Predictive repeated composition with retained port association¶
Four winding-one C9 organizations, grouped into two marked two-child parents, admit a useful four-coordinate parent description for one finite interaction. The exact registered restriction reuses the organized inheritance method. The additional result is a uniform predictive bound for independent fine deviations, with child and parent identity retained, and an explicit obstruction to exact parent closure when child-to-port association is discarded.
This is a conditional theorem for the supplied positive-loss sine law. The support, grouping, relative preparation and structural clock are supplied. The two parents interact through their actual boundary; no prescribed forcing trajectory, new solver or evaluated numerical response is used.
1. Fixed question, source and observations¶
The test fixes the following quantities before assessing the response:
| Item | Declared value or scope |
|---|---|
| Children and parents | Four marked C9 children; two children in each parent |
| Complete law | Sine potential, held unit capacities, weighted degree four |
| Structural clock | \(\tau=(1023/1024)t\), \(\gamma=1/(1023\pi)\) |
| Initial parent forms | \(X_0=X_1=0\) |
| Initial parent origins | \(\Phi_0=-1/40,\ \Phi_1=1/40\), in declared continuous lifts |
| Fine residual family | Parentwise mean-zero form and phase residuals, each coordinate at most \(\rho=1/100\) in magnitude |
| Interaction controls | Matching outer ports, or equal weights on offsets \(0,-1,1\) |
| Horizon | \(0\le\tau\le1\) |
| Selected readout | Parent-zero mean form increment \(X_0(1)-X_0(0)\) |
| Required prediction error | Less than \(10^{-7}\) in the declared form chart |
| Child identity | Winding one on every marked internal C9 |
| Parent identity | Its two children retain those identities and every same-slot intraparent phase gap has magnitude less than \(1/4\) |
The parent identity is a declared structural observation, not a new definition of every NFR or a parent winding number. Its selected interaction must remain predictable while its children differ. A stationary nested target alone would not answer this question. These are structural units and mathematical observations, without a laboratory clock, physical energy or sensor bridge.
2. Complete 36-node model and the two grouping maps¶
Label nodes \((g,a,j)\), with parent \(g=0,1\), child \(a=0,1\) and \(j=0,\ldots,8\), in order \(18g+9a+j\). Put \(\alpha=2\pi/9\) and \(\psi_j=\alpha(j-4)\).
Each child has its nine unit cycle edges. Within each parent, add the nine unit edges \((g,0,j)--(g,1,j)\). Between parents, join \((0,a,j)--(1,a,j+s)\), with indices modulo nine:
Both graphs have weighted degree four and total undirected conductance 72. The symmetric control has more distinct neighbors, but its weighted degree does not change. All diffusion and sine sums below consume conductance. This is not the native resultant-direction pressure or its neighbor-count normalization.
For the actual weighted Laplacian \(L\), define
Both rows acquire the factor \(1023/1024\) in the original clock \(t\). The signed form, primitive phase and held capacities retain the foundation contracts. In this chart the nodal pressure is independently supplied by the first row, not reconstructed from a measured derivative.
The complete storage and loss are
This follows from \(\nabla_xH=Lx,\ \nabla_\theta H=-S\) and cancellation of the reciprocal terms. The two global lifted means are conserved.
The first map takes each child's form mean and the mean of \(\theta_{g,a,j}-\psi_j\). Its exact family is
The second map averages the two child forms and the two registered child origins in each parent. Its exact family additionally imposes \(X_{g,0}=X_{g,1}=X_g\) and \(\phi_{g,0}=\phi_{g,1}=\Phi_g\). The composite observation therefore retains four coordinates from 72.
These phase means use declared lifts and marked slots. They are not angles of the fine first resultant, which vanishes for an ideal winding-one C9. Independent child relabeling with contacts held fixed is not a symmetry. The whole first and second families are invariant, as shown next; neither map is asserted to be an exact quotient outside its family.
3. Exact inherited rows, clock and storage¶
For \(\bar g=1-g,\ \bar a=1-a\), substitution in (2) gives
Internal cycle currents cancel. Every offset is a bijection on the marked cycle, and the symmetric offsets cancel the cosine harmonic. Thus both fine rows are constant within each child. Smooth uniqueness preserves (4). The first-stage storage is
The inner and outer phase coefficients differ. Replacing them by a single unweighted graph coefficient would change the inherited model.
On the second family put \(u=X_1-X_0,\ \Delta=\Phi_1-\Phi_0\). Then
These rows also preserve the second family. The pullback of (3) is
The complete normalized pair chart uses
In that clock both rows have the normalized reciprocal pair form: \(dZ_0/d\sigma=Z_1-Z_0+\gamma_{\rm eff}\sin\Delta\) and \(d\Phi_0/d\sigma=\gamma_{\rm eff}(Z_0-Z_1)\), with opposite rows for parent one. The normalized relative storage is \((H_4-B_\kappa)/(18\kappa)\). Changing only the form coefficient or only the clock would not give this equivalence. Fine capacity remains held at one; (9) does not select a new autonomous capacity law.
4. What the second grouping loses¶
Choose \(h=1/200\), retain zero forms, and set
Compare preparation A with \((\eta_0,\eta_1)=(h,h)\) and preparation B with \((\eta_0,\eta_1)=(h,-h)\). Both belong to the declared fine source. Their four parent coordinates agree. Each parent's unordered set of child origins agrees, as does its child-origin resultant magnitude \(\cos h\). Each has internal storage \(18(1-\cos\alpha)+9[1-\cos(2h)]\) and retains both child windings. This resultant concerns the child origins, not the vanishing fine C9 resultants.
The actual average outer current into parent zero is
It equals \(\kappa\sin\Delta\) in A and \(\kappa\sin\Delta\cos(2h)\) in B. Their parent-zero form rates differ by
Thus parent means, separate unordered child states and internal parent storage cannot define an exact four-coordinate law on this source. The missing information is the association between the states meeting at the marked ports. Complete outer storage is not claimed equal in A and B. This is a specific second-grouping obstruction, not an inference from overlapping error bounds. It remains compatible with an accurate approximate parent law because the missing current is quadratic in small internal offsets.
5. A full transverse domain and identity bound¶
At every time let \(X_g\) and \(\Phi_g\) be the actual parent means, and write
The residual sums vanish separately within each parent by definition. At preparation the fixed parent means are exact; all 68 remaining form/phase directions are allowed within the coordinate bounds. No child symmetry or first-stage-family membership is assumed.
Let \(\Pi\) average each parent and \(Q=I-\Pi\). The equitable weighted support gives \(L\Pi=\Pi L\). The exact transverse equations are
The ideal current is constant within each parent. Let \(r_x=\|\xi\|_\infty,\ r_\theta=\|\eta\|_\infty\). For the mean outer current, the Taylor-linear terms cancel separately for every offset: its bijection pairs two parentwise zero sums. Consequently
Indeed, each sine Taylor remainder is at most half the squared edge residual, and each residual difference has magnitude at most \(2r_\theta\). Internal currents cancel exactly in the parent mean.
The nodewise sine difference from the ideal family is at most \(8r_\theta\), since weighted degree is four. Subtracting the parent mean therefore gives \(\|QS\|_\infty\le8r_\theta+2r_\theta^2\). The heat semigroup generated by \(-L/4\) contracts the infinity norm. Duhamel's formula and \(\|L\|_\infty=8\) now give, while both residuals are at most \(R\),
Fix
Using \(\gamma<1/3200\), both right-hand sides of (16) are strictly below \(R\) for \(\tau\le1\). A first-exit argument proves the bound throughout the declared interval. The field is globally Lipschitz; no hidden continuation or phase-domain premise is needed for (2).
Every internal cycle edge has a continuous representative in \([\alpha-2R,\alpha+2R]\subset(0,\pi/2)\). The closing edge has raw carrier gap \(-8\alpha\); its retained branch integer adds \(2\pi\), giving the same representative \(\alpha\). Their oriented sum is \(2\pi\), because the residual differences telescope, so every child retains winding one. Every same-slot intraparent gap has magnitude at most \(2R<1/4\). This proves the declared parent identity for the entire fine source and horizon, not only the nominal stationary geometry.
6. Quantitative parent dynamics and resolved interaction¶
The actual parent means satisfy (7) with \(\kappa\sin\Delta\) replaced by \(J\) in the form rows. The diffusion and phase rows remain exact for all fine states. The four-coordinate unforced predictor is (7), initialized from the same parent observations.
To compare this predictor with the actual means, use conservation \(X_1=-X_0\). In the fixed form/phase chart \((X,\Phi)\), the error \(e=|X_0-\bar X_0|+|\Delta-\bar\Delta|\) obeys
The form inequality retains the nonpositive diffusion contribution; the phase inequality contributes \(\gamma|X_0-\bar X_0|\). Thus for \(\tau\le1\),
The last step uses \(e^z-1\le z/(1-z)\) for \(0\le z<1\) and \(\gamma<1/3200\). This is a controlled approximation despite the exact closure obstruction (12); it is not a four-coordinate exact quotient.
An explicit finite readout avoids solving even the reduced trajectory. Write \(\delta=1/20\) and define
Since the average outer current has \(|J|\le1\), variation of constants in the exact mean rows gives
Together with (15), the exact convolution for the form yields
At \(\tau=1\) this is less than \(1/60000000<10^{-7}\), uniformly over the fine source. Equation (22) includes both the internal-state defect and the change in parent phase; it is distinct from (19)'s error against the complete nonlinear reduced law.
The prescribed matching/symmetric control is resolved. Elementary strict bounds \(\gamma>1/3300\), \(1-\kappa_{\rm symmetric}>3/20\), \(\sin(1/20)>49/1000\) and \(1-e^{-1/2}>3/8\) imply
Subtracting both full-source errors proves
The result even allows independent members of the same admitted source family in the two controls; pairing identical sources is included. The prospectively allowed intervals \(P_\kappa(1)\pm10^{-7}\) are disjoint. This is a finite analytic discrimination with structural units, not an evaluated laboratory reading or a new numerical trajectory.
7. Carried acquisition and work at the second attachment¶
The broad residual domain above is independently supplied. It also contains two actual acquired parent families under additional stated preparation. For each parent, separately rebuild the existing matching-interface acquired-pair admission at deformation zero from its original primitive form/phase error allowances. The original \(10^{-10}\) allowances pass its isolated acquisition, fixed recovery dwell and first attachment admission. The all-future parent relative norm is bounded by \(15\cdot10^{-70}\), which is below \(\rho\).
The original parent's global means are conserved. Translating its entire phase by \(-1/40\) or \(1/40\) preserves its disconnected internal law and storage. These relative origins are additional supplied preparation information, not an autonomous choice or the unchanged all-aligned source. After this preparation, adding the outer contacts changes the degree from three to four. Equations (14)-(24), not the old degree-three dynamics, admit the ensuing interaction. No residual is reset to its target. The inherited long dwell remains a mathematical premise, not demonstrated laboratory preparation.
For any actual fine state the outer support-event work is exactly
Its nominal value at the fixed preparation is
The same zero-sum cancellation used in (15), Taylor's bound \(|\cos''|\le1\), and total outer conductance 18 give
Here the radii bound actual initial residuals. The exact observer retains their work, while acquired-source bounds retain the tiny correction separately from irrational nominal intervals. The controls have different attachment costs; equal degree is not equal preparation energy.
If support is supplied from preparation, this cost is already part of the initial storage. If it is added later, the actual jump needs an external work supply. No continuous loss is reused as an event reservoir. Neither this work identity nor successful maintenance selects the attachment event, forms the parent autonomously or establishes physical binding.
8. Executable evidence and limits¶
The detached owner rebuilds the fixed weighted geometry, full rows and work from primitive form/lift inputs. It encloses coefficients and finite analytic bounds with shared outward rational arithmetic. Source admission reconstructs the two parent reports from primitive allowances; cached verdicts are not premises. Within one static observation, identical admitted phase-gap intervals share their sine/cosine evaluations. This bounded local reuse preserves every outward-rounded operation and is discarded before the next observation. The independent controls compare direct weighted edge sums, both grouping stages, the port-association witness, bootstrap/resolution boundaries and original-source admission.
The accepted advance is finite predictive composition for perturbed organized parents, together with a concrete limit on discarded port information. It neither selects the sine law nor transfers this result to the native runtime, arbitrary events, other interfaces or indefinite horizons. Constituent formation, supplied parent preparation, conditional maintenance, autonomous assembly and physical identification remain distinct. The separate parent-acquisition theorem admits entry from four independently uncertain acquired children under the matching control, then predicts from the actual reached state. Its longer source-specific horizon retains correlations absent from the generic residual ball here; it does not extend this theorem by changing only its clock or radius.