Phase-nearest UM contact and joint identity maintenance¶
This comparison changes one existing UM policy parameter in the
actual organization-contact protocol:
UM_CANDIDATE_COUNT=3 replaces its unrestricted value 0. The source
family, ordered word, remaining parameters, actual generated weights,
continuous law and structural clock retain their declared premises.
The earlier unrestricted-support results remain valid for their own model.
1. Frozen comparison protocol¶
Before evaluating the actual word or its reserved static controls, freeze
the following comparison. Reuse the complete source and operator settings
in the original protocol,
including independently perturbed doubled-C5 components A and B, their
common prepared form mean and phase origin, z=1/512, the strict full-state
source balls, the actual target pair (0,1) and the ordered word
emission,coupling,silence. Change only UM_CANDIDATE_COUNT to 3;
retain proximity selection and the complete candidate inventory. No
receiver action, new operator, controller, input or continuous loss is
introduced. The actual event remains supplied.
The prospective support prediction is the old forty edges plus five: the internal A0 pair edge and all four A0-to-B0 edges. Every generated conductance must retain its actual value and endpoint association. Admit all actual operator preconditions and the full weighted continuation, including changed degree normalization and capacity. Do not substitute a pair-constant dynamics for generic boundary conductances.
The prospective common identity uses one twenty-node arithmetic projection
P=I-11^T/20, the original common-aligned twists theta_*, and
The phases in this expression are declared continuous lifts of the original twists. One common origin is removed; the relative A/B form and phase origins remain observable identity coordinates. The target is maintenance of this identity for the entire unchanged source family under the fixed post-word law, not just existence or linear stability of a symmetric reference. Establish a common stationary reference, exact balances and a strict energy barrier, or record precisely which obligation fails. Do not enlarge the identity radius, shrink the source or retune the word after examining a response.
Reuse the original split and unsplit nominal controls, source representation,
seed, timestamps, execution order and independent 100-digit static oracle
with ordinary response allowance 2^-46. Add independent exact algebraic
controls for support, mean weights, stationarity, coercivity, full-source
energy bounds and event-work accounting. Analytical bounds apply to the
exact family; represented controls re-admit their own input defects. No
continuous trajectory, time sweep, topology search or physical calibration
is part of this gate. No frozen response from the unrestricted-support
study is regenerated or reinterpreted as a result of this comparison.
2. Actual candidate selection and the five added edges¶
Write alpha=2*pi/5 and retain b=h=1/1024=z/2, where b is the AL
form increment and h the UM phase push. Before UM, each selected A member
has its four original A neighbors. The shared candidate reader excludes
the target and existing neighbors, applies the snapshot U3 phase gate,
then orders admitted missing neighbors by absolute circular phase gap,
using the retained node rank only to break ties.
The selected member's partner and the two B0 members have snapshot gaps
at most 2*z. Every member of B1 or B4 instead has gap at least
alpha-2*z>2*z; the distance-two A and B pairs fail the unchanged
pi/2 phase gate. Thus the three nearest candidates are exactly the
partner and both B0 members for every source in the unchanged family.
Their internal ranking can vary without changing the selected set.
The two target proposals create one common internal edge and four
cross edges, not duplicate undirected edges. No external candidate list
or new neighbor-selector law is supplied.
The selected candidate count changes only these link proposals. The existing-neighbor consensus, four bidirectional neighbor phase writes, capacity blend, selected stored-pressure relief and SHA reset are still the actual primary map. They read the immutable old support, not prospective new edges. B node state and execution history do not reset during this word. All inherited live form/capacity, circular phase, grammar and associated-history preconditions retain their original strict margins.
The final compatibility score uses the actual post-AL forms and merged
phases, with homogeneous Si=1/2 as before. A new cross gap is below
2*z+3*h*z<3*z, its form difference is below 2*z+b=5*z/2, and both
forms exceed 1/4. The shared score therefore satisfies
The internal edge retains its previous bound
omega>1-sqrt(2)*z>3/4. Every selected candidate therefore passes the
final phase and score gates. Every generated weight lies in (3/4,1]
and remains attached to its actual endpoints. New edges have the actual
weight attribute; no length or unit conductance is inserted by the
analysis.
The resulting connected support has 45 edges: forty retained unit edges,
the internal A0 edge, and the complete four-edge A0--B0 block. Selected
A0 members have bare degree seven, B0 members have degree six, and all
other nodes have degree four. Their post-word capacities are respectively
1/2,1,1. Consequently the actual mean weights are
The old unrestricted fan's degree normalization and mean weights do not
transfer to this graph. For the nominal post-word forms the weighted
form mean is m+7*b/26; arbitrary admitted deviations retain their own
weighted correlations rather than being replaced by that nominal value.
3. Complete law, reference and joint identity¶
Keep the conservative weighted-form/bare-phase completion
with e=0,w=beta=1, zero forcing, held actual support and conductances,
held post-word capacities and the original structural clock:
The independently specified pressure is p_i=S_i/(pi*d_i), evaluated
on the new support. It is distinct from the stored pressure relieved by
UM on the old support. The actual storage and conserved means are
Use the common-aligned reference x_*=m*1 and
theta_*(A_(k,s))=theta_*(B_(k,s))=k*alpha, on the original lifts.
Every old fine current cancels at that twist. All five new edges have
zero reference form and phase differences. The reference is therefore
stationary under the complete actual weighted law, with
Either common origin can be translated to match a particular source's conserved weighted mean, without changing H or the identity. The relative geometry uses one twenty-node arithmetic projection,
If eta=theta-theta_*, this identity separates exactly as
Here Z_A,Z_B are the earlier ten-node centered constituent norms.
Thus the joint identity retains relative form and phase origins rather
than quotienting them away independently. A bound on Z protects both
constituent geometries and their relative registration. The arithmetic
means used in this observation are not asserted to be conserved.
The whole reference has zero first circular phase resultant:
sum_i exp(i*theta_*,i)=4*sum_(k=0)^4 exp(i*k*alpha)=0.
There is no defined global resultant angle there. The arithmetic lift
projection is the declared geometric observation; it is not a new primitive
collective phase or a circular-mean reconstruction.
Generic actual cross conductances still require the member-to-boundary association identified by the earlier contact result. The proof below works on the complete fine state. Equal reference phases or equal bare neighbor counts do not turn the actual weighted evolution into a closed pair-symmetric model.
4. A full relative spectral bound for comparison matrices¶
Let L_0 be the unit Laplacian of the two disjoint original doubled C5
graphs, and N the unit Laplacian of the five added edges. The actual bare
Laplacian is L_U=L_0+N. Since all actual edge weights exceed 3/4,
This is a quadratic-form comparison, not a replacement of W in the law.
To lower-bound lambda_2(L_U), first omit the internal A0 edge from a
unit comparison graph. The remaining graph is the two-member doubling
of a ten-vertex base graph consisting of two C5 cycles joined by the
single A0--B0 base edge.
Only this unit comparison matrix is decomposed by its symmetries.
Fine pair-antisymmetric directions have eigenvalues four or six. On
pair-common directions its Laplacian acts as twice the base Laplacian.
Exchanging the two identical base cycles splits that base matrix into
the symmetric block L_C5 and the antisymmetric block
L_C5+2*e_0*e_0^T. The first block has one common zero mode and its
next eigenvalue is (5-sqrt(5))/2>1/5.
For the second block, five times its shift by I/5, in cyclic node
order, is the exact matrix
Its leading principal minors are
19,146,839,3901,909, all positive. Sylvester's criterion gives
lambda_min(L_C5+2*e_0*e_0^T)>1/5. Restoring the omitted internal
edge adds a nonnegative quadratic form. Consequently
The argument controls every relative fine direction, including independent member splits and the relative A/B origin. It imposes no symmetry on the actual W, source perturbations or future trajectory.
For the source upper bound below, two further simple facts suffice:
lambda_max(L_0)<=8 from its degree-four support, and
lambda_max(N)=4. The latter follows by restricting to the four A0/B0
members: N is the Laplacian of K4 with the B0 internal edge removed,
with eigenvalues 0,2,4,4; the other sixteen directions are zero.
5. Lower storage barrier on the unchanged joint radius¶
Take the frozen r=1/16. On Z<=r, any edge phase error from the
reference is at most sqrt(2)*r. Every old principal reference gap has
magnitude alpha and every added reference gap is zero. The old gaps remain acute, and their
cosines obey
Here cos(alpha)=(sqrt(5)-1)/4>3/10, sqrt(2)<3/2, and
alpha+sqrt(2)*r<pi/2. The added-edge cosines exceed the same lower
bound. Thus the phase Hessian throughout this whole radius obeys
L_(cos delta) >= L_U/5. The actual form block also dominates
L_U/5 because L_W >= 3*L_U/4.
Translate the common origins away and Taylor-expand about the stationary reference along the straight centered-state segment, which remains in the radius. The full Hessian comparison and Section 4 give
At the radius boundary this requires
The bound uses both complete rows and all fine directions. The same reference is a strict relative-storage minimum, but its existence alone would not establish retention of the actual post-word source.
6. Upper storage bound for the entire actual post-word family¶
Let v_0=(xi,eta) be the initial twenty-node deviation from the common
reference, before the supplied word. The independent constituent source
balls give
Let a=b*1_(A0) be the actual AL form increment and let d be the actual
UM phase increment, with all other phase entries zero. The previously
admitted consensus bounds give two selected increments below 3*h*z
in absolute value and four old-neighbor increments below 4*h/3.
Consequently
B phases are unchanged; SHA changes capacity only. The full post-word
deviation is exactly v_1=v_0+(a,d), with its retained phase lifts.
Centering is a contraction, so
Z_1<sqrt(2)*z+sqrt(11)*h<7*h<r. This admits the initial state to
the frozen radius before using an energy certificate.
A sharper storage estimate keeps the different form and phase curvatures.
The full post-word phase-error norm is below
sqrt(2)*z+3*h<6*h. Therefore every old-edge error is below
6*sqrt(2)*h<1/64. Along the straight segment from the reference to
this actual endpoint, the old-edge cosines are at most
Every new-edge cosine is at most one. Actual conductances are at most one. Define the positive-semidefinite comparison form
Taylor's formula at the stationary reference then yields
H_post-H_*<=Qcal(v_1). The use of 1/3 is local to this small source
segment; it is not asserted as an upper cosine bound throughout the
larger radius-r identity.
Since lambda_max(B)<=12, the original source contribution is
The AL increment has eight old boundary edges and four new cross edges; the internal edge has zero increment difference. Its exact comparison form cost is therefore
For the old phase part, the norm bound above gives
d^T*L_0*d/6<12*h^2. On the five new edges only the two selected
phase increments contribute, each of magnitude below 3*h*z. If those
increments are d_+,d_-, then
Hence the complete reset comparison cost satisfies
The square root of a nonnegative quadratic form is a seminorm, so its triangle inequality retains the worst allowed correlation between the initial source and the reset:
For the last strict algebraic inequality,
(sqrt(48)+sqrt(19))^2=67+2*sqrt(912)<128, since
4*912<61^2. No source correlation, selected-member asymmetry or actual
boundary conductance has been averaged away. The bound covers the entire
original source family with the frozen radius and word constants.
7. Joint identity is maintained for all structural time¶
The actual completed source has Z_1<r and storage excess below
1/8192, while any first radius exit would require storage excess at
least 1/6400. The complete fixed-support flow conserves H, so such
a first exit is impossible. Therefore
for every exact source in the frozen family, with time measured after the completed word in the original structural clock. All old and new fine edges remain acute, both original constituent winding neighborhoods and nonantipodal member charts persist, and the relative form and phase origins remain inside the joint bound. A common translation was removed; relative A/B motion was retained rather than assumed absent.
The conserved actual weighted means control the remaining common lifted coordinates. Together with the bounded relative state they prevent finite escape. Equivalently, the complete finite-dimensional field has bounded form rates and phase rates linear in the form range, giving global smooth continuation. The barrier is thus an all-time result, not a bound that silently stops at an unproved existence horizon.
This is conservative geometric maintenance. It does not imply convergence to the stationary reference, a period, asymptotic attraction, synchronization at a specified time, or acquisition of an initially absent organization. The two constituent organizations and their near-common registration were prepared, and their connection was supplied by the declared word.
8. Restoring relative-origin storage and complete event work¶
The connection contributes a joint geometric property absent from the
disconnected phase potential. Consider a configuration variation that keeps
each component's internal reference shape fixed but changes their relative
form and phase origins by Delta x,Delta theta. Its new cross storage is
exactly
The actual weight sum exceeds three, and the phase curvature at zero relative origin is four. Each old edge and the internal A0 edge retains its reference difference along this variation. Thus the joined storage penalizes relative registration changes that were flat directions for the disconnected storage. This is an exact potential restriction and local restoring curvature; actual weighted flow need not preserve rigid component shapes or close on these two origin coordinates. Common recentering can preserve both actual weighted means along this configuration variation without changing its storage formula.
The hybrid work uses the actual endpoints and generated weights. Write
X_(A,k) for an A pair's mean form, u_(A,0) for half its selected
form split and delta_(A,0)' for half the post-UM selected phase split
on the retained member chart. The
AL and old-edge UM terms are unchanged from the
whole-word accounting:
The actual attachment term now contains the internal edge and four cross edges only:
Thus H_after-H_before=DeltaH_AL+DeltaH_UM,old+DeltaH_attach exactly.
SHA and capacity synchronization change subsequent rates and conserved
mean weights, but add no direct storage term. Continuous conservation
does not make the event passive, pay its attachment work or select its
occurrence. The conductances remain held at their actual event values;
continuously recomputing compatibility weights would define another law.
9. What this policy comparison establishes¶
The existing phase-proximity selector, with the single declared count change, yields a distinct admitted contact whose entire unchanged source family has a strict joint storage barrier. The unrestricted thirteen-edge attachment still has its earlier stationary and invariant-level obstructions. Those results are not corrected or overwritten: they apply to a different support produced by the other supplied candidate policy.
The comparison isolates the importance of which boundary relations are created. A phase-compatible connection need not preserve a chosen identity, and more admitted links do not automatically strengthen its maintenance. Here the four-edge registered interface preserves a stationary joint geometry and a full-source barrier; the wider fan changes the stationary current constraints. Neither policy is selected as a universal natural rule by this comparison.
The result supplies conditional joint maintenance and relative-origin restoring storage using actual UM semantics and the explicitly admitted continuous completion. It leaves autonomous preparation and contact, acquisition of a new identity, sufficient reduced interaction coordinates, and independently calibrated physical observations as separate claims. Treating the maintained organization as a collective NFR retains its fine state, declared lifts, boundary associations and applicable execution context; it does not reduce it to one scalar node or establish closure under every registered operator.
The independent static controls reuse the existing complete-word executor and leave the earlier source, oracle and unrestricted-policy controls intact. The separate count-three fixture checks actual candidate membership, reset/context writes, every weight, full-field balances and event work. Exact full-matrix comparisons check the spectral and energy inequalities; the two represented preparations retain their own source defects and captured storage. These finite controls check implementation and proof ingredients, while the all-source and all-time claim follows from Sections 2-7. They do not evolve a trajectory or turn a detached field report into a maintenance certificate.
10. Frozen independent-registration protocol¶
This separately declared extension keeps Sections 1-9 and their evaluated controls unchanged. Before evaluating its new actual-word controls, freeze the open registration square
Here a is B's relative form origin in the declared model form units and g
is its relative primitive-phase origin in radians on the declared lifts.
A retains x_A=m+xi_A, theta_A=theta_*+eta_A; B now has
x_B=m+a+xi_B, theta_B=theta_*+g+eta_B. Retain the complete original
independent centered internal balls ||(xi_C,eta_C)||<z=1/512 and the
entire original A mean interval 9/32<m<11/32. Re-admit B on its translated
mean interval, contained in (9/32-h,11/32+h); do not silently restrict m
or demand that B remain in the original narrower absolute mean interval.
The two offsets are independently supplied preparation coordinates, not new
evolution laws or time-dependent inputs.
Keep the exact count-three proximity word, all other parameters, target
order, histories, timestamps, seeds, full candidate inventory and execution
path from Section 1. Keep actual generated conductances, held post-word
capacities, zero loss/forcing, original structural clock and the same
twenty-node identity Z<r=1/16 about the common-aligned reference. Do not
move the relative target to the offset source, remove its relative origins,
shrink the internal balls or enlarge r to obtain maintenance.
Predict the same five added edges, 45-edge support and mean weights
rho_A=60,rho_B=44,rho_total=104 throughout this square. A's old-neighbor
reset maps remain unchanged; B retains its shifted state during the word.
Recompute every cross conductance and attachment term from the actual
shifted endpoint. Admit the full weighted continuation and prove the uniform
source bound
or record which obligation fails without retuning the square. Reuse the
existing bare comparison metric, its exact old-spectrum upper bound
5+sqrt(5)<15/2, and the complete reset displacement bounds. The intended
discriminator is a quantitative open domain for both relative origins,
beyond aligned-source maintenance or continuity alone. It is a sufficient
domain, not a maximal capture region or a failure claim outside the square.
The reserved static controls use the original nominal split source and
the eight fixed offsets
(a,g)=(+s,+s),(+s,-s),(-s,+s),(-s,-s),(+s,0),(-s,0),(0,+s),(0,-s),
where s=3/4096=3*h/4. Shift only B's primitive form and phase by these
exact dyadic values before execution. Preserve the original dyadic twist,
seed, timestamps, complete word and 100-digit independent oracle with
ordinary response allowance 2^-46. Retain represented input defects
separately from the exact family. The previously frozen zero-offset split
and unsplit controls remain available without changing their producers.
For each fixed offset check actual selection, weights, primary/context writes, both full-field rows, new weighted means, complete hybrid work and the captured joint storage margin. Exact algebraic controls check the source-family inequalities and retained relative-origin coordinates. Axis controls distinguish the form and phase contributions to the declared rigid-registration storage restriction; that restriction is not a closed trajectory model. Use shared word execution, module fixtures and report readers. No trajectory, parameter sweep, event selector, extra input or physical identification belongs to this protocol.
11. Admission throughout the independent registration square¶
Keep the frozen notation a,g for B's relative form and phase origins.
Write f=b*1_(A0) for the AL displacement, to distinguish that vector
from the scalar a. The component-centered source deviations remain
v_0=(xi,eta) with ||v_0||^2<2*z^2, and b=h=z/2 as before.
The expanded B mean slab is part of this extension's admission. Every B source form satisfies
In particular B stays strictly between 1/4 and 1/2. A retains its
original mean slab, source ball and live event margins. Both source balls
remain complete independent balls; no correlation with a or g is imposed.
On the disconnected old graph, a uniform translation of one component's
form or primitive phase leaves its own complete continuous rows and
internal storage unchanged. Its previously established internal identity
therefore admits the translated mean leaf. The later operator score still
reads the translated B forms, so that readmission cannot be replaced by
an assertion that B's old absolute mean interval was preserved.
Before UM, the selected member's partner has phase gap below 2*z,
and the B0 members have gaps below
|g|+2*z<5*z/2. Every B1 or B4 member instead has gap above
alpha-|g|-2*z>alpha-5*z/2>5*z/2. The distance-two A pairs retain
their old exclusion, and the distance-two B pairs have gaps above
2*alpha-5*z/2>pi/2. The exactly three nearest admitted missing
neighbors are consequently the partner and both B0 members. Their
internal ordering need not be fixed. There is a strict gap between this
selected set and every other candidate throughout the entire square.
The primary phase, capacity and stored-pressure proposals read A's old
support and old neighbors. B's relative offsets enter none of those
reads. Thus the same A source produces exactly the same reset displacement
(f,d) as its zero-offset counterpart. In particular the two selected
phase increments satisfy |d_i|<3*h*z, the four old-neighbor increments
satisfy |d_i|<4*h/3, and B receives no state reset. Its unchanged
histories remain associated with the same B nodes; only the selected A
targets execute the declared lifecycle and word-context writes.
At the final link-score stage, every selected cross edge has phase gap below
and form difference below b+|a|+2*z<3*z. Both forms exceed 1/4.
Using the actual compatibility formula and its positive regularizer,
The internal A0 weight keeps omega>1-sqrt(2)*z>3/4. Thus every
selected link passes its final phase and score gates, the same five
edges are added, and all actual conductances lie in (3/4,1]. Cross
weights are recomputed from the shifted endpoints; they are generally
neither equal to one another nor equal to the zero-offset weights.
The resulting 45-edge support, capacities, degrees and rho values are
exactly those of Section 2. The full continuation of Section 3 is
therefore admitted with each preparation's own actual held W.
12. Retained origins, initial radius and conserved means¶
Let 1_B denote the indicator of B's ten fine nodes, and define the
registration displacement
For the same A/B internal deviations, the exact final deviation from the common-aligned reference is
No relative origin is absorbed into the target reference. Let
dbar_A=(1/10)*sum_(i in A) d_i. Since xi and eta are centered
separately in each source component, their post-word arithmetic relative
origins are exactly
Section 3's joint decomposition gives, writing P_10 for independent
ten-node centering only in this displayed identity,
Equivalently, relative to the otherwise identical zero-offset completed preparation,
The cross terms retain the reset/registration correlations. They cannot
be discarded by adding independent marginal radius budgets. For a simple
uniform initial admission, however, the joint projection is a contraction
and ||P o||^2=5*(a^2+g^2)<10*h^2, so
Here P acts on both form and phase blocks, as in Section 3. This admits the actual source to the unchanged identity radius before invoking any energy argument.
With rho_A=60,rho_B=44,rho_total=104, the actual conserved means
of the completed preparation are
Thus relative to its zero-offset counterpart, the two means change by
11*a/26 and 11*g/26, respectively. The nominal form contribution
is m+7*b/26+11*a/26; the fine perturbation correlations remain in
the actual formula. The common origins of the stationary reference can
be translated to match these two means without changing H, Z or the
relative target. Arithmetic component means are observations, not separate
conserved quantities after attachment.
These phase means use the declared continuous lifts. Even a perfect relative rotation of the two doubled twists leaves each component's first circular resultant zero. The means therefore do not supply a global resultant angle or a new scalar primitive collective phase.
13. Uniform storage margin for both independent offsets¶
The aligned reference remains stationary for every actual generated W:
all new reference differences vanish and all old reference currents cancel.
The reference storage is still H_*=40*(1-cos(alpha)). Since the actual
weights exceed 3/4, the full comparison gap and lower radius barrier
of Sections 4-5 apply without changing any coefficient.
For the upper bound, reuse the positive-semidefinite comparison metric
The old-edge cosine bound from Section 6 still holds along the straight
reference-to-source segment. The reason is edgewise cancellation: g
adds the same phase to both endpoints of every old B edge and adds
nothing to any old A edge. Thus every old-edge error along that segment
is exactly the corresponding zero-offset error. It remains below 1/64,
and the old-edge cosine upper bound remains 1/3. The former bound on
the full uncentered phase-error norm need not transfer to the shifted
source and is not used here. Every new-edge cosine remains at most one,
and all actual form conductances are at most one. Taylor's formula
therefore yields
A slightly sharper upper comparison than Section 6 makes room for both independent offsets. On either original doubled C5, fine pair-antisymmetric directions have eigenvalue four, while pair-common directions have twice the C5 spectrum. Consequently
Only these unit comparison matrices use their symmetry; the actual source
and W remain generic. It follows that lambda_max(B)<23/2, and hence
The exact AL comparison cost is still 6*h^2. Keeping the sharper
reset norm already established in Section 6 gives
The new-edge reset contribution is still d^T*N*d/2<36*h^2*z^2.
Thus the entire reset obeys
The last strict inequality uses the frozen z=1/512; it follows, for
example, from 117/(2*512^2)<1/9. The registration displacement has
no old-edge differences and no internal A0 difference. Its only four
nonzero edge differences are on A0--B0, so its comparison cost is exactly
This equality is in the unit upper comparison metric. The actual form storage uses the four actual W values and must not be substituted into the equality as if they were all one. Applying the seminorm triangle inequality once to all three contributions gives
The elementary bounds use 46<(34/5)^2 and 15<(39/10)^2.
Squaring establishes the frozen prospective inequality on the complete
source family:
In particular the storage gap to the unchanged radius boundary is
strictly greater than 51*h^2/20. This is a uniform analytic margin,
including the worst allowed source/reset/offset correlations, not a
margin fitted to the eight reserved preparations.
14. Maintenance on an open registration domain¶
For every exact source in the frozen independent square, Section 12
places the actual completed state inside Z<r, and Section 13 places
its conserved storage strictly below the radius-boundary requirement.
The same first-exit argument as Section 7 therefore gives
Time starts after the completed supplied word and uses the original structural clock. The held actual W, held capacities, both conserved weighted means and absence of forcing/loss are essential premises. The smooth full fine field has global continuation as in Section 7. Old and new edges remain acute, the two original winding neighborhoods persist, and both relative origins remain part of the retained joint identity. The prepared offsets are initial data; their subsequent relative coordinates are not held fixed by an additional law.
This supplies a quantitative open preparation domain around exact registration, with both independent internal balls unchanged. The square is sufficient and need not be maximal. Its boundary is not an inferred failure threshold, and the result does not establish a capture basin, autonomous contact, convergence to alignment, or asymptotic attraction. It establishes conservative maintenance of the prepared joint identity.
The proof needs total storage and its boundary barrier, not a uniformly
positive form-storage reserve. In particular the post-word arithmetic
form-mean difference is b/5-a, which can vanish inside this square.
A positive-reserve argument for the earlier equal-origin family cannot
be transferred merely by reusing that family's labels.
There is an admitted exact zero-form-storage example. Set a=b/5,g=0,
all initial phase deviations to zero, B's internal form deviations to zero,
and A's form deviations to -4*b/5 on its two targets and b/5 on its
other eight nodes. The A deviations sum to zero and their squared norm is
8*b^2/5<z^2. After AL every form is m+b/5; UM and SHA preserve those
forms. Therefore every weighted form difference and the entire post-word
form storage vanish, independently of the generated positive weights.
The phase storage and subsequent form evolution need not vanish. This
example invalidates a uniform positive form-reserve claim without affecting
the total-storage maintenance proof.
15. Actual event work and the registration storage restriction¶
Before attachment, uniform B origin translations leave every old-edge form and phase difference unchanged. A's AL term and old-edge UM work are therefore identical to those for the same zero-offset internal preparation. B's internal stored energy is also unchanged. The complete hybrid work identity from Section 8 still applies, but its four cross terms use the new endpoint differences and freshly generated weights:
Together with the unchanged AL and old-edge UM terms, this equals
H_after-H_before exactly for that preparation. The notation W(a,g)
records how the event output depends on its initial offsets; W is held
after the word. Recomputing it during continuous evolution would change
the model. Neither continuous conservation nor the positive storage
margin selects or pays for the supplied event.
For any one such held actual W, the rigid-component reference variation of Section 8 still has exact cross-storage restriction
This formula separates the two geometric storage contributions and can be evaluated on the declared form and phase axes. It does not equal the entire reset source excess in general: the prepared source also contains internal perturbations and the actual word displacement. Nor is it a closed trajectory law for two rigid objects. Generic actual conductances continue to couple the full constituent and boundary state.
16. Meaning and evidence boundary of the extension¶
Exact common preparation origins were sufficient in Sections 1-9, but are not necessary for their joint-maintenance mechanism. The unchanged count-three word admits a nonempty open square of independently different form and primitive-phase origins, while retaining every original internal perturbation, the original A mean interval and the same joint radius. B's translated absolute mean slab is explicitly re-admitted. The result uses actual selected edges, actual compatibility weights, all fine rows and the appropriate conserved means throughout.
The reserved controls in Section 10 separately check the implementation on fixed represented preparations. Their input defects and event outputs remain distinct from the exact family. The proof of the full-family, all-time claim is Sections 11-14; static samples cannot replace it. No reduced evolution, physical identification or closure under all registered operators follows. A collective TNFR description continues to retain the fine state, declared phase lifts, member-to-boundary associations and the applicable event and grammar contexts.
17. Frozen finite exchange comparison¶
This bounded comparison closes the response question for the admitted UM contact before the operator/pressure-handoff gate in the sole execution plan. It does not widen the registration square or introduce another contact policy. Freeze the following full-family prediction before evaluating the new paired discriminator and its algebraic controls.
Use the complete independent internal preparation of Section 10, with the
original A mean interval, z=1/512, held unit initial capacities and all
associated execution-state premises. Pair exactly the same internal form,
phase, pressure and history data in two preparations. Keep relative form
origin a=0; give B opposite phase origins +g,-g, where g=3/4096.
This lies strictly inside the proved registration square. Execute the same
count-three emission,coupling,silence word on (0,1) in both preparations,
with every parameter, seed, target order and timestamp unchanged. Each output
has its own generated conductances W_+,W_-; never substitute one for the
other. Continue each completed output under its own held weighted law from
Section 3, with zero forcing/loss and the original structural clock.
Observe the fixed linear form contrast
The weights are 14 on A0, 6 on B0 and 4 elsewhere in both preparations.
Define the paired response C(t)=D(x_+(t))-D(x_-(t)); its initial value is
zero because the completed forms agree. Retain continuous phase lifts
initialized by theta_+(0)-theta_-(0)=2*g*1_B. The frozen horizon is
T=1/8 after the word, with a readout window [1/16,1/8]. Predict, for
every matched source in the full internal family,
Establish the finite-window sign using both full evolution rows, their actual distinct conductances and the proved identity bounds. A nonzero initial derivative or a rigid-component oscillator approximation is insufficient. Keep the source correlations in the paired comparison; a uniform sign for either preparation's unpaired flux is not required or inferred.
For the null control, take the same completed primitive states, remove only
the four A0--B0 cross edges and retain the common internal A0 edge and actual
capacities. This is a detached alternative model, not an executed deletion
event or an unpaid support jump. Re-admit each connected component with
its own degrees and complete law. Retain exactly the same D readout as
above, without asserting that its joined weights are conserved in the
disconnected model. Predict C_disconnected(t)=0 for all time from
independent component phase-rotation symmetry.
The proof and observations are ideal mathematical ones. Reuse the two
previously declared pure-phase nominal split preparations from Section 10,
including their represented dyadic twist and input defects, as static
implementation controls. They are reused evidence, not new independent
response samples. Keep the existing 100-digit oracle and 2^-46 ordinary
static response allowance. Check actual paired primary state, both cut-current
balances, weight differences, detached component admission and the exact
finite-error inequalities. Share existing word/report fixtures rather than
executing additional source variants. Evaluate no trajectory, time sweep,
new input, laboratory readout or physical identification. Preserve all earlier
protocols and results regardless of the comparison's outcome.
18. Exact regional inventory balances on the joined support¶
Orient all four cross edges from A0 to B0. For one of the actual held weighted laws, define its two boundary currents
Define the regional form observations X_C=sum_C(rho_i*x_i)/rho_C
and, on the declared continuous lifts only,
Theta_C=sum_C(rho_i*theta_i)/rho_C. On the joined support
rho_A=60,rho_B=44 and rho_i*M_i=1/pi. Antisymmetry cancels
every internal edge separately in each regional sum. The complete
fine equations therefore give exactly
Consequently, for D=X_B-X_A and Y=Theta_B-Theta_A,
These identities express exchange of the law's weighted inventories. They do not identify those inventories with a measured physical charge, or the lifted phase mean with a primitive collective phase. They also do not close an evolution law for D and Y: J and K retain all four boundary endpoint associations, the actual W and the evolving internal state. The complete fine evolution remains the model used below.
19. Matched preparations retain their distinct actual weights¶
Use exactly the matched preparations and fixed g=3/4096 of Section 17.
Section 11 gives identical A reset displacements and no B reset in the
two words. Thus the completed states satisfy
The support, capacities and degree normalization M are common. Every old
weight remains one, and the internal A0 weight is identical in the two
outputs. Only the four cross weights can differ. On a given cross edge,
the completed phase gaps are phi_e+g and phi_e-g, on the admitted
small-gap lifts, while the two form differences and their score denominators
agree. The compatibility formula therefore implies
Both weights lie in (3/4,1] and each retains its actual value. Equality
of the forms and the initial phase displacement does not justify using
a common weight matrix. Write Delta W=W_+-W_-; the two independent
continuations hold their respective event outputs fixed.
By Section 14, both complete trajectories stay in Z<r=1/16 for all
time. Hence each trajectory has full form range below sqrt(2)*r, and
every new-edge phase gap has magnitude below sqrt(2)*r. In particular,
The phase gaps here use the retained lifts, with zero cross-edge reference gap. This bound concerns edge differences and therefore is unaffected by a common phase origin. It applies throughout the finite comparison window, not only to the initialized state.
20. A full-law bound on the finite phase-displacement remainder¶
Let
These norms keep all fine directions and common coordinates. In particular
A(0)=0,B(0)=2*g; no independent component-origin projection is used.
The exact difference equations are
The sine function is globally one-Lipschitz. Since every capacity is at most one and every held edge weight is at most one, rowwise triangle inequalities give
The remaining term accounts for the actual difference between the two
laws. Only A0 and B0 rows meet changed edges, with exactly two such
edges per row. A B0 row has M_i=1/(6*pi); an A0 row has
M_i=1/(14*pi). Section 19's weight and form-range bounds thus give
The A0 bound is smaller, g/(70*pi^2), and all other rows have zero
weight-difference forcing. This estimate uses the actual cross incidence
and new degree normalization; it does not treat all seven neighbors as
changed weights or suppress the weight difference.
The upper right Dini derivatives of the two norms consequently satisfy
Equivalently these inequalities hold almost everywhere for the locally
Lipschitz norms, which is enough to integrate them. For s=A+B, use
pi>3 to obtain
An integrating factor gives, for 0<=t<=T=1/8,
Here e^(1/12)<1/(1-1/12)=12/11, by comparison of the positive
exponential and geometric series. Thus the estimate requires no
numerically sampled exponential bound. Integrating the form inequality
now gives A(t)<3*g*t/2 for positive t. Finally integrate the full
phase-row norm, rather than the derivative of B, to keep its initial
displacement explicitly:
The remainder includes every internal form/phase displacement and the separately generated held weights. No pair-symmetric trajectory, rigid component assumption, small-oscillation closure or numerical time step enters this estimate.
21. Uniform signed exchange throughout the finite window¶
For any cross edge oriented A to B, define
phi_(+,e)=theta_(+,B)-theta_(+,A) and similarly for the minus
preparation. The full phase-displacement estimate implies
At zero time the difference is exactly 2*g, so the same lower bound
also holds there. Both phase gaps remain between -1/10 and 1/10.
Throughout the interval between them, cos(u)>9/10, for example from
cos(u)>=1-u^2/2. Integrating the cosine between the two actual gaps
gives, for every one of the four cross edges,
The full boundary-current contrast therefore satisfies
This is a paired inequality; arbitrary admitted internal phase deviations
need not give either unpaired current a prescribed sign. With the fixed
readout from Section 17, C(0)=0 and the exact regional balance gives
Hence the prospective finite-time statement holds uniformly over the complete internal source family:
For the final rational inequality, pi<22/7 implies
7*kappa>637/7260>1/12. Substituting the frozen g=3/4096 gives
These are ideal observation bounds in the declared model form units and original structural clock. They are not laboratory-unit claims. The proof retains all source/reset correlations and controls the finite response; it does not infer that response from an initial derivative alone. Both preparations retain their joint geometric identity for all time by Section 14, although this signed contrast is only proved on the frozen finite window.
22. Exact disconnected null with the same fixed observation¶
Construct the detached control exactly as declared: retain the two completed primitive states, capacities and internal A0 weight, and remove only the four cross edges. The remaining 41-edge graph has two connected components. Its bare degrees are five on A0 and four on every other node. Its own complete-law mean weights are consequently ten on A0 and four elsewhere, with totals
In particular the disconnected normalization on A0 is 1/(10*pi)
and on B0 is 1/(4*pi). Keeping the joined normalization would be a
different comparison law. The remaining weights in the plus and minus
controls agree exactly: old unit edges and the common internal A0 weight.
With these correctly re-admitted laws, independent uniform phase rotation
of either disconnected component is an exact symmetry. The two initial
control states have identical forms and differ only by 2*g*1_B in
phase. Existence and uniqueness for the smooth complete finite-dimensional
field therefore give
for all time. Bounded form rates and phase rates linear in the finite form range ensure global continuation. Applying the same fixed linear readout D from the joined protocol yields the exact null
The joined readout weights 14,6,4 and divisors 60,44 remain the
observation here; they are not asserted to be conserved disconnected
inventories. An individual disconnected D need not be constant for this
argument: the two form trajectories coincide, so their contrast vanishes.
This construction compares two explicit models at the same primitive
states. It executes no support-deletion event and transfers no event-work
or passivity claim between them.
23. What the finite comparison adds¶
The admitted joined organization supports a robust finite response that the corresponding disconnected preparations cannot produce under the same observation. Opposite small prepared phase origins cause a uniformly signed difference in regional weighted form exchange, while both joined trajectories keep the proved geometric identity. The complete internal source balls remain unchanged, and each trajectory consumes its own actual member-associated conductances.
This is autonomous continuous exchange after a supplied word. It does not select that word's occurrence, prove acquisition from an absent organization, infer a unique fundamental law, or establish a closed scalar collective oscillator. The matched preparation is an explicit experimental design within the mathematical model; it does not infer the sign of either unpaired trial from an incomplete observation.
The frozen static controls reuse the earlier pure-phase preparations and the shared complete-word and weighted-field readers. They check actual state matching, both current balances, weight-difference bounds, the alternative support admission and the exact finite-remainder inequalities. Their represented input defects and arithmetic allowances remain distinct from the exact family. Sections 18-22 supply the analytical finite-window and null claims; no numerical trajectory or new independent response sample is needed or asserted.