Capacity and temporal access inside the maintained contact

A supplied common capacity increase can make genuine three-sample histories cross the default THOL acceleration threshold without changing the maintained organization's relative identity. This is a conditional accessibility result for an open part of the existing source family. It does not supply an event that realizes the capacity increase, a measured history, or admission of THOL. The original source's no-crossing theorem remains valid at its original capacities. The research program owns the research status.

1. Source, capacity family and unchanged clock

Retain the exact aligned count-three preparation in the proximity-contact owner and the birth-obstruction owner. Its actual AL/UM/SHA word has twenty nodes and 45 edges: forty original unit edges, the internal A0 edge and four A0--B0 edges. Retain every actual generated conductance, its endpoint association, the declared phase lifts, zero forcing and continuous loss, and the original structural time 7 after the word. The held capacities are nu_0=nu_1=1/2, with all others one. Bare degrees are seven on A0, six on B0 and four elsewhere.

The complete continuation, with e=0,w=beta=1, is

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

The new constitutive family holds c*nu, with a supplied constant c>0, instead of nu. The support, conductances, initial x and theta, pressure map, clock and threshold are unchanged. This is an analytical comparison of complete laws. It inserts no VAL action, elapsed flow during an event, or autonomous capacity selector. Put y=(x,theta) and write y_1 for the original flow. The uniform-capacity identity then gives the exact relation

\[ y_c(7+s)=y_1(7+cs),\quad \dot x_c(7+s)=c\dot x_1(7+cs),\quad \ddot x_c(7+s)=c^2\ddot x_1(7+cs). \tag{2} \]

Both evolution rows transform. The relation compares different capacities at the same structural clock; it does not transform the threshold policy. The latter remains the represented default THOL_BIFURCATION_THRESHOLD=0.1, denoted tau, with no override. A change of clock coordinates for one fixed model would instead transform both the acceleration and tau by the same inverse squared factor, preserving its decision.

For every c, the storage H, reference H_star and initial storage excess are unchanged. The weights rho_i=d_i/nu_i acquire the common factor 1/c, so their normalized means also remain unchanged. The existing all-source barrier therefore retains the exact same identity

\[ Z^2=\|Px\|_2^2+\|P(\theta-\theta_*)\|_2^2<r^2, \qquad P=I-11^T/20,\qquad r=1/16, \tag{3} \]

for all future time. Equivalently, (2) transfers its already proved all-time maintenance and global continuation. Nothing here identifies capacity with mass or converts structural time into laboratory time.

2. A nonzero signed acceleration at the exact unsplit source

Use an exact unsplit member of the existing family, with 9/32<m<11/32, alpha=2*pi/5 and b=h=1/1024. Before the word, form is m everywhere and each pair k has phase k*alpha in both components. This is an exact-real source. It is not a relabeling of the earlier binary64 nominal control, whose dyadic twist and execution residuals remain with its owner.

After the actual word, the complete primitive state is:

• x_0=x_1=m+b; every other form is m.
• The A0 phases and B0 phases are zero. The two A1 phases are (1-h)*alpha, the two A4 phases are (4+h)*alpha, and A2/A3 retain 2*alpha,3*alpha. Every B pair retains k*alpha.
• Capacities and degrees are those in Section 1. Old edge conductances are one; the internal A0 edge has conductance one. Each of the four actual cross edges has the same generated conductance

\[ \omega=1-\frac{b}{4(2m+b+\epsilon_E)},\qquad 3/4<\omega<1, \tag{4} \]

where epsilon_E is the represented positive implementation regularizer 1e-9. It is retained, not replaced by zero. The phase formulas follow from the old neighbor consensus: its resultant is 1+4*cos(alpha)=sqrt(5)>0, so its angle is zero. The bidirectional UM writes move A1/A4 toward that angle on their declared local charts. No new B neighbor participates in this old-support consensus.

Put gamma=(1-h)*alpha and K=cos(gamma)>cos(alpha)>3/10. The weighted form gradient on this actual source is

\[ q_0=q_1=(4+2\omega)b,\qquad q_j=-2b\ (j\in A_1\cup A_4),\qquad q_j=-2\omega b\ (j\in B_0), \tag{5} \]

with every other entry zero. The internal A0 edge has zero form difference; the four cross contributions in (5) retain omega. Under (1), the exact complete-law acceleration is

\[ a_i:=\ddot x_i=M_{ii}\sum_{j\sim i} \cos(\theta_j-\theta_i)(M_{jj}q_j-M_{ii}q_i). \tag{6} \]

The A0 partner contributes zero to a_0; the four old A neighbors have cosine K, and the two B0 neighbors have cosine one. Thus

\[ a_0(7)=-\frac{b}{14\pi^2} \left[4K\left(\frac12+\frac{2+\omega}{7}\right) +2\left(\frac\omega3+\frac{2+\omega}{7}\right)\right]. \tag{7} \]

Using K>3/10 and omega>3/4, the bracket exceeds 33/14. Since pi^2<10 and b=1/1024,

\[ a_0(7)<-\frac{33}{1960\cdot1024}<-2^{-16}. \tag{8} \]

This is a signed instantaneous identity and bound. A finite observed acceleration needs the whole-window argument below; (8) alone is not a history record or an event trigger.

3. An open source neighborhood retains the signed margin

Inside the original admitted source family, take centered constituent form and phase deviations whose joint norm in each ten-node component is less than

\[ \varepsilon=2^{-18}<z=1/512. \tag{9} \]

This specifies an open subset of the original prepared coordinates. It does not change their allowed common form origin, capacities, rails, word, candidate policy or identity, and it makes no crossing claim for every member of the larger source family. Compare each output to the exact unsplit output at its own common m. The following bounds keep the complete actual operator map.

Each input coordinate differs by less than epsilon. The selected old-neighbor complex resultant differs from its nominal positive value sqrt(5) by at most 5*epsilon. Its real part exceeds 2-5*epsilon; consequently its angle differs from zero by less than

\[ \frac{5\varepsilon}{2-5\varepsilon}<3\varepsilon. \tag{10} \]

Every relevant wrapped consensus displacement stays in its original nonantipodal chart. A selected output phase is the affine interpolation between its own phase and its own consensus; each old-neighbor output is the corresponding interpolation with the mean of the two consensus writes. Since h is 1/1024, the output phase deviations are below (1+2*h)*epsilon<2*epsilon. B phases do not change. The AL form increment and SHA capacity reset are the same as before, with inactive form clipping. Thus the output forms differ by less than epsilon, and its phases by less than 2*epsilon, in the fixed lifts.

The original support theorem still gives the same five new edges and actual conductances in (3/4,1]. On a new edge, both forms exceed 1/4. For N=abs(x_i-x_j) and D=x_i+x_j+epsilon_E, use N/D<1, D>1/2, abs(delta N)<=2*epsilon and abs(delta D)<=2*epsilon. The change in N/(4D) is at most 2*epsilon. A cross-edge phase gap changes by less than 3*epsilon, and an internal-edge gap by less than 4*epsilon. The phase part of its score therefore changes by less than 2*epsilon/pi. Combining them gives the common bound

\[ |\Delta W_{ij}|<3\varepsilon \tag{11} \]

on every new edge; old weights are unchanged. The internal nominal form difference is zero, and every nominal cross difference has magnitude b. Writing q-q_nom=L_W(x-x_nom)+L_(W-W_nom)x_nom now yields

\[ |q_i-q_{i,\rm nom}| \leq d_i(2+3b)\varepsilon<3d_i\varepsilon, \qquad |(Mq)_i-(Mq_{\rm nom})_i|<\nu_i\varepsilon. \tag{12} \]

All nominal form differences are at most b, so each nominal phase rate is at most nu_i*b/pi in magnitude. At an edge from parent 0, its nominal phase-rate difference is therefore less than b/2. Its changed phase-rate difference is less than 3*epsilon/2, and its cosine changes by less than 4*epsilon. Subtracting (6), using the perturbed cosine for the changed-rate term, proves

\[ |a_0(7)-a_{0,\rm nom}(7)| <\frac16\left(2b+\frac32\right)\varepsilon <\frac\varepsilon3. \tag{13} \]

Together with (8), this gives a_0(7)<-3*epsilon on the whole open neighborhood (9). The argument keeps changed actual weights and fine constituent phases. It does not impose twin symmetry on the neighborhood.

4. Whole-window acceleration and genuine histories

First use the original capacities, c=1. The already proved identity (3) implies an edge difference bound sqrt(2)*r for both form and phase error. The reference has zero S on the actual bare support. Since sine is one-Lipschitz and W_ij<=1,

\[ |\dot x_i|,|\dot\theta_i| <\frac{\nu_i\sqrt2 r}{\pi}<\frac{\nu_i}{32}. \tag{14} \]

Here sqrt(2)<3/2 and pi>3. Capacities are at most one, so theta_ddot=M L_W x_dot gives

\[ |\ddot\theta_i|<\frac{\nu_i}{48}. \tag{15} \]

For example, every summand in L_W x_dot has magnitude below 1/16. Differentiating (6), with fixed support, weights and capacities, gives

\[ x_i'''=M_{ii}\sum_{j\sim i} \left[-\sin(\theta_j-\theta_i)(\dot\theta_j-\dot\theta_i)^2 +\cos(\theta_j-\theta_i)(\ddot\theta_j-\ddot\theta_i)\right]. \tag{16} \]

At parent 0, (14)--(15) imply the uniform, deliberately conservative bound

\[ |x_0'''|<\frac16\left(\frac1{256}+\frac1{24}\right) =\frac{35}{4608}<\frac1{128}. \tag{17} \]

This bound uses whole-interval identity, not a three-endpoint membership check. Set T=2^-11. For 0<=s<=T, equations (13) and (17) imply

\[ a_0(7+s)<-3\varepsilon+\frac{T}{128} =-2\varepsilon=-2^{-17}. \tag{18} \]

For any true samples of the same solution at 7<=t_0<t_1<t_2<=7+T, the nonnegative unit-mass kernel identity expresses the shared three-sample acceleration as an average of a_0 over the whole window. Therefore

\[ \widehat A_{0,1}<-2^{-17}. \tag{19} \]

For the capacity family, (2) proves the corresponding signed result on every window wholly inside [7,7+T/c]:

\[ \widehat A_{0,c}<-c^2 2^{-17}. \tag{20} \]

In particular, the exact sufficient accessibility interval is c>sqrt(2^17*tau). The simple interval c>=128 satisfies it with c^2*2^-17>=1/8>tau. These factors are derived from the analytical margins; no trajectory or gain scan selected them. A larger c reduces the guaranteed physical duration proportionally. For homologous triples, their observed accelerations scale exactly by c^2; a longer unchanged observation horizon cannot be substituted into (20) without another signed-window proof.

The small-capacity obstruction is also retained: the earlier all-time bound scales to abs(A_hat_0,c)<c^2/128. Thus c^2/128<=tau suffices to exclude parent threshold crossing on every event-free window of the complete original source family. The gap between this sufficient obstruction and (20) remains unclassified. Neither bound identifies a sharp critical capacity.

5. Observation precision is a separate obligation

Freeze homologous equal-spacing triples

\[ (t_0,t_1,t_2)=\left(7,\ 7+\frac{2^{-12}}c,\ 7+\frac{2^{-11}}c\right), \qquad h_c=2^{-12}/c. \tag{21} \]

The samples in the theorem are actual exact values of the same solution. No numerical producer or measured history is evaluated here. If each materialized endpoint has an independently certified form error at most epsilon_obs at the exact timestamps, the exact endpoint contribution to the reported acceleration is bounded by

\[ E_{\rm samples}\leq\frac{4\epsilon_{\rm obs}}{h_c^2} =c^2 2^{26}\epsilon_{\rm obs}. \tag{22} \]

Retaining the prior illustrative allowance epsilon_obs=2^-40 gives E_samples=c^2*2^-14. This sufficient error allowance is eight times the guaranteed true magnitude c^2*2^-17; it cannot certify crossing using the present bounds. Raising c does not repair that ratio. This is failure of this sufficient observation certificate, not proof that a true history fails (20), that actual endpoint errors attain their allowance, or that no other observation protocol could distinguish the response.

If a future producer also bounds clock, materialization and reader arithmetic contributions by E_other, a sufficient represented crossing condition is

\[ c^2\left(2^{-17}-2^{26}\epsilon_{\rm obs}\right) -E_{\rm other}>\tau. \tag{23} \]

In particular, before any additional errors are admitted it requires

\[ \epsilon_{\rm obs} <\left(2^{-17}-\frac{\tau}{c^2}\right)2^{-26}. \tag{24} \]

As an explicitly different, hypothetical precision budget, take epsilon_obs<=2^-52 and 128<=c<=256. Use the same dyadic absolute times (7,7+2^-20,7+2^-19) for every c in this interval. Their rescaled span is c*2^-19<=T, so (20) applies. These common physical-time triples are subwindows of the guaranteed domains, rather than a single homologous-window comparison. Their spacing is exactly h=2^-20, and the sample allowance is 4*epsilon_obs/h^2<=2^-10.

The existing form-range proof retains every exact form strictly between 1/8 and 1/2; common capacity scaling leaves its normalized conserved mean and identity unchanged. The hypothetical represented sample error also leaves these bounds intact. At the specified dyadic times, a binary64 subtraction of two such forms has rounding error at most 2^-55; multiplying by 2^20 to obtain a slope is exact, so each slope error is at most 2^-35. Each slope has magnitude below 3*2^17, so their difference has magnitude below 3*2^18 and additional rounding error at most 2^-34. The inherited two errors and this subtraction total at most 2^-33; the final exact factor 2^20 gives reader arithmetic error at most 2^-13. The timestamps and their differences are exact; no timestamp jitter is admitted in this illustration. All nonzero computed intermediates remain normal and finite. Thus,

\[ |\widehat A_{\rm binary64}|> \frac18-2^{-10}-2^{-13}>\tau \tag{25} \]

would follow if the true endpoint error were independently certified. No calculation here supplies that endpoint accuracy or actual records. At the retained 2^-40 endpoint allowance the same fixed dyadic window instead permits sample error four, which again cannot certify crossing from these bounds. A current derivative, fabricated quadratic samples or a synthetic runtime fixture cannot supply the missing observations. Non-dyadic triples in (21) retain their separate timestamp representation and jitter obligations.

The shared sample-jet budget owner provides independent endpoint, timestamp and finite-window derivative allowances where its own hypotheses are admitted. The full-law third derivative scales as c^3/128 in (17). When using the exact positive-kernel lower bound (20), do not also subtract a Taylor truncation allowance merely for that same finite-window averaging; such a term belongs to a different comparison against an instantaneous derivative. Reader rounding and sampling provenance remain separate in either comparison.

6. Constitutive access does not complete canonical event access

The complete VAL map has a supported EDGE_AWARE_ENABLED=False branch that leaves form fixed while multiplying capacity. Applying its ideal primary proposal to every node with common factor c would have the desired physical endpoint: no phase or support change, and zero H work. A binary64 endpoint still requires represented ratio checks. This map identity is not a live admission or an executed word. With edge awareness enabled, actual form scaling and projection must instead remain in the complete map; that event cannot silently replace the pure capacity comparison in (2).

Under the selected fresh pressure handoff (1), reciprocal bare support gives

\[ \sum_i d_i p_i=\frac1\pi\sum_i S_i=0. \tag{26} \]

The inherited strict VAL precondition requires stored pressure at least its positive unoverridden minimum VAL_MIN_DNFR=1e-6 at every targeted node. After the declared constitutive refresh, (26) prevents simultaneous admission at all twenty targets. Thus this direct global pure-capacity VAL stage is blocked under the unchanged policy. Unrefreshed old stored pressure has different provenance and cannot be substituted for (1) to remove this obstruction. Disabling strict checks or altering the minimum changes the selected execution contract; no such change is made here. This statement does not exclude every local, staged or differently prepared canonical word.

The temporal trigger also does not settle THOL's own strict positive-pressure condition. At the exact unsplit source, the old sine contributions at parent 0 cancel and the new gaps are zero, so p_0(7)=0. For the original capacities, p_dot_0=a_0/nu_0=2*a_0; (18) therefore makes pressure strictly negative for every 7<t<=7+T. Under (2), the homologous pressure values are unchanged. The nominal member hence fails the strict THOL positive-pressure gate despite its above-threshold acceleration at the larger supplied capacities. The open-neighborhood acceleration theorem alone makes no positive-pressure assertion for its other members.

Finally, the inherited AL/UM/SHA history supplies no recent registered destabilizer for THOL's independent U4b context. A capacity number appends no glyph or genuine EPI history. An actually admitted VAL action would supply such a registered destabilizer, which is a useful conditional connection between capacity access and grammar; its own remaining U2 debt and closing requirements would still need an admitted word. The blocked direct global stage cannot claim that history effect. Hierarchy depth, metabolic child construction, a later parent-target UM proposal, and successor identity each keep their own applicable contracts; none is discharged by (20). There is no child event or cross-birth U6 reference in this study.

7. Shared calculation, controls and scope

The weighted sine support owner owns form_acceleration_bounds, a complete-law static row calculation from admitted primitive state. It does not turn a current derivative into temporal evidence. The acceleration controls check its actual weighted acceleration and capacity scaling against independent arithmetic. The temporal-access controls check the analytical source, margins, polynomial homology and precision/strict-admission distinctions. The exact finite-window kernel and its independent controls remain with the obstruction owner. The new controls' synthetic histories and runtime states are controls, not sampled records of this maintained source. The capacity-action controls separately check the pure proposal, its strict pressure admission and its represented endpoint limits.

No frozen source producer, evaluated research response or continuous trajectory is regenerated. The mathematical result is a nonempty open temporal-access domain inside an already maintained organization under a different supplied capacity family. It separates true dynamics, observation accuracy and canonical event admission. It establishes neither autonomous capacity selection nor birth, physical identification, or a universal capacity threshold for TNFR.