Degree-aware composition of reduced sine-class ports¶
Prospective composition claim and frozen protocol¶
The two-component reduction admits one central contact per component. This gate asks whether the derived component description composes on a finite simple undirected unit contact graph, with the actual contact degrees, an exact joined storage balance and a uniform finite-window approximation of the complete fine law. It does not ask for another donor-class response comparison.
Freeze this single control before evaluating its certificate:
- Use three simple unit C9 components, in component order \((0,1,2)\) and local node order \(0,\ldots,8\), with winding classes \((1,2,1)\). The supplied contact edges are \((0,1)\) and \((1,2)\), each joining the local central nodes \(4\). Their contact degrees are \((1,2,1)\); the joined fine port degrees are \((3,4,3)\).
- Retain unit capacities, beta one, \(e=1023/1024\), \(w=1/1024\), the structural clock \(\tau=et\), and \(\gamma=1/(1023\pi)\). The complete rows on each live support are
[ x'=-KLx+\gamma KS(\theta),\qquad \theta'=\gamma KLx, ]
where \(K\) is inverse fine degree, \(L\) is the unit Laplacian, and \(S_a(\theta)=\sum_{b\sim a}\sin(\theta_b-\theta_a)\). No edge weights, loss changes, inputs or occurrence law are inferred. - Retain the original class-specific form sources \(k_i m(j-4)\), \(m=(2046/9)(355/113)^2\). Nominal initial phases are constant on each component, with supplied origins \((o_0,o_1,o_2)=(0,1/1000,0)\). Each component has independent per-coordinate initial form and continuous phase-lift errors at most \(10^{-10}\), subject to zero sum separately in each channel. Retain all sixteen relative source coordinates per component and its original storage ceiling \(10^9\); source costs need not agree. - Freshly admit formation of both classes at \(\tau=100\) and relative radius \(r=1/12\). Continue the actual unprobed trajectories for \(D=10^{13}\). Require Euclidean form and target-phase error bounds \(\epsilon=10^{-32}\), separately on every component, using the exact decay bound \(2^{-512}\) only if \(512\le\kappa D\le4096\). The original common origins are retained; no target reset is made. - At \(\tau_c=100+D\), add the two unit contacts simultaneously, leaving every state coordinate unchanged. Declare total event-work allowance \(2\,10^{-6}\), the sum of two declared per-edge allowances. Hold the joined support and complete law afterward. Require full joined-family acute identity retention and recovery on its actual postevent conserved-mean leaf. - Derive the thirty-coordinate surrogate, ten coordinates per component, from the actual postcontact degrees. Retain the class tangent only for internal cycle currents and the exact nonlinear bridge currents. The full comparison has fifty-four coordinates. The nominal surrogate starts at zero form and zero displacement from the limiting twists and supplied origins; actual fine-state errors remain in the approximation bound, not in a reset. - Require the maximum absolute error in each fine form and phase coordinate, relative to the lifted surrogate, to stay strictly below \(2\epsilon=2\,10^{-32}\) throughout \(0\le\tau-\tau_c\le h=1/100\). This is a full-state mathematical approximation allowance in the fixed normalized coordinate units, not a sensor error or a new response contrast. - Retain the exact degree-normalization control: with all phase deviations and origins zero, middle port form one and every other retained form zero, the middle form rate is \(-1\). Reusing the one-contact denominator three gives \(-4/3\) and a nonzero actual-degree-weighted form-charge rate \(-4/3\). Require the degree-aware assembly's exact storage/loss and charge identities, rather than a tolerance-based pass for this rational control. - Preserve source and protocol before the first reserved control evaluation. Use admitted exact primitives and outward rational interval bounds. Keep the tiny squared handoff bounds and decay as exact rationals. No trajectory integration, topology search, horizon/precision sweep or refitted component coefficient is admitted. Pass only if fresh formation, every actual handoff, contact work, full identity, algebraic controls and the strict whole-window approximation allowance all pass unchanged.
The reusable theorem below concerns fixed finite simple undirected unit central-contact graphs. It does not extend the law to weighted support, arbitrary port locations or changing contacts during the certified window. Supplying contacts and organized sources does not derive their occurrence. The long structural dwell and source accuracy have no laboratory bridge.
Component coordinates, contact degree and origins¶
Let \(\mathcal G\) be a fixed finite simple undirected unit graph on \(m\) components. Give component \(i\) class \(k_i\in\{1,2\}\), contact degree \(d_i\), and one declared real common phase origin \(o_i\). Each contact joins the central nodes, so it changes only their fine degrees, from two to \(2+d_i\). With \(\alpha=2\pi/9\), use the centered reference twist \(\Theta_{i,j}=k_i\alpha(j-4)\) and \(c_i=\cos(k_i\alpha)>0\).
Use the reflection projection and lift \(S,T\) on the five layers \((4),(3,5),(2,6),(1,7),(0,8)\). They obey \(ST=I_5\), \(TS=(I+R)/2\), and have maximum-norm operator norm one. For a fine state, the coordinate map is
The reduced trajectory defined below is a surrogate, generally different from the projection of an evolving nonlinear fine state. Its lifted phase is \(\Theta_i+o_i\mathbf1+T\bar y_i\). In particular, the common origin is not also included in \(\bar y_i\).
Let \(L_0=2L_{P_5}\), twice the five-node path Laplacian. The edge between fine nodes zero and eight has zero difference on an even lift. The exact reduced mobility mass and internal operator are
Thus the first row of \(A_i\) is \((2,-2,0,0,0)/(2+d_i)\). The other four rows are unchanged from the one-contact reduction. The class coefficient is inherited from the cycle; mobility additionally depends on the declared interface. Treating the one-contact matrix as independent of contact degree would change the complete law, rather than reuse its reduction.
For each component aggregate the two definite input channels
With \(e_0\) the central layer vector, the component equations are
Both rows use the same postcontact degree. The state has ten coordinates per component rather than eighteen. Disconnected contact graphs and an isolated component also define these equations; a single global recovery certificate will separately require connected support.
All edge origins must come from the same vector \(o\). Their signed sum around every contact cycle is zero as a real lift. Independent edge offsets with a nonzero cycle sum would introduce a different frustrated model. Common shifts of all \(o_i\) are immaterial to rates. Replacing \(o_i\) by \(o_i+a_i\) and \(\bar y_i\) by \(\bar y_i-a_i\mathbf1\) preserves each reconstructed phase and all rates, since \(L_0\mathbf1=0\). Wrapping arbitrary phase deviations before applying \(L_0\) is not this coordinate change.
Exact assembled storage, dissipation and charges¶
Stack the retained vectors, set \(M=\operatorname{diag}(M_i)\), and let \(b_{ij}\) be the retained vector with entries plus one and minus one at the two port layers. Define
The first is exactly \(T^{\mathsf T}L_{\rm fine}T\) for the block lift. Internal potentials are tangent quadratics; bridge potentials remain the original nonlinear sine potentials. Equation (2) becomes
Consequently the reduced storage
satisfies, for every retained state and every fixed admitted contact graph,
The two mixed terms cancel by symmetry of \(M^{-1}\). This is a derived balance of this surrogate, not an assertion that its storage equals the full sine storage away from the reference geometry.
Both \(\mathbf1^{\mathsf T}M\bar x\) and \(\mathbf1^{\mathsf T}M\bar y\) are constant: the gradients and Laplacian have zero total sum. The individual component charges instead obey
They cannot be held separately fixed after attachment.
The observed port values are not automatically energy-conjugate outputs. For internal storage alone, put \(a_i=L_0\bar x_i\) and \(b_i=c_iL_0\bar y_i\). Its open-port supply term is
The internal dissipation is \(-a_i^{\mathsf T}M_i^{-1}a_i\). Summing internal storages alone omits bridge form storage as well as bridge potential. Equation (4) includes both and provides the joined balance; it does not identify \(u_x\bar x_0+u_\theta\bar y_0\) as physical power.
A rational control rejects the unchanged one-contact assembly¶
For the frozen three-component path, set middle port form to one and every other retained form to zero, with all deviations and origins zero. The middle port has four neighbors, so its exact form rate is \(-1\). The adjacent internal layer has rate \(1/2\), and each endpoint component's port has rate \(1/3\). These give actual-degree-weighted charge rate \(-4+2+1+1=0\).
Keeping denominator three at the middle port produces \(-4/3\). Its other rates above remain unchanged, so the actual weighted charge rate becomes \(-16/3+2+1+1=-4/3\). The corresponding phase-charge rate is \(4\gamma/3\), rather than zero. The correct storage loss at this state is \(-17/3\); the erroneous mobility gives \(-7\). Thus nonincreasing storage under some changed metric would not validate the declared model. This exact algebraic control is independent of a numerical trajectory or the reserved formation-family bounds.
Uniform error against the full nonlinear fine law¶
Extend equation (2) to all nine coordinates per component solely as a proof device: linearize internal cycle sine currents about their own twist, retain exact bridge sine currents, and keep the actual fine mobility. Independent reflections fix every central port. This extended surrogate preserves their even subspace, and projection/lift of that subspace gives (2) exactly. The full nonlinear sine law does not generally preserve the same even subspace: quadratic internal terms can generate odd deviations. No such invariance is assumed below.
Write the full ideal comparison as \(x=\gamma v\), \(\theta=\Theta+o+y\), with \(v(0)=y(0)=0\), and put \(\eta=\gamma^2\), \(A=KL\). The diffusion semigroup \(\exp(-At)\) is a maximum-norm contraction, \(\|A\|_\infty\le2\), and the normalized sine field is globally Lipschitz with constant two. Its reference forcing is supported at the central ports. Define
This retains the algebraic sum of incident currents, rather than fitting a forcing from the evaluated response. Suppose
Duhamel's formula and \(y'=\eta Av\) give
For \(B_h=\sup_{0<t\le h}\|v(t)\|_\infty/t\), these imply \(B_h\le\sigma+(2/3)\eta h^2 B_h\). Smoothness at zero and global existence justify the finite supremum. Therefore
On each internal edge the first-order sine remainder has magnitude at most half the squared deviation gap. Each gap is at most \(2\|y\|_\infty\), and internal degree divided by actual fine degree is at most one. Hence the full normalized internal-field defect is at most \(2\|y\|_\infty^2\), irrespective of the number of components or central contacts. Bridge fields have no reduction defect. In physical form coordinates the complete-row defect is thus at most \(2\gamma^5\sigma^2t^4/C_h^2\).
Both full and extended surrogate fields are globally Lipschitz in the joint maximum norm with constant at most \(2+2\gamma<3\). Variation of constants for the error, or the integral Gronwall inequality, bounds the fine ideal-to-lifted-surrogate discrepancy by
For example, take the supremum of the error on \([0,h]\): its integral inequality is at most the integral of the stated defect plus \(3h\) times that supremum. This proves (8) without evaluating an exponential and covers both form and phase errors, including feedback from generated odd modes. The restriction \(h<1/3\) belongs to this sufficient bound, not to existence of either flow.
Every actual formation-family endpoint has fine maximum-norm distance at most \(\epsilon\) from its ideal comparison state. Global Lipschitz comparison separately bounds its actual full trajectory's distance from the ideal full trajectory by \(\epsilon/(1-3h)\). In particular, arbitrary admitted odd initial errors and their independent component correlations are retained. The whole-window full-state error certificate is
It applies to every fine coordinate after lifting the surrogate and restoring each reference twist and origin. At \(h=0\) it is the initial uncertainty bound. If all origins agree, \(\sigma=0\), so the ideal zero-deviation comparison is stationary; actual preparation errors still remain. The bound is uniform in graph size because of normalized degrees, not because the graph or initial state was omitted.
Actual source handoff, support work and joined identity¶
Reuse the unprobed nonlinear handoff with every original formation source re-admitted. Common phase origins are exact isolated-cycle symmetries. Thus its per-class proof transfers to each component independently without rotating a reached state or modifying any source-family width. The report must establish the actual endpoint bounds before equation (9) becomes a claim about that family.
For the following joined certificate require \(m\ge2\) and connected \(\mathcal G\), and put \(\Delta=\operatorname{diam}(\mathcal G)\). Any fine node lies within four cycle edges of its center, so the joined fine diameter is at most \(\Delta+8\). For \(n=9m\), every mean-zero real vector \(z\) satisfies
The first inequality is the bounded-range variance inequality; the second uses a shortest path and Cauchy--Schwarz. Consequently
This proof also permits cycles in the contact graph. No tree-only stability theorem is being transferred.
The aligned joined target has each component's class twist, a single common phase origin, zero bridge phase gaps, and common form. Internal sine forces cancel and bridge currents vanish, so it is critical for the joined support. In its Euclidean quotient ball of radius \(r\), with \(0<r\le1/12\), every continuous representative of a reference edge gap changes by at most \(\sqrt2r\). The fixed closing-edge integer is retained. Thus all circular gaps remain acute and their cosines are at least
The acute-chart coercivity argument gives full sine storage excess at least \(\kappa_*\|z\|_2^2\), where
Here \(z\) removes the arithmetic common form and phase origins for geometric comparison. Those arithmetic means need not remain constant; the conserved actual-degree-weighted means are retained below.
At contact write \(x_i\) and \(u_i=\theta_i-\Theta_i-o_i\mathbf1\) for the actual component deviations. Their Euclidean norms are at most \(\epsilon\), and their component sums are exactly zero, inherited from the isolated degree-two law. If \(\bar o=m^{-1}\sum_i o_i\), the quotient squared distance obeys
The cross terms vanish because the phase residual sum is zero on each component. Replacing the actual correlated family by unrelated absolute boxes would lose this equality.
The cycle Laplacian has largest eigenvalue at most four. Its form and phase storage excesses are each at most \(2\epsilon^2\) per component; the phase gradient at the twist is zero and its Hessian norm is at most four. The added bridge form storage is at most \(d_{\max}\sum_i|x_{i,4}|^2\le d_{\max}m\epsilon^2\). Using \(1-\cos a\le a^2/2\) on each new bridge gives
The exact event work is the sum of the added edge storages,
and independently
This supplied work is not a continuous loss reserve. The subsequent full-law balance is exactly
If \(Z_0^2<r^2\) and \(B_H<\kappa_*r^2\), a first-exit argument traps the actual full family in the acute quotient ball for all future uninterrupted joined flow. All target cycle periods are retained, including the individual C9 windings and the zero periods on the aligned contact cycles. Strict convexity in this chart and the largest invariant zero-loss set give recovery to its unique target geometry on each actual conserved-mean leaf, by the whole-state recovery argument. This identity/recovery conclusion uses the actual full sine storage, not the surrogate balance (4) or its finite error estimate.
Total degree mass is \(M_{\rm tot}=18m+2|E|\). Since each target twist is centered and its central phase is zero, the exact postevent conserved origins are
Each residual fraction has magnitude at most \(2|E|\epsilon/M_{\rm tot}\). Component origins are not conserved separately after joining. The final common form is \(\mu_x\), and the final common phase origin relative to the aligned twists is \(\mu_\theta\), using the retained continuous lifts.
Frozen-control sufficiency and scope¶
For the declared path, \(m=3\), \(|E|=2\), \(d_{\max}=2\), and \(\Delta=2\), so \(\lambda_*=2/135\) and \(\kappa_*r^2>r^2/2700\). The reference forcing in (6) is \(\sigma=\sin(1/1000)/2\): the two middle currents add, while each endpoint current is divided by three. In (12), the origin contribution is \(6(1/1000)^2\). Equations (13)--(14) retain both contacts and every actual endpoint error. The source, handoff and outward strict-margin computations remain separate prerequisites of the reserved certificate; these formulas do not substitute a cached prior verdict for any of them.
The result is a context-aware composition rule with one scalar contact degree per component, not a context-free reuse of the old denominator. It supplies an exact reduced balance and controlled finite full-state approximation under the declared unit law. It neither proves a minimal realization nor an exact nonlinear quotient. It does not install a support event, select component birth, infer pairwise physical forces, or establish a new many-body response observation. Failure of a sufficient finite error or trapping bound must be reported as unavailable, unless an independent obstruction is proved. No sensor or laboratory clock is identified by this mathematical coordinate-error allowance.
Retained evaluation of the unchanged composition protocol¶
The first reserved evaluation, performed after preserving the protocol,
proof and producing source, returned certified_sine_port_composition
with no unavailable reasons. The frozen stopping rule passed unchanged.
Fresh formation and both class handoffs passed; all three components
therefore retain their independent original source uncertainties. The
joined identity/recovery, supplied-work and uniform approximation flags
are all true. These are analytic interval certificates, not observations
of a propagated or measured trajectory.
The saved upper bounds have the following decimal summaries:
| Quantity | Retained value, approximately | Declared comparison |
|---|---|---|
| Whole-window full-coordinate error | \(1.0339346076859747\,10^{-32}\) | Strictly below \(2\,10^{-32}\) |
| Ideal nonlinear reduction defect | \(3.006772634428256\,10^{-35}\) | Included once in the total error |
| Initial joined quotient distance squared | \(6\,10^{-6}\) | Strictly below \(r^2=1/144\) |
| Full sine storage excess | \(1\,10^{-6}\) | Below the barrier \(2.9141691574402282\,10^{-6}\) |
| Total supplied contact work | \(1\,10^{-6}\) | At most the declared \(2\,10^{-6}\) |
The tiny positive endpoint-error contributions in the last three rows remain in the exact report; the decimal summaries do not round those contributions away for certification. The following integers divided by \(2^{128}\) are the exact retained outward lower margins:
| Strict margin | Lower numerator over \(2^{128}\) |
|---|---|
| Approximation allowance minus total error | 3287350 |
| Squared radius minus initial quotient bound | 2361030298304991476603765637298244192 |
| Full storage barrier minus excess bound | 651358011580819424763819880101342 |
| Work allowance minus event-work bound | 340282366920938463463374607418156 |
The exact algebraic control also passed: the middle port rate is \(-1\), the actual weighted charge rate is zero, form storage is two, and storage rate is \(-17/3\). Reusing denominator three gives middle rate \(-4/3\) and weighted charge rate \(-4/3\), as predicted before evaluation. The control rejects incorrect contact normalization without claiming a new receiver-response observation.
Retain the frozen protocol, complete report, producing source archive and evidence manifest. The archive preserves this owner's pre-evaluation protocol and derivation; this retained-result section was added afterward. Exact arithmetic checks against the saved report confirmed primitive-input equality, the stated margin endpoints and stopping inequalities without rerunning its producer.
The shared implementation exposes evaluate_sine_port_composition and
assess_sine_port_composition, with SinePortCompositionState and
SinePortComposition, in
relational_sine_port_composition.py.
The contract
and usage guide
retain admission, availability and coordinate conventions. The independent
composition controls
check fine-node projection, mobility, storage/supply, origins, actual-family
bounds and strict numerical boundaries; they do not replace the all-time
proof or execute a trajectory.
The result establishes degree-aware reduced composition under the supplied unit sine law, with finite full-state accuracy and separate actual identity retention. The structural dwell remains \(10^{13}\), and the approximation allowance remains \(2\,10^{-32}\); neither has acquired an empirical clock, preparation or sensor interpretation. No runtime speedup, autonomous support selection, minimal state or uniquely selected physical law follows.