One source-preserving enclosure of the organized receiver

This owner defines one prospective finite enclosure for the unchanged two-donor preparation. It asks whether the initially unprepared receiver acquires winding one while both donors retain that identity and the complete state enters the joined capture region. It also admits a certified donor-loss outcome. Failure of a sufficient enclosure or exhaustion of its frozen budget is unavailable evidence, not proof that receiver acquisition is impossible.

The slow-admission theorem supplies the source-inclusive phase inclusion. The numerical reference below is an instrument for enclosing the actual complete law; it is not a replacement phase-only model and does not discard form or preparation errors. The research program remains the sole task queue.

Complete model and retained source

Use exactly the source and support of the receiver-mechanism theorem: three labelled C9 regions, unit internal weights, and reciprocal contacts from the oriented region pairs \((0,1),(1,2),(2,0)\) with offsets zero and one, each of weight one half. All 27 nodes have weighted degree four. Support is present from preparation; capacities remain one. With \(A=L/4\),

\[ F_i(\theta)=\frac14\sum_jw_{ij}\sin(\theta_j-\theta_i),\qquad x_\tau=-Ax+\gamma F(\theta),\qquad \theta_\tau=\gamma Ax, \quad \gamma=\frac1{1023\pi},\quad \tau=\frac{1023}{1024}t. \]

The law is detached reciprocal sine dynamics, without forcing, events, resets or changing conductances. It does not execute a registered operator.

The original form ramps remain on regions zero and one only:

\[ x_{r,j}(0)=m(j-4)+\eta_{x,r,j}\quad(r=0,1),\qquad x_{2,j}(0)=\eta_{x,2,j},\qquad \theta_{r,j}(0)=\eta_{\theta,r,j}, \quad m=\frac{2046}{9}(355/113)^2. \]

Every form and phase error has magnitude at most \(10^{-10}\), with separate exact zero sums in each original region. They need not share the nominal symmetry. Only the global form and phase means are conserved during the connected evolution. Retain the continuous phase lifts and the original storage, including the initial contact cost; no recovered-source certificate is substituted.

The earlier theorems certify donor winding one at \(\tau=200\), its retention through \(\tau=400\), and receiver winding zero at release. Set

\[ u=\gamma^2(\tau-400),\qquad P=\alpha(q,q,0),\quad \alpha=2\pi/9,\quad q_j=j-4. \]

For every allowed original source, the freshly reconstructed release bounds supply

\[ \|\theta(0)-P\|_2\le b_0,\qquad \theta_u=F(\theta)+r(u),\quad \|r(u)\|_2<\rho=11/10^6, \qquad \|x(u)\|_2\le X_0. \tag{1} \]

Here \(b_0\) and \(X_0\) are the outward source bounds at structural time 400, not newly chosen preparations. The global means of \(\theta\), \(F\) and \(r\) are zero. The actual defect \(r\) retains the form memory specified by the exact second-order law. The inclusion merely bounds that defect; it does not supply an arbitrary forcing to the original model.

Frozen numerical policy and observations

The finite policy is fixed before evaluating any reference response:

Item	Declared value or rule
Elapsed slow horizon	\(u=32\), starting at \(\tau=400\)
Step and maximum count	\(h=1/8\), at most 256 steps
Taylor order	Eight, with order-nine whole-tube remainder
Arithmetic	Shared outward 128-bit dyadic rational intervals
Reference	Thirteen nominal reflection/sign coordinates
Complete-state uncertainty	One Euclidean phase ball on the full 26-dimensional global mean-zero leaf, plus the carried form norm \(X_0\)
Velocity discrepancy	Fixed \(\rho=11/10^6\), with original-source admission rebuilt
Local growth search	Fixed bracket \([-1,2]\), eight bisections; no bracket expansion
Cosine majorant	Downward 20-bit dyadic bounds, minimized on reflected edge orbits
Prospective phase radius	\(5/4\) times the global one-step envelope plus \(2^{-80}\) inside that factor
Uncertainty budget	Stop if the carried endpoint phase-error radius exceeds \(1/2\)
Adaptation	No step refinement, horizon extension, source retuning or alternative route
Positive observation	Continuous donor retention and one common endpoint passing full target-cell and storage-capture conditions
Negative observation	A robust endpoint winding different from one for at least one donor
Other stops	Validation failure, uncertainty budget or horizon: unavailable for the requested acquisition claim

The original-time endpoint corresponding to the full horizon is \((1024/1023)(400+32/\gamma^2)\). This is a structural time, not laboratory seconds. The single frozen run retains the protocol, source/configuration, ordered support, exact bounds, partial valid prefix, response and verdict. A protocol identity associates those artifacts; it does not authenticate an external physical preparation.

A nominal symmetry that does not restrict source errors

Let \(R\) be the node permutation

\[ R(0,j)=(1,8-j),\quad R(1,j)=(0,8-j),\quad R(2,j)=(2,8-j). \]

It preserves the complete weighted support, has thirteen two-node orbits and one fixed node \((2,4)\), and satisfies \(R^2=I\). Because sine is odd and the support is invariant, \(F(-Rz)=-RF(z)\). Thus the antisymmetric space \(\mathcal V_-:=\{z:Rz=-z\}\) is invariant for the reference equation \(y_u=F(y)\), and \(P\in\mathcal V_-\).

Choose one node \(a_k\) from each pair and define \(C_-e_k=e_{a_k}-e_{Ra_k}\). The nominal reference can be integrated in thirteen coordinates, with lift \(y=C_-v\) and

\[ C_-^TC_-=2I,\qquad \|C_-v\|_2^2=2\|v\|_2^2. \tag{2} \]

The full initial-error family is not projected onto this space. Its transverse directions stay in the radius carried under (1). Choosing a symmetric reference changes the cost of the calculation, not the quantified source.

At the initial reference, materialize a rational midpoint of the interval enclosure of each retained coordinate of \(P\). Its lifted outward rounding error is added to \(b_0\). At every later step, the rational reference center comes from the preceding Taylor endpoint enclosure. Its complete lifted recentering error is added to the carried radius. No recentering resets the actual uncertainty to zero.

Validated reference and full-state error transport

At one step, let \(c\in\mathcal V_-\) be the rational reference center and let \(e\) bound the Euclidean difference between the actual phase and \(c\). The shared metric Taylor owner is called on the thirteen coordinates with metric \(2I\), initial radius zero, growth bound two, duration \(h\) and order eight. Strict Picard inclusion gives a whole-time reference tube; the normalized solution jets and whole-tube order-nine remainder give its endpoint enclosure. The metric is exactly the lifted norm (2).

Radius zero applies only to this local exact reference solution. The actual source error \(e\) remains external and is propagated by the following comparison. In particular, the reference validation is not evidence that the actual full law starts at its center.

The global Jacobian bound \(\|DF\|_2\le2\) holds on all phase lifts. Writing

\[ \phi(\mu,s)= \begin{cases}(e^{\mu s}-1)/\mu,&\mu\ne0,\\s,&\mu=0,\end{cases} \]

it gives the prospective global discrepancy bound

\[ G=e^{2h}e+\rho\phi(2,h),\qquad R_*=\tfrac54(G+2^{-80}). \tag{3} \]

All arithmetic in (3) is outward. The convex sum of the lifted reference Picard box and the mean-zero ball of radius \(R_*\) contains both local solutions throughout the step. The strict margin follows from \(R_*>G\), not from presuming the validity of a future local growth estimate.

For each fine edge, its reference phase difference is bounded from that box; the complete discrepancy enlarges the edge interval by \(\sqrt2R_*\) on both sides. This factor is the norm of the edge incidence vector. It retains arbitrary mean-zero errors, including errors outside \(\mathcal V_-\).

Let \(\ell_e\) be a lower bound for cosine on the enlarged edge interval. Round it downward to the declared dyadic grid, with the exact global floor \(-1\), and take the minimum over the reflected edge orbit. For every point in the prospective convex domain,

\[ DF=-\tfrac14 B\operatorname{diag}(w_e\cos\delta_e)B^T \preceq J_*:=-\tfrac14B\operatorname{diag}(w_e\ell_e)B^T. \tag{4} \]

This Loewner bound follows term by term from \(w_e(\cos\delta_e-\ell_e)b_eb_e^T\succeq0\), including edges with negative cosine. Independent interval matrix entries are not mistaken for an exact Jacobian.

The orbit minimum makes \(J_*\) commute with \(R\). Both \(J_*\) and the actual Jacobians annihilate the common-phase vector. To certify an upper logarithmic growth rate on the entire mean-zero leaf, the calculation uses both reflection sectors. The antisymmetric sector has basis \(C_-\) and Gram matrix \(2I\). A basis for the symmetric mean-zero sector is

\[ C_+e_k=e_{a_k}+e_{Ra_k}-2e_{(2,4)},\qquad C_+^TC_+=2I+4\mathbf1\mathbf1^T. \tag{5} \]

These orthogonal sectors each have dimension thirteen and together span all 26 mean-zero directions. Hence the two exact rational PSD tests

\[ \mu C_\pm^TC_\pm-C_\pm^TJ_*C_\pm\succeq0 \tag{6} \]

prove \(DF|_{\mathbf1^\perp}\preceq\mu I\). Testing only the antisymmetric block would omit source directions and is insufficient. The upper search endpoint two is valid because \(\ell_e\ge-1\) and \(J_*\preceq L/4\preceq2I\). The lower search endpoint minus one is a search policy, not an asserted lower bound on the actual logarithmic norm. Each retained rate is checked by exact rational semidefinite elimination.

Pairing the actual-reference difference equation with its normalized error and using (1), (4)--(6) yields

\[ D^+\|\theta-y\|_2\le\mu\|\theta-y\|_2+\rho. \]

Therefore a whole-step discrepancy bound is

\[ E(s)=e^{\mu s}e+\rho\phi(\mu,s),\quad 0\le s\le h, \qquad E_{\max}=\max\{e,E(h)\}. \tag{7} \]

The scalar solution is monotone or constant on a step, so its maximum is at an endpoint. Outward scalar exponential enclosures handle either sign of \(\mu\). Strict inclusion \(E_{\max}<R_*\) is checked explicitly. An unavailable inclusion stops the validated prefix; it cannot be replaced by a favorable endpoint alone.

Let \(d\) be the full \(2I\)-metric Taylor endpoint recentering error. The next exact rational reference center and carried full phase radius obey

\[ e_{\rm next}=E(h)+d. \tag{8} \]

No observation box is converted back into a smaller radius, and neither the Taylor error nor the complete-law defect is discarded. The reference may change at the step boundary; (8) accounts for its entire jump. The original complete trajectory itself has no reset or discontinuity.

Identity, capture and the meaning of a failed margin

For continuous donor retention, use the release winding-one integers on each fixed internal donor edge. The whole reference tube, enlarged by \(\sqrt2E_{\max}\), must keep every registered edge strictly within \((-\pi,\pi)\). The previously proved time-200 to time-400 bridge covers the interval before this calculation starts. Receiver branch crossing is allowed: imposing the receiver's original winding-zero cell at every step would exclude the requested acquisition by construction.

A failed sufficient donor margin is recorded permanently for the positive whole-history claim, but does not end the enclosure. The actual state may still be enclosed at later steps, and its donor winding may become decidable there. For an endpoint, freshly determine principal-difference integers with strict branch margins for every internal edge. Their signed sum gives the integer winding; do not infer it from a rounded phase plot.

If a donor's entire endpoint ball has winding different from one, the continuous full law must have lost that donor identity after its certified acquisition. Since the ball encloses every original admitted source, such a verdict is a whole-family donor-loss result. It refutes the requested donor-retaining acquisition claim for this preparation and law. It does not prove that the receiver can never acquire winding, that other preparations cannot do so, or that all compatible TNFR laws share the outcome.

For positive joined capture, the endpoint center \(c_*\) and radius \(e_*\) must pass the same complete-state contract:

\[ \max_{\text{fine }e}|\delta_e(c_*)|+\sqrt2e_*<2, \tag{9} \]

\[ U(c_*)-U(\Theta)+\|\nabla U(c_*)\|_2e_* +4e_*^2+4X_0^2<196/625,\qquad \Theta=\alpha(q,q,q). \tag{10} \]

The fine-edge integers in (9) are those registered to \(\Theta\), not freely chosen local seams. The potential, gradient and form contribution refer to the same complete weighted support and same endpoint. The linear gradient term in (10) remains present. Joined capture plus certified continuous donor history gives the requested positive result and invokes the existing future trapping and convergence theorem.

An endpoint capture inequality without that history may establish an endpoint property, but does not repair unavailable donor retention. Likewise, a reference donor losing winding without the complete error margins proves nothing about the actual source family. Numerical validation failure, phase-error budget exhaustion, unresolved winding or a horizon without capture remain explicit unavailable outcomes; none is silently promoted to nonformation.

Evaluated result: loss of both donor identities

The first response to the frozen protocol certifies whole-family donor loss. The response archive preserves the first outcome JSON and every original step JSON byte-for-byte at their recorded repository-relative paths. Its outcome contains the complete source admission, ordered step manifest and terminal observation. This lossless transport does not replace or regenerate any evaluated value. The freeze record and source archive preserve the evaluated implementation and policy. There were 76 accepted steps, with no refinement or retry. The declared donor-loss stopping rule ended the calculation at

\[ u_*=19/2,\qquad \tau_*=400+\frac{19}{2\gamma^2},\qquad t_*=\frac{1024}{1023}\left(400+\frac{19}{2\gamma^2}\right). \]

The final carried Euclidean phase radius is approximately \(0.0031002376916330595<0.00311\). Every source in the original error family has endpoint windings \((0,0,0)\). Each donor's principal branch margin at that endpoint exceeds \(1/20\), including the complete source error and velocity defect. This is a validated property of the actual complete law, not a winding read from the nominal reference alone.

The retained history distinguishes the loss proof from the earlier failure of a sufficient corridor:

Retained endpoint	Actual winding enclosure	Continuous donor-history evidence
Step 74, \(u=37/4\)	\((1,1,0)\)	Both donors retained through this entire prefix
Step 75, \(u=75/8\)	\((1,1,0)\)	The whole-step donor corridor is unavailable; its endpoint remains certified
Step 76, \(u=19/2\)	\((0,0,0)\)	Both donors have definitely lost winding one

Thus, for each original source and each donor, a winding-one to winding-zero transition occurs between \(75/8\) and \(19/2\). The first donor loss is enclosed by \([37/4,19/2]\): retention is proved through the former time, and loss is established by the latter. The narrower endpoint interval does not establish the first or unique crossing, because the step-75 interior margin is unavailable. That unavailable margin was not treated as a negative verdict; validation continued to the strict endpoint loss certificate.

A further consequence follows from the retained full-time tubes without a new trajectory evaluation. Reconstructing their original receiver edge differences and enlarging by the certified \(\sqrt2E_{\max}\) gives a strict winding-zero branch margin greater than \(13/10\) on every one of the 76 intervals. The receiver therefore remains winding zero throughout \(0\le u\le19/2\), including the donor losses. This is an additional retained-tube consequence, distinct from the prospectively declared terminal donor-loss verdict. Together with the earlier initial-layer exclusion, the receiver has not acquired winding one anywhere from preparation to this stop.

The requested acquisition with donor retention and joined capture is therefore false for this full source family under this complete law. Initial motion toward the donor phases, reciprocal interaction, monotone complete storage and the existence of the joined attracting target are insufficient to establish that claim. The loss concerns the declared regional winding identity; winding zero does not imply that the phase state is uniform or that every possible organization has disappeared. Neither future receiver acquisition nor global synchronization is decided by this finite result.

The three-ramp acquisition theorem remains valid under its different preparation. This result supplies a counterexample to transferring its acquisition verdict to a locally unprepared receiver, even though support, constitutive law and attracting target are shared. Supplied support, donor preparation and structural clocks remain premises; no physical identification or unique TNFR law follows. The execution plan owns the subsequent foundational question.