A THOL birth obstruction inside the maintained contact

The existing count-three contact is a maintained organization under its declared conservative sine law. Its continuous dynamics nevertheless cannot supply a true three-sample EPI acceleration above the unchanged default THOL threshold while retaining that preparation, capacity and clock. The obstruction holds for every event-free sampling window after the contact word, not only for one snapshot or a small-window limit.

This is a source-specific result about the child-creation condition. THOL also has a pressure action that does not require crossing the child threshold. The research program owns research status; the hierarchy contract owns the operator's temporal observation and separate admission requirements.

1. Frozen source, identity and temporal question

Select the exact aligned post-word family already admitted by the proximity-contact protocol. Before its supplied emission,coupling,silence word, there are two independently perturbed doubled C5 components with pairs A_k=(2*k,2*k+1) and B_k=(10+2*k,11+2*k), k=0,...,4. Each has its original twenty unit edges and unit capacities. Retain alpha=2*pi/5, the common form origin 9/32<m<11/32, phase origin zero, and independent centered constituent deviations of joint norm less than z=1/512. The complete preparation, original rails, target order (0,1), associated execution state and word parameters remain with that protocol.

In particular, retain AL_boost=UM_theta_push=1/1024, SHA_vf_factor=1/2, UM_vf_sync=UM_dnfr_reduction=1/8, all four UM flags, compatibility threshold 3/4, and the already selected proximity candidate count three. No new invocation or changed candidate policy is evaluated here. Its proved post-word family has twenty nodes and 45 edges: the forty old unit edges, the internal A0 edge and the four A0--B0 edges. Retain each actual generated conductance, with

\[ 0<W_{ij}\leq1,\qquad \nu_i=\begin{cases}1/2,&i=0,1,\\1,&\text{otherwise},\end{cases} \qquad d_i=\begin{cases}7,&i\in A_0,\\6,&i\in B_0,\\4,&\text{otherwise}. \end{cases} \tag{1} \]

Actual new conductances are greater than 3/4; none is replaced by a unit weight or an averaged boundary association. Each source realizes its own weights once, then holds them fixed. The source family includes full fine perturbations; exact twin symmetry is not required.

Mark node 0 as the prospective parent. Keep signed scalar form, primitive circular phase with the original continuous lifts, positive held capacities, zero forcing and zero continuous loss. The complete continuation is

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

The selected coefficients remain e=0,w=beta=1. The original structural clock places the completed word at absolute time 7; operator-stage counts are not elapsed continuous time. After that instant there are no events, inputs, support changes, resets or capacity changes in this question.

Use the unchanged joint identity, with P=I-11^T/20 and the original common-aligned doubled-twist reference theta_*:

\[ Z^2=\|Px\|_2^2+\|P(\theta-\theta_*)\|_2^2<r^2, \qquad r=1/16. \tag{3} \]

This observation retains relative constituent form and phase origins. The arithmetic means removed by P are not assumed to be conserved. The conserved means of (2) use the actual weights rho_i=d_i/nu_i instead.

For a fixed prospective window, choose the absolute timestamps

\[ (t_0,t_1,t_2)=(7,29/4,15/2),\qquad h_1=h_2=1/4. \tag{4} \]

The sample at time 7 means the continuous initial state after the completed word. The interval contains none of that word's jumps. The samples in the mathematical question are the true values x_0(t_k) of the same solution, not stored operator-step records. No samples are numerically produced here. The theorem below also handles every nonuniform window with 7<=t_0<t_1<t_2 under these unchanged premises.

Retain the shared resolver's unoverridden default THOL_BIFURCATION_THRESHOLD=0.1, with no explicit threshold or generic BIFURCATION_THRESHOLD_TAU override. Write its represented value as tau; it is greater than the exact rational 1/10. Its unit is form per squared structural time. This is a supplied operational threshold, not a constant derived from the nodal identity. The strict child trigger is abs(observed_acceleration)>tau.

2. Why the entire interval remains in the original identity

The existing all-source proof already establishes Z(7)<r and

\[ H(7)-H_*<1/8192<1/6400,\qquad H=\frac12x^TL_Wx+ \sum_{\{i,j\}\in U}(1-\cos(\theta_j-\theta_i)). \tag{5} \]

For this actual support, weights and common reference, its boundary barrier gives H-H_*>=Z^2/25 throughout Z<=r. A first boundary hit would therefore require storage excess at least 1/6400, contradicting the exact conservation of H under both rows of (2). The retained positive capacities, conserved weighted means and bounded relative state give global smooth continuation, as proved by the maintenance owner. Consequently,

\[ Z(t)<1/16\qquad\text{for every }t\geq7. \tag{6} \]

This reused theorem supplies whole-interval membership, including (4). Checking three endpoints alone would not supply it. No conclusion is borrowed from the later NUL/IL/SHA continuation, whose different completed source leaves this identity. The conservative maintained contact, that finite departure and the P2 hierarchy-to-contact source remain distinct preparations.

3. A uniform complete-law acceleration envelope

For fixed support, conductances and capacities, differentiation of both rows of (2) gives the complete-law acceleration

\[ \ddot x_i=m_i\sum_{j\sim i}\cos(\theta_j-\theta_i) (m_jq_j-m_iq_i). \tag{7} \]

This uses the actual phase velocity. It does not infer a pressure from the observed response or replace temporal evidence by a current RHS value.

For any two nodes, (3) and Cauchy--Schwarz imply

\[ |x_i-x_j|=|(e_i-e_j)^TPx| \leq\sqrt2\|Px\|_2<\sqrt2r. \tag{8} \]

Using the actual nonnegative conductances bounded by one,

\[ |q_i|\leq\sum_{j\sim i}W_{ij}|x_i-x_j| \leq d_i\sqrt2r, \qquad |\dot\theta_i|\leq\frac{\nu_i\sqrt2r}{\pi}. \tag{9} \]

Keeping the two phase-velocity contributions in (7) and using abs(cos)<=1 yields

\[ |\ddot x_i| \leq B_i:=\frac{\nu_i\sqrt2r}{\pi^2} \left(\nu_i+\frac1{d_i}\sum_{j\sim i}\nu_j\right) \leq\frac{\nu_i(\nu_i+\nu_{N_i}^{\max})\sqrt2r}{\pi^2}. \tag{10} \]

No form/phase independence, symmetry or tangent approximation is assumed. The envelope is conservative and need not be attained; discarding a useful correlation enlarges an upper bound without inventing a reachable state. The bare neighbor count remains separate from the conductance weights.

At parent 0, nu_0=1/2 and every neighbor has capacity at most one. Using sqrt2<3/2 and pi>3 gives

\[ |\ddot x_0|\leq\frac{3\sqrt2}{64\pi^2} <\frac1{128}<\frac1{10}<\tau. \tag{11} \]

All nodes have capacity at most one, so a second uniform corollary is

\[ |\ddot x_i|\leq\frac{\sqrt2}{8\pi^2} <\frac1{48}<\frac1{10}<\tau\qquad\text{for every node }i. \tag{12} \]

These strict rational margins apply for all structural time by (6). The parent bound does not require increasing precision, waiting for a different phase, choosing another sampled subinterval or averaging an organization into a new scalar parent. Equation (12) separately excludes switching the marked target to another constituent node within this same preparation and policy. It is not a coarse-grained THOL contract.

4. Exact three-sample windows are positive averages of acceleration

Let f(t)=x_i(t) along any solution admitted above, with 7<=t_0<t_1<t_2, h_1=t_1-t_0, h_2=t_2-t_1, and H_t=h_1+h_2. The true timestamped observation uses the same convention as the shared acceleration reader:

\[ \widehat A_i= \frac2{H_t}\left[ \frac{f(t_2)-f(t_1)}{h_2} -\frac{f(t_1)-f(t_0)}{h_1}\right]. \tag{13} \]

For a twice continuously differentiable f, define the piecewise-linear kernel

\[ K(s)=\begin{cases} \dfrac{2(s-t_0)}{h_1H_t},&t_0\leq s\leq t_1,\\[6pt] \dfrac{2(t_2-s)}{h_2H_t},&t_1\leq s\leq t_2. \end{cases} \tag{14} \]

Its values agree at t_1, it is nonnegative, and its two areas are h_1/H_t and h_2/H_t. Integrating K(s)f''(s) by parts on each subinterval cancels the two terms involving f'(t_1) and gives exactly

\[ \widehat A_i=\int_{t_0}^{t_2}K(s)\ddot x_i(s)\,ds, \qquad \int_{t_0}^{t_2}K(s)\,ds=1. \tag{15} \]

Thus this finite-window observation is a positive normalized average of the full acceleration, without a small-window approximation. Equations (11)--(12) imply

\[ \boxed{|\widehat A_0|<1/128<\tau,\qquad |\widehat A_i|<1/48<\tau\quad\text{for every node }i.} \tag{16} \]

This holds for the frozen window (4) and for every later unequal-spaced window wholly on the same continuation. Neither a change of sampling times nor arbitrarily fine exact sampling can create a crossing. A window through a supplied form, phase, capacity or support jump has different premises; (15) cannot be applied to it as a smooth trajectory of (2).

5. Represented observations need a separate error allowance

The result above concerns exact source states and true trajectory values. It is not a claim that every floating-point record accepted by a history reader is a certified sample of that trajectory. Source materialization, solver enclosures, timestamps, arithmetic and target/clock association retain their separate obligations.

At fixed exact timestamps, let each recorded form differ from its true value by at most epsilon_k. Linearity of (13) gives the exact endpoint-error bound

\[ E_{\rm samples}=\frac2{h_1+h_2} \left(\frac{\epsilon_0+\epsilon_1}{h_1} +\frac{\epsilon_1+\epsilon_2}{h_2}\right),\qquad |\widehat A_{\rm recorded}-\widehat A_{\rm true}| \leq E_{\rm samples}. \tag{17} \]

For (4), a separately certified common endpoint allowance epsilon_k<=epsilon=2^-40 would give

\[ E_{\rm samples}\leq64\epsilon=2^{-34}. \tag{18} \]

This is an illustrative observation budget, not an evaluated solver accuracy or a new trajectory. Any time and arithmetic errors must also be bounded. If their combined effect on the reported acceleration is E_other, a sufficient no-crossing check at parent 0 is

\[ \frac1{128}+2^{-34}+E_{\rm other}\leq\tau. \tag{19} \]

For this particular fixed window, the reader's binary64 arithmetic also admits a separate bound under explicit represented-input premises. Immediately after the source word, every form lies in (m-z,m+1/1024+z). The conserved positive-rho weighted form mean lies in the same interval. At any later time, (8) bounds the difference between a node's form and that weighted mean by sqrt2*r. Hence every true future form satisfies

\[ \frac18<\frac{95}{512}<x_i(t)<\frac{451}{1024}<\frac12, \tag{20} \]

where the outer rational comparison uses sqrt2*r<3/32, the original strict form-origin range and the unchanged AL increment. Recorded binary64 samples with total materialized endpoint error at most 2^-40 therefore remain in [1/8,1/2].

The represented timestamps in (4), their differences 1/4, and their span 1/2 are exact. Each subtraction of two such form samples has magnitude at most 3/8 and absolute round-to-nearest error at most 2^-55. Division by 1/4 is exact binary scaling, so each computed slope differs from its exact slope on the represented samples by at most 2^-53. The slope difference has magnitude at most three; its additional rounding error is at most 2^-52. The two inherited errors and this subtraction contribute at most 2^-51 before the final exact factor four. Thus 2^-49, and in particular the conservative allowance 2^-48, bounds the final reader arithmetic error. Every nonzero intermediate remains normal and finite; this argument does not cover another representation or an arbitrary window. With the assumed endpoint allowance, it follows that

\[ |\widehat A_{\rm binary64}|<\frac1{128}+2^{-34}+2^{-48}<\frac1{10}<\tau. \tag{21} \]

No sample producer has supplied the endpoint allowance in this calculation; (21) is a conditional observation-transfer result. Source and temporal provenance remain premises. Very short windows instead amplify endpoint errors; a large computed divided difference then need not represent a physical acceleration crossing. A current derivative, absent history, cached D2_EPI value or manufactured quadratic record cannot substitute for the missing true samples.

6. Operator, grammar and clock consequences

For this source and a genuine admitted observation, the public SelfOrganization proposal does not enter its abs(d2_epi)>tau child-creation branch. Other child prerequisites cannot repair that failed necessary condition. There is no newborn to submit to the hierarchy-to-contact composition, so no UM candidate evaluation or cross-birth U6 reference is activated by this gate.

The conclusion does not reject every possible THOL pressure action. The optional live precondition gate warns, rather than rejecting solely because the acceleration is below tau; pressure action, birth, live structural preconditions and grammar remain distinct. The actual recent word AL,UM,SHA also contains no registered destabilizer, so it does not supply THOL's independent recent U4b context. Waiting or sampling does not append a glyph. Even a hypothetical repair of that grammar context would leave (16) in force if it preserved all premises of the same future flow; no such intervention is inserted here.

The clock contract also matters. Under a coordinate change t'=a*t, with a>0, every evolution row transforms and the three-sample acceleration becomes A_hat'=A_hat/a^2. The same threshold policy has tau'=tau/a^2, so its decision is unchanged. Keeping the numerical threshold fixed while rescaling time instead changes that policy. Increasing capacities at the original fixed clock changes the selected model and its preparation; it is not a resolution within the present frozen source.

7. Shared calculation and verification scope

The shared weighted sine support owner provides conditional_form_acceleration_bounds from freshly admitted primitive report data. Its general comparison retains the actual conductances, bare degrees, capacities and coefficients, including the w^2/beta factor outside the w=beta=1 specialization here. A supplied centered-form radius is a conditional whole-interval premise; checking the source against it alone does not prove later maintenance. In this theorem, that missing temporal premise is supplied by (5)--(6), not by a report label or an instantaneous observation.

The operator and analytical controls check the nonuniform finite-difference convention, rational margins, represented error allowances, missing history and the effect of very short windows on manufactured endpoint errors. Manufactured polynomial histories serve only as independent mathematical reader controls. They are not relabeled as records of the maintained sine organization. The static comparison tests exercise shared complete-law bounds and their domain. The existing hierarchy controls separately exercise pressure action, child creation and grammar boundaries. No THOL event, source producer, completed research response or continuous trajectory is replayed to establish (16).

The maintained contact therefore answers the frozen birth-evidence question negatively with a strict analytical margin. Its maintenance theorem and its valid operator contracts remain intact. The separate open P2 family in the composition owner still admits sufficiently short genuine above-threshold histories under its own larger capacities and different preparation. Neither result establishes that every maintained organization can or cannot generate children, nor selects an autonomous event law or a physical identification.