theta'=D\nabla_X H,\qquad H'=0. \]

The factors are inherited from the two equal-multiplicity receiver and leaf coordinates and their full degrees. This identity is a restriction of the actual conserved storage and nodal field, not a new auxiliary Hamiltonian or an added law.

Where all principal branches are unambiguous, the receiver period is

\[ W=\frac{2\operatorname{wrap}(v-u) +2\operatorname{wrap}(-v) +\operatorname{wrap}(2u)}{2\pi}. \]

The corresponding raw gaps telescope to zero. Integer branch offsets, rather than rounded division by 2pi, establish the winding. Reflection alone reverses the oriented period, and phase sign reversal also reverses it. Their combination preserves it. Consequently this invariant family can contain unit-winding states; it does not have the zero-winding obstruction of an ordinary reflection-fixed preparation.

The full saddle's unique hyperbolic pair lies in this family

At

\[ X_*=0,\qquad \Theta_*=(-2\pi/3,-\pi/3,-2\pi/3,-\pi/3), \]

all contacts have zero phase difference. The five principal receiver gaps are (pi/3,pi/3,pi/3,pi/3,2pi/3), and each oriented sine current is the same. The full state is therefore an equilibrium with H=7/2 and W=1. It is not acute: its closing gap is 2pi/3.

The full form Hessian restricted to X and the phase Hessian restricted to Theta are, respectively,

\[ G=\begin{pmatrix} 8&-2&-2&0\\-2&6&0&-2\\-2&0&2&0\\0&-2&0&2 \end{pmatrix},\qquad B=\begin{pmatrix} 1&-1&-2&0\\-1&4&0&-2\\-2&0&2&0\\0&-2&0&2 \end{pmatrix}. \]

The leading principal minors of G are 8,44,64,80, so G is positive definite. Eliminating the two contact differences makes B congruent to the receiver block [[−1,−1],[−1,2]] and two positive entries 2. The receiver block has negative determinant, so B has three positive directions and one negative direction, with no zero direction. The linearized reduced system is

\[ \dot X=-D B\Theta,\qquad \dot\Theta=D G X,\qquad \ddot\Theta=-D G D B\Theta. \]

Because D G D is symmetric positive definite, its product with B is similar to a real symmetric matrix with the same inertia as B. Thus the eight-dimensional tangent has one real stable/unstable pair and three purely imaginary pairs. These are exact eigenvalue-type counts, not a nonlinear attraction or stability claim.

The full graph has the same single negative phase direction. Its contact differences contribute five positive squares. For the four successive receiver path differences d_k, its remaining Hessian quadratic form is

\[ \frac12\sum_{k=0}^3d_k^2-rac12\left(\sum_{k=0}^3d_k\right)^2. \]

Writing d=m*1+r, with sum(r)=0, gives ||r||^2/2-6m^2: three positive directions and one negative direction, besides the global phase origin. Hence the complete phase Hessian has inertia (8 positive, 1 negative, 1 zero). Removing the conserved global form and phase origins makes the form Hessian positive definite. The full twenty-dimensional linearization has one real pair, eight imaginary pairs and two semisimple zero modes for those origins. The odd reduction already contains the only real pair; the complementary even subspace contains five imaginary pairs and the two origin modes.

For a reproducible algebraic identification, let lambda=sigma^2>0 be the positive eigenvalue of -D G D B. Its characteristic polynomial is

\[ P(\lambda)=324\lambda^4+1404\lambda^3 +1527\lambda^2+61\lambda-15. \]

Descartes' sign rule gives exactly one positive root. Exact rational signs at 19709/250000 and 78837/1000000 isolate it between those endpoints. Set

\[ b_\lambda=\frac{4\lambda+1}{18\lambda^2+43\lambda+5},\qquad v_\lambda=\left(1,b_\lambda, \frac{7-b_\lambda}{6\lambda+8}, \frac{-1+10b_\lambda}{6\lambda+8}\right). \]

Substitution into (-D G D B-lambda I)v_lambda gives zero, using P(lambda)=0. The corresponding form vector is X_lambda=-D B v_lambda/sigma. The tangent eigenvectors are (X_lambda,v_lambda) and (-X_lambda,v_lambda), with rates +sigma and -sigma in scaled time, or +sigma/pi and -sigma/pi in original time. Their closing-gap derivative is nonzero because the first phase component is one. This gives a direction transverse to the wider-sector boundary; crossing that boundary still does not mean crossing the principal phase seam or reaching W=0.

The shared assess_sine_c5_leaf_saddle owns this exact conditional saddle analysis. Its mathematical phase turns must remain distinct from rational approximations to radians in a numerical preparation. Rounded phases cannot be declared an exact critical point. In its report, odd_phase_hessian is B above, odd_phase_from_form is D G, and odd_negative_form_from_phase is D B; the rate blocks must not be confused with the unweighted storage Hessians.

A finite nonlinear envelope for the tangent comparison

Let z_* denote the exact saddle in full coordinates, and write

\[ F(z_*+h)=Jh+\mathcal R(h). \]

The phase row is linear, so R_phase=0. For each sine edge, the global Taylor bound |sin(delta+eta)-sin(delta)-cos(delta)*eta|<=eta^2/2, together with |eta|<=2||h_phase||_infinity, yields

\[ \|\mathcal R_{\rm form}(h)\|_\infty \le 2\|h_{\rm phase}\|_\infty^2. \]

The complete field is globally Lipschitz with constant two in the joint form/phase maximum norm in tau, and ||J||_infinity<=2. The same bounds hold for the exact reduced field because its signed reconstruction is an isometry in that norm. For ||h(0)||_infinity<=r, Gronwall and Duhamel therefore give the fully nonlinear comparison

\[ \boxed{ \|h(\tau)-e^{J\tau}h(0)\|_\infty \le r^2 e^{2|\tau|}\bigl(e^{2|\tau|}-1\bigr). } \]

Indeed, ||h(s)||<=r exp(2s) for positive s; integrating exp(2(t-s))*2r^2 exp(4s) gives the displayed bound. For negative time, apply the same argument to -F and -J. This requires no acute chart and no phase normalization. It is useful only where its retained error is small enough for the intended directional comparison; it does not turn a tangent eigenvector into a remote acquisition certificate.

A nonlinear local sector passage with an explicit error budget

The envelope proves a small but nontrivial passage under the complete law, without evaluating a trajectory. Normalize the eigenvector as above, with receiver phase component v_lambda[0]=1, and choose

\[ 0<\varepsilon\le2^{-24},\qquad h(0)=\varepsilon(3X_\lambda,-v_\lambda). \]

This is an exact, correlated preparation around the saddle. Its algebraic eigenvector and exact phase turns specify a mathematical family, not a measured preparation or an independent twenty-coordinate uncertainty box. The rational eigenvalue bracket implies 1/10<b_lambda<1/5, 79/100<v_lambda[2]<83/100, 0<v_lambda[3]<b_lambda and sigma>7/25. Substitution in X_lambda=-D B v_lambda/sigma then gives ||v_lambda||_infinity=1 and ||X_lambda||_infinity<1. Thus ||h(0)||_infinity<=3*epsilon.

The phase component of the tangent solution is exactly

\[ h^{\rm lin}_\Theta(\tau) =\varepsilon\bigl(e^{\sigma\tau}-2e^{-\sigma\tau}\bigr)v_\lambda. \]

Initially its first component is -epsilon, so the closing principal gap is 2pi/3-2epsilon. The state is strictly inside the wider sector Omega_1 of Section 42. At tau=3,

\[ e^{3\sigma}>e^{21/25} >1+\frac{21}{25}+\frac12\left(\frac{21}{25}\right)^2>2, \qquad e^{3\sigma}-2e^{-3\sigma}>1. \]

The full nonlinear comparison error at that time is less than

\[ 9\varepsilon^2 e^6(e^6-1) <9\cdot729\cdot728\,\varepsilon^2<\varepsilon, \]

using e<3 and 9*729*728<2^24. The actual first phase perturbation is therefore positive at tau=3: its closing gap has crossed above 2pi/3. The orbit has exited Omega_1 within that fixed finite time.

This is not a hidden winding or acute-entry claim. Throughout [0,3], ||h_phase||_infinity<2187*epsilon<pi/12. The four path gaps remain in (pi/6,pi/2) and the closing principal gap remains in (pi/2,5pi/6). No principal branch changes; the winding remains 1 and the closing gap remains nonacute throughout the passage. Negating this perturbation gives the opposite local sector passage. Neither construction proves travel from the earlier numerical checkpoint, zero-winding preparation, or a distant acute pattern. It establishes that the unique full-system hyperbolic direction permits a quantitatively controlled nonlinear crossing of the local sector boundary.

What the reduction does not remove

A nonzero-width independent box in all twenty form and phase coordinates is not confined to Fix(T). Exact symmetry does not justify dropping its even perturbations. If S q_nom(tau) is an exact reconstructed reduced solution and an actual full initial state differs from it by at most delta in the joint maximum norm, the same complete-field Lipschitz bound gives

\[ \|z(\tau)-S q_{\rm nom}(\tau)\|_\infty \le\delta e^{2|\tau|}. \]

A validated error bound for a numerically computed reduced nominal solution must also be retained. This supplies a route for using a reduced nominal calculation with full transverse uncertainty; it does not assert that the uncertain members stay symmetric. The frozen rational checkpoint and its time-reversed center belong to the exact family, whereas their outward endpoint boxes contain independent full-state perturbations. The absence of an additional real tangent eigenpair is not a nonlinear stability theorem.

The exact saddle remains stationary. No distinct orbit reaches that exact equilibrium at a finite time under the same smooth law. Nearby states and transverse tangent directions can be studied without making that false finite-entry claim. Finally, a trajectory leaving the 2pi/3 protected sector can still have W=1 and be nonacute. The same-orbit construction supplies the additional whole-state passage and finite-retention proof under its explicit preparation and uncertainty hypotheses; the local result alone does not imply those conclusions.

49. A directed nonlinear corridor reaches the winding seam

Retain Section 48's complete conservative unit law, C5/private-leaf support, signed invariant family and clock tau=t/pi. The following argument uses its actual nonlinear rows throughout a finite phase corridor. It does not extrapolate a saddle eigenvector or replace the environmental nodes by a force. The result connects a nonacute unit-winding state outside the protected sector to zero winding. Reversing it does not by itself connect zero winding to an acute finite-retention target.

The principal seam has a lower minimum than the sector saddle

At a principal seam, one cycle edge has potential 2; the remaining four oriented gaps sum to pi modulo 2pi. Divide those four gaps into two pairs. If the first pair sums to A modulo 2pi, their cosine sum is at most 2|cos(A/2)|; the other pair contributes at most 2|sin(A/2)|. Cauchy's inequality therefore bounds the four cosines by 2sqrt(2), with equality when each remaining principal gap is pi/4 up to orientation. Consequently

\[ \min_{\text{principal seam}}V_R=6-2\sqrt2<\frac72. \]

This minimum is attained within the signed family: take u=-pi/2, v=-pi/4, matched leaf phases p=u,q=v, and zero form. The four path gaps are pi/4 and the closing gap is at its seam. Thus the signed reduction introduces no stronger static seam barrier. The 7/2 saddle separates the wider sector from its exterior; it is not the minimum cost of changing the integer winding.

The closed-budget result nevertheless proves that a finite outside-to-acute orbit needs strictly H>7/2. At equality its sector boundary can only contain full equilibria, which a distinct orbit cannot reach in finite time. An energy-compatible seam configuration and a local unstable direction do not override this obstruction.

Exact transverse storage and a directed momentum

Use the real lifted coordinates

\[ s=v-u/2,\qquad r=p-u,\qquad h=q-v,\qquad C=\cos(u/2). \]

The inherited storage is H=F+V, where F=X^T G X/2 and

\[ \begin{aligned} V(u,s,r,h) &=4[1-C\cos s]+1-\cos(2u) +2[1-\cos r]+2[1-\cos h],\\ V_0(u)&=5-4\cos(u/2)-\cos(2u). \end{aligned} \]

Thus V>=V_0 whenever C>0. The transverse receiver and contact coordinates are still dynamic; setting them to their minimizing values in a bound does not set their actual rows to zero.

Define the full-state linear momentum

\[ P=6a+3b+2c+d. \]

Adding the eight inherited rows with these coefficients cancels the explicit contact currents and gives the exact identity

\[ \boxed{P'=-2[\sin(u/2)\cos s+\sin(2u)].} \]

This cancellation retains the contact state in P and in the subsequent evolution of u,s; it does not remove environmental feedback. Equivalently, the constant mechanical mass in phase coordinates is (D G D)^(-1). After the displayed coordinate change, its first row is (53/2,17,6,5), so

\[ P=\frac{53}{2}u'+17s'+6r'+5h'. \]

In particular, positive or increasing P alone does not mean that u increases at every instant. The transverse velocities must be retained.

A uniform force bound throughout the corridor

Choose declared real endpoints and a full-storage upper bound satisfying

\[ -\frac{2\pi}{3}<u_L<u_0<-\frac\pi2<u_R<0, \qquad H\le\overline H<\min\{4,B(u_L)\}, \]

where

\[ B(u)=4+24\cos^4(u/2)-8\cos^2(u/2). \]

On this interval C>0. The nonnegative form and contact stores give

\[ \cos s\ge\frac{5-\cos(2u)-\overline H}{4C}. \]

Substituting this lower bound in the exact momentum row, with sin(u/2)<0, yields

\[ P'\ge-\frac12\tan(u/2)[B(u)-\overline H] \ge m, \qquad m=-\frac12\tan(u_R/2)[B(u_L)-\overline H]>0. \]

Here B is strictly increasing on (-2pi/3,0) because cos^2(u/2)>1/4, while -tan(u/2) is positive and decreasing. This is a nonlinear, full-state bound valid while u remains in the closed corridor; it uses no local truncation. V_0 is strictly decreasing on the same interval. Indeed V_0'=2sin(u/2)[1+4cos(u/2)cos u], and the bracket increases from zero when expressed as 1+8C^3-4C, with C>1/2.

A directional exit condition keeps the lower face closed

Let l=(6,3,2,1) and k=(4,-1,-1,0)/3, so P=l^T X and u'=k^T X. The exact inherited form matrix gives

\[ l^{\mathsf T}G^{-1}l=\frac{53}{2},\qquad k^{\mathsf T}G^{-1}k=\frac29,\qquad k^{\mathsf T}G^{-1}l=1. \]

Consequently P^2<=53F. For a prospective lower-face exit, a sharper bound follows by setting l_perp=l-(9/2)k:

\[ l_\perp^{\mathsf T}G^{-1}l_\perp=22,\qquad P=\frac92u'+l_\perp^{\mathsf T}X. \]

At a first lower-face exit u'<=0. If P>0, Cauchy's inequality therefore gives P^2<=44F. Suppose the actual initial momentum obeys

\[ \boxed{P_0>0,\qquad P_0^2>44\max\{0,\overline H-V_0(u_L)\}.} \]

Since P increases while the orbit remains in the corridor, this inequality excludes a lower-face exit: there F<=overline H-V_0(u_L) would contradict the necessary exit bound. If the latter upper bound is negative, that face is already excluded by storage.

Meanwhile the entire closed corridor gives

\[ P\le P_{\max} =\sqrt{53[\overline H-V_0(u_R)]}. \]

Combining this finite upper bound with P'>=m proves that the orbit cannot stay in the corridor indefinitely. It must reach the upper face within

\[ \boxed{T_{\rm exit}\le\frac{P_{\max}-P_0}{m}.} \]

This bound concerns scaled time and the actual complete flow, not a prescribed path in phase space. Its endpoints, energy and strict momentum margin are prospective premises.

Why this exit changes winding, and what it leaves open

Require initially that the four raw path gaps v-u,-v,-v,v-u lie in (-pi,pi). Since 2u_0 lies in (-4pi/3,-pi), the closing edge has one branch offset and the initial winding is +1. Each of the path gaps occurs twice in the inherited potential, so reaching a principal path seam would cost at least 4, which overline H<4 excludes. The same bound excludes a contact seam, since each nonzero contact coordinate also occurs twice. All these branch offsets therefore remain fixed. At the upper face 2u_R lies in (-pi,0): the closing seam has been crossed, the raw gaps still telescope, and the winding is 0. No assumption of monotone u was used.

The sufficient class is nonempty without solving an orbit. For example, choose any permitted u_0, take s=r=h=0, and set

\[ X=U(2,2,3,5/2),\qquad U>0. \]

These are supplied initial data, not maintained constraints. They give F=53U^2/4, P_0=53U/2 and P_0^2=53F; since V_0(u_0)<V_0(u_L), the strict lower-exit test follows automatically. Any such preparation satisfying the declared upper-energy bound supplies the directed passage. Because V_0(u_0)<7/2<B(u_L), this family includes energies strictly above 7/2; a saddle-level or sub-saddle budget is not required.

The retained rational illustration uses u_L=-2, u_0=-9/5, u_R=-3/2 and U=1/12: its reduced form is (1/6,1/6,1/4,5/24), its phase is (-9/5,-9/10,-9/5,-9/10), and its full storage is V_0(-9/5)+53/576, about 3.50233. The shared assess_sine_saddle_corridor re-admits the full state and signed family before applying these bounds; lower_phase and upper_phase declare the two corridor faces. It adds no solver, law, damping or live support event. A failed sufficient inequality does not prove trapping. The proof concerns an exact correlated signed-family source. A full independent source box additionally needs the transverse error control of Section 48 and sufficient endpoint branch margins; it is not contained in the eight-dimensional family.

This establishes a genuine nonlinear connection from the outer saddle corridor to the winding seam. Its initial closing gap exceeds 2pi/3, so the source is outside Omega_1 and is nonacute. Reversing the result connects zero winding back to that outer corridor, not across the saddle into an acute target. Joining this passage to the local sector crossing and a finite-retention region requires one compatible full-state orbit; separate existential passages cannot simply be concatenated. The stronger formation-to-retention obligation is not settled by this corridor alone. Section 50 supplies a compatible same-orbit construction under additional quantitative premises.

50. Conservative formation and retention on the same full-state orbit

There is a conditional existence construction for the strengthened formation-to-retention question. It uses the same complete conservative unit law and supplied C5/private-leaf support throughout. One exactly specified near-saddle preparation connects both nonlinear corridors, rather than treating their separate examples as one trajectory. Time reversal then provides an initially zero-winding source, an actually reached target, and acute retention for T=1 in tau=t/pi. Positive uncertainty widths cover every form and phase coordinate of the ten-node system.

This is an analytic existence theorem. The source and target centers below are defined through the unique mathematical flow and finite hitting events; their numerical coordinates have not been computed. It is neither an evaluated frozen prediction nor a claim that a source is autonomously selected, easy to prepare or physically identified.

One exact preparation supplies both corridor admissions

Use the exact saddle and algebraic eigenvectors of Section 48, with v_lambda[0]=1, and set

\[ 0<\varepsilon\le2^{-32},\qquad z(0)=z_*+\varepsilon(3X_\lambda,-v_\lambda). \]

Let z(tau) be this one complete solution. Its real phases and forms are exactly in the signed invariant family; no rounded value of pi or an independent box around an eigenvector is substituted for that preparation. Write Q=X_lambda^T G X_lambda and P_lambda=l^T X_lambda, with the matrix and momentum row already defined. The exact rational root enclosure of Section 48 implies

\[ 1<Q<\frac65,\qquad P_\lambda>\frac{53}{10},\qquad \|v_\lambda\|_\infty=1,\quad\|X_\lambda\|_\infty<1. \]

The saddle is critical and v_lambda^T B v_lambda=-Q. The initial form and quadratic phase terms therefore give 4Q epsilon^2 above its storage. The third-order remainder of each of the ten edge potentials is at most (2epsilon)^3/6. Consequently the actual conserved storage obeys

\[ H=\frac72+4Q\varepsilon^2+R_H, \quad |R_H|\le\frac{40}{3}\varepsilon^3, \quad \boxed{\frac72<H<\overline H:=\frac72+5\varepsilon^2 <\frac{35001}{10000}.} \]

For either time sign at |tau|=3, the full nonlinear comparison error of Section 48 is at most

\[ \rho=9\cdot729\cdot728\,\varepsilon^2<\frac{\varepsilon}{512}. \]

Put E=exp(3sigma). Since sigma>7/25, the cubic lower Taylor sum for exp(21/25) gives the rational inequalities

\[ E-2/E>7/5,\qquad E+2/E>79/25, \qquad E^{-1}-2E<-4,\qquad E^{-1}+2E>9/2. \]

The tangent expressions, with the retained error rho, now certify the following properties of actual states of the same solution:

State	Reaction coordinate	Directed momentum
z(3)	u>u_*+epsilon	P>(33/2)epsilon
R z(-3), where R(x,theta)=(-x,theta)	u<u_*-epsilon	P<-23epsilon

For example, the outer momentum is bounded below by epsilon*(53/10)*(79/25)-12rho>(33/2)epsilon; the coefficient 12 is the sum of the absolute entries of l. The analogous bound with 9/2 gives the inner negative momentum after time reversal. The global bound ||z(tau)-z_*||_infinity<2187epsilon for |tau|<=3 also places these states strictly inside their declared coordinate strips and preserves all their initial path branches. Both remain nonacute at these local times.

At u_*= -2pi/3, V_0'=0, V_0''=-3/2, and |V_0'''|<=17/2. Taylor's inequality gives

\[ V_0(u_*\mathbin{\pm}\varepsilon)\ge\frac72-\varepsilon^2, \qquad 44[\overline H-V_0(u_*\mathbin{\pm}\varepsilon)] \le264\varepsilon^2. \]

This last upper bound is strictly smaller than both retained momentum squares. Thus the actual local states pass the directional face tests; the argument does not reset their form, contact phase or conserved storage to a new example.

Outer and inner nonlinear connections

For z(3), use Section 49 with u_L=u_*+epsilon, u_R=-3/2. On [u_*,-pi/2], B'(u)>=sqrt(3), so B(u_L)-overline H>epsilon. The positive force is greater than epsilon/3. The inherited Cauchy bound gives |P|^2<=53H<196, hence |P|<14. The same orbit reaches the upper face u=-3/2, and therefore W=0, within a further 42/epsilon scaled time units. Denote one such finite upper-face time by t_o.

For the inner direction, evolve the complete reversed solution w(tau)=R z(-tau), starting with its actual state at tau=3. Use the strip

\[ u_A=-5/2\le u\le u_* -\varepsilon. \]

Here cos s<=1 in the exact momentum law gives P'<=-V_0'(u)<0. The function V_0' is concave on this strip: V_0'''=-sin(u/2)/2-8sin(2u)<0. Its left endpoint satisfies V_0'(-5/2)>1/100>epsilon; at the right endpoint, V_0''<=-1 near u_* gives V_0'(u_*-epsilon)>=epsilon. Thus P'<=-epsilon throughout the strip.

At a prospective upper-face exit u'>=0, the same orthogonal decomposition P=(9/2)u'+l_perp^T X gives P^2<=44F when P<0. The local momentum bound excludes that exit. Since |P|<14, the solution must instead reach u_A=-5/2 within a further 14/epsilon time units. Let t_d be its first such deep hit. The paired path phases remain on their inherited branches because H<4; contacts remain fully dynamic.

The resulting forward formation orbit is

\[ \chi(s)=R z(t_o-s),\qquad s\ge0. \]

It begins with W=0, passes the same near-saddle preparation, and reaches the deep state at s=t_o+t_d. Its elapsed time to any earlier checkpoint is less than 6+56/epsilon<64/epsilon. This is a conservative analytic upper bound, not a computed hitting time or a proposed simulation horizon.

An explicit acute band supplies a unit of retained evolution

Consider the closed phase band

\[ -13/5\le u\le-12/5, \qquad H\le35001/10000. \]

It contains the deep hit u=-5/2. A paired receiver gap reaching its acute face has minimum cycle potential at fixed u

\[ A(u)=\frac72+2[\sin u+1/2]^2. \]

This follows by setting one of the two distinct path gaps to pi/2 in the exact potential; a negative path face is incompatible with the other path gap remaining acute on this lifted branch. On the displayed band the minimum is at u=-13/5. A rational Taylor lower bound sin(13/5)>103/200 proves

\[ A(u)-H>\frac1{3000}. \]

Since |partial V_R/partial s|<=4, each paired gap has acute margin greater than 1/12000 throughout the admitted band component. The closing gap is 2pi+2u and its acute margin is at least 24/5-3pi/2>1/12000. These estimates both prove acuteness at the deep state and prevent an acute face from being crossed before a band exit. The path to the deep state inherits this component: between the saddle neighborhood and u=-5/2, its paired acute-face cost also exceeds H.

While acute, V_R>=V_5=5(1-cos(2pi/5)). The exact form metric gives

\[ |u'|^2\le\frac49F\le\frac49(H-V_5)<\frac1{49}. \]

For example, sqrt(5)<161/72 and H<=35001/10000 reduce the last strict inequality to the rational comparison 13/288+1/10000<9/196. Starting at the deep state, either band face is 1/10 away. A first exit in either time direction must therefore take more than 7/10 scaled time units. In particular, the complete nominal orbit on the interval from one half-unit before the deep hit to one half-unit after it is acute with every edge margin greater than 1/12000.

Define the target checkpoint at S=t_o+t_d-1/2 along chi. It is later than the initial zero-winding source, and its next full scaled unit stays inside that acute band.

Positive widths include all twenty form and phase coordinates

Let the target be the exact real maximum-norm ball of radius

\[ \delta_T=2^{-18} \]

around the actual state chi(S), with support and unit capacities fixed. The complete field is globally Lipschitz with constant two, regardless of whether an uncertain state belongs to the signed family. Over T=1, each full-state error is at most delta_T exp(2), and an edge phase error is at most twice that. Since

\[ 2\delta_T e^2<18\delta_T<\frac1{12000}, \]

every independent target member retains strict acute winding +1 for the full unit interval. This includes form and phase perturbations at all receiver and environmental nodes; it does not impose symmetry on them.

At the initial source chi(0), the closing gap has principal-seam margin pi-3>1/10. A paired path gap within 1/10 of a seam would cost 2[1+cos(1/10)]>H, so each of those margins also exceeds 1/10. Take the full independent source ball of exact radius

\[ \boxed{\delta_S=2^{-19}\exp(-128/\varepsilon)>0.} \]

Every source member still has W=0. Since S<64/epsilon, its error relative to the same nominal orbit at the target checkpoint is strictly less than 2^-19, which lies inside the radius-delta_T target. Thus one full-width zero-winding source reaches one full-width acute-retention target under the unchanged complete law. The target ball need not equal the reachable image; inclusion is the proved claim.

The source-width bound is extremely small even at the largest admitted epsilon. It is a formal positive robustness radius, not evidence of practical preparation tolerance. Its logarithm or exact analytic expression must be retained; replacing an underflowed floating value by zero would discard the theorem's premise. The unknown numerical hitting states must likewise not be replaced by nearby hand-selected snapshots.

The shared assess_sine_saddle_formation reports this scoped analytic family, and assess_sine_saddle_retention_band owns the admitted band estimate. The complete source law and support are re-admitted; a supplied snapshot is not automatically either constructed hitting state. No trajectory is evaluated by this proof reader.

This closes the mathematical conditional existence obligation for formation followed by T=1 finite retention on this supplied support. It does not select an autonomous preparation or microscopic law, establish indefinite lifetime, compute an operational source/target pair, or identify the resulting pattern with a physical constituent. Those are distinct questions from the existence established here.

51. Rational preparation and retained-metric propagation

Section 50 establishes a formal source-to-target connection. Making that construction operational requires actual represented preparation coordinates and a validated propagation that preserves enough information to reach its hitting states. These are numerical obligations under the existing complete law, not missing pressure terms or new constitutive parameters. Increasing arithmetic precision and proving tolerance to physical preparation error are separate questions.

A rational preparation must retain the same finite gates

The shared prepare_sine_saddle_state constructs a rational midpoint of an enclosure of Section 50's correlated algebraic preparation. It re-admits that actual rational center, including its complete form, phase, support and conserved storage. The input source anchors the law and support; it is not relabeled as the constructed state.

If the rational center differs from the exact correlated preparation by at most r in the full maximum norm, the complete-field Lipschitz bound adds at most exp(6)r<729r to the local comparison at either tau=3 or tau=-3. Thus the total normalized local error becomes

\[ \eta_{\rm loc} =9\cdot729\cdot728\,\varepsilon +\frac{729r}{\varepsilon}. \]

The two phase-displacement and two momentum inequalities of Section 50 must be recomputed with this retained error. Exact signed-family membership, initial winding and branch admission are checked again; a component interval containing the desired value is not itself an exact symmetry proof. The actual center's energy must independently satisfy 7/2<H<7/2+5epsilon^2. Using the shared outward rational trigonometric arithmetic preserves the small difference above the saddle budget; rounding that difference away with a binary floating cosine would not establish either inequality.

If the retained precision cannot separate the actual center's storage from 7/2, preparation admission remains unavailable even when the ideal analytic family exists. The reader does not replace that unresolved represented state by the exact theorem's preparation.

This supplies usable rational coordinates for the near-saddle preparation and retains its conditional forward and backward corridor gates. When those checks pass, the same-orbit corridor and finite-retention argument applies to the actual rational preparation, with its retained errors and budget; the proof does not continue from an ideal center instead. The preparation reader alone does not supply the zero-winding source coordinates, a computed hitting time or a validated future response. The rounding allowance r concerns computational representation; it must not be silently substituted for a laboratory preparation tolerance.

Why independent coordinate boxes lose essential cancellations

At the exact saddle, in scaled time, the full twenty-coordinate Jacobian is

\[ J=\begin{pmatrix}0&-K H_\theta\\K L&0\end{pmatrix}, \]

where H_theta is the actual sine-potential Hessian. Its only expanding eigenvalue is sigma<1/3; it also has the oscillatory pairs and two conserved-origin modes identified in Section 48. The current componentwise Taylor comparison replaces off-diagonal derivatives by absolute bounds. Since the diagonal entries of J vanish, its exact-saddle twenty-coordinate comparison block is |J|.

Take the positive comparison vector with all form entries 1 and all phase entries sqrt(2). Every row of |K L| sums to 2. The row sums of |K H_theta| are 1 at the two receivers touching the negative-cosine edge, 4/3 at the other receivers, and 2 at the leaves. Consequently

\[ |J|\begin{pmatrix}\mathbf1\\\sqrt2\,\mathbf1\end{pmatrix} \ge\sqrt2 \begin{pmatrix}\mathbf1\\\sqrt2\,\mathbf1\end{pmatrix}, \qquad \rho(|J|)\ge\sqrt2. \]

Thus a rigorous independent-box comparison can grow much faster than the actual tangent. The common-origin vectors are an especially clear example: the signed field cancels their contributions, while its absolute comparison does not. This is a limitation of the enclosure representation, not evidence of an additional unstable nodal mode or an error in the completed finite forecasts. Their retained enclosures and negative results remain unchanged.

An exact rational metric retains all relative coordinates

No arbitrary transverse error may be dropped by replacing the full state with the eight-dimensional signed family. Instead retain both conserved weighted means and all nine relative coordinates for each field. Let d=K^(-1)1 be the full degree vector, and choose a rational full-rank matrix U whose columns span d^perp. For example, with a private leaf last, take U_i=e_i-(d_i/d_last)e_last for the other nine nodes. Write

\[ x=\bar x\mathbf1+Ua,\qquad \theta=\bar\theta\mathbf1+Ub, \qquad \bar x=\frac{d^{\mathsf T}x}{d^{\mathsf T}\mathbf1},\quad \bar\theta=\frac{d^{\mathsf T}\theta}{d^{\mathsf T}\mathbf1}. \]

Both origins are retained constants, not discarded observations. Define

\[ G=U^{\mathsf T}LU,\qquad B=U^{\mathsf T}H_\theta U,\qquad D=(U^{\mathsf T}K^{-1}U)^{-1},\qquad M=(DGD)^{-1}. \]

These are the full relative matrices, distinct in dimension from the four-coordinate matrices of Section 48. The relative tangent is exactly a'=-DBb, b'=DGa. All matrices are rational at the exact saddle. The following symmetric positive-definite metric requires no fitted eigenvector basis or numerical Lyapunov-equation solution:

\[ W_{\rm rel}=\operatorname{diag}\left(G,\,B+\frac16M\right). \]

For r=1/3, the Schur complement of 2rW_rel-(J_rel^T W_rel+W_rel J_rel) is

\[ \frac23\left(B+\frac5{48}M\right). \]

Exact rational positive-definiteness checks therefore prove

\[ J_{\rm rel}^{\mathsf T}W_{\rm rel} +W_{\rm rel}J_{\rm rel}\preceq\frac23W_{\rm rel}. \]

Changing J_rel to -J_rel changes the off-diagonal Schur factors' signs but not their product, so the same inequality holds for reverse time. Add a positive unit weight for each conserved origin and pull the metric back through the exact coordinate transformation. This gives a rational positive-definite matrix W on all twenty original coordinates. It contains the even perturbations, both origins and every live contact. The induced error norm is ||h||_W=sqrt(h^T W h). This metric controls numerical sensitivity; it is not an added conserved storage, physical observable or constitutive law.

A finite-neighborhood nonlinear bound

Suppose each primitive phase stays within rho radians of the declared saddle phase lift. Every edge phase difference then departs by at most 2rho. Since cosine is globally one-Lipschitz, the change Delta B in the full relative phase Hessian obeys

\[ -2\rho G\preceq\Delta B\preceq2\rho G. \]

The numerical owner verifies the further exact matrix inequalities

\[ 2G-GDG\succeq0,\qquad B+\frac1{12}M\succeq0. \]

The first is the normalized-Laplacian bound in these coordinates. The second, together with the definition of W_rel, gives W_phase>=M/12. Factor Delta B=G^(1/2) C G^(1/2), where ||C||_2<=2rho. Then

\[ \|G^{1/2}DG^{1/2}\|_2\le2,\qquad \|G^{1/2}W_{\rm phase}^{-1/2}\|_2\le\sqrt{48}. \]

Only the form-from-phase block of the Jacobian changes. Its induced metric norm is consequently at most

\[ 2(2\rho)\sqrt{48}=16\sqrt3\,\rho<28\rho. \]

The same full-state metric therefore yields the local nonlinear logarithmic growth bound

\[ \boxed{\gamma_W\le\frac13+28\rho.} \]

For instance rho=1/1000 gives a bound below 0.362 in scaled time, while retaining all coordinates. The radius is a declared proof domain, not a new physical constant. This is not a global hyperbolic-rate theorem: the relevant whole-time phase enclosures must stay in that neighborhood. For two complete flows whose phases remain there, the segment between them also stays in the phase box, giving ||h(tau)||_W<=exp(gamma_W*|tau|)||h(0)||_W. A numerical center with a residual requires its separate inhomogeneous error integral; the homogeneous estimate does not cover truncation error by itself.

Conversion to primitive coordinates and execution boundary

The exact full metric supplies conservative conversion factors

\[ \|h\|_W\le \sqrt{\sum_{i,j}|W_{ij}|}\,\|h\|_\infty, \qquad |h_i|\le\sqrt{(W^{-1})_{ii}}\,\|h\|_W. \]

The shared assess_sine_saddle_sensitivity re-admits the law and support and checks the rational matrix premises, both time signs and these conversions. The captured source is a support anchor, not evidence that a trajectory stays near the saddle. This reader neither advances state nor produces an ellipsoid-based validated forecast.

A validated step retains the metric radius

The shared validated_metric_taylor_step now propagates the exact real set

\[ \mathcal E(c,r)=\{c+e:e^{\mathsf T}We\le r^2\}, \]

with a rational center c, rational positive-definite W and nonnegative radius r. Its enclosing coordinate box uses outward bounds on sqrt((W^(-1))_ii)r. That box is used for strict Picard inclusion and whole-time derivative and domain bounds; it does not replace the retained metric ball. In particular, the next step does not reconvert that enclosing box into a new metric radius.

Let h>0 be the step duration in the declared execution clock. Strict Picard inclusion must enclose the exact center flow and every flow starting in E(c,r). The model-specific growth inequality must hold throughout the resulting convex tube, including the segments between those flows. If its logarithmic bound is gamma, their separation at the endpoint obeys

\[ \|\Phi_h(c+e)-\Phi_h(c)\|_W\le e^{\gamma h}r. \]

Compute an interval Taylor polynomial from the exact center and its full order-p+1 remainder over the Picard tube. Let C_i be the resulting enclosure of the exact center endpoint, let c_i^+ be its exact rational midpoint and let a_i bound the distance from that midpoint to both materialized endpoints. Thus a_i includes Taylor truncation, interval arithmetic and any outward materialization of a nondyadic initial center. The local endpoint error satisfies

\[ \|\Phi_h(c)-c^+\|_W^2 \le\sum_{i,j}|W_{ij}|a_i a_j\le\ell^2 \]

when ell is chosen as an outward upper bound on the displayed square root. The triangle inequality therefore proves the retained-ball update

\[ \boxed{\Phi_h\bigl(\mathcal E(c,r)\bigr) \subseteq\mathcal E\left(c^+,\,e^{\gamma h}r+\ell\right).} \]

The local remainder is an endpoint error, so it is added after propagation of the initial radius. Every scalar exponential, square root and final radius is rounded outward. The current exact exponential evaluation admits abs(gamma*h)<=1; this is a numerical work limit, not a restriction of the mathematical inclusion. The generic kernel checks positive definiteness, Picard inclusion and the arithmetic. The caller remains responsible for the supplied logarithmic-norm theorem on the whole tube.

The retained endpoint ball can contain unreachable states or extend beyond the previous Picard tube. The next step therefore checks its whole initial projection again. Intersecting an observation box with an earlier tube would not justify shrinking this independent ball. Failed domain or remainder admission leaves only the already validated prefix available.

Sine execution with retained full-state uncertainty

forecast_sine_saddle_metric re-admits the complete unit-capacity C5/private-leaf law, reconstructs the metric and advances the actual captured source with the shared sine field. It retains all twenty coordinates in source-node order: first every form, then every primitive phase. The common origins remain present, and no exact signed-family assumption is imposed on the uncertainty.

The API declares elapsed original structural time t. In its fixed saddle-domain mode, tau=t/pi gives the outward original-clock growth bound

\[ \gamma_t=\frac{1/3+28\rho}{\pi_{\rm lower}}. \]

Both forward and reverse field directions use the corresponding proved metric inequality. In that mode, every whole-time phase interval is compared with an outward enclosure of its exact saddle lift 2*pi*turn_i; neither a rounded value of pi nor an implicit phase wrap substitutes for this real-lift domain check. The positive domain margins apply to the nominal and perturbed flows together.

An independent initial coordinate cube is enclosed in the metric ball once. Successive steps carry only the exact recentered state and retained radius in the same metric. Coordinate endpoint intervals are observations of that ball, not the uncertainty representation consumed by the next step. Declared step, order, domain policy and work budget remain fixed: a failure does not authorize a larger domain or an undeclared retry.

Certifying growth on a full Picard tube beyond the saddle neighborhood

The metric W itself is constant and positive definite on the entire twenty-coordinate state space. Only the previously supplied bound 1/3+28rho depends on remaining near the saddle. A direct Jacobian inequality on each complete Picard tube can replace that local bound without changing the metric, law or uncertainty representation.

Let J(z) be the Jacobian of the declared execution field on such a tube. Enclose every entry of the symmetric matrix

\[ S(z)=J(z)^{\mathsf T}W+WJ(z) \]

by symmetric rational intervals. Write their exact midpoint matrix as M and their symmetric nonnegative radius matrix as R. For every vector v,

\[ \begin{aligned} v^{\mathsf T}[S(z)-M]v &\le\sum_{i,j}R_{ij}|v_i v_j|\\ &\le\sum_i\left(\sum_jR_{ij}\right)v_i^2. \end{aligned} \]

Consequently the single exact matrix inequality

\[ \boxed{2\gamma W-M- \operatorname{diag}(R\mathbf1)\succeq0} \]

proves S(z)<=2gamma W at every point of the whole tube. It is a nonlinear flow bound because all possible Jacobians on that tube are enclosed, rather than just the Jacobian at its midpoint. Interval correlations discarded in forming the majorant can weaken the estimate, but cannot strengthen it without proof.

The shared step can select this route with growth_rate=None, a declared growth_rate_bounds bracket and a fixed growth_bisections count. The sine adapter's default bracket is (0,1) with eight exact positive-semidefiniteness bisections. The upper endpoint must first pass the matrix test; failure leaves the step unavailable, without expanding the bracket or changing the time step. Each selected rate and its matrix premises belong to the retained step evidence. This deterministic numerical policy is fixed before evaluating the reserved response; it does not fit or alter the nodal law.

For this sine model, W is block diagonal in form and phase, while J and hence S have only off-diagonal form/phase blocks. Conjugation by diag(I,-I) leaves W and the diagonal radius inflation unchanged and changes M to -M. Thus a successful matrix test also supplies the corresponding reverse-field inequality. This shortcut uses the admitted block structure; it is not a property of an arbitrary supplied Jacobian.

The sine adapter selects the tube-derived mode with phase_radius=None. It rebuilds the same full-state metric, then admits arbitrary primitive phase lifts under the globally smooth, fixed-support conservative sine law. Every actual tube still needs strict Picard inclusion and the matrix certificate. The saddle's primitive-phase radius is no longer a consumed growth premise. Since the shared automatic differentiation now acts on the original-time field, its J, S and certified gamma already contain the clock factor 1/pi; dividing the selected rate by pi again would be incorrect. The earlier fixed-domain mode keeps its separate clock conversion.

Across successful steps, the propagated part of the uncertainty is bounded by exp(sum_k gamma_k h_k) in the same metric, with each local endpoint error added by the retained-radius recurrence. There is no need to switch metrics or turn a coordinate observation box into an independent source when the trajectory leaves the small saddle neighborhood.

Retained finite connection and its preparation boundary

A forward endpoint is wholly at zero winding only if every receiver edge has a certified unique principal branch and the branch sum gives W=0. On a backward leg from the same admitted preparation, acute retention requires a contiguous collection of whole-time tubes with all receiver gaps strictly in (-pi/2,pi/2), fixed winding +1 or -1, and total original-time length at least an outward upper bound on pi. A sampled event midpoint cannot establish the required scaled duration T=1. Time reversal relates those two legs along the same complete flow, including each admitted perturbed preparation.

The frozen declaration uses the rational preparation with epsilon=2^-32, independent errors 2^-100 in all twenty coordinates, both time directions, original-time horizon 256, unit steps, Taylor order 16, and the declared (0,1) growth bracket with eight bisections. Its first-endpoint/first-window rule was fixed before evaluation. The retained evidence bundle and manifest preserve the source, protocol, full response and verdict.

Evaluated obligation	Retained result in original structural time
Forward zero-winding endpoint	The first wholly certified W=0 endpoint occurs at t=237
Forward declared horizon	The prefix through t=240 is admitted; the next step fails the declared growth-bracket upper endpoint 1
Backward declared horizon	All 256 units are admitted
Backward whole-time acute identity	The first retained W=+1 window is [230,234], with acute margin greater than 0.01001717 radians
Combined verdict	Connection is certified on the validated prefixes; completion of both declared horizons is false, so the protocol verdict is passed=false

Source checks before and after evaluation agree, and no evaluation error is recorded. The forward stopping reason is declared_growth_upper_endpoint_not_certified. It does not invalidate earlier tubes or prove a dynamical singularity; the declared sufficient matrix bound was unavailable on the next tube. The bracket, horizon and preparation were not retuned after that result.

Let Phi_t denote the original-time flow and R(x,theta)=(-x,theta) its reversing symmetry. For every initial state z in the admitted preparation, the source R Phi_237(z) is at zero winding. Its forward evolution obeys

\[ \Phi_s R\Phi_{237}(z)=R\Phi_{237-s}(z). \]

Consequently it retains acute winding +1 throughout s in [467,471]. The certified duration is 4/pi>1 in scaled time. This is a finite mapped source-family connection under the supplied complete law, with all environmental coordinates retained. The initial independent cube was prepared near the winding-one saddle; its image is not an independently prepared zero-winding coordinate ball.

An inverse-inclusion test separates an image from a preparation

The same retained step evidence supplies a sufficient inverse-inclusion test without a new trajectory. Write g_k for each outward metric growth factor and ell_k for its local endpoint error. Starting with P_0=1 and eta_0=0, propagate

\[ P_{k+1}=g_kP_k,\qquad \eta_{k+1}=g_k\eta_k+\ell_k. \]

Rounding these nonnegative products and sums upward retains valid upper bounds. Thus eta_F bounds the exact nominal endpoint's distance from the retained rational endpoint center c_F. The two-sided metric inequality also gives ||Phi_A(z)-Phi_A(c)||_W >= ||z-c||_W/P_F for admitted trajectories. On the boundary of the initial metric ball of radius r, the image is therefore at distance at least r/P_F from Phi_A(c).

The complete sine field is globally Lipschitz on the full real lift, so its finite-time flow is a global diffeomorphism. Its image boundary is the image of the initial boundary. A segment leaving that image from Phi_A(c) must meet this boundary; the distance bound forbids such a crossing inside the radius-r/P_F ball. Closure includes its boundary. The triangle inequality then proves

\[ \boxed{\mathcal E(c_F,q)\subseteq\Phi_A(\mathcal E(c,r)), \qquad q=\frac r{P_F}-\eta_F>0.} \]

This argument uses the already admitted forward trajectories and global invertibility, rather than assuming a candidate inverse trajectory stays inside an unverified domain. The reversing symmetry is an isometry of the block-diagonal metric. A positive q would therefore supply an explicit reflected source ball, and a coordinate cube of radius at most q/sqrt(sum_ij|W_ij|) inside it.

The solver propagates an enclosing initial metric ball, which is larger than the originally declared coordinate cube. To require preimages inside that cube of radius delta, use instead its admitted metric ball of radius r_in=delta/max_i sqrt((W^(-1))_ii), with an outward denominator. These are different preparation claims even though the same tubes can bound both.

The exact postprocessing record binds the retained response digest and recurrence, without a new forecast. Postprocessing the retained data through the first zero-winding endpoint gives outward bounds approximately P_F=7.60523305384e10 and eta_F=7.36366081855e-13. Both the enclosing-ball and inscribed-ball expressions for q are negative. Thus this sufficient test supplies no independent zero-winding preparation ball. It neither disproves the mapped connection nor establishes that an independent source ball is dynamically impossible. The finite connection, the incomplete forward horizon and the unavailable operational source width remain separate conclusions.

The local-agreement/global-storage discriminator keeps this frozen response intact while testing which constitutive premises its winding passage consumes. Agreement with the preparation, local saddle motion or acute geometry does not alone select a global pressure law.

52. A constitutive change preserves equilibria but blocks the same source's formation

The declared comparison eta=1/100 gives an energetic obstruction, not merely an unavailable trajectory-error estimate. The same source that forms and retains acute winding under sine lies below the changed law's acquisition barrier. This result reuses the retained sine response; it requires no new trajectory, altered preparation or coefficient search.

The complete changed law and fixed source

The cubic-storage completion keeps the C5/private-leaf support, held unit capacities and phase row:

\[ \begin{aligned} j_\eta(\delta)&=\sin\delta+\eta\sin^3\delta,\\ U_\eta(\delta)&=1-\cos\delta+ \eta\left(\frac23-\cos\delta+\frac{\cos^3\delta}{3}\right),\\ x'&=KS_\eta(\theta),\qquad \theta'=KLx, \qquad S_{\eta,i}=\sum_{j\sim i}j_\eta(\theta_j-\theta_i). \end{aligned} \]

Primes denote tau=t/pi; there are no inputs, events or discarded coordinates. For eta>=0, the changed nonnegative storage H_eta=x^T Lx/2+sum_e U_eta(delta_e) is conserved by its own reciprocal rows. The sine storage is not conserved by this changed field in general. The fixed scale eta=1/100 bounds the current change by one percent of normalized unit current. It is a declared robustness question, not a physical constant or a replacement engine default.

Use exactly the retained source family

\[ p_z=R\Phi^{\sin}_{237}(z),\qquad R(x,\theta)=(-x,\theta), \]

where z belongs to the frozen intermediate preparation. Flow times in this source definition are original structural time. Both laws start at this same exact p_z, whose zero winding is certified by the retained endpoint enclosure. Its sine reference path is

\[ y_z(s)=R\Phi^{\sin}_{237-s}(z),\qquad 0\le s\le471. \]

The whole sine interval [467,471] is acute with winding one, of scaled length 4/pi>1. Its retained tubes are forward steps 236,...,0 traversed in reverse and reflected, then backward steps 0,...,233 reflected. Preparing R Phi_eta,237(z) instead would change the source. An independent rectangular source ball or positive inverse-inclusion radius is unnecessary for this comparison of the same exact mapped states; the comparison does not supply either admission.

The exact changed acquisition barrier

Keep the entire sector from Section 42:

\[ \Omega_\sigma=\{W=\sigma,\ |\delta_i|<2\pi/3\text{ for all five edges}\}, \qquad \sigma\in\{-1,1\}. \]

Every acute state of winding sigma belongs to this sector. For every eta>=0, the map s -> s+eta*s^3 is strictly increasing. Consequently j_eta(t)-j_eta(pi/3) is nonpositive on [-2pi/3,pi/3] and nonnegative on [pi/3,2pi/3]. Integrating gives the global tangent bound on this whole interval,

\[ U_\eta(t)\ge U_\eta(\pi/3)+j_\eta(\pi/3)(t-\pi/3). \]

On the positive-winding boundary with one gap 2pi/3, the other four sum to 4pi/3. Their linear terms therefore cancel, proving

\[ \boxed{B_\eta:=\min_{\partial\Omega_\sigma}\sum_iU_\eta(\delta_i) =U_\eta(2\pi/3)+4U_\eta(\pi/3) =\frac72+\frac{47}{24}\eta.} \]

Equality is attained by the five permutations of (2pi/3,pi/3,pi/3,pi/3,pi/3). A gap -2pi/3 forces the other four to be 2pi/3, and has the strictly larger cost 5U_eta(2pi/3). Evenness gives the negative-winding result. No nonnegative-gap restriction or small-eta approximation enters this minimum.

For each finite eta, the periodic current is bounded, so form grows at most linearly on bounded time intervals; the linear phase row then also remains finite. The smooth full flow therefore exists in both time directions. All other full-system storage is nonnegative. Thus H_eta<B_eta forbids sector-boundary crossing in either time direction. Equality also forbids a finite crossing: it forces constant form and zero contact storage, while all cycle currents at the minimizing geometry are equal. The entire state is then stationary, and uniqueness excludes reaching or leaving it at finite time from a different state. Therefore a source outside both sectors with H_eta<=B_eta cannot acquire acute winding +1 or -1 at any time. This does not prohibit nonacute winding outside these sectors or identify every possible future state.

Applying the barrier to the retained source

Define the nonnegative added phase storage on all ten edges,

\[ A(\theta)=\sum_e\frac{(1-\cos\delta_e)^2(\cos\delta_e+2)}3. \]

Sine conservation and form reversal give the exact source identity

\[ H_\eta(p_z)=H_0(z)+\eta A(\theta(p_z)). \]

Here H_0(z) is bounded from the original preparation, and A is bounded from the complete retained endpoint at original time 237. Neither term uses the changed law's unknown response. The retained evidence contains the initial full-state enclosure and forward endpoint 236. The constitutive comparison record binds those retained premises without modifying the frozen response. Outward evaluation gives, conservatively rounded,

\[ 1.71785519306<A(\theta(p_z))<1.71785519309, \]

\[ H_{1/100}(p_z)<3.51717855194 <B_{1/100}=\frac72+\frac{47}{2400}, \qquad B_{1/100}-H_{1/100}(p_z)>0.00240478139. \]

The full source enclosure has winding zero. Thus the declared cubic law cannot acquire either acute unit-winding identity from any member of this same source family, although sine certifies the retained finite window. This is an all-time acquisition obstruction under the changed complete law, not evidence that an already formed identity disappears, that no other preparation can form it, or that the sine result was incorrect.

The sharp budget comparison can also be written

\[ B_\eta-H_\eta(p_z)= \eta\left(\frac{47}{24}-A(\theta(p_z))\right) -\left(H_0(z)-\frac72\right). \]

It explains the sensitivity: the sine construction uses a very small excess above its own saddle barrier, while the changed law raises that barrier more than it raises this fixed source's storage. A positive budget margin above a law's barrier is necessary for crossing, not a sufficient formation theorem. This identity supplies no instruction to retune the source until a preferred outcome passes.

The complete critical geometry and its inertia remain unchanged

The effect is not removal of the critical patterns. On this connected support, equilibrium requires constant form. Each private leaf has one neighbor and must have contact phase 0 or pi modulo 2pi. Cycle stationarity then requires one common oriented value of j_eta(delta). Its strict monotonicity as a function of sin(delta) makes this exactly the sine condition: all oriented cycle sines have the same value s. Thus the entire critical phase set is unchanged for every eta>=0.

At any such critical state the cycle Hessian weights are

\[ U_\eta''(\delta_i)=(1+3\eta s^2)\cos\delta_i, \]

while contact Hessian weights remain +1 or -1. Change phase coordinates to the receiver phases and the five leaf-minus-receiver contrasts. The phase quadratic splits into a positive scalar 1+3eta*s^2 times the original cycle quadratic plus the five unchanged signed contact squares. This is a congruence argument, so the full phase Hessian has the same inertia, including nullity. The full Hessian is not generally a common scalar multiple: the cycle and contact blocks scale differently. The form quadratic and mobility are unchanged, but individual dynamical rates and frequencies need not be.

This extends the prepared-geometry result to the entire critical set on the present unicyclic support. It explains why matching equilibrium geometries or their stability types cannot establish matching accessibility from one fixed preparation.

The sufficient error-transport method has a separate role

A finite comparison can also write the original-time fields as F_eta=F_0+eta D, where only the form rows of D are nonzero and each has absolute value at most 1/pi. For the retained block metric,

\[ \|D\|_W\le\frac{\sqrt{\sum_{i,j}|(W_x)_{ij}|}}{\pi_{\rm lower}}\le M. \]

On a common convex tube with admitted original-time sine logarithmic rate gamma, the paired-state error starting at e_0 obeys

\[ e(s)\le e^{\gamma s}e_0+|\eta|M\varphi(\gamma,s),\qquad \varphi(\gamma,s)=\begin{cases}(e^{\gamma s}-1)/\gamma,&\gamma\ne0,\\ s,&\gamma=0.\end{cases} \]

Initially the paired error is zero because the source is identical. The retained Picard image initial_box+[0,h]*direction*F_0(tube) gives strict clearance of the reference path inside each tube, also after reflection and reverse traversal. Keeping every coordinate projection sqrt((W^-1)_ii)*max e(s) below that clearance closes the common-domain premise by first exit. On target tubes, subtract the sum of both endpoint phase-error bounds from every strict acute margin. The block symmetry from Section 51 supplies the needed direction change, and its retained rates already contain the original clock factor 1/pi.

Numerical enclosure error and intermediate preparation uncertainty enter those retained domains and margins; they are not extra constitutive forcing. A failed sufficient bound would only leave the comparison unavailable. Here the stronger conserved-budget obstruction already settles the fixed-scale question, so no 471-step defect-transport campaign, coefficient reduction or new trajectory is needed. Neither this negative comparison nor critical-geometry agreement selects a fundamental law or establishes physical identification.

The detached research reader assess_sine_constitutive_robustness(evidence_directory) returns SineConstitutiveRobustness. It re-admits the frozen source, preparation, support and required endpoint evidence before rebuilding the changed storage and sector barrier. It neither installs the changed law nor executes a new forecast.