Hierarchy-to-contact composition under an explicit intermediate law

This owner studies actual THOL birth followed by parent-target UM. The isolated prefix uses an explicitly supplied frozen-isolate completion; the connected sine reader is not silently widened. The result concerns a conditional mechanism on an open connected-state family, not the formation or maintenance of the previously studied winding organizations.

The research program owns status. The hierarchy owner, actual UM owner and native birth/transport comparison retain their contracts. Their laws and earlier finite responses are not substituted for the model below.

1. Protocol fixed before event evaluation

The analytical source is the ordered two-node graph (0,1) with one unit edge, parent target 0, and independent coordinates

\[ \frac{31}{64}<A=x_0<\frac{33}{64},\quad \frac{47}{64}<B=x_1<\frac{49}{64},\quad \frac{31}{16}<\nu_0,\nu_1<\frac{33}{16},\quad \theta_0=\gamma,\quad\theta_1=\gamma+\phi,\quad \frac7{64}<\phi<\frac9{64}. \tag{1} \]

Phases use one declared lift; gamma is a common origin. This is a regular acute connected-state family, not a claimed formed composite identity. The represented nominal control is (A,B,nu_0,nu_1,gamma,phi) equal to (1/2,3/4,2,2,0,1/8).

The complete word is OZ -> THOL -> UM -> SHA, always targeting only the original parent. Use initialized nonzero form, supplied AL,IL history and zero initial debt; validate the whole word and every live prefix with the core grammar and GRAMMAR_REJECTION_MODE="raise". Enable strict operator preconditions. Begin with empty hierarchy and no child records. The original parent and neighbor have sense index zero through the declared existing default; the newborn has the same default. No learned sense observation is claimed. Pressure is refreshed from the selected complete law at boundaries.

Keep THOL's default positive acceleration gain, default threshold 0.1, metabolism enabled with its default gains and the default child-capacity damping. The source has an admitted three-sample parent history with abs(A_hat)>tau; this is an independent event-admission premise. The static runtime control supplies [(0,A),(1,A-1/4),(3,A)], whose acceleration is exactly 1/4, and structural time 3. These are manufactured reader inputs, not observed samples from the selected law. Section 3 separately proves that genuine sufficiently short same-law windows are possible in (1).

All four UM flags are true: bidirectional phase updates, capacity blending, stored-pressure relief and functional links. Fix all three active UM factors to 1/8. Keep the default compatibility threshold pi/(pi+1), phase limits DELTA_PHI_MAX=UM_MAX_PHASE_DIFF=pi/2, candidate count zero (all candidates), mode proximity, execution seed 17 and no restricted _node_sample. The candidate inventory is rebuilt after birth. No parent-child edge or conductance is externally installed.

Use SHA factor 1/2, SHA_MIN_VF=1/8, EPI_LATENT_MAX=1, form rails [0,1] and hard clipping. Inputs remain interior where those rails are consumed. Other consumed configuration retains its shared canonical defaults. Retain node order, actual aliases, event records and phase lifts. The frozen-isolate law below supplies the intermediate refresh; the ordinary connected reader must still reject the full born graph.

The prospective discriminator is an actual connected child with negative phase rate, positive form rate and positive form acceleration, against a separately specified null with UM_FUNCTIONAL_LINKS=False. The null uses the same word, source, history, old-neighbor phase/capacity reset and child construction, but retains the isolated child under the frozen law. It is not a failed coupling attempt relabeled as an interaction.

No continuous trajectory, retained producer, parameter search or experimental physical comparison is part of this protocol. Exact-real family statements and finite represented endpoint checks remain distinct. Operator occurrence, history provenance and a physical clock are supplied, not inferred.

2. Prefix law and the connected restriction

On each connected component with at least two nodes, retain the selected weighted-form/bare-phase completion

\[ q=L_Wx,\quad S_i=\sum_{j\sim i}\sin(\theta_j-\theta_i),\quad m_i=\frac{\nu_i}{\pi d_i},\qquad p_i=\frac{S_i}{\pi d_i},\qquad \dot x_i=m_iS_i,\quad\dot\theta_i=m_iq_i. \tag{2} \]

Support is finite, simple, loop-free and undirected. Conductances and held capacities are nonnegative; the constructive family has positive ones. There is no forcing or continuous loss. Bare degree counts edges, including zero-conductance support; it is not weighted strength. At every isolate explicitly declare p=xdot=thetadot=0, held form, phase and capacity. This is the supplied frozen-isolate completion, not a value assigned to the undefined quotient 0/0 in (2).

The shared detached owner keeps bound_relational_sine_support_exchange restricted to connected support and exposes the distinct opt-in bound_relational_sine_frozen_isolates comparison. Such a report supplies instantaneous fields for the declared law; it installs no integrator, event selector or autonomous hierarchy mechanism.

Use edge-only storage

\[ H=\frac12\sum_{\{i,j\}}W_{ij}(x_i-x_j)^2 +\sum_{\{i,j\}}[1-\cos(\theta_j-\theta_i)]. \tag{3} \]

Each nontrivial component conserves H under (2); isolates contribute no term. An intrinsic child storage would change the birth budget and is outside this completion. With positive capacities, inverse-capacity means are defined on each nontrivial component using rho_i=d_i/nu_i. Frozen child coordinates are independent constants; assigning them zero mean weight merely ignores them and is not a complete observation of the enlarged state.

This declaration admits the original graph, the actual isolated birth prefix and the connected UM successor under one explicit piecewise state-domain contract. On the final connected support its fields are exactly those of (2). No elapsed evolution is inserted between events by this admission theorem.

3. True history compatibility and every prefix's live domain

Write u=B-A>0. Before any event, the parent has positive selected pressure sin(phi)/pi and its true form acceleration from the complete two-row law is

\[ \ddot x_0=\frac{\nu_0(\nu_0+\nu_1)}{\pi^2}\,u\cos\phi. \tag{4} \]

In (1), nu_0,nu_1>19/10, u>1/5, cos(phi)>99/100 and pi^2<10. Thus (4) exceeds 35739/250000>7/50>1/10. This has a strict margin on the whole open source box. Smooth local backward flow exists in its fixed finite-dimensional law. For sufficiently short three-sample windows ending at each source, the divided difference converges to (4), so genuine same-law acceleration evidence can exceed the default THOL threshold. The window can remain inside the form rails and phase sector by continuity. No particular sampling interval, numerical record or source acquisition is proved by this existence statement.

The shared history reader still has to admit the actual supplied record. The synthetic nominal acceleration 1/4 is not the nominal constitutive value 2*cos(1/8)/pi^2. Their agreement on a strict threshold decision does not authenticate the supplied record as a trajectory. Timestamped history, the structural clock, signed form and explicit event records remain separate execution inputs.

OZ preserves source form, phase, capacity and support. Its stored-pressure write is erased by the declared refresh, so the physical source and (4) are unchanged when THOL is considered. THOL's strict form/capacity minima, positive-pressure gate, nonzero degree, temporal history and depth conditions all have the stated margins. Its threshold crossing proposes one child c. The default metabolic input has one neighbor, hence exact-real circular phase variance zero and form gradient u. The unclipped child is

\[ z=x_c=\frac3{10}A+\frac{u}{2\pi},\qquad \nu_c=\frac{19}{20}\nu_0,\qquad \theta_c=\gamma,\qquad p_c=0. \tag{5} \]

This is the ideal form of the actual construction; the runtime materializes its declared coefficients, circular variance and endpoint arithmetic. It creates no edge. The source interval gives

\[ \frac{113}{640}<z<\frac{129}{640},\qquad D=A-z>\frac{181}{640}>\frac7{25}. \tag{6} \]

For example, 1/7<1/(2*pi)<1/6 proves these bounds from (1). They keep the child strictly inside the clipping rails. The old component is unchanged apart from stored pressure and context; the explicit isolate law supplies the newborn's zero rows and pressure. A connected-only refresh at this prefix would reject rather than advance to UM.

UM must target the original parent. Its old neighbor 1 passes U3, both before and after the phase proposal. The child as the first target has no existing neighbor and fails the mandatory U3 gate, irrespective of its equal phase. Including it in an all-node atomic UM stage rejects the stage as well. Parent targeting is therefore a substantive prefix condition, not a label.

All live nonneighbor candidates are enumerated after birth. For this P2 source the actual child is the parent's only nonneighbor; no other potential edge is suppressed. A stale prebirth node sample would omit it, and an otherwise valid UM call would not create a connected successor. Pre-event candidate phase and post-merge phase tests both apply. The hierarchy association itself does not bypass either test or the compatibility score.

The complete word provides the OZ context needed by THOL, uses THOL's stabilizing action before UM and closes with SHA. Live preconditions remain independent of whole-word validity. U5's hierarchy record is not a numerical parent/child coherence assessment; a newborn has no measured change-rate observation, so a numerical U5 assessment lacks a required input. U6 is a separate configured monitoring contract: its current comparison requires the same complete node support in both observations. At birth the before-state has no child observation, so that comparison is unsupported. Supplying an invented zero baseline would not repair the missing observation. The actual word and stage gates pass; neither these gates nor equation (4) supplies the missing U5/U6 evidence or selects an occurrence.

4. The actual parent-target UM map and its connected domain

The old compatible neighborhood contains only node 1. The bidirectional consensus phase of the parent and its old neighbor is gamma+phi/2 on the declared lift. The child was not a neighbor when proposals were read. With the supplied push and capacity factor 1/8, ideal endpoints are

\[ \delta=\frac{\phi}{16},\quad \theta_0^+=\gamma+\delta,\quad \theta_1^+=\gamma+\frac{15\phi}{16},\quad \theta_c^+=\gamma,\qquad n:=\nu_0^+=\frac{7\nu_0+\nu_1}{8}. \tag{7} \]

The other capacities remain nu_1,nu_c; every form remains A,B,z. The old edge gap contracts to 7*phi/8; the prospective new gap is delta. Both are strictly acute. The child's capacity must not be inserted into the same-call neighbor mean in (7).

The parent stored-pressure relief uses the old compatible neighbor and its final gap: its multiplier is 1-(1/8)*(1-(7*phi/8)/pi). This remains a stored-channel reset; the selected post-event pressure is rebuilt from the new support.

With equal zero sense indices, the actual scalar compatibility formula has the exact-real form

\[ \omega=1-\frac{\delta}{2\pi} -\frac{D}{4(A+z+\epsilon)},\qquad\epsilon=10^{-9}. \tag{8} \]

The kernel evaluates its EPI ratio through the shared scaled arithmetic. Since 3*z>A follows from (6) and A<33/64, the last ratio before its factor 1/4 is below 1/2. Also delta<1/100, so

\[ \omega>\frac78-\frac1{600}>\frac45> \frac\pi{\pi+1}. \tag{9} \]

Thus both U3 tests and the default compatibility threshold pass with strict margins. UM creates exactly the edge {0,c} with the generated positive conductance omega. It is neither externally imposed nor replaced by one. The final bare degrees are (2,1,1) in order (0,1,c), and the weighted form gradients are

\[ q_0^+=-u+\omega D,\qquad q_1^+=u,\qquad q_c^+=-\omega D. \tag{10} \]

The whole successor is now in the connected selected-law domain. Parent-only SHA subsequently changes n to n/2, keeps phase/form/support fixed and supplies its lifecycle context; the other capacities are unchanged. Its output capacity is positive and above the declared floor. No symmetry, winding or old identity tube is inferred from this small graph's connectedness.

All general endpoint identities below are exact for admitted represented endpoints viewed as real coordinates. Equations (5), (7) and the simplified family bounds describe the ideal event map. Production coefficients, circular variance, consensus, modulo and multiplication can have represented defects. Finite controls re-admit actual endpoints and check the margins directly; they are not a uniform proof of every floating-point realization of (1). No conclusion requires forcing the generated conductance to its ideal value.

5. Complete event work, degree changes and mean leaves

For a general admitted UM stage that preserves form, let F be its actual set of newly committed edges, E the old support, and theta^+ its complete phase endpoint, including old-neighbor writes. Old conductances are unchanged. The exact storage jump is

\[ \begin{split} H^+-H^-={}&\frac12\sum_{\{i,j\}\in F}\omega_{ij}(x_i-x_j)^2\\ &+\sum_{\{i,j\}\in E} [\cos(\theta_j-\theta_i)-\cos(\theta_j^+-\theta_i^+)]\\ &+\sum_{\{i,j\}\in F}[1-\cos(\theta_j^+-\theta_i^+)]. \end{split} \tag{11} \]

Ignoring the middle line would omit bidirectional UM's work on old edges. Capacity is absent from this storage, so its reset has no additional term. THOL birth has zero edge-storage jump under the explicitly selected isolate convention; this is not a claim of zero physical creation cost.

For the source family, (11) becomes

\[ \Delta H_{\rm UM}=\frac12\omega D^2 +\cos\phi-\cos(7\phi/8)+1-\cos(\phi/16). \tag{12} \]

The old-edge phase term is negative. Nevertheless (6), (9), phi<3/20, and 1-cos(phi)<=phi^2/2 give

\[ \Delta H_{\rm UM}> \frac25\left(\frac7{25}\right)^2- \frac12\left(\frac3{20}\right)^2 =\frac{2011}{100000}>\frac1{50}. \tag{13} \]

The final SHA leaves H unchanged. OZ and refreshed THOL also leave the old edge storage unchanged. Thus the complete word's positive storage jump is accounted for by (12), not claimed as energy conservation or an unbudgeted autonomous gain. An event reservoir or selection law has not been derived.

Degree normalization changes both rows. For a general old node, define a_i=sum_{new j}omega_ij*(x_i-x_j), and let Delta S_i include the phase change on old edges and all new sine terms. With m_i^+=nu_i^+/(pi*d_i^+),

\[ \dot x_i^+-\dot x_i=(m_i^+-m_i)S_i+m_i^+\Delta S_i, \qquad \dot\theta_i^+-\dot\theta_i=(m_i^+-m_i)q_i+m_i^+a_i. \tag{14} \]

These are fresh-law comparisons, not the raw pressure-relief map. For a former isolate use its explicitly zero old rows and the full new rows; there is no old nu/(pi*0) to substitute into (14).

The old P2 means use weights (1/nu_0,1/nu_1) and total R. Let mu_x and mu_theta denote those means before birth. Immediately after UM set r_0=2/n, r_1=1/nu_1, r_c=1/nu_c, and R^+=r_0+r_1+r_c. Then

\[ \mu_x^+-\mu_x= \frac{(r_0-1/\nu_0)(A-\mu_x)+r_c(z-\mu_x)}{R^+}, \tag{15} \]

\[ \mu_\theta^+-\mu_\theta= \frac{(r_0-1/\nu_0)(\gamma-\mu_\theta) +(r_0-r_1)\delta+r_c(\gamma-\mu_\theta)}{R^+}. \tag{16} \]

The new child's weight and the changed parent degree/capacity are all retained. The isolated prefix did not already possess this connected mean. For the final SHA endpoint replace r_0 by 4/n; all other terms are unchanged. Each ensuing fixed-support segment conserves its own new means, not the old metric's values. Phase means require the declared continuous lifts.

6. A dynamically active child and its disconnected null

Let k=1 immediately after UM and k=1/2 after parent-only SHA, so the parent capacity is k*n. Equations (7), (10) give the child's full rows

\[ \dot x_c=\frac{\nu_c}{\pi}\sin\delta>0,\qquad \dot\theta_c=-\frac{\nu_c\omega D}{\pi}<0. \tag{17} \]

The instantaneous form acceleration follows by differentiating the declared two-row law with held support/capacity, not by inferring it from telemetry:

\[ \ddot x_c= \frac{\nu_c\cos\delta}{\pi^2} \left[\frac{k n}{2}(-u+\omega D)+\nu_c\omega D\right]. \tag{18} \]

For both k values, dropping a nonnegative contribution gives

\[ \frac{k n}{2}(-u+\omega D)+\nu_c\omega D >\frac95\frac45\frac7{25}-\frac{21}{20}\frac3{10} =\frac{441}{5000}. \tag{19} \]

Here nu_c>9/5, n<21/10, u<3/10 and D>7/25. With cos(delta)>99/100 and pi^2<10,

\[ \ddot x_c> \frac{(9/5)(99/100)(441/5000)}{10} =\frac{392931}{25000000}>\frac3{200}. \tag{20} \]

The parent phase gradient need not be positive throughout the source box; (19) does not assume it. The child's negative phase rate and the resulting relative phase velocity suffice. This accounts for the actual UM parent capacity blend and for its later SHA reduction.

In the disconnected null, the child construction, phase and capacity are the same, and the UM phase/capacity/pressure proposals on old nodes are identical: functional candidates are not members of that snapshot's old neighborhood. Disabling only functional links therefore leaves the child isolated under the declared extension. Its form rate, phase rate and form acceleration are identically zero. A stale prebirth candidate inventory is a second nonattachment control, not a proof that the live-inventory protocol fails.

This discriminates a connected dynamical degree of freedom from an added hierarchy record. It does not establish that the enlarged system is a persistent composite, that the old organization is retained, or that nature selects either event. Rates and local derivatives provide no finite-time response amplitude or lifetime without further analysis.

7. Scope of the composition result and checks

Under the declared state, history, grammar, configuration and intermediate law premises, the actual operator mechanisms admit a THOL/UM composition: birth supplies an isolated coordinate; parent-target UM admits the actual child candidate, generates a positive edge and returns the full graph to the connected weighted-sine domain. The ideal source family has strict birth, candidate, acute-domain and dynamical-response margins. Actual represented endpoints must satisfy their own admitted arithmetic and checks.

The runtime and algebra controls use the shared operator stages and both explicit sine readers. They check live prefix context, connected-reader rejection before UM, parent versus isolated-child targeting, the actual candidate/weight, metabolic construction, degree and mean changes, full work and child rows/acceleration, and the disconnected controls. Manufactured temporal data and constitutive derivative predictions are separately identified. Finite same-node-support U6 checks pass for OZ, UM and SHA; the cross-dimension birth comparison rejects. There is no whole-word U6 certificate, and equations (11)--(20) do not claim universal U6 confinement. The fixture's initial [-1,1] rail transcription was corrected to the frozen [0,1] protocol and rerun; all consumed forms were positive interior values in both, so this correction changed no selected source, event map or analytical discriminator.

The frozen-isolate reader controls check component reuse, explicit zero isolate rows, primitive admission and the unchanged connected-reader boundary. Testing owns suite selection. Neither report installs the supplied law as a native flow.

For a previously formed organization, (11), (14) and the typed prefix contract remain available methods. One must re-admit its actual histories, candidate inventory, phase/capacity map, generated edges and identity conditions. This small connected example does not transfer those premises or prove composite formation for the marked winding contact. No new trajectory or physical identification is part of the result.