THOL compatibility: temporal evidence, hierarchy and the complete-law domain

THOL has a pressure action on the existing nodes and, on its public or atomic route, a conditional construction of a child. These have different continuation contracts. A pressure-only event leaves the physical sine state unchanged and can return to its constitutive pressure by refresh. Public child creation adds an isolated coordinate; it is a valid runtime hierarchy event but leaves the connected-support domain of the selected complete law.

This owner closes that distinction for general admissible states and for the marked collective contact as a conditional control. It does not require the EN/IL/SHA/NAV sequence, supply missing research histories or identify a child with an autonomously maintained organization. The research program owns status. The registered action, THOL pressure owner and native birth and transport owner retain their respective definitions and earlier conditional results.

1. State and selected continuous law

Let U be a finite simple loop-free connected undirected support with at least two nodes. Retain signed scalar form x, primitive circular phase theta, finite nonnegative conductances W, held nonnegative capacities nu and a declared structural clock. Uniform-real BEPI uses the same signed scalar admission; complex/nonuniform BEPI and magnitude projection do not replace it. The shared weighted-form, bare-phase law is

\[ q=L_Wx,\quad S_i=\sum_{j\sim i}\sin(\theta_j-\theta_i),\quad d_i=|N_i|>0,\quad M=\operatorname{diag}(m_i),\quad m_i=\frac{\nu_i}{\pi d_i}, \qquad p_i=\frac{S_i}{\pi d_i},\qquad \dot x=MS,\quad\dot\theta=Mq. \tag{1} \]

Bare neighbors include supported zero-conductance edges. Pressure is specified before evaluating the form response. There is no forcing or continuous loss. The shared reader admits nonnegative capacities and conductances; positivity is an additional premise where an inverse-capacity mean or a nontrivial weighted coupling is needed. At zero capacity both rows of this particular completion vanish at that node; the general nodal identity alone would freeze only its form row.

With declared phase lifts, the selected storage is

\[ H=\frac12x^TL_Wx+ \sum_{\{i,j\}\in U}\bigl(1-\cos(\theta_j-\theta_i)\bigr). \tag{2} \]

It is conserved by (1), since the form and phase work cancel. When all capacities are positive, both means with weights rho_i=d_i/nu_i are also conserved. These are mathematical storage and lifted-coordinate identities, not laboratory energy or a physical identification.

In addition to this physical state, THOL consumes target-associated temporal history, stored pressure, grammar context, hierarchy and child identifiers, configuration and execution order. Metabolic construction consumes actual bare neighbor forms and phases. Its acceleration and child diagnostics do not replace any of these primitive inputs.

2. Acceleration predicted by this law and acceleration actually observed

Let C(theta) be the bare cosine Laplacian: an edge has coefficient cos(theta_j-theta_i), irrespective of W. Direct differentiation of (1), with held capacity and support, gives

\[ \ddot x=-M C(\theta)M L_Wx, \qquad \ddot x_i=m_i\sum_{j\sim i}\cos(\theta_j-\theta_i) (m_jq_j-m_iq_i). \tag{3} \]

Thus the selected completion does have structural acceleration. It is neither an independently selectable extra state variable nor the cached acceleration telemetry. In any specified state,

\[ |\ddot x_i|\leq m_i\sum_{j\sim i}|\cos(\theta_j-\theta_i)| (m_j|q_j|+m_i|q_i|). \tag{4} \]

There is no universal finite acceleration ceiling obtained from capacity alone on unrestricted signed form space. For a constructive example, take theta_i=0, every other phase phi with 0<phi<pi/2, and forms x_i=b+u, x_j=b elsewhere. If nu_i>0 and w_i=sum_{j~i}W_ij>0, then

\[ p_i=\frac{\sin\phi}{\pi}>0,\qquad \ddot x_i=-u m_i\cos\phi \left(d_i m_i w_i+\sum_{j\sim i}m_jW_{ij}\right). \tag{5} \]

The parent acceleration has either sign and exceeds any supplied finite magnitude threshold for a sufficiently large form contrast. A common shift b can independently place its form above a supplied minimum. All edges are initially acute; this is a zero-winding preparation, not the marked winding-one family. The strict inequality persists on a neighborhood of admissible positive-capacity/positive-conductance preparations. Finite form rails, specified mean or storage leaves and particular preparations require their own bounds; (5) does not show that they contain this example.

Smooth local flow through a state with abs(ddot x_i)>tau implies the existence of sufficiently short genuine three-sample windows with the same strict magnitude inequality. This follows from convergence of the three-point divided difference to the derivative. It supplies a compatibility theorem, not an executed window, recorded history, target selector or grammatical context. For one unit edge, capacities one, phase difference pi/3 and x=(3,1), equation (3) gives ddot x_0=-2/pi^2, whose magnitude exceeds 0.1. No trajectory is computed to establish this algebraic example.

Actual THOL uses the shared acceleration reader. For its last three active timestamped samples, it computes

\[ h_1=t_1-t_0>0,\quad h_2=t_2-t_1>0,\quad s_1=\frac{x_1-x_0}{h_1},\quad s_2=\frac{x_2-x_1}{h_2},\qquad \widehat A=\frac{2(s_2-s_1)}{h_1+h_2}. \tag{6} \]

The timestamped source is authoritative when selected; its endpoint EPI must match current signed form. Legacy histories instead use unit operator steps and do not authenticate the endpoint or a continuous-flow interval. Fewer than three active samples give explicit unavailability in the observation; the compatibility numerical function returns zero in that case. Neither this fallback nor a cached D2_EPI establishes observed zero acceleration.

The reader checks supplied sample values and ordering, not an independent live physical clock or producer provenance. A record can have a matching endpoint value while referring to a different interval. The execution protocol must supply temporal continuity, units, pressure law, event record and clock authority before (6) can be interpreted as evidence from (1). Operator-step records and a structural-time record are different inputs.

For endpoint errors bounded by epsilon_0,epsilon_1,epsilon_2 at fixed exact times, the observation error obeys

\[ |\delta\widehat A|\leq\frac{2}{h_1+h_2} \left(\frac{\epsilon_0+\epsilon_1}{h_1} +\frac{\epsilon_1+\epsilon_2}{h_2}\right). \tag{7} \]

Arithmetic and time/solver errors require additional budgets. Primitive nonzero values lost to materialization and nonfinite arithmetic are not evidence of a zero response. A computed threshold decision is not an exact derivative theorem; a strict true margin must dominate the observation error before its decision transfers.

3. Admission, grammar and the actual event branches

The public proposal reads the active history and applies the checked stored-pressure jump

\[ p_i^+=p_i+a\widehat A_i,\qquad x_i^+=x_i,\quad \nu_i^+=\nu_i,\quad\theta_i^+=\theta_i, \qquad a=\mathrm{THOL\_accel}>0. \tag{8} \]

The configured default is 1/(4*pi). A supplied finite nonnegative threshold tau selects the child proposal only when abs(A_hat)>tau; equality produces no child. Reaching the configured hierarchy depth also suppresses birth while retaining (8). Missing acceleration gives the compatibility zero update if optional strict preconditions are disabled, without turning missing evidence into a measured zero. A graph-backed primitive dispatch performs this pressure proposal without hierarchy construction; a graphless primitive instead has an explicitly supplied acceleration contract. These routes are not interchangeable.

When enabled, the shared optional strict gate additionally requires admitted form and capacity minima, positive stored pressure, configured connectivity and at least three active samples. It does not require the birth threshold to be crossed. Full proposal validation separately checks hierarchy references, depth, child-domain bounds, metabolic configuration and representable outputs. Signed scalar admission is general; the configured nonnegative THOL minimum restricts the stricter execution domain without redefining signed EPI. The shared resolve_thol_min_epi policy resolver supplies both strict runtime and detached admission, with unchanged core default 0.2; a present invalid override rejects. The actual child's temporal result shows why above-threshold acceleration and positive fresh pressure do not replace this independent form requirement or acquired history.

Grammar is independent: THOL is a stabilizer, a bifurcation handler and a transformer that requires a recent destabilizer under U4b. It is not a closure. OZ -> THOL -> SHA is a valid complete word with existing nonzero form when its live conditions pass. This is an example, not a compulsory evolution law. The synthetic tests execute it on an independently prepared no-child case. Their manufactured temporal records are reader/event controls, not observations of a retained organization. Neither passing grammar nor (5) supplies those records to an actual research object.

The atomic stage proposes every target from one immutable snapshot, allocates collision-free child identities in snapshot-node rank and validates the merged hierarchy before committing. Histories and diagnostics retain requested target order. Public single-target execution has its own transaction; sequential public calls do not become one simultaneous stage. No callback external-effect rollback or future stability follows from graph-state atomicity.

4. Pressure-only continuation and its complete accounting

When no child is committed, all physical coordinates consumed by (1), bare support, conductances and capacities remain fixed. Consequently

\[ \Delta H=0,\quad\Delta\mu_x=\Delta\mu_\theta=0,\qquad p_{\rm fresh}^+=p_{\rm fresh},\quad \dot x_{\rm fresh}^+=\dot x_{\rm fresh},\quad \dot\theta_{\rm fresh}^+=\dot\theta_{\rm fresh}. \tag{9} \]

The mean statements require positive capacities. These identities hold for one or simultaneous targets, with no pressure cross term in H because H does not consume stored pressure. History, debt and diagnostics can still change; the contextual event is not the identity map on sufficient execution state.

The raw stored pressure in (8) generally lies off the constitutive relation p=S/(pi*d). Immediate selected-law refresh erases its direct contribution. Holding the changed pressure for a later interval instead changes the complete model: its form row is no longer the refreshed row of (1). Pressure-only event compatibility therefore includes an explicit handoff; it does not silently promote stored pressure to an independent evolving coordinate. A finite-duration held-pressure response would require its own law, clock, phase row and budget.

The separate scalar diagnostic satisfies

\[ \frac12\bigl[(p_i+a\widehat A_i)^2-p_i^2\bigr] =a p_i\widehat A_i+\frac12a^2\widehat A_i^2, \tag{10} \]

which can have either sign. THOL's stabilizer role is not universal pressure contraction. Subsequent OZ, SHA or other word stages have their own resets; (9) is the THOL stage budget, not a whole-word identity.

5. Child creation changes the state dimension and law domain

If its threshold, depth and proposal domains pass, public THOL adds a child c with the actual represented endpoints

\[ \nu_c^+=0.95\,\nu_i,\qquad \theta_c^+=\theta_i\pmod{2\pi},\qquad p_c^+=0, \qquad x_c^+=\mathcal B_i(x_i,\{x_j,\theta_j:j\sim i\}). \tag{11} \]

The products and wrapping denote the admitted runtime results, not exact decimal identities after binary64 evaluation. The amplitude map is owned by metabolism: before final configured child bounds, it clips 0.3*x_i + g*(mean_neighbor_x-x_i) + k*phase_variance to [0,1] when metabolism is enabled, and 0.3*x_i otherwise. Its inputs use bare neighbor multiplicities and circular phase variance; conductance-weighted averages and a simple cell graph do not substitute for them. The configured EPI interval must intersect [0,1]. Acceleration selects eligibility and is recorded; it is not a second amplitude multiplier.

The event writes the parent/child association, hierarchy path and level, sub_nodes, sub_epis and graph hierarchy. It leaves every old form, capacity, phase and edge unchanged. The child has a single legacy EPI sample, so it does not inherit the parent's observed acceleration. No parent amplitude is removed and no neighbor form is metabolically depleted; this construction is not an additive mass split. A common form shift need not commute with the bounded child construction, even though the old continuous law has that symmetry.

The whole-graph continuation obstruction is exact. THOL creates no edge. The child has d_c=0; the resulting graph is disconnected and the denominators in (1) are undefined there, even if its capacity happens to be zero. The existing shared sine reader correctly refuses this successor. The pressure value zero and nesting metadata do not resolve 0/0, provide a child phase law or create coupling. Optional strict THOL connectivity applies to the target before the event; it does not certify the newborn's continuous-law domain.

Restricting the actual successor to the old component gives exactly the same fresh (1), H and inverse-capacity means as before the event. This is a restriction result, not a whole-successor certificate. In particular a stage whose final callback demands the existing full-graph sine refresh rejects and rolls back the birth and contextual writes; it cannot return an admitted selected-law successor by silently ignoring the child.

An explicit comparison extension, not an installed law

One mathematically consistent extension would declare (1) on every connected component of size at least two and, on each isolate, explicitly set

\[ p_c=0,\qquad\dot x_c=0,\qquad\dot\theta_c=0. \tag{12} \]

This is a supplied frozen-isolate completion, not a consequence of dividing by zero or an implementation change. It has a smooth vector field on each fixed-support stratum. Choosing the same edge-only storage (2) makes its old component conservation proof apply verbatim; isolates have no storage term. With that explicit convention, isolated birth has zero selected edge-storage jump and leaves the old component means fixed. Assigning an intrinsic child storage would change the birth budget and require its own premise.

For positive capacities, the old component weights remain rho_i=d_i/nu_i. Setting an isolate's weight to zero only defines an observation that ignores it; it is not a complete mean of the enlarged organization. Any additional positive fixed child weight kappa would also give a conserved weighted mean under (12), but birth shifts that mean by kappa*(x_c-mu_x)/(sum(rho)+kappa) for one child. No particular kappa is selected by (1). The frozen child's coordinates are independent constants, not a coupled higher-scale dynamical mode.

Other isolate laws or actual connections are possible supplied completions. They are not selected by the hierarchy record. Connecting a child changes degrees, both rows, phase pressure, edge storage and mean weights; it needs its own admitted support event and work accounting. The native birth/transport results cannot transfer their diffusion law or coefficients to this sine completion. This owner installs no extension or child-coupling policy.

The later hierarchy-to-contact composition explicitly selects this comparison for its intermediate prefix and supplies a separate shared detached reader. It retains the connected reader's rejection, the additional isolate-law premise and the absence of native flow dispatch. Its small connected source family does not establish birth or maintenance of the marked winding contact studied here.

6. Collective descent, U5 and clock compatibility

For the marked twenty-node, forty-five-edge contact, the contextual eleven-cell chart retains singleton ports 0 and 1 and nine physical twin pairs. It must also retain per-member temporal samples/units, grammar histories and debt, hierarchy and identifiers, support multiplicities, construction parameters, stored pressure, semantic data and ordered targets. A mean history or unordered list of histories does not determine which parent crosses tau or which child is constructed. The full form origin matters to the child amplitude.

Pressure-only events at these singleton ports leave physical twin equalities and the continuous reduction intact after refresh. Birth at both ports keeps the old twins but adds two associated singleton children: the physical chart has thirteen cells and twenty-six form/phase coordinates, plus its hierarchy and temporal context. Its support is disconnected; describing that chart is not admission to (1). More general twin-target events require compatible target sets, histories, configuration and child association before a descended event is defined. Physical equality alone is insufficient.

U5 has separate structural and numerical obligations. The proposal validates a coherent hierarchy record. The detached U5 assessment additionally compares C_parent >= alpha*sum(C_child) for an explicitly supplied alpha and tolerance. It requires explicit finite stored-pressure and change-rate inputs for every assessed parent and child; missing channels reject rather than become zero observations. A newly constructed child has no supplied change-rate observation, so birth alone does not establish a numerical U5 certificate. With actual or explicitly supplied diagnostic inputs, valid nesting can pass one alpha and fail another. Neither a single pass nor the variance-based sub-EPI amplitude alignment proves subsequent maintenance.

Under an affine clock change s=c*t+b, c>0, one way to retain the form of both rows in (1) is nu_s=nu_t/c, with pressure unchanged. Then

\[ \widehat A_s=\widehat A_t/c^2,\quad \tau_s=\tau_t/c^2,\quad a_s=c^2a_t. \tag{13} \]

Transforming capacity minima and all history timestamps consistently preserves the same event decision and stored-pressure increment. Keeping numerical tau and gain fixed instead defines a different policy. If capacity units are held fixed and pressure is instead rescaled, the phase row and gain need their corresponding transformations as well. A nonlinear time change also adds the usual acceleration term -xdot*s''/(s')^3; finite differences must be rebuilt from the transformed sample intervals. Synchronization or operator count does not supply such a clock transformation.

7. Verification and remaining scope

Independent synthetic controls exercise the complete no-child word, the two public branches and primitive distinction, actual metabolic multiplicities, unchanged old-component fields, isolated-child rejection and graph rollback, independent U4/U5 conditions, signed temporal evidence and affine clock scaling. The law-acceleration control compares independently differentiated edge equations with shared instantaneous fields. The supplied quadratic records are explicitly manufactured reader inputs; no continuous trajectory or retained research response is executed.

The shared history, proposal and transaction owners retain their broader tests; Testing owns selection. These contracts establish a scoped partial action and a precise complete-law obstruction. They establish neither an observed THOL birth under (1), autonomous occurrence, a recursively coupled organization, future maintenance nor physical identification.