Sine law and constitutive information boundaries¶
The complete global sine comparison; common exchange geometry, form balance, storage alternatives, first-moment restrictions and form-origin information premises.
Part of Native relational exchange admission. Section numbers are stable across the collection; hypotheses and model changes remain local to each result.
11.6 Global closure obstruction and a normalized-sine comparison¶
The finite-time boundary audit
requires distinguishing a limited-domain law from a globally defined one.
The pressure owner
already proves that principal-argument pressure has no continuous extension
through either an antipodal branch or a zero neighbor resultant. With
w*nu_i>0 the form row inherits that discontinuity. Changing only phase
evolution, storage or locality cannot repair it while leaving that form row
unchanged. The reflected boundary
additionally has path-dependent phase-rate limits from positive resultants.
A regular invertible chart change removes neither obstruction.
The incompatibility is stronger than the local-completion theorem¶
Take P2 with r=x_0-x_1!=0, delta=theta_1-theta_0 in (-pi,pi),
c=nu_0+nu_1>0 and arbitrary phase velocities F_0,F_1. The native form
row and current storage give, before any locality assumption,
Preserving the prescribed exact loss forces
This relative rate diverges as delta->pi at fixed nonzero form contrast.
It is unaffected by a common phase clock. Dropping capacity separability or
allowing nonlocal phase rows therefore cannot supply a bounded completion
with the same pressure, storage and loss.
Even weakening exact loss to total passivity is insufficient. If F were
bounded near this antipodal state, its phase-work term would tend to zero,
leaving lim Edot=c*(-e*r^2+w*r). For e>0 choose 0<r<w/e; for e=0
any r>0 suffices. The limit is positive, contradicting passivity.
This argument concerns all-state work identities, not an assertion that
these fixed-form paths are trajectories. Actual finite-time collisions have
their separate proof. The bounded rho*Ng and fixed-form nonlinear phase
corrections above retain their regular-domain results but cannot cure this
global obstruction.
| Retained requirement | What it forces or leaves open |
|---|---|
| Native principal-argument form pressure on the same circular full state | No globally continuous extension; phase-only changes cannot resolve it |
| That pressure, cosine storage and exact global loss | Unbounded relative phase response near antipodal P2, without needing locality |
| That pressure, cosine storage and only total passivity | Still incompatible with bounded phase completion on all finite form states |
| Restricted invariant preparations | Remain legitimate limited-domain models; the sufficient storage theorem and protected basins apply |
| Changed pressure, enlarged state, event or generalized solution convention | Separate model obligations; none is supplied by assigning a numerical value at zero |
Reuse an existing phase reading with an explicit changed premise¶
The repository already distinguishes J_phi/pi from argument pressure.
Write S_i=sum_j sin(theta_j-theta_i), q=Bx and D=diag(d_i) on
fixed connected simple unit support. The following comparison law uses
the same signed form, circular phase, held nonnegative capacity and clock:
Its phase-to-form pressure is s=J_phi/pi and its response metric is
H_s=pi*D. This replaces both native phase-pressure response and its
matching exchange row. It is not the native law with a different domain
setting, and not the separately supplied Kuramoto/U3 sine phase clock:
here sine enters form pressure, while form contrast drives phase.
Existing current telemetry does not select its use as a dynamical law.
On the native regular domain,
Thus the existing geometric relation directly relates the two responses. The comparison retains cancellation strength as well as direction, and has no division by a resultant or its argument. It remains defined when phases cancel. No new primitive variable, cutoff, fitted gain or boundary event is introduced. Choosing this pressure is nevertheless an additional premise.
Complete balance, continuation and inherited geometry¶
With the unchanged storage E=E_D+beta*V_phi, grad V_phi=-S gives
The exchange cancels exactly, leaving the same loss -e q^TND^{-1}q.
On any fixed finite graph the complete vector field is globally Lipschitz
on the form/phase lift: its form dependence is linear with constant
coefficients, and sine has bounded derivative. Hence every finite initial
state has a unique global solution, descending to circular phases.
This mathematical theorem concerns the comparison, not the native executor.
For strictly positive held capacities, reciprocity supplies two identities:
The second uses a chosen continuous phase lift; it is not in general a single-valued scalar observation on the phase torus. At zero capacity these weighted expressions are unavailable, although the smooth law and work balance remain valid. The native law generally conserves neither mean.
With all capacities positive, equilibria are exactly q=0,S=0. All native
regular equilibria therefore survive, but the complete set grows: zero and
negative-real resultants with S=0 are now allowed. The seven-equilibrium
classification must not be transferred as exhaustive.
At each equilibrium the existing
full-network stiffness proof applies with
H=pi*D: positive quotient cosine Hessian gives local exponential recovery
for e,w,beta>0, and negative stiffness gives growing real modes.
Consensus has exactly the native linearization. Away from consensus, equal
geometry and stability signs do not give equal rates, memory or basins.
For e>0 and positive capacities, storage and conserved weighted form mean
give precompactness on form times the phase torus. On the largest invariant
subset with zero loss, q=0 and
BND^-1 S=0, so ND^-1 S=c*1. Reciprocity sum_i S_i=0 forces c=0.
LaSalle therefore gives approach to the equilibrium set. This does not
prove convergence to consensus, to one equilibrium, or to a physical entity.
Conditional selection and remaining freedom¶
There is a more structural selection route than assuming a constant metric. Keep cosine storage, the same diffusion channel and exact loss on the full phase torus for every finite connected simple unit graph and every nonnegative capacity vector. Require zero-capacity phase freezing, own-capacity separability and a finite continuous phase law. The additional pressure premise is normalized pairwise superposition,
with one continuous, periodic, odd kernel f across supports. On a star with
only the root active and arbitrary q_i!=0, exact loss gives
beta*S_i*F_i=w*nu_i*q_i*g_i. Thus every finite family with
sum_j sin(delta_j)=0 must also have sum_j f(delta_j)=0.
All phases, including cancellations and antipodal leaves, are admitted by
the proposed global completion; no regular-domain padding is required.
Reuse the balanced-star algebra
with a different object being classified. The two gaps
delta,-arcsin(sin(delta)) give
f(delta)=f(arcsin(sin(delta))). Put h(s)=f(arcsin(s)). Three gaps with
sines a,b,-(a+b) give h(a+b)=h(a)+h(b) whenever all arguments lie in
[-1,1]. Continuity implies h(s)=k*s and therefore f(delta)=k*sin(delta).
Matching the native consensus tangent fixes k=1/pi.
Capacity isolation now forces
F_i=w*nu_i*q_i/(beta*pi*d_i) whenever S_i!=0; own-capacity separability
transfers this row to general capacities. Nonzero S_i states are dense
by perturbing a neighbor phase, including near zero resultants, so continuity
extends the row everywhere. Varying only the root phase would not justify
that density at a zero neighbor resultant.
This derives the metric and phase independence instead of assuming them. It does not derive pairwise superposition, which the native collective Arg source does not obey. A separate sufficient selection premise is to require the phase row itself to be phase-independent while leaving the phase-only pressure initially unrestricted: the same isolated-capacity identity and consensus tangent then also force the displayed comparison.
Neither selection premise follows from smoothness, storage or circular symmetry. For example, the comparison family
retains smoothness, positive mobility, exact loss, the same equilibria and the same consensus Jacobian. It supplies a counterexample to unique selection, not another production setting. The frozen pairing comparison reuses its zero-loss members to separate a robust local grouping direction from different relative response rates. The family generally violates the common pairwise-superposition premise. The original storage classification assumed native pressure; changing that premise means cosine storage is retained here explicitly, not rederived by that old theorem.
An exact discriminator before any trajectory¶
On a three-leaf star, take center phase zero, leaf phases (0,0,pi/2),
uniform form and positive held capacities. All native resultants are regular.
Put W=sum_i d_i/nu_i and let M be the degree/inverse-capacity weighted
form mean. At this same initial state,
The sign follows from atan(1/2)<1/2<pi/6. Both initial phase rates are
zero, and both laws dissipate zero storage initially; the mean response
still distinguishes them. This is an exact prospective initial-rate
comparison with identical preparation, coefficients and clock, not a fitted
response or physical validation. The older same-mean/different-spread
pressure test remains useful; no frozen acquisition is rerun.
There is also a purely spectral distinction at the same maintained C5
geometry. For a uniform acute cycle twist with c=cos(kappa)>0, common
capacity nu and nonzero combinatorial Laplacian eigenvalue ell, write
z=2*s/(nu*ell) for the scaled temporal exponent. The two characteristic
polynomials are
For winding-one C5, e=w=1/2,beta=1 and c=cos(2*pi/5)<1/3.
The native discriminant 1/4-1/(pi^2*c) is negative, whereas the comparison
discriminant 1/4-c/pi^2 is positive. Both recover locally, but one has
damped oscillatory linear modes and the other distinct negative real modes.
Local pulse character therefore depends on the constitutive response, even
when the stable geometry and storage agree. This is a tangent theorem,
not a computed nonlinear response or a transfer of earlier memory records.
bound_relational_sine_exchange
is a detached static comparison with explicit law identity and exact
represented-state trigonometric bounds. It reuses shared graph/scalar
admission and report projection, while preserving native phase-domain
rejection. Its computed interval work residuals are distinct from the
analytic cancellation identity. It installs no dynamics, boundary extension
or automatic model selector. Frozen transit, capture, memory and formation
responses remain evidence of their original law.
11.7 Common exchange geometry, composition and a controlled common regime¶
The two complete laws above belong to one mathematical family. This is an
identity between declared models, not a derivation of a uniquely selected
microscopic law. Keep the same support, clock, held capacities, coefficients
and storage. Write M=diag(m_i) with finite positive entries on its stated
domain. Then
gives Edot=-e*q^T*N*D^-1*q for every such mobility. In fact
grad_x E=q and grad_theta E=-beta*S, so the two exchange work terms
cancel by symmetry of NM. No derivative of M enters this first storage
derivative, even if M depends on the state. Existence, regularity, symmetries
and higher derivatives still require their own hypotheses. Equal loss
functions do not imply equal storage histories along different trajectories.
The native choice is m_i=1/H_i, while the normalized-sine comparison uses
m_i=1/(pi*d_i). The common skew work structure therefore leaves precisely
the constitutive mobility being investigated. It does not by itself give a
Poisson bracket: the earlier Jacobi audit
distinguishes that extra requirement from work cancellation.
Geometric attenuation is not generally a change of clock¶
On the native regular domain put z_i=R_i*exp(i*alpha_i). The exact ratio is
Here R_i<=d_i and 0<sinc(alpha_i)<=1. Equality requires all neighbors
to have the center's phase modulo a turn. The sine law scales both nodal
exchange terms by a_i and leaves form diffusion unchanged. Different
nodes generally have different attenuations.
Even P2 with positive diffusion rules out a common clock interpretation.
Set r=x_0-x_1, delta=theta_1-theta_0 in (-pi,pi) and
c=nu_0+nu_1>0. In these two relative coordinates the native and sine
velocity vectors are
Thus
when e>0,r!=0,0<|delta|<pi. A regular common clock change multiplies the
whole vector field by one positive scalar, so it cannot identify these
fields, even after removing common offsets. For e=0 on P2 only,
F_s=sinc(delta)*F_n: there is a restricted orbit equivalence. It is not
general on larger graphs; the preceding star's weighted-mean discriminator
also applies at e=0 and forbids a positive common multiplier there.
A controlled near-alignment regime¶
Suppose all wrapped incident gaps satisfy |delta_ij|<=epsilon<pi/2.
Then C_i>=d_i*cos(epsilon) and |alpha_i|<=epsilon. Since
H_i/pi=C_i*tan(alpha_i)/alpha_i>=C_i with the removable value at zero,
These are instantaneous bounds at a shared state, not trajectory enclosures.
The phase discrepancy is O(epsilon^2*|q_i|); the complete discrepancy
is cubic only when form contrasts are also O(epsilon). A future error
estimate must additionally bound the field Lipschitz constant and preserve
the neighborhood throughout the comparison horizon. The common small-signal
regime is derived; the ratio does not supply an autonomous switching rule.
Composition exposes an independent premise¶
At a fixed root state let P(A) be the phase source of a nonempty finite
multiset of incident gaps A; equal entries mean different neighbors with
the same phase, not parallel edges. Independent normalized composition
would require, for every partition into disjoint incident groups,
Induction over singleton groups makes this condition equivalent to
P(A)=sum_{delta in A} f(delta)/|A| with f(delta)=P({delta}).
It is exactly the added superposition premise used in the preceding
classification, rather than a consequence of that classification.
Both candidates pass equal replication: duplicating every incident
coordinate multiplies d,S,H,q by the same factor and preserves their
complete nodal rates. That test is strictly weaker than independent
composition. For example, take A=(0,0), B=(t) with 0<t<pi/2.
Native source composition would require
Arg(2+exp(i*t))=t/3. Instead
The merger defect is nonzero although all groups are regular. Sine sources obey the merger identity exactly. The smooth collective counterfamily in section 11.6 also preserves replication while generally violating merger. Neither nodal locality nor replication establishes independent composition.
The stationary mediator reduction
answers the associated scale question constructively: even a primitive
pairwise sine law induces collective visible interaction after eliminating
one node. It shares the native conditional minimum and inherited storage,
but transmits a different pressure. An emergent Arg coordinate is therefore
not a derivation of native Arg pressure. Finite hidden capacity retains a
tracking defect and memory; its controlled fast limit has explicit premises.
The composition controls exercise merger, replication, the local discrepancy bounds and clock non-equivalence using existing field owners. The detached mediation report integrates the justified reduction without selecting a new primitive law. Together these results identify a common exchange geometry and a common local regime, while leaving the deeper mobility-selection hypothesis open.
11.8 Closed form balance can select primitive pressure composition¶
The composition premise has an equivalent whole-graph interpretation within an explicit information class. This supplies another conditional selection route; neither conservation nor that information class follows from the nodal identity. Here a closed graph means that the entire fixed support is included, with no external source or event. Degree-one vertices are allowed.
Keep finite connected simple unit support with at least two nodes, held
positive capacities, the support Laplacian B=D-A and q=Bx from section 1,
and the form row
Require one common phase-source family across supports,
g_i=P_{d_i}(A_i), where A_i is the multiset of circular incident gaps
theta_j-theta_i. The source consumes only these gaps and their count. It
does not consume form, capacity, neighbor degrees, connections between
neighbors, graph identity, or hidden state. Equal gaps can represent distinct
neighbors. This is stronger than unspecified one-hop locality: replacing
the root's environment by a star must preserve its source. No phase row or
storage identity is needed for the next equivalence.
Diffusion fixes the weight of a possible linear form invariant¶
For e>0, suppose a nonzero state-independent vector a, allowed to depend
on the held graph and capacities, makes a^T*x invariant at every form and
phase state. Varying form at fixed phase removes the phase-only source and
gives
Connectedness gives ker(B)=span{1}, hence
a_i=c*d_i/nu_i for one nonzero constant c. Thus the only possible
nontrivial linear form invariant has, up to scale, the weights
This identifies the weight if such an invariant is required; it does not
establish its existence. For e=0 this uniqueness argument is unavailable.
For the displayed weight and any e>=0, the exact accounting identity is
The positive-capacity premise matters: these inverse-capacity quantities are unavailable at a zero-capacity node.
Whole-graph balance is equivalent to an odd pairwise source¶
In the gap-only source class, require sum_i d_i*g_i=0 for every phase
state on every finite connected support. It suffices to require this on
every star. Put f(delta)=P_1({delta}). On the single-edge graph the two
root gaps are delta and -delta, so balance gives
f(delta)+f(-delta)=0. On a star with root phase zero and arbitrary leaf
phases delta_1,...,delta_d, the root has source P_d(A) and each leaf
has source f(-delta_j). Therefore
Conversely, summing this formula over any reciprocal graph cancels the two orientations of each edge. Whole-graph balance is therefore equivalent to normalized pairwise superposition with one odd circular kernel, and to the preceding independent normalized-composition identity together with singleton oddness. No continuity or differentiability is used. Requiring the balance at one positive capacity assignment per star is enough because the source is capacity-independent and the capacities cancel from the accounting identity.
The support and information quantifiers cannot be discarded. Testing only
regular graphs permits g_i=a_{d_i}*sum_j sin(delta_ij)/d_i with arbitrary
degree-dependent gains. Consuming neighbor degrees instead permits
which conserves the same weighted form on every graph by edge cancellation, but has no common singleton kernel across supports. It lies outside the stated gap-only information class. Ordinary locality, relabeling covariance or equal replication does not supply the missing restriction.
Cosine exchange selects sine, with a separate phase-row premise¶
Combine this conservation premise with the global comparison domain and
storage assumptions of section 11.6:
continuous circular pressure on the full phase torus, the retained storage
E_D+beta*V_phi, its exact loss -e*q^T*N*D^-1*q, and finite phase
velocities that freeze at zero own capacity. The complete law must admit
all nonnegative held capacity vectors; weighted form itself is used only
on the positive-capacity part of that domain.
Activate only a star's root capacity and choose q_root!=0. All other
phase rows freeze, so exact loss leaves
Whenever sum_j sin(delta_j)=0, finiteness forces
sum_j f(delta_j)=0. The balanced-star argument in section 11.6 then gives
f(delta)=k*sin(delta); matching the native consensus tangent fixes
k=1/pi. Thus conservation plus the stated global exchange premises
selects the normalized-sine pressure without independently assuming
pairwise superposition. This reuses the existing classification rather
than deriving cosine storage or conservation from it.
Own-capacity separability is an additional requirement for selecting the
complete phase row. With that premise and continuity of the phase law,
capacity isolation at each node first forces the row where S_i!=0,
transfers it to arbitrary capacity vectors, and extends it by continuity
to S_i=0:
Without the extra phase-row premise, the pressure selection does not remove all globally work-null phase velocities. The required conservation law, cosine storage, prescribed loss, capacity behavior, full phase domain and consensus normalization remain independent constitutive assumptions.
The combined premises also cover induced-neighborhood pressure¶
The sine-pressure conclusion survives a larger information class when both
conservation and exact cosine exchange are imposed. Allow pressure to consume
the phases and topology of the induced rooted ball G[{i} union N(i)],
including edges between neighbors. Require one rule invariant under rooted
isomorphism and common phase rotation across all supports, independent of
form and capacity. External neighbor degrees, remote state and hidden
support-specific coefficients remain excluded. This does not extend the
conservation-alone equivalence to arbitrary local sources.
First isolate one active capacity in the global storage identity. For every
node, S_i=0 forces g_i=0: choose q_i!=0, use finiteness of its phase
row and freezing of all other phase rows. This implication holds at every
capacity assignment because pressure is capacity-independent. On P2,
whole-graph balance again makes the common singleton kernel f odd.
Adapt the induced-neighborhood completion.
Retain an arbitrary root ball with all its internal edges and phases. For
each root neighbor j and every retained incidence j-k, attach one private
leaf to j with phase 2*theta_j-theta_k. No new leaf touches the root, so
its entire source information is unchanged. Every retained neighbor now has
S_j=0, and hence g_j=0, because each old sine contribution has its
opposite at the reflected leaf. Unlike the native regular-domain completion,
no aligned padding is needed: the full phase torus admits zero or negative
resultants.
Assign strictly positive capacities throughout this completed graph before
using whole-graph form balance. A reflected leaf at j-k contributes
f(theta_k-theta_j). Leaves associated with the two incidences of an edge
between root neighbors cancel by oddness. The remaining leaves correspond
to root edges and contribute -sum_{j~i}f(theta_j-theta_i). Therefore
The original root inherits this identity by uniform rooted locality. The preceding balanced-star classification now selects normalized sine. No locality or capacity separability of the phase row was used to select this pressure; uniqueness of its complete phase row retains the earlier extra premise. The proof never evaluates inverse-capacity form weights at zero capacity. These completion graphs are mathematical comparison states, not support events. The locality controls check the retained root ball and reflected-leaf cancellation separately from the existing native-domain completion.
Branching tests information that paths and cycles cannot determine¶
Even conservation, composition, smoothness, the consensus tangent and exact
cosine-storage loss on every path and cycle do not select sine. For
epsilon>0, take the common odd kernel
On graphs of maximum degree two, let s_j=sin(delta_j) at a root and use
the finite positive mobility
The identity s_1^3+s_2^3=(s_1+s_2)(s_1^2-s_1*s_2+s_2^2) gives
P_d=m_d*S. Consequently F_i=(w/beta)*nu_i*m_i*q_i supplies a smooth
full-state exchange completion with the prescribed exact loss, zero-capacity
freezing and own-capacity separability on this restricted support class.
Its tangent is the normalized-sine tangent, but its nonlinear pressure
differs.
A degree-three star rejects that same kernel. Choose the acute leaf gaps
(arcsin(1/4),arcsin(1/4),-pi/6). Their sine sum is zero while their sine
cubes sum to -3/32, giving
With only the root capacity active and nonzero root form contrast, its phase work is zero and its source work is nonzero. No finite phase row with the stated freezing can supply the prescribed loss. The obstruction occurs strictly inside an acute phase chart, not only at a zero resultant or an antipodal boundary. Degree-three branching is therefore indispensable to this countercontrol; further path or cycle trajectories cannot resolve the same constitutive ambiguity.
The fixed cosine-storage premise is essential to that obstruction. The same cubic pressure has an all-degree completion if its phase potential is changed explicitly to
For c=cos(delta) this is
(1-c)+epsilon*(1-c)^2*(c+2)/3>=0, with U_epsilon(0)=0.
Moreover U_epsilon'=sin(delta)+epsilon*sin(delta)^3 and
U_epsilon''(0)=1. On any admitted finite support define
Then grad_theta E_epsilon=-beta*tilde S, so the same constant
reciprocal Poisson tensor and form-diffusion block reproduce both rows,
and dot E_epsilon=-e*q^T*N*D^-1*q. Odd edge currents preserve weighted
form balance at positive held capacities. The pressure is smooth and
pairwise, and its complete consensus Jacobian equals the normalized-sine
one, although its nonlinear phase potential and pressure differ.
At the degree-three witness the original sine sum vanishes, but the changed phase gradient does not; the exchange therefore has the missing phase work. This countermodel shows why conservation, Jacobi, smoothness and the consensus response still do not select sine without a phase-storage premise. It changes that premise, rather than evading the fixed-cosine selection theorem. Cosine-storage capture certificates and the earlier native-pressure-derived storage classification cannot be transferred to it without checking their changed hypotheses. It supplies no new default law, runtime setting or physical storage identification.
Exact local agreement does not select global winding passage¶
The restricted-domain classification already exhibits a smooth periodic addition invisible on acute gaps. The following complete-law comparison reuses that mechanism to test the specific C5 saddle preparation and retained connection. It is a constitutive counterexample, not a newly selected TNFR pressure or a rerun of that frozen response.
An explicit smooth continuation. Define
The function exp(-1/y) and every derivative tend to zero at y=0+.
Thus U is C-infinity, even, nonnegative and 2*pi-periodic, with
U(0)=0, U''(0)=1 and U(pi)=4. Its added term and all its derivatives
vanish for |wrap(delta)|<=3*pi/4. The associated odd edge current is
This is smooth but not real analytic at the cutoff. Requiring global real analyticity together with exact equality on an open interval would exclude this continuation by analytic uniqueness. Such an analyticity requirement is an additional premise; it is not supplied by the nodal identity. Finite observations or equality of a finite number of derivatives would also be weaker than exact agreement on an open interval.
The complete law and its balance. On the same fixed simple unit graph,
hold every capacity at one and take no forcing or events. In tau=t/pi,
with K=diag(1/d_i) and q=Lx, declare
The original-time rows are these rows divided by pi. In particular the
supplied pressure is S_U/(pi*d), rather than a pressure reconstructed from
the response. Since grad_theta H_U=-S_U,
Odd pair currents preserve both degree-weighted lifted origins. Periodic
smoothness bounds j', so the full real-lift field is globally Lipschitz
and has unique solutions in both time directions. Form reversal remains
a reversing symmetry. These conclusions use the same constant reciprocal
exchange construction as the preceding storage family. More generally its
balance holds for an even C2 periodic potential with its derivative used
as the current; nonnegative storage is a further hypothesis for the barrier
below. This does not install an arbitrary-potential runtime or admit the
changed law under the sine identifier.
Whole-window agreement for the actual preparation. Let the ideal
near-saddle preparation be z_*+epsilon*(3X_lambda,-v_lambda), let r
bound the actual rational center's full-coordinate error to it, and let
delta be the independent preparation radius in every form and phase
coordinate. The sine field has full maximum-norm Lipschitz constant two
in scaled time. Its exact saddle is an equilibrium, and the ideal initial
perturbation has norm at most 3epsilon. Hence for every |tau|<=3,
Every saddle edge has principal gap at most 2*pi/3. Therefore the strict
check
keeps every cycle and contact edge in the region where both complete fields agree, throughout the window. Uniqueness transfers the entire trajectory there, rather than merely its initial derivatives. The nominal rational preparation consequently retains the same finite local directional gates. The independent full-state cube retains equality of the two flows; it does not acquire the nominal center's exact signed-family symmetry.
The original preparation uses epsilon=2^-32 and delta=2^-100.
Its representation error must be rebuilt, and this stronger whole-window
condition checked: an earlier, weaker phase-radius flag is insufficient.
All initial edge additions then vanish exactly, so the full alternative
storage equals the sine storage on the admitted preparation cube. Rebuild
its upper enclosure and require H_U^upper<4. One independent sufficient
check, with m full-support edges, is the sine-storage Taylor bound
where the initial cube is wholly in the common-potential region. The two
quadratic remainders use |e_j-e_i|<=2delta and |cos|<=1.
For the present support m=10. Initial winding and branch admission remain
separate from the storage test.
A different global outcome for the same preparation. Winding on a
fixed cycle can change only when an edge meets its principal seam. At such
a state that edge has potential U(pi)=4; every other potential and the
form quadratic are nonnegative. Conservation and H_U<4 therefore forbid
any seam, for all positive and negative time. An admitted initially
winding-one cube remains winding one under this alternative.
The retained sine response instead reaches winding zero from that same intermediate preparation. No alternative trajectory is needed to decide this discriminator. The result does not prohibit formation from other preparations, establish permanent acute geometry, or show that the mapped sine winding-zero source fails under the alternative: that later source can have a different alternative storage. Local maintenance geometry and global accessibility are different obligations.
Which selection premises differ. The current remains additive over the full incident-gap multiset, but it cannot discard all information beyond the first resultant. Consider the two four-leaf stars with gap lists
Both have degree four, Z_1=2 and zero sine sum. Set
y_a=(sqrt(3)/2-c)/(1-c)>0. Only the first gap of A activates the
addition, giving
Thus the first-moment sufficiency theorem excludes this law if its information restriction is independently required. At the first star's root, native argument pressure is zero while the alternative potential gradient is nonzero; all star resultants are regular. No finite diagonal metric can provide global alignment with the unchanged native pressure there. This is outside, rather than a counterexample to, the full-domain native-storage classification. Fixing cosine storage globally would likewise exclude the altered cost by premise.
The shared
assess_saddle_storage_discriminator
keeps this alternative's identity distinct, rebuilds the rational preparation
and checks its common-region, full-storage and branch hypotheses. It is a
detached static comparison: the input source anchors the support and reference
law, rather than being relabeled as that generated preparation. Its local
gate observations refer to the nominal
preparation; its all-time winding exclusion consumes the separately admitted
full cube. Neither it nor the existing classifications selects which global
constitutive or information premise nature satisfies.
What restricting primitive phase information selects¶
The alternative potential above leaves a precise information question: must the primitive phase source retain the whole incident-gap multiset, or can it discard everything except degree and the first complex circular moment? This is an additional constitutive premise, not a consequence of the nodal identity, reciprocity or the shared consensus tangent.
Let a bounded real kernel j(delta) be defined on the full phase circle;
ordinary continuity suffices for boundedness. For any finite nonempty list
of incident gaps, retain
Here j is the unscaled edge current, so the preceding normalized edge
kernel is j/pi. Repeated phases represent distinct neighboring nodes;
simple stars realize these multisets without parallel edges. The information
premise says that equal d,Z_1 imply equal P for all such lists.
Degree is retained separately; this is not a direction-only Arg(Z_1)
contract, and it does not add nonlocal data or future observations.
Classification. This premise holds if and only if
If the edge current is odd, it follows that j(delta)=kappa*sin(delta).
Consequently g=kappa*Im(Z_1)/(pi*d). The value of kappa still needs a
normalization or constitutive scale; no coefficient is fitted to a response.
Elementary proof. Every point z of the closed disk |z|<=2 is the sum
of two unit phasors. Degree-two sufficiency makes their current sum a
well-defined function F_2(z), including different antipodal decompositions
of zero. Comparing two degree-four lists gives
Thus G(z)=F_2(z)-F_2(0) is bounded and locally additive. If |z|<=2/n,
additivity yields G(nz)=n*G(z); boundedness makes G continuous at zero.
Integer subdivision, rational homogeneity and continuity give
G(z)=a*Re(z)+b*Im(z) throughout the disk. Evaluating the repeated pair
at z=2*exp(i*delta) yields the displayed classification, with
c=F_2(0)/2. Conversely its sum is exactly cd+a*Re(Z_1)+b*Im(Z_1).
Oddness eliminates the constant and cosine terms. No differentiability or
pressure dynamics is used in this information theorem.
If the premise is initially restricted to lists with nonzero Z_1, two
equal-zero lists can each be padded with one aligned entry. Their resultants
become one and their degrees remain equal; canceling the common contribution
proves the needed equality at zero. The same proof then applies. This
comparison does not assign a direction or a native pressure at zero resultant.
Degree two alone is insufficient to select sine. A nonzero resultant already
determines its unordered pair of unit phasors; zero leaves the antipodal pair
(delta,delta+pi) undetermined. Cancellation on every such pair requires
the additional half-turn condition j(delta+pi)=-j(delta), not merely
oddness j(-delta)=-j(delta). The non-sine cubic current above and
sin(3*delta) satisfy both conditions and therefore pass every degree-two
comparison. The classification also compares degree-four compositions;
its premise ranges over all finite lists, not just pairs.
For example, j_2(delta)=sin(2*delta)/2 is odd, has unit consensus slope
and derives from the even nonnegative potential U_2=sin(delta)^2/2.
Nevertheless its sum on an antipodal pair is sin(2*delta), which depends
on the unresolved orientation. Thus some odd currents can be distinguished
already at degree two. For a smooth odd periodic current, half-turn
antisymmetry removes the even sine harmonics; oddness alone does not. The
all-degree information premise then removes the remaining higher harmonics.
These are distinct restrictions on the supplied law, not interchangeable
consequences of reciprocity or an even phase potential.
Storage consequence and independent premises. When the current is the
derivative of a periodic potential, j=U', integrating the odd result and
normalizing U(0)=0 gives U=kappa*(1-cos(delta)). Nonnegative storage
requires kappa>=0; unit consensus curvature selects kappa=1. Alternatively,
applying the classification directly to an even additive potential whose sum
is determined by d,Z_1 gives the same normalized shape. These are conditional
selection statements, not a derivation of first-moment sufficiency itself.
The two central premises are independent. Native argument pressure consumes the first resultant but is not an additive sum of one common edge current; the existing composition boundary retains that distinction. Conversely the smooth cubic-sine current is additive, odd and conservative under its supplied companion law, but consumes information beyond the first moment. The next control exposes exactly that discarded information, rather than merely comparing two unequal laws.
Balanced geometry does not independently select the phase current¶
Consider the structural proposal that every finite phase neighborhood whose unit vectors sum to zero must exert zero net phase current:
For a bounded additive odd kernel on the full phase circle, this proposal is equivalent to first-moment sufficiency, not an independent deduction of it. A two-member balanced list first gives \(j(\delta+\pi)=-j(\delta)\). For any equal-resultant lists \(A,B\), append the half-turn shift of \(B\) to \(A\). The combined resultant vanishes, so (BC) gives \(\sum_A j=\sum_B j\). Applying the preceding classification and oddness selects \(j(\delta)=\kappa\sin\delta\); conversely that current obeys (BC). Without oddness, (BC) permits a cosine term as well. It forbids the constant term which degree-aware first-moment sufficiency alone permits.
Thus calling the first phase resultant a geometric balance does not prove that it contains all the information used by pressure. The smooth cubic alternative above supplies an exact control. On the rotated balanced triad \(\delta_a=\pi/6+2\pi a/3,\ a=0,1,2\), both the resultant and sine sum vanish, but \(\sum_a\sin^3\delta_a=-3/4\). Its normalized phase pressure is therefore \(-\epsilon/(4\pi)\). Both complete smooth laws admit this configuration. The native argument-pressure law does not: its zero-resultant direction is undefined. This test does not extend that native domain by convention.
Checking only pairs and balanced triads is insufficient. For any \(\eta>0\),
is odd, has unit consensus slope, and derives from a nonnegative even potential. It cancels on every antipodal pair and every rotated balanced unit triad, but not on a regular pentagon rotated by \(\pi/10\): its current sum there is \(5\eta/(1+5\eta)>0\). A balanced unit triad is necessarily an equilateral phase arrangement; the fifth harmonic survives its symmetry. These are phase-space configurations, not physical spatial polyhedra. This counterexample is an algebraic admission control, not a new engine law.
The premise audit consequently closes with a nonselection result: (BC) is a meaningful candidate constitutive restriction, but its unrestricted version restates the information choice and its pair/triad version leaves other smooth conservative potentials. None of the nodal identity, reciprocity or existing persistence results establishes (BC). An independent structural principle or observation is still required.
Equal first moments conceal a higher-harmonic response¶
Use phase turns and fix u=1/6. On a degree-three star, set the root phase
to zero and compare leaf-gap lists
Both preparations have strictly acute edges. Their root resultants are
nonzero with positive real part, and each degree-one leaf is also regular.
Thus the whole supplied star is admitted; it need not be an equilibrium.
For Z_k=\sum exp(2*pi*i*k*gap_turn), direct phasor algebra gives
Since sin(delta)^3=(3*sin(delta)-sin(3*delta))/4, the existing
j_epsilon=sin(delta)+epsilon*sin(delta)^3 has
Cosine storage and the sine root source agree between these lists. At the
fixed choice alpha=1/24, the cubic differences are respectively
-epsilon*sqrt(2)/(8*pi) and epsilon*(sqrt(2)-2)/8. Forms, capacities,
coefficients and clock can be held identical on the same star; these are
initial source and storage identities, not sampled future predictions or
equilibrium certificates.
The shared phase-response reader
provides PhaseMomentInformationAssessment through
assess_phase_moment_information(left, right, epsilon=...). Its primitive
inputs are exact rational unit phasors (cos(delta),sin(delta)), retaining
first and third moments, both candidate sources and both potentials. It does
not decode arbitrary roots of unity or install a new runtime law. One exact
rational control uses three copies each of (1,0),(3/5,4/5) on the left,
and five copies of (4/5,3/5) plus one (4/5,-3/5) on the right. Both have
degree six and Z_1=(24/5,12/5). Their right-minus-left cubic source and
storage differences are -14*epsilon/(125*pi) and -24*epsilon/125.
The information controls
retain exact equality and admission, alongside the existing
composition controls.
Such finite controls exercise this reader; the classification is the proof
above, not an inference from those examples.
Scope under composition and observation. The
stationary mediator result
already gives a collective phase cost k-|Z_1|. This is nonpairwise but
still a function of the first resultant. It therefore violates primitive
edge additivity, not first-moment sufficiency by itself. Its retained finite
capacity also brings memory and hidden initialization; that earlier proof
must not be replaced by a claim that every coarse law needs higher harmonics.
Likewise, sufficiency for the root's instantaneous source or storage does not
make d,Z_1 a sufficient full state or future predictor. Neighbor dynamics,
form, capacities and hidden state retain their separate contracts. The result
identifies what a primitive information restriction would select and what the
known alternative retains. Why that restriction should hold fundamentally
remains open; no physical selection, autonomous support mechanism or new
default law follows.
A finite observation of the information restriction¶
The existing exact degree-six preparations give a prospective test without reconstructing pressure from an evaluated derivative. For each preparation, use a seven-node star with fixed undirected unit support, unit held capacities, zero loss, unit phase weight and storage coefficient, and no inputs or events. Set all ideal forms to zero and the root phase to zero. The six leaf phasors are the rational lists above: three copies each of \((1,0),(3/5,4/5)\) for \(A\), and five copies of \((4/5,3/5)\) plus one \((4/5,-3/5)\) for \(B\). They have the same first resultant and degree. They need not have equal storage under the alternative law.
In the declared common clock \(\tau=t/\pi\), the complete supplied family is
The phase row and the associated \(U_\epsilon\) are retained from the complete storage family above. This is not a held-phase pressure experiment. Compare \(\epsilon=0\) with one prospectively fixed \(\epsilon>0\). The observation is the difference of the two root form readings at the same finite time \(h>0\), \(D_\epsilon(h)=x_{0,B}(h)-x_{0,A}(h)\). Its ideal initial derivative is \(-14\epsilon/125\).
Let \(J_\epsilon=1+\epsilon\) and \(L_\epsilon=1+3\epsilon\); these globally bound \(|j_\epsilon|\) and \(|j_\epsilon'|\). Admit independent initial errors of at most \(\rho\ge0\) in each form and lifted phase coordinate around each ideal preparation. The laws, capacities, support and clock are held exact. Then, on the full nonlinear star and for \(\tau\ge0\),
Integrating the Lipschitz bound on each root's phase current, including its initial form error, gives
If each endpoint root reading additionally has error at most \(\sigma\ge0\), the measured difference belongs to
These bounds use every node's evolving form and phase; no numerical trajectory, linearization or discarded leaf state enters the proof. Absolute current bounds give at most linear form growth and exclude finite-time escape, so the complete smooth solutions exist over the declared window. The bound is sufficient, not an optimized noise budget.
For the declared example \(\epsilon=1,h=1/16,\rho=\sigma=0\), the two prediction intervals are
Their separation is \(521/128000\). Allowing the separately declared \(\rho=\sigma=1/100000\) still leaves the exact positive separation \(25453/6400000\), before outward numerical materialization. These are structural-coordinate budgets, not a claim about laboratory precision.
Equal initial information is not equal future response. Writing \(\mu=\operatorname{mean}_a j(\delta_a)\), the full zero-form star rows give \(\delta_a''(0)=-j(\delta_a)-3\mu\) and
For the sine member, the two values are \(-6/5\) and \(-32/25\). Consequently \(D_0'''(0)=-2/25\); in particular the future sine contrast is not identically zero. The finite error interval, rather than an assumed zero baseline, is what permits the law comparison.
The shared phase-response owner supplies assess_phase_information_response. It reuses exact phasor admission and the source comparison, propagates declared preparation/readout budgets and exports both predictions through the SDK. Exact unit phasors specify generally irrational ideal angles; floating angle conversion error belongs in the preparation budget. The controls compare independent full-star evolution and include inconclusive budgets. This report neither installs a law nor admits physical data. Disjoint predictions distinguish the two supplied models, not every possible kernel; agreement with a sine interval cannot prove universal first-moment sufficiency. Independent physical preparation, observation and clock identification remain the missing admission.
The finite receiver collision uses this same complete kernel family for a separate inverse question. One exact endpoint can coincide for different law coefficients and internal orientations; this does not contradict selection of sine when first-moment sufficiency is independently required.
Phase information must retain its pairing with motion¶
The existing resultant chain rule
already differentiates the first moment. Its higher-harmonic version connects
the preceding static source comparison to response and work without adding a
primitive coordinate or changing a law. On fixed support, let each relative
gap have a declared angular rate r_a=dot(delta_a) in one specified clock, and
define
For the existing cubic-sine current, put A=1+3*epsilon/4 and
B=epsilon/4. Its source and incident potential are
pi*d*g=A*Im(Z_1)-B*Im(Z_3) and
V_inc=d*(1+2*epsilon/3)-A*Re(Z_1)+(B/3)*Re(Z_3). Consequently
These identities assume fixed coefficients and support. The potential rate counts root incidences once; it is not a whole-network storage balance. Phase rates are supplied with their provenance or derived from the complete chosen law, never reconstructed from the source response being predicted.
A complete-law pairing control. On the three-node star 0--1,0--2, use
unit capacity, e=0,w=beta=1 and the conservative sine law in tau=t/pi.
Keep the labeled relative phasors fixed at (3/5,4/5) and (4/5,3/5) and
compare
The full phase row theta'=D^{-1}Bx gives the common root rate -1/4
and the respective relative-rate lists (1,0) and (0,1), where B is
the unnormalized graph Laplacian and primes here use tau. Both states
have the same labeled phases, hence every unweighted circular moment, the
same form and rate multisets, the same degree-weighted form mean 1/8
and the same full storage 73/80. Nevertheless
Thus even all phase moments together with a separate rate multiset discard information needed for the next response: which motion belongs to which phase. The example changes that association, not the pressure law. The different incident work is compatible with conservation of total storage.
The shared
derive_phase_moment_motion
returns PhaseMomentMotion from exact unit phasors and declared gap rates,
reusing the static moment admission. The existing sine comparison's
resultant_kinematics()
derives its first-moment rates from rebuilt full-law state. The
native joint-response owner
instead consumes the argument-source Jacobian; its law must not be replaced
by this sine or cubic derivative.
Retaining M_m still does not establish an autonomous reduction. Where the
rates are differentiable,
Accelerations and further phase/rate correlations therefore enter unless an admitted invariant family or observation closure determines them. This is the same information-retention issue addressed by the existing complete-law and hidden-memory owners, not a reason to introduce free fitted moments.
There is a scoped sufficient-state exception, already connected to the
unordered pair and its temporal reconstruction.
For the same three-node star, the
regular moment chart
reconstructs the full relative state from (Z1,M1) when 0<|Z1|<2,
and derives its exact evolution. It retains four real degrees of freedom;
it does not close generic moment statistics. Coincident phases hide internal
form variance. Antipodal phases with nonzero M1 retain the unordered phase
pair but hide its mean form relative to the root. Those boundary controls
are part of the same retained-state argument, not new primitive variables.
Existing acute-star observations separate the premises¶
Reuse the preceding composition control's incident-gap preparation: a star
with root phase zero and leaf phases (0,0,t), 0<t<pi/2. The balance
control fixes t=1/2, x=(0,0,0,0), capacities (1,2,3,4) in root-first
order, e=0, w=beta=1, no Gamma or events, and epsilon=1 for the
current-squared comparison. The source laws, clock and exact preparation
are declared before this static evaluation; no trajectory is acquired.
Reuse of the earlier gaps is distinct from evaluating these whole-form
observations. No earlier frozen response or producer is regenerated.
More generally, at uniform form and arbitrary positive held capacities, all three laws have zero phase velocity and zero instantaneous storage loss at this star. Their weighted-form mean rates nonetheless differ:
The native sign is the already proved merger defect. For the existing
current-squared family, the root contributes
(1+epsilon*sin(t)^2/9)*sin(t)/pi to sum_i d_i*g_i, and the nonaligned
leaf contributes -(1+epsilon*sin(t)^2)*sin(t)/pi. Their difference gives
the displayed expression. These are exact conditional initial-rate
identities; the shared static readers check represented-state observations
and intervals separately. The comparison reuses the existing laws and
preparation rather than installing a new dynamics or claiming physical
selection from a computed response.
The composition controls exercise these balance and branching boundaries. Whole-graph form balance concerns the held-support continuous law. Changing degrees or capacities changes its weights, and hybrid events need their own jump accounting. The invariant is not identified with physical mass, energy or a substrate origin. The conservation-only equivalence concerns the gap-only information class; the induced-neighborhood extension also requires exact cosine exchange. Neither conclusion selects a universal law or changes the native runtime and retained evidence.
11.9 Form balance, common origins and the primitive information contract¶
The conservation premise in section 11.8 has different content on a relative state and on a primitive nodal law. Fix a connected simple unit graph with at least two nodes, held strictly positive capacities, the original clock and an unforced complete field
Assume both rows are invariant under x -> x+s*1 for constant real s.
The native and sine exchange rows have this property on their respective
phase domains. Use their local existence/uniqueness domains; no global
continuation through a native phase singularity is assumed. Put
The weights are held, so P^2=P, P*1=0 and a^T*P=0.
Changing capacities, support, clock or inputs requires the separate
continuous/event accounting.
Exact relative equivalence and its retained reconstruction coordinate¶
The unique correction along the common form direction that makes this Q constant while preserving the phase row is
It satisfies a^T*Fhat=0. Every form difference has the same derivative,
Fhat_i-Fhat_j=F_i-F_j; both full fields induce the same phase derivative.
Because both rows depend only on the form quotient, this is a trajectory
equivalence on the quotient, not just an instantaneous identity. With the
same initial complete state, throughout their common existence interval,
The notation for h allows the general shift-invariant case; the displayed
phase-only pressure makes h a function of phase and held capacities alone.
Every fixed state-observation map invariant under common form shifts has
the same history. Model-dependent pressure and raw-rate reports use different
maps for the two fields and need not agree. In particular, grad_x E=Bx=q
and 1^T*q=0, so the original storage E=E_D+beta*V_phi and its total
exchange work are unchanged by this correction, including along the paired
trajectories. An exact loss identity already satisfied by the selected
original law is therefore preserved; the arbitrary phase row displayed above
does not establish it by itself. Nodal form-work contributions instead change
by -h*q_i, whose sum is zero. This statement
compares a law with its own projection, not native trajectories with sine
trajectories.
The complete coordinate description (y,m)=(Px,Q/W) instead has
Dropping m loses the absolute common form origin. Keeping its initial value
but imposing mdot=0 changes the full law whenever h is nonzero. A
trajectory-dependent moving frame must retain c and its reconstruction row;
it is not a proof of conservation of the original Q. A time-independent
invertible origin chart requires the separate derivative equation and
domain hypotheses of the
asymptotic-origin construction.
Constant translation symmetry alone supplies neither that chart nor Q balance.
The balancing pressure consumes global state and capacity ratios¶
On positive capacities the corrected nodal pressure is
Unless sigma vanishes identically on the admitted domain, this is generally
outside the primitive information class in section 11.8: it reads the whole
support's phase-source sum and the capacity ratios in nu_i W. Uniformly
rescaling all capacities cancels from this pressure correction, but changing
one remote capacity generally does not. Conservation can therefore be
enforced on a chosen relative model without deriving a primitive-local,
capacity-independent conserving law.
There is a stronger uniqueness restriction. Suppose two pressures p and
p_tilde in the same original form chart are independent of capacities.
If their nodal rows N*p and N*p_tilde give the same relative-form rates
at every state for every independently varied positive capacity vector, then
they are equal. Indeed, writing r=p_tilde-p, quotient equality requires
nu_i*r_i=nu_j*r_j for every pair. Vary nu_i alone while retaining all
other capacities and the same form/phase state: r_i=0, then r_j=0.
Thus no nontrivial common-direction freedom survives these primitive
premises across the capacity family. This is not a derivation of conservation
or capacity independence. It distinguishes those added premises from an
unobserved common offset at one fixed capacity assignment.
Zero-capacity rates and pressures have different limits¶
The correction need not diverge at a freezing boundary. For positive capacities,
At fixed regular form/phase state, h tends to zero whenever any capacity tends to zero. If the original full field is continuous there, the projected full field extends continuously by that original field on each zero-capacity face. Its form row still freezes at every zero-capacity node. This extension does not define inverse-capacity Q on the face.
Pressure has a subtler boundary. If only nu_i -> 0 while all other
capacities stay positive, its correction tends to -w*sigma/d_i; the
corrections at the other nodes vanish. If two capacities satisfy
nu_i=r*epsilon, nu_j=epsilon, with fixed r>0 and every other capacity
positive, then
For nonzero sigma these limits depend on r. There is no continuous single-valued pressure extension through that corner, although the full velocity extension exists. The nodal product at zero capacity cannot resolve that pressure ambiguity. These claims hold at retained regular states; they do not control a simultaneous phase singularity or a changed capacity law.
Frozen static discriminators and existing consumers¶
The origin controls
fix a native acute K1,3 preparation before evaluation: root-first phases
(0,0,0,t), t=atan(3/4), uniform zero form, capacities (1,2,3,4) and
e=0,w=beta=1, admitted through the regular-domain reader. A separate
shift/work control uses form (0,1,-1,2), a common shift of 8 and
e=w=1/2,beta=1. There is no trajectory or parameter fit. Write
alpha=atan(3/14). Since sin(t)=3/5, cos(t)=4/5 and
tan(3*alpha)=1737/2366<3/4 on the same acute branch,
Projection conserves Q but preserves the native difference between the
nonaligned and an aligned leaf, -4*w*t/pi. Sine instead gives
-4*w*sin(t)/pi; hence the projection does not recover sine even after
removing the common origin. At this preparation an originally zero form
rate on an aligned leaf also becomes -h!=0, excluding a positive common
clock rescaling. All phase rates and the instantaneous storage loss vanish
because form is uniform; zero storage loss does not certify zero full motion.
A second fixed support has edges 0-1,0-2,0-3,3-4,4-5,4-6, zero form,
unit capacities and phases (0,0,t,0,0,0,u). Changing the remote u=0 to
u=t preserves the complete induced root ball at node 0, but doubles the
nonzero sigma with W=12 unchanged. The original root pressure stays the
same while its projected pressure changes. Holding u=0 and changing only
the capacity at node 6 from 1 to 2 instead preserves sigma and changes
W from 12 to 23/2. Exact symbolic identities
and the shared native reader are checked separately from binary64 rounding;
no support or evolution law is installed by these tests.
Existing relative-state reports already retain moving-reference rates and reconstruction information. Recovery admission refuses absolute mean-slab membership from relative observations, while replica membership consumes the absolute mean when its declared family requires it. These consumers cannot silently inherit a quotient-only conservation convention.
Nor can two disconnected patterns be recentered independently before a join: their bridge form gap contains the difference between their two origins. One common translation preserves it; two independent translations do not. The internal quotient descriptions must retain that relative offset or explicitly leave contact observations unavailable. The same information obligation holds for relative phase origins. A native attachment observer does not transfer its complete law to a sine composition.
Conclusion. Common-origin symmetry permits a closed relative description and a conserving lift when its extra information is admitted. It does not derive conservation of the original primitive Q. Primitive locality and capacity independence make that balance an additional restriction, not a free relabeling. Together with section 11.8 this closes the equivalence question without selecting a unique microscopic law, justifying cosine storage, changing an executor or revising any retained response.