Local capacity and joint temporal admission in the maintained contact¶
A capacity-only VAL action at the fixed first A1 member admits a nonempty open domain in which the maintained contact retains its identity and fresh positive parent pressure while genuine three-sample histories exceed the unchanged THOL child threshold. This is a conditional mathematical admission with a separate observation-precision obligation. It supplies neither measured samples nor an executed child event. The research program owns status and subsequent work.
1. Fixed source, target and actual capacity-only action¶
Retain the aligned twenty-node, 45-edge source of the proximity-contact protocol. The original AL/UM/SHA word acts on A0, whose members are nodes 0 and 1. The prospective local VAL and THOL target here is node 2, the first A1 member. This target was fixed before the present derivation. No target-0 history or threshold verdict is transferred to it.
Retain the original support, each actual generated conductance, source origin
9/32<m<11/32, twist alpha=2*pi/5, increments b=h=1/1024, lifts, rails,
candidate policy and execution state. The original independent centered form
and phase deviation balls have joint radius z=1/512 in each ten-node
component. Use the following open subset in those same prepared coordinates:
This shrinks the admitted source domain without changing the contact word or
retuning its output. The old weights are one; every actual new weight lies in
(3/4,1]. Before the new action, capacities are one except
nu_0=nu_1=1/2. Degrees are seven on A0, six on B0 and four elsewhere.
Apply the supported capacity-only VAL branch
with EDGE_AWARE_ENABLED=False, at node 2 alone. Keep strict operator
preconditions and the unoverridden VAL_MIN_DNFR=1e-6. Fresh pressure is the
constitutive evaluation in (3), admitted below. Write c for the actual
post-action capacity of node 2, whose prior capacity is one. The analytical
factor interval is
This interval follows from the estimates below; no trajectory or gain scan
selects it. A requested factor must produce this represented endpoint.
Multiplication of the original unit capacity by a represented c is exact;
other representations or a changed initial capacity need their own endpoint
check. The unchanged strict VAL_MAX_VF=10 policy checks the incoming
capacity one; it is not a ceiling on the proposed endpoint. The source form
also exceeds the unchanged VAL_MIN_EPI=1/(2*pi). A second VAL immediately
after this one would fail that incoming-capacity check. This action does not
establish unrestricted repeatability or a physical upper-capacity law.
The complete VAL primary endpoint retains form, primitive phase, support,
conductances and stored pressure. No elapsed continuous time occurs in the
action. Its secondary execution records remain associated with node 2. With
the inherited empty glyph history there, the actual local VAL produces
['VAL'] and U2 debt one; it supplies a registered destabilizer for THOL's
U4b context. It supplies no EPI history. Neither this prefix nor a prospective
THOL event is a completed word with its closing obligations discharged.
The continuing law, at the same structural clock with the completed action at absolute time 7, is
Here nu_2=c, the other capacities retain their preceding values, and all
capacities are held. Retain e=0,w=beta=1, no forcing, no continuous loss,
fixed conductances and bare support. There are no events on the subsequent
observation interval. Pressure is independently specified by (3), not
reconstructed from an observed derivative. The local change is not the global
time conjugacy used in the common-capacity study.
Both actual rows and the new mean weights are retained.
2. Zero event work, changed means and all-time identity¶
Use the original storage and relative identity,
The capacity-only action has exactly zero H work and leaves Z unchanged.
The existing source and barrier proof
gives Z(7)<r, H(7)-H_*<1/8192 and
H-H_*>=Z^2/25 throughout Z<=r. These statements depend on the actual
support, weights and state, not on a common value of the positive capacities.
Equation (3) conserves H for the nonuniform diagonal M as well. Consequently
the same first-exit argument gives
The conserved means after VAL use rho_i=d_i/nu_i. The original total weight
is 104; after the event it is R_c=100+4/c. For either form or the declared
continuous phase lift v, the mean change is exactly
These new means, rather than their old values, are conserved during the continuation. Positivity of their weights and (5) bound the common coordinates and give global smooth continuation, as in the maintenance argument.
The live form gate can also be kept explicitly. Put w=rho/R_c. Then
sum rho_i^2=704+16/c^2<=720, R_c>=100, and
Every initial form is between m-epsilon and m+b+epsilon, so the conserved
positive weighted mean is in the same interval. Since
x_i-mu_x=(e_i-w)^T Px, (5)--(7) give, for every node and every future time,
In particular 1/5<x_i(t)<1/2, retaining the strict THOL form gate and the
weaker (1/8,1/2) range used for reader arithmetic below. This is an all-time
identity and form-range result, not a claim that the parent pressure keeps its
sign forever or that the organization remains maintained after child birth.
3. Exact nominal pressure and nonuniform acceleration¶
Use the exact unsplit source at its own common m. Its actual post-word coordinates and generated cross weight are derived in the preceding source calculation. Write
where epsilon_E is the unchanged represented regularizer 1e-9. Node 2 has
phase gamma and form m. Its four neighbors are the two A0 nodes, with phase
zero and form m+b, and the two A2 nodes, with phase 2*alpha and form m.
Consequently
For the inequality, use cos(alpha)>3/10 and
sin(h*alpha)>(3/4)*h*alpha. The latter follows directly from
sin u>=u-u^3/6 and 0<u=h*alpha<1/100. Thus this is a positive-pressure
candidate before any capacity change, independently of c.
The weighted gradients are q_2=-2b, q_0=q_1=(4+2*omega)b, and zero on
A2. Differentiating both rows of (3), with capacities held, gives
Keeping the two distinct neighbor contributions at node 2 yields
This is a positive linear-plus-quadratic expression in the local c, not a
globally scaled old acceleration. Using cos(h*alpha)>15/16,
cos(alpha)>3/10, pi^2<10 and b=256*epsilon gives
4. Full-source perturbations retain both strict signs¶
The existing open-source calculation
retains the complete actual contact map on (1): each output form differs from
its own unsplit nominal by less than epsilon, each output phase by less than
2*epsilon in the retained lifts, and each new weight by less than
3*epsilon. Old weights remain one. These estimates follow from the nonzero
old-neighbor resultant, its circular chart, the actual score denominator and
the unchanged AL/SHA endpoints. No twin symmetry is imposed on (1).
For example, the resultant changes by less than 5*epsilon from
sqrt(5)>2, so its angle changes by less than 3*epsilon; the old-neighbor
interpolation then gives the stated phase bound. On each new edge, both forms
exceed 1/4; the unchanged score denominator exceeds 1/2, which bounds
the form part of the weight change by 2*epsilon and the phase part by
2*epsilon/pi. This supplies the stated 3*epsilon weight bound without
discarding the actual generated conductance.
Sine is one-Lipschitz. At node 2 the four changed phase gaps therefore imply
The represented policy value VAL_MIN_DNFR=1e-6 is also strictly below this
rational margin. VAL leaves phase and support unchanged, so its refreshed
pressure has the same value. Both pre-action and post-action positive-pressure
statements refer to (3), not to an older stored-pressure label.
An implementation must additionally admit the represented pressure supplied
to its live precondition: its evaluation interval and materialization error
must retain the strict margin to the unchanged policy threshold. An interval
midpoint is a represented observation with that separate accuracy obligation,
not an exact-real pressure or an automatically installed continuation law.
Writing q-q_nom=L_W(x-x_nom)+L_(W-W_nom)x_nom, all nominal edge form
differences have magnitude at most b. Therefore
The same nonuniform capacities are used in each side of the comparison. At an
edge incident to node 2, the changed phase-rate difference is less than
(c+1)*epsilon. Its nominal magnitude is less than b*(c+1)/6, since the
parent phase rate is -c*b/(2*pi) and an A0 neighbor's is
(2+omega)*b/(7*pi). The cosine changes by less than 4*epsilon.
Subtracting (11), retaining the changed cosine with the changed-rate term,
gives
Combining (13) and (16) proves the uniform open-domain margin
These are instantaneous model calculations. The next section supplies the whole interval needed to infer a true sample acceleration.
The same mathematical bounds admit a useful direct-output corollary. A
separately declared state need not be asserted to lie exactly in the image of
the ideal contact producer if it independently satisfies all consumed output
premises: the same associated support and capacities; old unit weights and new
weights in (3/4,1]; coordinate deviations less than epsilon in form and
2*epsilon in the declared phase lifts from an exact nominal (9) at an
allowed m; new-weight deviations less than 3*epsilon from that nominal;
and Z(7)<r, H(7)-H_*<1/8192 for its own actual weights. The structural
barrier comparison in Section 2 applies to those weights. Equations
(8)--(29) then follow from exactly the premises used in their derivations.
This permits separate admission of a represented initial state and its exact
continuing law, without identifying rounded coordinates with exact alpha or
authenticating the state as a generated source. Execution context and any
claimed relationship to a producer remain independent obligations.
5. A complete observation window with positive pressure¶
The all-time identity (5) bounds every edge form difference and every edge
phase error from the reference by sqrt(2)*r. The bare reference has zero S.
Since the conductances are at most one, both rows of (3) obey
Moreover, theta_ddot=M L_W x_dot gives
For (19), every capacity is at most c, so a form-rate difference is less than
c/16; apply the weighted row sum bound and pi>3. All four neighbors of
node 2 still have capacity at most one. Differentiating (11) gives
The local neighbor capacities in (18)--(19), rather than a global time conjugacy, now imply
The rational inequality follows from c+1<=129*c/128. It is valid throughout
the maintained identity, not only at the initial snapshot. For
integration of (21) from (17) gives
Because p_dot_2=a_2/c for this held-capacity law, pressure strictly increases
on the same window. Thus
The form statement also holds beyond this window by (8). These inequalities retain the prospective THOL pressure, form, active-capacity and neighbor requirements with their unchanged thresholds. They do not select when to apply the event.
For any genuine samples of this same event-free solution at
7<=t_0<t_1<t_2<=7+T_c, the
nonnegative unit-mass kernel identity
expresses the shared three-sample acceleration as an average of a_2 over
the entire window. Consequently
where tau is the represented unoverridden THOL_BIFURCATION_THRESHOLD=0.1.
Freeze the common absolute timestamps
For every c in (2), c*2^-19<=2^-11, so this triple is wholly inside (22).
The local c-law trajectories are not asserted to be reparameterizations of
one another. Equation (25) holds separately for each admitted nonuniform law.
Every sample must be taken after the capacity event; a secant straddling VAL
would require a different argument.
6. Observation and reader error remain separate from access¶
The true-history statement (25) uses exact values of the same continuous
solution. No endpoint values or trajectories are produced here. An implemented
history must independently certify its own relation to those true samples.
As a sufficient, explicitly hypothetical budget, require each materialized
endpoint at (26) to have absolute form error at most epsilon_obs=2^-52.
No timestamp jitter is admitted; the absolute dyadic times and their
differences are exactly representable in binary64.
With spacing h_t=2^-20, the three-sample functional has coefficient absolute
sum 4/h_t^2. Its endpoint-error contribution is therefore at most
The strict form range (8), including this tiny endpoint allowance, lies inside
(1/8,1/2). Under ordinary binary64 round-to-nearest arithmetic, subtraction
of two such represented forms incurs error at most 2^-55; multiplying by
the exact slope factor 2^20 gives at most 2^-35. Each slope magnitude is
below 3*2^17. Their subtraction has magnitude below 3*2^18 and adds at
most 2^-34 rounding error. The inherited two slope errors and this rounding
sum to at most 2^-33. The final exact factor 2^20 gives
All nonzero intermediates remain finite and normal in this estimate. It applies to the shared two-slope reader at these exact timestamps, not to every backend or arbitrary rearrangement. Combining independent endpoint and reader allowances with the whole-window lower bound gives
The earlier illustrative endpoint allowance 2^-40 would instead allow
sample error four at these same times and cannot certify (29). That is a
failure of this sufficient precision certificate, not a failure of the true
access theorem. Increasing a capacity without controlling observation error
cannot supply the missing evidence. Numerical solver error, initial-state
representation, materialization and timestamp uncertainty must be assigned
to their actual independent budget before claiming a measured crossing.
The bound (23) already controls the actual finite-window average. No additional Taylor truncation term is needed merely to compare that average with its own lower bound. Such a term is required for a different comparison against an instantaneous derivative. Current RHS values, fabricated quadratic samples and synthetic runtime fixtures cannot establish the observation premise.
7. Canonical and scientific scope¶
The actual strict capacity-only VAL prefix is compatible with the open source pressure margin and leaves the full physical state unchanged except at the target capacity. Its target-associated history supplies THOL's recent destabilizer context; its U2 debt remains explicit. The mathematical continuation supplies the prospective live-state margins and a genuine above-threshold history domain. Actual temporal evidence must still end at the live state, use the declared time units, and satisfy the independent sample and error admissions. None of these facts installs a pressure refresh law in the general operator runtime or selects the occurrence of either event.
An eventual THOL proposal still retains its own hierarchy, metabolic, history,
grammar and commit contracts. No child proposal, birth, parent-target UM stage
or closing word is executed here. No newborn dEPI baseline, cross-birth U6
reference or successor-maintenance certificate is inferred. In particular,
the capacity-independent barrier proved above applies before any support
change; it does not transfer through child creation or attachment without
their complete event accounting and new-law admission.
The original fixed-capacity obstruction and global VAL fresh-pressure obstruction remain valid. This result concerns the fixed node-2 action and the open subdomain (1) under the stated nonuniform continuation. It establishes neither a universal capacity threshold nor autonomous hierarchy formation, a unique TNFR law or physical identification. No frozen source producer, evaluated research response or continuous trajectory is regenerated.