Retention mechanisms and the complete-law envelope boundary¶
Scope: mechanism admission and one local finite-input variational evaluation under the existing positive-loss cycle law. Exact balances and cubic coefficient identities are proved below; envelope profiles solve an auxiliary modulation equation. A full-state residual theorem and a relative-storage comparison specify how to validate such a reference against the actual law, retaining near resonances. The first correction panel is evaluated in section 17; the full horizon, nonlinear envelope maintenance and interaction-generated composites remain unproved. The original source and strict-passage result remain unchanged. The research program alone selects further work and owns reactivation of the deferred variational dependency.
1. Retained law and internal relations¶
Keep the complete two-row model:
signed real form x, a continuous real phase lift theta, unit cycle
conductances and lengths, held unit capacities, beta one, no input or events,
and the clock tau=w*t/pi. For e,w>0, sigma=pi*e/w, A=L/2,
Writing z=x+i*theta retains two real channels. It gives
This is neither complex EPI nor an identification with a quantum state.
The actual positive-loss tangent J_sigma has the existing contractive
Euclidean semigroup. A proposed envelope is an approximate full-state
history, not an invariant subspace or a replacement for either row.
Supplied symmetry does not establish acquisition¶
Reflection Jz(j)=z(-j) commutes with (1). The original nominal joint
preparation is reflection symmetric, so that relation is preserved exactly.
The same-source linear control also preserves it. The relation is supplied
initially and does not discriminate a newly formed organization.
For primitive errors of norm at most epsilon each, the initial antisymmetric
part has norm at most 2*epsilon. Within the original joint norm bootstrap
||z||,||Jz||<2a, the shared derivative estimate ||DR||<=4S^2 with S=2a
gives
This robust retention is still retention of a prepared symmetry. It cannot replace initial absence and discriminating acquisition in the formation contract.
The exact period-four family
reduces to independent damped pendula u and v. With
E_u=u'^2/2+1-cos(u) and similarly for v,
There is no transfer term between these two modal energies. For any pendulum
solution q and nonnegative time shift delta, u(tau)=q(tau) and
v(tau)=q(tau+delta) give an exact pair. Such a relative registration can be
supplied; these equations do not select it through mutual exchange. Decay
toward zero also destroys an oscillation's phase observability. Localized
modulation and its boundary cannot be removed by invoking this reduction.
2. What can accumulate locally under this law?¶
The regional balance owner
requires boundary exchange. Here an exact cycle-specific expression makes
the consumed phase information explicit. Orient cycle edges consistently,
put y=Dx, u=Dtheta, and let edge indices j and k be neighbors on the edge
cycle. Their rows are
For h_j=y_j^2/2+1-cos(u_j), define
Direct differentiation of h and substitution of (5) prove (6). Summing over
all edges recovers H'=-(sigma/2)||Lx||^2. On a fixed subset, internal
currents cancel; an increase requires inward boundary current to exceed
local dissipation. Form gaps and sine phase gaps determine that current,
so intensities or centers alone cannot infer it. Positive total loss does
not exclude transient regional accumulation. Conversely, positive inflow
does not prove retention or select a boundary. Moving membership requires
its separate transport accounting; (6) is for fixed sets.
3. The cubic balance selects a different candidate regime¶
Let an on-grid carrier satisfy kN in 2*pi*Z, and write
Use a positive bookkeeping scale varepsilon, distinct from the original source
error epsilon, and slow variables
X=varepsilon*(j+v*tau), S=varepsilon^2*tau, phi=kj+lambda*tau.
The candidate carrier travels with site velocity -v.
Exact cubic harmonics and a two-channel reconstruction¶
The cubic part of R is R_3(theta)=D^T(Dtheta)^3/12. At locally constant B,
substituting theta=Im(B*exp(i*phi)) gives exactly
For example, a forward phase gap has coefficient
(exp(ik)-1)B/(2i); cubing and applying the backward incidence row gives
all four terms. Keeping only the resonant first term would discard an
actual part of the form row.
Cancellation through cubic order uses the auxiliary equation
and the full reconstruction
Require lambda>0 and nonzero displayed denominators. The operator
partial_tau-iA multiplies a constant harmonic exp(i*l*phi) by
i*(l*lambda-lambda(l*k)); (11) cancels the three nonresonant terms of (8).
Envelope derivative errors remain and are bounded in section 5. Both real
coordinates are reconstructed as x_app=Re(z_app),
theta_app=Im(z_app); neither row is declared exact by (9).
The carrier k=pi/4 is already mode 128 of the unchanged C_1024 support.
It has positive curvature and no third-harmonic resonance:
This is an analytical candidate, not a searched optimum. For general k, third-harmonic resonance and spatial aliases must be checked; spatially aliased terms can still have distinct temporal frequencies.
The actual damping does not disappear¶
Equation (9) is a conservative comparison equation. All actual damping can
be retained in the residual of (10), as in section 5. Alternatively, writing
gamma=sigma/varepsilon^2 yields the leading damped modulation
Its counterrotating damping corrector is
-i*gamma*conj(B)*exp(-i*phi)/4. This follows by multiplying its coefficient
by -2i*lambda, giving the omitted damping coefficient
-gamma*lambda*conj(B)/2. For a periodic envelope,
d/dS integral |B|^2=-gamma*lambda*integral |B|^2 under (13).
Fixed positive sigma therefore cannot be suppressed in a
varepsilon -> 0 limit or replaced with zero to certify a desired profile.
The residual bound below applies to the conservative reference (9)--(11);
using (13) and its additional corrector requires redoing that bound.
4. Which relation can the auxiliary balance maintain?¶
For d,g>0, (9) admits the exact line profile
This follows from sech''=sech-2*sech^3. The relation locks the envelope
shape, height, width and slow internal phase in the comparison equation;
the corresponding carrier frequency is lambda-varepsilon^2*Omega.
It is not a freely chosen stationary shape of the original nodal law.
The sign restriction is substantive. For a smooth real decaying profile f,
d*f''+g*f^3=Omega*f has first integral
d*f'^2/2=Omega*f^2/2-g*f^4/4. Exponential decay requires Omega/d>0,
and a nonzero turning amplitude requires Omega>0. Thus this bright-profile
balance requires d>0. At the original k=pi/2, d vanishes and this
quadratic-dispersion mechanism is unavailable. This does not exclude every
higher-order modulation or nonperturbative mechanism at that carrier.
Periodic references and the source scale¶
The line sech cannot be sampled across a cycle seam and treated as a smooth periodic solution. An exact periodic alternative of (9), with kappa and rho strictly positive, is defined by
The solution oscillates smoothly between rho*sqrt(1-m) and rho. Separation
of variables gives period 2K(m)/kappa, where
K(m)=integral_0^(pi/2) (1-m*sin(u)^2)^(-1/2) du.
Choose that period to equal varepsilon*N; this supplies a genuinely
periodic reference with one maximum per cycle. Its trough, concentration,
derivative bounds and error allowance still need a quantitative admission.
No value of m, width, horizon or source tolerance has been evaluated here.
For fixed N and m, that period condition fixes the site-scale height
varepsilon*rho=2*K(m)*sqrt(2*d/g)/N and inverse width
varepsilon*kappa=2*K(m)/N. Changing varepsilon alone only reparametrizes
this reference; the derivative bounds change with it. It is not a free
parameter for reducing the physical residual on fixed support.
The line-profile scale already explains why the original small preparation
cannot simply be renamed. Write physical site width ell=L/varepsilon and
height b=varepsilon*rho; (14) implies
For the literal sampled sech with integer center and 1<=ell<=N/4, at
least ell sites have distance at most ell. Since sech(1)^2>1/4, its norm
squared exceeds b^2*ell/4>=K_*^2/N. On C_1024 its norm therefore exceeds
1/4, whereas the entire original joint tube has norm below 1/2048.
This is a scale incompatibility of this particular reference; it is not a
theorem excluding all small nonlinear localized states. The full corrected
periodic reference must have its own source admission.
A single retained envelope is not capture of two incoming patterns¶
For patterns to remain within a common width O(varepsilon^-1) over a
slow interval O(varepsilon^-2), their leading group velocities must differ
by at most O(varepsilon), unless a different, nonperturbative mechanism
changes them. Near pi/4, nonzero cosine makes this require carrier
separation O(varepsilon). Replacing the original carriers with +/-pi/4
still leaves velocity difference sqrt(2): focusing alone does not capture
that counterpropagating pair. Distant carriers can share a velocity; they
require a separately derived multiple-envelope system. The
equal-velocity carrier control
derives the local cubic channels for pi/4 and 3*pi/4, retaining temporal
labels within spatial aliases. Its modulation restrictions do not provide
a full-law formation or maintenance certificate.
A prepared profile satisfying (14) or (15) already contains its phase/shape relation. Even successful full-law maintenance would not establish its acquisition from two initially independent patterns. That claim additionally needs initial absence, a compatible relative preparation, retained boundary state, entry and a discriminating continued response under the same law.
5. An explicit conditional bridge to the complete law¶
Here is a sufficient finite-time validation method, rather than an appeal to
a formal asymptotic symbol. Let B solve (9), be periodic in X with period
varepsilon*N, and have continuous bounded derivatives through order three
in X and order one in S on the full proposed slow interval. Define uniform
bounds b_j>=sup|partial_X^j B|, j=0,...,3, and
b_S=|d|*b_2+g*b_0^3>=sup|B_S|. All bounds include intervening X values,
not just graph samples. Put
Every phase gap of (10) has magnitude at most G. For
r_app=z_app'-F_sigma(z_app), a uniform per-site bound is
The terms have separate derivations:
- Taylor expansion of the leading nearest-neighbor row leaves
varepsilon^4*b_3/6after (9). Corrector transport differs from the leading group velocity and is bounded using|partial_X F_l(B)|<=3*b_0^2*b_1. - Freezing B in the cubic phase gaps changes each gap by at most
varepsilon^2*b_1. The cubic difference inequality supplies the last term in the first bracket. - The corrector's slow derivative and second spatial remainder use
|partial_S F_l|<=3*b_0^2*b_Sand|partial_X^2 F_l|<=6*b_0*b_1^2+3*b_0^2*b_2. - Its addition changes each phase gap by at most
2*varepsilon^3*A_0*b_0^3, giving the cubic correction term. The global Taylor inequality|sin(u)-u+u^3/6|<=|u|^5/120gives the fifth-order term. - The final term bounds the entire actual damping row, using the
nearest-neighbor bound
|A Re(z_app)|<=2*sup|z_app|.
The Euclidean sqrt(N) factor is retained; localization or cancellations
cannot silently replace it. The residual is a pair of real-row errors,
including the phase row, not a fitted effective pressure.
Let the initial full-state error be at most delta>=0 and propose a strictly
positive error radius r and horizon T over 0<=tau<=T.
In that radius every actual and intervening phase
gap is bounded by G+2r. Since
variation of constants under the actual contractive J_sigma and Gronwall
give the sufficient bootstrap condition
A first exit at norm r contradicts this estimate, so the full-state error
stays at most E_* throughout the interval. A required acute phase chart can
be admitted separately by G+2r<pi/2. Reference norm and concentration
floors must exceed their error allowances before any retained identity is
certified. If (20) fails, this sufficient method is inconclusive; no
nonexistence or dynamical instability follows.
This is a conditional theorem with explicit finite constants. It has not
been instantiated for a prepared envelope. In particular, at
T=S_*/varepsilon^2, the fixed positive sigma contribution must still be
evaluated. A formal small residual or a profile of (9) alone cannot certify
that window. Using localization-aware norms or a damped reference may
improve the method, but each needs its own proved bound.
A scoped obstruction to this coarse certificate¶
For the proposed periodic family (15), one can decide a limitation without
selecting a profile or evaluating a response. Fix the actual original law
coefficients e=2^-20, w=1-e, so sigma=pi/(2^20-1), retain N=1024,
and take k=pi/4. Put b=varepsilon*rho and define one slow internal-phase
time by
Consider the particular certificate requiring both (20) and the acute-chart
condition G+2r<pi/2, with r>0. Since
1/4<lambda<1/3, g<1/72, D_k>1/2 and pi<4,
this condition forces r<1 and b<4. Consequently
Here the second inequality uses the actual positive damping term
r_*>=2*sigma*b; b_0 is at least the profile maximum rho. Also
Even at zero initial error, the published sufficient majorant therefore obeys
using exp(u)>=1+u+u^2/2 for nonnegative u. The contradiction persists for
any T>=T_phi. Thus this coarse acute-chart certificate cannot validate
the periodic conservative reference for a full nonlinear phase time.
Changing width, refining source error or increasing arithmetic precision
cannot repair this bound within the stated class.
This is a theorem about a sufficient method, not an obstruction to actual nonlinear maintenance. It does not exclude (20) without the additional acute condition, shorter intervals, other reference histories or other error estimates. A structure-sensitive stability argument or an explicitly damped reference with its own residual would change the method's premises. The present calculation does not establish that either alternative succeeds.
The existing moving-reference relative-storage identity is a reusable method for retaining reactive cancellations. Its earlier conservative graph, exact reference and quantitative constants do not transfer here. Positive loss, two-row reference residuals, moving phase curvature and the uncontrolled common means must all be re-admitted before using such storage to bound this full-state error. Relative storage is not automatically decreasing along a moving reference.
6. Relative storage retains the complete error balance¶
Let (x_*,theta_*) be a differentiable, independently specified reference
on the same cycle and clock, with residuals in both actual rows:
Use matched continuous phase lifts, u=x-x_*, eta=theta-theta_*,
a=Dtheta_*, b_e=(Deta)_e, and define
Direct differentiation and substitution of both rows give the exact identity
The form/phase exchange cancels. The moving-curvature term does not cancel and has no fixed sign. This extends the earlier moving-pattern identity to the actual positive loss and an approximate reference; it does not transfer that owner's graph, exact reference or constants.
Means, coercivity and the retained coordinates¶
Let P project off the constant nodal vector. All inverse powers of L below
act only on its mean-zero subspace. Put
Suppose ||y||<=rho and ||a||_infinity+rho<=chi<pi/2. With
kappa=cos(chi)>0, Taylor's integral formula gives
The missing means have exact independent equations
bar(u)'=-bar(r_x), bar(eta)'=-bar(r_theta). They need their initial
values and residual integrals. Constant form or phase errors have zero
relative storage; neither a full-state estimate nor a zero-mean residual
may be inferred from storage alone. The factor ell_1^-1 is the actual
finite-cycle coercivity cost, not a carrier-band approximation.
Taylor expansion of the moving term in (26) gives
For the second bound, write the sine difference as t_e b_e, where
t_e=int_0^1 cos(a_e+s*b_e) ds and |t_e|<=1, and apply Cauchy--Schwarz
to the two residual rows together. No pressure is reconstructed from a
response and no error mode is discarded.
7. A conditional comparison that retains near resonances¶
There is a precise alternative to taking the absolute value of all of M before integration. Define the skew principal tangent on the mean-zero coordinates by
This is a comparison operator inside the proof. Actual damping stays in (26) and in the error equation; the physical law is still (1).
For the reconstruction (10), its leading edge wave can be written
a_0=Im(c(tau)*exp(i*lambda*tau)), where c is the complex edge amplitude
obtained by applying D to varepsilon*B(X,S)*exp(i*k*j).
Its coefficients vary with the envelope. With omega=2*lambda, set
Indeed -(a_0*a'_{0,fast})/2=Re(i*lambda*c^2*exp(2i*lambda*tau)/4).
All envelope derivatives, generated harmonics and the difference between
sine and its argument remain in M-M_fast. This is an exact decomposition
of (28), not a declaration that its remainder vanishes.
The finite-dimensional homological equation¶
Choose an orthonormal complex eigenbasis J_0 v_alpha=i*mu_alpha*v_alpha,
with all nonconstant cycle modes and both frequency signs. For a complex
symmetric matrix H use the congruence coefficients
Hhat_alpha,beta=v_alpha^T H v_beta. The transpose is intentional: these
are quadratic forms, not Hermitian expectation values. The congruence
preserves the Frobenius norm. On these coefficients the operator
i*omega*C+J_0^T*C+C*J_0 has divisor
i*(omega+mu_alpha+mu_beta).
Fix a positive threshold Delta before any bound evaluation. Retain every
entry with |omega+mu_alpha+mu_beta|<=Delta in H_near, including zero
divisors. Let H_far=H-H_near, and define the symmetric matrix C by
Reconstruct C by inverse congruence and set K=Re(exp(i*omega*tau)*C).
The matrices K and Re(exp(i*omega*tau)*C') are real symmetric. Direct
differentiation, with the same fixed spectral split, proves
In particular, conservative sufficient bounds are
Actual operator-norm bounds may be sharper. A changing cutoff would add
terms not covered by (31). The split does not delete perturbations: near
entries remain in
M_ret=M-M_fast+Re(exp(i*omega*tau)*H_near).
Positive metric, actual loss and a full-state bootstrap¶
On a prospective interval admit uniform bounds
Define E_c=E+y^T K y/2. Equations (27) and (31) imply
For completeness, the exact error equation in these coordinates is
y'=J_0*y+f_sigma+f_nl-rhat, with
||f_sigma||<=2*sigma*||y|| and ||f_nl||<=chi^2*||y||.
The latter follows from
sin(a+b)-sin(a)-b=(int_0^1 cos(a+s*b) ds-1)*b,
||L^(1/2)D^T||=4, and 1-cos(q)<=chi^2/2 in the admitted chart.
Here rhat=(L^(1/2)r_x,L^(1/2)r_theta); it is not an additional forcing law.
Substitution in the derivative of E_c proves
Thus q=sqrt(2*E_c) has the comparison majorant
At zeros of E_c use a regularized square root and pass to the limit.
If the initial chart holds and Q(T)<rho*sqrt(kappa-k_*), a first exit
from ||y||<rho contradicts (33)--(35). Equations (27) and the independently
integrated means then control the complete nodal error. This is a
conditional theorem; every bound is uniform over the declared interval and
source ball. Small residuals alone do not supply k_*<kappa or that bootstrap.
Actual positive damping has not been removed: (34) retains its negative storage production and bounds its interaction with K explicitly. A better use of the negative term requires an additional proved estimate. A damped envelope reference also needs its own residual under (25). Neither is supplied by selecting a cutoff. The bound improves on (20) only when its admitted constants and full-state margins demonstrate that improvement.
8. Which curvature terms cannot be averaged away?¶
Two exact coefficient calculations delimit the preceding construction. They concern the frozen leading carrier in the homological equation, not a solved localized reference. They are not trajectory observations.
The carrier block has a nonzero resonant source¶
Temporarily write the leading reference as
theta_*=b*sin(phi_j), phi_j=k*j+lambda*tau, with real amplitude b,
and take a same-carrier error
z-z_*=(alpha+i*beta)*exp(i*phi_j). This complex notation still represents
two real errors. Exact spatial summation on C1024 at k=pi/4 gives
The brackets denote the principal fast average, with frozen error
quadratures, not averaging an evaluated nonlinear response. The term can
have either sign. Its homological divisor is zero:
2*lambda-lambda-lambda=0. A bounded periodic quadratic correction cannot
have a nonzero constant derivative along this periodic principal motion.
Therefore this source cannot be removed by such a correction.
Its maximum magnitude relative to E_0 is 2*g*b^2. In the conservative
auxiliary envelope's carrier-rotating frame, linearization about constant
amplitude gives alpha'=0, beta'=-2*g*b^2*alpha in the fast clock at this
order: amplitude error changes the phase rate. The resulting shear has
linear-in-time entries, not an exponential-instability proof. Over the
formal phase time (21), the Euclidean logarithmic-norm bound for this
auxiliary shear has exponent g*b^2*T_phi=2/(2-m), of order one.
This coefficient calculation does not
bound all other modes or certify the localized envelope. In particular,
the actual loss, spatial modulation and residual remain in (34)--(35).
Adjacent modes invalidate an exact-resonances-only shortcut¶
For distinct modes q=k+h, r=k-h, the principal divisor is exactly
It is nonzero but small. Use unitary Fourier coefficients z_q,z_r,
d_q the incidence symbol with |d_q|^2=2*lambda(q), and
A_b=|d_k|*b. Put
P_qr=d_q*d_r*z_q*z_r*exp(-2i*(lambda*tau+arg(d_k))).
The principal moving production on this pair is
-A_b^2*lambda*Im(P_qr)/8; along J_0, P_qr'=i*delta_h*P_qr.
Consequently its compulsory zero-homogeneous correction is
The supremum is over nonzero pair errors; equality uses equal weighted modal magnitudes and the appropriate relative phase. For the amplitude scale imposed by periodic family (15), this ratio is
It exceeds one half even as m tends to zero. Shrinking the bookkeeping
varepsilon at fixed physical profile cannot make it vanish. The continuous
elliptic integral K increases from pi/2 to infinity, so the class contains
an amplitude with K(m)=pi. For that analytic witness R_h>2, while
b=2*pi*sqrt(2d/g)/1024<1/16 because sqrt(2d/g)<9 and pi<7/2.
Its leading phase gaps are small. Yet the canonical principal correction
has a direction below -2*E_0. The relative-storage Hessian itself is at
most E_0 since cos(a)<=1; adding this correction loses positivity.
This is a scoped obstruction to eliminating all nonzero divisors around J_0 while retaining only exact resonances with a small correction over that frozen-carrier amplitude class. It does not exclude (30), which can retain these adjacent modes, or all positive time-dependent metrics. Adding a large homogeneous metric changes the normalization and its error bounds. Including the slow phase rate or loss in a different inverse changes the divisors and requires a separate calculation. The actual localized envelope has spatially varying coefficients and additional couplings; (39) is not a nonexistence or instability theorem for it.
9. The spatially varying periodic reference has explicit finite bounds¶
The following calculation applies to the whole periodic family (15), not
its frozen carrier. Fix the analytical rule Delta=lambda/2 for (30).
Retain the original support and loss, N=1024, sigma=pi/(N^2-1), and
k=pi/4. No profile parameter, source tolerance or response is selected
or evaluated. The formulas below are uniform in the continuous translation
of the profile and in time.
Physical profile and derivative budgets¶
Write the site-scale profile as F(s)=varepsilon*f(varepsilon*s) and put
The full reconstruction (10) is exactly
z_app,j=F(j+v*tau)*exp(i*psi_j)+F(j+v*tau)^3*sum_l a_l*exp(i*l*psi_j).
The bookkeeping varepsilon has disappeared. The first integral and its
derivative give the uniform bounds
For example, set s=1-(F/b)^2 in [0,m]. Then
F'^2=zeta^2*b^2*s*(m-s), whose upper bound follows from
s*(m-s)<=m^2/4. Also F''=zeta^2*F*(2s-m) and
F'''=zeta^2*(-4-m+6s)*F', proving the other two inequalities.
The exact slow derivative of the profile factor is -i*omega_s*F.
It must not be omitted when differentiating the moving metric.
Let A_h=sum_l |l*a_l|, with the three harmonic coefficients from (11).
For the leading edge amplitude c of (29) and the harmonic contribution to
the phase gap, define
The fundamental theorem of calculus across one site proves
||c||_infinity<=C_0, ||c'||_infinity<=C_1.
Differentiating every harmonic, including its spatial transport, gives
the bounds G_h and V_h. Consequently ||a||_infinity<=G and
||a'||_infinity<=a_* for the complete reference.
To specify exactly the inherited scalar residual budget under audit, put
S_0=d*B_2+g*b^3 and rewrite (18) without varepsilon:
Here S_0>=omega_s*b; it deliberately retains the already proved scalar
budget. No lower bound on the actual residual is asserted by any positive
term of (42). Sharper use of the exact slow derivative, harmonic phases or
damping row would define a different budget needing its own proof.
All-mode bounds without a dimension factor from the Frobenius norm¶
For a smooth N-periodic function use its continuous Fourier coefficients
with normalization 1/N, and let W be their absolute sum. Parseval and
Cauchy--Schwarz, with sum_(n!=0) 1/n^2=pi^2/3, give
The second bound uses the zero mean of a periodic derivative.
Absolute convergence makes sampling
legitimate: discrete Fourier coefficients sum aliased continuous
coefficients, so their absolute sum cannot increase. Translation changes
the phases of the continuous coefficients, so the same alias bound holds
uniformly over shifts; discrete aliased coefficients need not acquire one
common phase. Multiplication obeys W(f*g)<=W(f)*W(g) and the
one-site difference has norm at most two. Thus the sampled edge amplitudes
satisfy
This is a norm of Fourier coefficients, not a restriction to selected
carrier modes. In the normalized eigenbasis of J_0, each congruence entry
of H has magnitude lambda*|hat(c^2)(-q-r)|/8. The factors from U have
modulus one on every nonconstant mode. Summing over both frequency signs
bounds every absolute row and column sum by lambda*W(c^2)/4.
The symmetric near/far mask cannot increase these sums. The Schur bound,
unitary invariance of matrix norms and (30) therefore prove, for the fixed
Delta=lambda/2,
For the last remainder, write
sin(a)*a'-a_0*a'_(0,fast)=(sin(a)-a_0)*a'+a_0*(a'-a'_(0,fast)).
The harmonic gap is at most G_h and |sin(a)-a|<=G^3/6, giving the
second term of m_*. Thus (44) includes spatial transport, slow phase,
generated harmonics and sine curvature, with no frozen-envelope assumption.
Metric positivity still requires k_*<cos(chi) separately. These are
explicit sufficient constants, not a claim that this cutoff is optimal.
The inherited scalar defect cannot close this bootstrap¶
Use the constant residual allowance
mathfrak r_*=2*sqrt(N)*r_* in (35), justified by
||L^(1/2)||=2, and the natural derivative budget a_* from (41).
Even with zero initial error, the resulting majorant cannot close its
edge-error ball over T>=T_phi=1/omega_s, for any 0<m<1 and
positive radius rho satisfying the chart premises. This conclusion does
not depend on the size of the retained-block bound or on the cutoff.
To prove it, set c_5=D_k^5/120. The published allowances obey
If metric positivity fails, there is already no certificate. Otherwise
alpha>=a_*rho/(6*delta_kappa)>0. The constant-residual term in (35) and
exp(x)-1>=x^2/2 imply
For the fixed carrier, lambda>1/4, D_k>3/4, g<1/72 and
c_5>1/512; the original loss satisfies sigma>3/N^2.
Using u+v>=2*sqrt(u*v) for positive u,v gives
Thus Q(T_phi)>rho, whereas (35) requires
Q(T_phi)<rho*sqrt(delta_kappa)<=rho. Larger horizons or initial errors
only increase this particular majorant. Neither choosing another width
nor eliminating every other growth allowance repairs it.
There is a separate obstruction if one attempts only to improve the cubic error term while retaining the same scalar forcing allowance and global edge-to-node conversion. Even granting zero growth, zero initial error and zero mean error, the resulting full-state allowance U_T obeys
Here G+rho<pi/2 implies b<4; A_0<1/16 gives
1+A_0b^2<2, and sqrt(ell_1)<8/N. The same arithmetic-geometric-mean
bound proves the final inequality. The denominator in (48) is already an
upper bound on the reference norm. This allowance therefore cannot certify
an error smaller than the signal itself, even before resonance growth.
It is not a lower bound on actual error or a failure of the exact flow.
Method decision. Equations (40)--(44) supply the previously missing spatially varying constants, but the inherited scalar defect treatment fails before their optimization could matter. Equations (47)--(48) exclude that specified combination of residual, nonlinear remainder and norm conversion, not all relative-storage methods or all conservative references. A damped reference has not been proved necessary. Retaining the defect's phase, spatial support and two-row correlations in its transport, or a different justified residual/observation estimate, changes the premises. Merely bounding the retained block more tightly does not.
10. Signed defect transport retains phase, aliases and both rows¶
The next comparison changes the failed allowances, rather than optimizing
(42). Let h_0=2*pi/N, omega= lambda-omega_s, and expand the continuous
periodic profiles in Fourier coefficients normalized by 1/N. The real
coordinates of the complete reference in section 9 are
The profiles are evaluated at j+v*tau. Retain spacetime labels (n,l)
with n an integer and l=1,3 initially, their conjugates, and all labels
generated by the nonlinearity. Define
Set v_nl=0 outside the reference harmonics l=+/-1,+/-3; nonlinear
residuals can still have nonzero coefficients at those other labels.
Spatial aliasing does not identify temporal frequencies: replacing n by
n+N leaves q unchanged but changes Omega by 2*pi*v. Nor does a nonzero
carrier ensure constant reference means: some aliased labels have q zero
and Omega nonzero. Both effects must remain in a full-state comparison.
Take d_q=1-exp(i*q) for the oriented incidence. With
a_nl=d_q*(v_nl)_theta, convolution * on the integer label lattice, and
S(a)=sin(a)-a, the exact coefficient formulas are
These are the signed residuals of both real rows under the actual law. Smooth periodic profiles give absolutely summable coefficients and first temporal derivatives. The absolutely convergent sine series justifies the convolutions and interchange of sums. Cancellations in (51) occur before taking a norm; independently bounding its derivative, tangent and sine terms would generally destroy the small defect one is trying to assess.
Actual positive-loss harmonic transport¶
For a real temporal frequency Omega, define the entire matrix kernel
When its inverse is admitted, the equivalent expression is
K_q=(exp(i*Omega*tau)*I-exp(J_q*tau))*(i*Omega*I-J_q)^-1.
For q nonzero, the determinant is
lambda_q^2-Omega^2+i*sigma*lambda_q*Omega, which cannot vanish for
positive sigma and real Omega. The entire expression remains the owner of
the zero-mode and small-denominator limits. At q zero, K is
(exp(i*Omega*tau)-1)/(i*Omega)*I, with value tau*I at Omega zero.
For the actual 0<sigma<2, put s_sigma=sqrt(1-sigma^2/4) and
a_q=sigma*lambda_q/2. Direct squaring of
C_sigma=[[-sigma/2,-1],[1,sigma/2]] gives
The branch factors in (52) are
[exp(i*Omega*tau)-exp((-a_q +/- i*lambda_q*s_sigma)*tau)] /
[a_q+i*(Omega -/+ lambda_q*s_sigma)]. Their corresponding spectral
projectors are not orthogonal and cannot be replaced by unit weights.
At all modes the full matrix satisfies ||K_q||_2<=tau, since the actual
semigroup is contractive. Thus no uniform nonzero detuning or unproved
number-theoretic condition on aliases is needed. Resolved nonresonant
labels may have a much smaller effect than this general bound.
The signed linear defect response is
For Parseval, sum the separate time-dependent contributions within each
spatial mode first, then use ||h||_2^2=N*sum_q ||hat(h)_q||_2^2.
No inverse Laplacian occurs. In particular,
K_q*(i*Omega*I-J_q)*v=(exp(i*Omega*tau)*I-exp(J_q*tau))*v;
the linear part of (51) and the nonlinear coefficient must still be
combined before replacing (54) by an absolute bound.
Finite truncations need their own tail allowance¶
The exact representation is not a license to omit small coefficients.
For any scalar N-periodic profile P with ||P'''||_infinity<=C_3,
integration by parts gives, for an integer cutoff M at least one,
Use C_3=B_3 for F and
C_3=6B_1^3+18bB_1B_2+3b^2B_3 for F cubed. Equation (55) controls
both the omitted coefficients and their temporal derivative, including
translation. The factor |l*omega| for each harmonic must also be retained.
For phase-gap coefficient sums bounded by A, truncating the sine series
after degree 2P+1 leaves the separate absolute-sum allowance
provided the displayed denominator is positive. A phase-gap truncation of
absolute-sum size epsilon additionally changes S by at most
(cosh(A)-1)*epsilon, where A bounds every interpolating coefficient norm.
The incidence factor |d_q|/2<=1 does not enlarge this form-residual sum.
If the complete omitted two-row residual has coefficient-norm sum at most
epsilon_r, its contribution to (54) is at most sqrt(N)*T*epsilon_r.
This explicit tail treatment replaces the fixed fifth-order allowance;
it is not permission to reduce a tail without a declared finite budget.
11. A response-corrected reference has an exact phase row¶
Equation (54) solves h'=J_sigma*h-r_app, h(0)=0, where r_app is
the signed vector residual of (51), distinct from the scalar allowance
r_* in (42).
Define the independent corrected reference z_c=z_app+h. Its full-law
residual is exactly
Indeed z_c'=J_sigma*z_c+R(theta_app). Its phase row is exact, and both
reference means remain at their initial values. Aliased mean drift in the
original reference is canceled by the zero-mode part of (54); it was not
assumed absent. This is a prescribed response correction, not a trajectory
of the nonlinear law and not a supplied pressure replacing that law.
Equation (57) is generally first order in h. For example, at a
nonconstant reference with gaps a in (-pi,pi), take h_theta=t*theta_app.
Then the inner product of r_c,x/t with theta_app tends to
-sum_e a_e^2*(1-cos(a_e))/2<0. A quadratic-error claim would be false.
A genuinely quadratic residual would instead require solving the full
variational correction
h'=J_sigma*h+(DR(theta_app)*h_theta,0)-r_app, with its actual time-dependent
Jacobian. That is a different linear problem and needs a separate bound.
For that full variational correction h_v=(u_v,eta_v), the exact remaining
form source is R(theta_app)+DR(theta_app)*eta_v-R(theta_app+eta_v).
If its phase-gap norms are bounded by H_v0 and H_v2 in infinity and
Euclidean norm, respectively, Taylor's theorem gives
Indeed the edge remainder is sin(a+b)-sin(a)-cos(a)*b, bounded by
|sin(a)|*b^2/2+|b|^3/6, and ||D^T||/2=1. This is genuinely
quadratic in h_v and needs no acute-chart assumption. It does not supply
the time-dependent coupled-mode propagator or a quantitative bound on h_v.
For (54), if ||Dtheta_app||_infinity<=G_app,
||Dh_theta||_infinity<=H_0, and ||Dh_theta||_2<=H_2, the exact sine
secant gives the conditional source allowance
It starts at zero when h does and depends on the signed transported defect.
It has neither the standalone damping nor the fifth-order floor in (42).
The actual damping remains in (52)--(54). Bounding (58) by a large constant
can still be inconclusive; the exact residual (57) and time dependence
remain available. The total error to the original reference is h+e,
where e=z-z_c; the correction h must always be reported.
12. Full-state metric without a spectral-gap conversion¶
A second constructive change removes the ell_1^-1/2 factor in (48).
For a reference (x_*,theta_*), let xi=(u,eta) be the full nodal error,
including its means, and start with E_0=||xi||_2^2/2.
Write a=Dtheta_*, B=DR(theta_*), and
M(B)=[[0,B/2],[B/2,0]]. Its exact quadratic production is
xi^T M(B) xi=u^T B eta. The real errors have no Fourier-band restriction.
Use the original leading edge wave a_0=Im(c*exp(i*lambda*tau)), retaining
||c||_infinity<=C_0 and ||c'||_infinity<=C_1 from (41). Thus
||a_0||_infinity<=C_0. Admit ||a-a_0||_infinity<=H_gap and
G=C_0+H_gap.
For z_app, H_gap=G_h; for z_c one may use H_gap=G_h+H_0.
The exact splitting is
Taylor's inequality |1-cos(a)-a^2/2|<=|a|^4/24 and ||D||=2
prove (59). There is no moving-curvature a' term in this energy.
An acute chart is not a positivity premise for the metric below.
Static and oscillating corrections preserve every coordinate¶
On the mean-zero space solve A S+S A=B_0, and set S zero on constants.
The positive Sylvester integral and 0<=B_0<=C_0^2*A/4 give
With K_0=diag(S,-S), direct multiplication yields
J_0^T K_0+K_0 J_0=-2M(B_0). The incidence factors remove the apparent
small denominator; no inverse spectral-gap bound is used.
For the oscillating matrix H=M(B_f) use the fixed
Delta=lambda/2 homological split (30). In eigenvectors
(f_q,-i*s*f_q)/sqrt(2), s equal to plus or minus one, opposite signs
give zero congruence entries. Same-sign entry magnitudes are
|d_q*d_r|*|hat(c^2)(-q-r)|/16. The inequality
|d_q*d_r|<=lambda_q+lambda_r bounds the far inverse. Only the
negative/negative block can be near resonant. Its real part has half the
complex operator norm: it joins two orthogonal conjugate eigenspaces.
The real far inverse is bounded by half the sum of its two branch norms.
The row-sum estimates of section 9 consequently give, for K=K_0+K_f,
Here d_* bounds the uncanceled coefficient derivative
K_0'+Re(exp(2i*lambda*tau)*C_f'), not the complete derivative K'.
The latter includes the canceled fast terms. The value m_* bounds the
retained quadratic production after those cancellations. Means remain
under the identity metric, since every correction has zero constant rows
and columns. These are new Euclidean constants, not transferred Bregman
constants with the spectral factor removed by assertion.
Nonlinear remainder, actual loss and a prospective bootstrap¶
Assume ||xi||<=rho, k_*<1, and set
Taylor's theorem, ||Deta||_infinity<=sqrt(2)||eta||_2 and ||D||=2
give ||R(theta_*+eta)-R(theta_*)-B*eta||<=n_rho*||eta||.
The baseline nonlinear work then costs at most
n_rho*||xi||^2/2, using ||u||*||eta||<=||xi||^2/2.
For E_c=xi^T(I+K)xi/2, both actual rows imply
The terms 2*sigma and G^2 in c_* bound the interaction of K with the
actual damping and reference Jacobian, respectively. Thus, for uniform
admitted constants, initial full-state error delta, and alpha=c_*/(1-k_*),
The strict condition Q(T)<rho*sqrt(1-k_*) proves the whole interval by
first exit. No Poincare factor or unobserved mean coordinate remains.
If an acute phase chart is separately needed, require
G+sqrt(2)*rho<pi/2. For the response-corrected reference, insert (57)
or its proved allowance (58), and include ||h|| in the final error to
z_app. Time-dependent source bounds must not be replaced by their
endpoint values without a proof; h starts at zero but can vary thereafter.
This changes both mechanisms behind (47)--(48): the residual is transported before its nonlinear remainder is bounded, and the error metric controls the full nodes directly. It supplies a conditional comparison with no inherited scalar-defect floor. It does not prove that (63) closes on the periodic family. The remaining quantitative obligation is a certified bound on (54), (57), their Fourier/sine tails and the nonlinear allowance over a phase time, with a nonzero localization and signal margin.
13. A canonical-profile preflight rejects a positive coefficient budget¶
The signed comparison above remains conditional. Its first quantitative
preflight fixes the self-complementary profile parameter m=1/2, exact
nominal initial state and T=T_phi=1/omega_s. Keep the same C1024 law and
carrier. The declared finite budget is profile cutoff |n|<=12, sine
degree seven and the shared 128-bit outward arithmetic. The preflight
stops at a failed necessary condition before constructing those finite
sums. No profile search, prepared nonlinear response or trajectory is used.
The particular allowance tested here replaces each signed kernel vector
by a positive full-two-row norm, then uses that norm to bound its phase
component. For an invertible M_nl=i*Omega_nl*I-J_q, set
At the singular mean label use t*||r_nl||. Contractivity and (52)
justify both branches of (64). Sum within each spatial alias class before
Parseval, giving the explicit positive allowances
Here G_app=C_0+G_h is specifically the published allowance (41), and
S^+ instantiates (58). Finite outward enclosures retain positive tail
allowances. These operations are justified upper bounds, but discard
vector directions and temporal cancellation inside the alias sums.
The lower bounds below apply to this chosen allowance, never to the
norm of the signed sum (54), the actual correction or its exact residual.
A necessary source moment eliminates a radius search¶
More generally, use any nonnegative residual allowance S(t) in (63).
For uniform k_*<1, gap allowance G and
s_G=min(1,G), its exponent satisfies
Since exp(x)>=x for nonnegative x, even at zero initial error the
bootstrap implies the necessary condition
If the displayed lower bound is at least one, no positive choice of rho closes this majorant. This is a necessary test for a sufficient error method, not a necessary condition for actual retention. It detects the interaction of positive residual accumulation with the conversion from full nodal error radius to pointwise phase-gap error in n_rho.
One retained coefficient already forces failure¶
Let f_0=hat(F)(0). The first integral (15), with
F/b=sqrt(1-m*sin(phi)^2) and
ds=dphi/[zeta*sqrt(1-m*sin(phi)^2)], gives the period averages directly:
Thus no Fourier quadrature is needed for the retained label (0,1).
At m=1/2, put z_0=3b^2/4, and define
X=f_0*(1+a_-1*z_0), Theta=f_0*(1-a_-1*z_0) and
Omega=lambda-omega_s. Its reference coefficient is (X/2,-i*Theta/2).
The profile is even, and the complete phase row is odd under simultaneous
reflection of the envelope and carrier arguments. The actual sine
divergence preserves that oddness: the two neighbor shifts exchange under
reflection. Its spacetime coefficients are therefore purely imaginary.
Consequently, for some real beta_r, the exact signed residual has the form
No bound on beta_r or sine truncation is used. The exact resolvent numerator
has imaginary first component Omega*A_r and real second component
lambda*A_r+sigma*lambda*C_r=sigma*lambda*Omega*Theta/2. Hence, writing
D_0=lambda^2-Omega^2+i*sigma*lambda*Omega,
All required inequalities follow from elementary fixed-profile bounds.
The elliptic integral definition gives
pi/2<K(1/2)<=pi/sqrt(2). At the unchanged carrier,
1/4<lambda<1/3, 1/4<d<1/2, 8<sqrt(2d/g)<9, and D_k>3/4.
Using 3<pi<22/7 and pi^2<10 gives
Also G_app<1/24: for this fixed m, (40)--(41) give
C_0<(513/512)b and G_h< b^3/8. These loose bounds suffice.
Equation (69) implies Omega>=lambda/2, while |a_-1|<1/16
implies X>=f_0/2. Since
X^2+Theta^2>=2*f_0^2 and |D_0|<=3*lambda*omega_s, (68) yields
Indeed the first branch in (64) is at least sigma*lambda*f_0*t/4;
the second is at least sigma*lambda*f_0*T/(3*sqrt(2)), larger than
sigma*lambda*f_0*t/5. The conjugate label has the same bound and lies
at the distinct spatial mode -k. Thus the chosen H_2^+ is at least
sqrt(2N)*D_k*B_01, and
For the corrected reference G is at least G_app. Because G_app<1,
min(1,G)>=G_app even if the corrected gap allowance exceeds one.
If the metric is positive, alpha>=G_app*rho/sqrt(2) for every radius.
Equations (63) and (71), again using exp(x)>=x, give
For the last strict rational bound use (69), G_app>=D_k*b>18/N,
D_k>3/4, lambda>1/4, T>N^2/15 and sqrt(N)=32.
The bootstrap requires Q(T)/rho<sqrt(1-k_*)<=1. If metric positivity
fails instead, there is already no certificate. Nonzero initial error,
additional positive mode allowances, tails and outward arithmetic can
only worsen this particular bound.
Preflight result. The declared positive coefficient budget fails at the canonical profile before its finite Fourier/sine sums need evaluation. It cannot be repaired by radius selection, extra modes or precision. This does not exclude other profile parameters, the exact signed phase-projected transport, a different nonlinear work estimate, or actual maintenance. In particular, (72) is neither an observed error nor a lower bound on an actual response. The next comparison must retain information lost between (54), (64) and (65), or admit a genuinely different correction law; repeating their positive-norm chain is not a quantitative advance.
14. A finite signed variational representation has controlled tails¶
Keep the canonical m=1/2 reference and full phase interval of section 13.
Write V(t)=[[0,B(t)],[0,0]], with B(t)=DR(theta_app(t)), so that
h_v'=(J_sigma+V(t))*h_v-r_app and h_v(0)=0. The first-order feedback
is included in this linear comparison, not in a modified constitutive law.
V couples modes; separate constant two-by-two tangent propagators do not
by themselves propagate this equation.
The declared finite representation uses profile cutoff 24, the exact cube of that finite profile, sine degree seventeen, cosine degree sixteen and 256 chronological Jacobian insertions. Retained time integrals and their coefficients are exact mathematical objects here. The result below bounds analytical truncation error; no retained sum, prepared response or trajectory is evaluated. Shared rational/128-bit interval controls on constants do not certify a future sum's accumulated arithmetic error.
Chronology, resonance and the factorial tail¶
Let S(t)=exp(J_sigma*t). Define operator terms for s<=t by
The integrals preserve matrix order and signed vector addition. With
L=G_app^2 and R_*>=sup||r_app||, contractivity of S and induction over
the ordered simplex give
The series converge uniformly on the whole interval even when LT>1.
Their sums solve the full variational initial-value problem. This is not
an assertion that its one-step Volterra map is a contraction.
For a nonnegative integer P, beta>=LT and beta<P+2, summing the decreasing
factorial ratios bounds the omitted orders j>P by
These positive bounds apply only to omitted orders. Replacing the retained terms in (73) by (74) would discard the phase information being preserved.
In the spatial Fourier basis, write f=1-cos(a), a=Dtheta_app. The
coupling is
Each chronological path changes spatial mode by Q_alpha and retains its
own temporal frequency. Spatial aliases cannot merge those frequencies.
The d factors annihilate constant rows and columns. Thus terms h_j with
j>=1 have zero means; h_0 alone cancels the original reference's mean
drift. The exact corrected reference still has a complete exact phase row.
Successive integrations of exponential factors are entire divided differences of the exponential function. For nodes z_0,...,z_d, their ordered-simplex representation is
The simplex measure has volume 1/d!. Along a variational path the nodes
are actual damped tangent eigenvalues plus accumulated imaginary forcing
frequencies; the final source node is purely imaginary. Their real parts
are nonpositive, giving modulus at most t^d/d!. Coincident nodes use the
entire limit, for example t^d*exp(t*z)/d! when all nodes equal z.
There is no lower-detuning premise. If branch factors are used, retain the
actual nonorthogonal projectors from section 10. Chronological products
of different B(t) generally do not commute.
The full linear propagator has a moderate uniform gain¶
The direct metric of section 12 applies at zero error radius to
J_sigma+V(t), using the original reference gap G=G_app, H_gap=G_h.
Put c_lin=m_*+d_*/2+k_*(2*sigma+G_app^2) and
alpha_lin=c_lin/(1-k_*). Its homogeneous energy bound gives
Here is a fixed-profile verification rather than an assumed conditioning
constant. Equations (40)--(43) and (69) imply
W_0<11b/8, W_1<b/256, C_0<b, Z_0<9b/8, and k_*<b^2<1/625.
Also
using lambda>2/7, D_k>3/4, and b^2T=4/(3g).
For the metric terms, C_1*T<25/4, Z_1*T<37,
C_0<33/800 and Z_0<9/200 give d_*T<4/3.
Equation (79) and D_k<4/5 also give G_app<b, and G_h<b^3/8.
The near-block term obeys
5*lambda*Z_0^2*T/64<945/256; the remaining terms of m_* have sum
at most b^4*T/6<1/20. Hence m_*T<81/20.
Finally sigma*T<1 and (79) give
k_*(2*sigma+G_app^2)*T<78/625<1/8. Therefore
Since exp(1)<3, their product is less than (21/20)*3^5<256, proving
(78). This controls all modes, including means; it does not bound the
inhomogeneous signed correction by a small observed value.
Finite profile, source and Jacobian inputs¶
At m=1/2, the elliptic nome is q_e=exp(-pi)<1/20. The
Jacobi dn Fourier formula, with the
nome convention, gives
DLMF's elliptic modulus is sqrt(m), distinct from the carrier k.
The normalization follows from (67) and the period N; all coefficients
are positive. Thus W(F)=b and W(F^3)=b^3. These are properties of the
admitted profile, not evaluated trajectory data. Let F_M retain |n|<=24.
Its cube is formed exactly, so its support extends to |n|<=72.
The geometric tail allowance
uses f_0<1/32 and h_0<1/160. Also
W(F')<=4*f_0*h_0*q_e/(1-q_e)^2<2^-14.
Products give W(F^3-F_M^3)<=3b^2E_F; differentiating the product
gives a bound [6bW(F')+3b^2/4]E_F<b^2E_F for its derivative error.
Use the complete reconstruction (49) with F_M to obtain w_M and its gap a_M. Positivity of the retained profile coefficients bounds every convex interpolation of a and a_M in coefficient absolute-sum norm by
Let S_17 be the sine polynomial through degree seventeen minus its linear
term, and C_16 the polynomial for 1-cos through degree sixteen. Define
This preserves the signed source terms before their final addition. The source and Jacobian have different truncation allowances. In particular,
Here 8*sinh(1/16)<1; the second term is less than 2^-122.
For the source, use W(x)+W(theta) to bound its two-row coefficient norm.
The profile substitution costs less than 3E_F in state and 3E_F in
time derivative; ||J_sigma|| in that sum norm is below four. The sine
substitution costs less than E_F, while its separate series tail is
A^19/19!/(1-A^2/(20*21)). Including the nodal factor sqrt(N)=32 yields
These deliberately loose bounds control only removed input terms. They do not substitute absolute coefficient sums for the retained solution.
Combined analytical error and remaining computation boundary¶
Let h_tilde solve the finite-input linear equation
h_tilde'=(J_sigma+V_tilde)*h_tilde-r_tilde, with zero initial value.
Equations (18) and (42) give R_*<32: the bounds
b,B_1,B_2,B_3,C_0,G_app,S_0,sigma<1/16, A_0<1/16 and A_1<1/8
already imply r_*<1/32<1 by direct substitution. Equation (69) gives
T<2^19. Set M_*=2^8
from (78). Variation of constants for the input difference, using
sup||h_v||<=M_*R_*T, gives
Since M_*T*epsilon_B<1/2, this small input-perturbation term can be
absorbed. It is not a contraction assumption on the full V evolution.
Consequently
For the finite inputs, (79) and (84) give
sup||V_tilde||*T<76; also sup||r_tilde||<33<64, so its product with
T is below 2^25. Use their signed chronological recurrence (73) through
order 256, calling the finite sum h_tilde^[256]. Exact rational factorial
comparisons in (75) give
The second, forced coefficient has an additional factorial from the source integration. It implies
This passes the declared 2^-40 analytical representation target. It is
not a smallness claim about h_v itself or the nonlinear error to the
reference. Every retained term is a finite signed sum of entire ordered
kernels, so (87) is constructive in exact arithmetic. Naive path
enumeration is nevertheless very large: the truncated Jacobian alone can
reach |n|<=1152, |l|<=48, and the source |n|<=1224, |l|<=51.
These are label bounds, not permission to truncate later generated modes.
Efficient accumulation, certified compression and numerical roundoff still
need an implementation and their own resource admission. A symbolic
finite representation is not an executed 128-bit solver.
A defect certificate can avoid enumerating chronological paths¶
The finite inputs have the direct action
V_tilde(t)*(u,eta)=(D^T[C_16(a_M(t))*Deta]/2,0). Evaluating the retained
profile and its derivative on the N nodes costs O(NM) by direct summation;
the Jacobian action then costs O(N). The polynomial nonlinearities are
evaluated on those same nodes, without deleting spatial aliases or their
actual time dependence. This representation does not build a lifted path
matrix or change the finite-input equation.
Duhamel's formula and (78) bound its exact propagator by
M_*exp(M_*epsilon_B*(t-s))<2^9. For a continuous, piecewise continuously
differentiable approximation h_hat on a finite partition, define its complete
equation defect
d_hat=h_hat'-(J_sigma+V_tilde)*h_hat+r_tilde. Variation of constants gives
This is an alternative evaluation certificate for the same exact h_v. It compares directly with h_tilde, so no chronological truncation term is needed in (88). A numerical implementation must enclose coefficient, arithmetic, derivative and between-sample errors in d_hat; pointwise solver tolerances do not establish this integral. The small input tails and the large retained forcing must remain separate. The first-panel result evaluates one local approximation; no full-interval evaluator is resource-admitted.
An evaluated approximation h_hat must additionally certify its error to
h_tilde^[256], or supply the defect certificate (88). Only then may its
combined error allowance enlarge phase-gap bounds by
sqrt(2)*epsilon_h in infinity norm and 2*epsilon_h in Euclidean norm,
and its full-state bound by epsilon_h. Those enlarged bounds describe the
exact h_v used in the quadratic residual of section 11. A finite-input
approximation is not itself an exact-phase-row reference unless its own
source and representation defects are retained.
The remaining nonlinear bootstrap, initial-error ball, signal/localization margin and same-source linear discrimination are still unproved. This result closes finite analytical representability and its truncation budget; it does not establish maintenance or evaluate the whole representation.
15. Matrix-free inputs and whole-interval validation requirements¶
The finite-input equation (83) admits a direct nodal implementation without
enumerating chronological paths. Its fixed domain remains N=1024, m=1/2,
M=24, sine degree seventeen and cosine degree sixteen. Evaluation of these
prescribed inputs is distinct from integration of their forced linear
equation, and neither is a nonlinear prepared response.
Enclosing the finite profile and its signed source¶
The elliptic constant is enclosed using the
arithmetic-geometric mean identity.
Start with a_0=1, g_0=sqrt(1/2) and apply eight simultaneous updates
The exact sequences bound their common limit by g_8<=AGM<=a_8.
Outward interval arithmetic therefore encloses the complete elliptic
integral by pi/(2*[g_8.lo,a_8.hi]); agreement of rounded iterates is not
a stopping criterion or an assumed exact value. DLMF uses modulus
sqrt(m), while this owner uses parameter m. The nome is evaluated as
exp(-pi/4)^4, within the existing unit-decay enclosure's domain.
These constants determine all the coefficients of F_M, the slow frequency,
and an interval containing the exact T. The numerical input evaluator may
use the rectangle [0,T.hi]; that is an enclosure of the physical horizon,
not a changed model clock. Time and both action vectors are admitted before
conversion or arithmetic. The factory consumes no cached bounds or verdicts.
Evaluate F_M and its spatial derivative at j+v*tau. Differentiate (49)
with both translation speed v and carrier frequency lambda-omega_s.
Form the exact cube of the retained profile at the same node, then compute
both signed source rows in (83) before taking a norm. Spatial harmonic
recurrences reuse fixed roots and two time rotations; they do not merge
distinct temporal frequencies merely because their spatial samples alias.
For any admitted real rows u and eta, the complete matrix-free action is
The incidence convention is Dy_j=y_j-y_(j+1) and
D^Tz_j=z_j-z_(j-1). Thus both constant directions and both row means
are retained. Every output uses the same immutable input evaluation;
editing a displayed report must not change its captured action law.
No dense 2048 x 2048 matrix is needed. Direct profile evaluation costs
O(NM) and a Jacobian action costs O(N). The original recurrence retained
O(N+M) live storage; the reusable spatial basis below explicitly trades
O(NM) storage for fewer interval operations per evaluation. These are
finite input costs, not a bound on a complete solution's work.
Reusing fixed spatial work in the same finite inputs¶
Equation (80)'s finite Fourier profile already separates spatial and
temporal factors. With C_jn=cos(n*h_0*j), S_jn=sin(n*h_0*j),
c_n=cos(n*h_0*v*tau) and s_n=sin(n*h_0*v*tau), write
These are exact identities for the same retained profile, including its
derivative. Spatial roots may use (j*n) mod N; n remains unreduced in
the temporal rotation and derivative. Distinct temporal frequencies do
not merge when their spatial samples alias. Preweight immutable spatial
rows once, then contract them against the M temporal harmonics at each
time. The carrier k=pi/4 has only eight spatial classes, so its rotated
first and third harmonics also need only eight reconstructions per time.
The exact pointwise cube and its derivative 3*F_M^2*F'_M remain in (49).
An identity for the untruncated elliptic profile cannot replace this finite
cube without accounting for the difference.
The shared dyadic linear reduction
retains interval uncertainty during these contractions and the Lobatto
transform. For scale S=2^128, represent an admitted interval by integer
center numerator C and radius numerator R over 2S. For a dot product,
accumulate the signed integer center and a conservative radius
The interval (C_*+[-R_*,R_*])/(4S^2) encloses every product sum;
applying an exact rational scale and one final outward projection retains
that enclosure. This does not assume independent errors or remove their
widths. Public consumers admit all primitives before preparing or using
encoded rows. Immutable fixed bases can be reused; evaluated states,
report verdicts and time-dependent phases are not authoritative cache keys.
This reuse changes arithmetic ordering and enclosure widths, not the finite functions or the analytic interpolation tails. Rational harmonic and derivative controls must establish the algebra independently; overlap between old and new interval boxes alone would not prove correctness. Existing floating FFTs can propose coefficients but cannot replace these outward reductions. The older Fejer-source spectral factory and its reflection shortcut have different preparation hypotheses. Direct full spacetime convolution is also unnecessary here: the finite sine/cosine polynomials retain their existing nodal Horner evaluation and aliases.
Why a componentwise box propagation does not transfer¶
The existing general comparison kernels have finite dimension and derivative
order contracts; increasing their limits would not prove suitability here.
There is also a mathematical obstruction to their coordinatewise absolute
Jacobian treatment over this horizon. Write
A=I-(P+P^T)/2 on the cycle. The one-sided diagonal/absolute-offdiagonal
majorant of J_sigma maps spatially uniform coordinate radii by
The negative form diagonal cancels its two positive neighbor terms;
|A| has row sum two. This radius comparison grows as exp(2*tau)
along its positive eigenvector, although the actual tangent is contractive.
For the finite-input Jacobian, 1-C_16(a_M)>1/2 throughout the admitted
gap domain. Its coordinate majorant still dominates the uniform-radius
block [[0,1],[2,0]], with positive growth sqrt(2). Hence replacing the
full-state cancellation by this majorant is unsuitable for the declared
small-error target on T. This excludes a comparison method, not the actual
dynamics, other norms or every possible interval solver.
A continuous residual budget includes joins and small-norm arithmetic¶
For piecewise continuously differentiable h_hat on a finite partition,
let j_i=h_hat(t_i+)-h_hat(t_i-) be its interior joins. Applying the
same full-interval propagator once to each source and join gives
The partition covers the whole physical interval; a final enclosure through T.hi is sufficient. No repeated gain of 512 is imposed at every step. Continuous pieces with exactly shared endpoints have zero joins. Recentring or rounding an endpoint is otherwise an additional source, not free error removal. The initial term is computed from the approximation even though the exact correction starts at zero.
The prospective numerical target is 2^-24, separate from the previous
2^-40 analytical representation target. A sufficient numerical budget is
that the entire bracket in (91) is at most 2^-34: its contribution is
2^-25, and adding the proved input tails stays below 2^-24.
With zero initial error and joins, a uniform complete defect below 2^-53
is sufficient since T<2^19. This is an accuracy policy for the linear
comparison, not a proved nonlinear retention margin.
Small defect norms require scaling before squaring. For nonnegative exact
coordinate bounds b_i, set s=max(b_i) and use
s*sqrt(sum((b_i/s)^2)), with the zero case handled exactly. Rounding an
unscaled square to the absolute 2^-128 grid can introduce a norm floor
of 2^-64, independently of the actual defect's size. A 128-bit grid
therefore does not by itself certify 128-bit relative norm accuracy.
The fixed input preflight evaluates normalized times 0,1/2,1, enclosing
the latter two with the same T interval, and requests a midpoint-radius
Euclidean allowance below 2^-80 for the state, derivative, gap and source.
This checks input implementation and arithmetic only. Even exact state and
defect samples cannot prove (91): take a unit constant spatial direction w,
annihilated by the actual full variational matrix, and
h_hat(t)=a*P(t/T)*w, with
P(s)=s^2*(s-1/2)^2*(s-1)^2. In the homogeneous zero-initial comparison,
both h_hat and its defect vanish at those three samples, while
||h_hat(T/4)||=9*|a|/4096. No pointwise tolerance can replace the
between-sample derivative and residual enclosure.
The remaining implementation obligation is a bounded-cost continuous approximation with a whole-panel signed residual and admitted join errors. Point input enclosures, finite representability and a small coefficient rounding error do not supply that approximation or evaluate h_v.
A direct Taylor realization is sufficient but computationally large¶
The finite inputs are entire functions of complex time. In a unit complex
disk around any real expansion time, every reconstruction frequency has
absolute value below 3/2. Since exp(3/2)<6, each real row's coefficient
absolute-sum bound is at most 6*(b+A_0*b^3)<1/4. Consequently the full
complex state norm is below 16, its derivative norm is below 24 and its
phase-gap coefficient norm is below 1/2. The cosine polynomial implies
||V_tilde||<1/2, while ||J_sigma||<17/8. The complete generator norm
is therefore below three. The nonlinear sine remainder has norm below
32*(sinh(1/2)-1/2)<1, giving ||r_tilde||<73<96 on the disk.
Admit a local expansion center with norm at most 2^35. Variation of
constants along a radius of this disk bounds its analytic solution by
27*(2^35+96)<Y=2^40. For its exact Taylor coefficients c_n, Cauchy's
estimate gives ||c_n||<=Y. Let P_16 be its degree-sixteen polynomial,
R the omitted series and A(t)=J_sigma+V_tilde(t). Its complete signed
equation defect is -R'+A*R, so a panel of length delta less than one has
Indeed integrating the derivative tail cancels each coefficient's factor n;
the remaining term is bounded by three times the integrated state tail.
For a fixed grid with panel lengths at most 1/64, summing (92) gives
This leaves room inside the declared 2^-34 budget for finite coefficient
arithmetic and joins, but does not bound those additional errors. With
exact coefficients, initialize each panel at the preceding polynomial's
endpoint so joins vanish. The exact forced solution satisfies
sup||h_tilde||<512*64*2^19=2^34; the defect bound and first-exit induction
therefore keep these exact polynomial centers below 2^35. Finite
implementations must check their centers and pay their extra defects.
This realization uses step exactly 1/64 except for its final shortened
panel. It requires more than 2^22 panels and at most 2^25, using
T>N^2/15 and T<2^19. Seventeen coefficients
in each of 2048 real coordinates already entail more than 2^37 scalar
coefficient entries across the run, before series products and validation.
That count excludes a direct full-grid Python rational campaign from the
present implementation route. It is neither a complexity lower bound for
the variational problem nor a failure of its dynamics. A practical method
must reuse time dependence across panels or retain exact tangent transport,
and certify its changed remainder before any full-interval run. Raising
the existing derivative-order or dimension limits alone does not do so.
16. Signed Chebyshev panels retain the continuous defect¶
The next representation uses panels of length at most sixteen, with degree-96 input and candidate polynomials. It preserves the same complete variational equation and clock. The finite polynomial validator retains the continuous defect. Canonical input interpolation and a candidate are evaluated separately in the first-panel result; a manufactured polynomial test alone does not evaluate that correction.
Source bounds sharpen without a response evaluation¶
The bound R_*<32 was deliberately loose. Substitute the primitive upper
bounds b<=81/(2N), zeta<=9/(2N), D_k<=49/64, d<=1/2, g<=1/72,
A_0<=1/16, A_1<=1/8 and sigma<=16/(5N^2) into the positive expression
(42), with B_1=zeta*b/4, B_2=zeta^2*b/2, B_3=9*zeta^3*b/8 and
T<8N^2/27. Exact rational inequalities give
This improves a supplied upper allowance, without evaluating the signed source or inferring any lower bound on the correction. It does not close a nonlinear maintenance estimate.
Uniform input interpolation on the whole panel¶
Map a panel to s in [-1,1], with t=t_c+(Delta/2)s, Delta<=16.
Use 97 Chebyshev-Lobatto nodes and the Bernstein ellipse with parameter
rho_E=1+sqrt(2). Its imaginary time semiaxis is at most eight. On that
strip, q_e*exp(8h_0|v|)<1/19, because h_0<1/160 and
exp(1/20)<20/19. The positive Fourier coefficients then give
W(F_M)<(1/32)(1+4/18)<1/24. Each reconstructed real row has coefficient
norm at most 27/24+3^8/(16*24^3)<5/4, so the full state norm is below
64 and its derivative norm below 96. The gap norm is below 5/2.
The finite cosine and sine polynomial majorants consequently imply
For example, cosh(5/2)-1<13 bounds the cosine polynomial, and
sinh(5/2)-5/2<11 bounds the sine remainder; the two incidence factors
give the operator factor two. The reconstruction frequency bound 3/2
controls its derivative. These analytic input extensions do not extend a
physical solution past its admitted horizon.
The Chebyshev interpolation bound
is 4M*rho_E^-d/(rho_E-1) for degree d and an analytic norm bound M
on the ellipse. It also follows directly by bounding the Chebyshev
coefficients using Cauchy's formula and the aliases at Lobatto nodes;
the same triangle inequality applies to vector and operator norms.
Since (1207/500)^96>2^122 and sqrt(2)>707/500, (95) gives the safe
degree-96 interpolation allowances
Interpolate the scalar edge multiplier before applying D and D^T, and both
signed source rows on the same affine clock. Enclose all sample values and
transform coefficients; (96) bounds analytic interpolation error only.
A floating transform's output is not an outward coefficient enclosure.
Point evaluations already supplied by the input owner can be reused under
these premises. A practical builder may share the fixed spatial roots and
local time offsets. Advancing rectangular interval rotations indefinitely
is not justified: their widths can grow under |cos|+|sin| even when the
exact rotation is an isometry. Re-enclose each panel's center rotation or
prove a separate propagation bound.
Reconstruct the signed equation before bounding its norm¶
Let the candidate p, signed source r_poly and scalar edge multiplier C_poly have degree at most 96 in ordinary Chebyshev coefficients, including the full constant coefficient. Form the complete polynomial residual
The identity T_j*T_k=(T_(j+k)+T_|j-k|)/2 retains every generated
coefficient through degree 192. In particular, projecting the residual
back to degree 96 would silently delete part of the equation being checked.
Differentiate the polynomial in its affine panel coordinate and retain
both mean directions. The shared finite-polynomial owner computes signed
products by exact packed integer convolution of dyadic Laurent coefficients,
then projects each result outwards with the existing interval type.
It neither changes the shared Taylor-jet order limit nor discards a tail.
Explicit factor-degree and packed-bit limits bound its work.
For coefficient vectors p_k and d_k, define
P_i=sum_k||p_k||_2 and R_i=sum_k||d_k||_2, using scaled norms of
the outward coefficient boxes. Since |T_k(s)|<=1,
The panel owner re-admits primitive coefficient tables, duration, sigma and error premises, and rebuilds (97), coefficient norms and both endpoint enclosures. Internal algebra reuses the locally normalized rows; dyadic integer norm reductions preserve the exact scaled rational sum before the shared square root. Its local result is conditional on the supplied polynomial inputs enclosing the actual source and Jacobian on the same panel. Default zero input errors assert exact polynomial inputs; they do not authenticate a canonical preparation. The validator supplies no response, propagator gain or retention flag. Initial mismatch and joins still belong to (91).
For example, if every panel satisfies P_i<=2^35 and R_i<=2^-54, (96)
and T<2^19 give a total integrated defect below
2^-35+2^-59<2^-34. With zero initial error and joins, (91) then meets
the declared 2^-24 numerical target. Nonzero initial error and joins must
share the same budget. Neither these policies nor small analytic input
tails show that a candidate passes them.
Candidate generation and the remaining resource boundary¶
On a real panel, keep the exact contractive S(s)=exp(J_sigma*s) and
iterate only the small Jacobian term in (73), starting at the carried
endpoint. The primitive gap bound and (84) give
Delta*sup||V_tilde||<16/625<1/32. This local Volterra map is a contraction;
the same estimate does not establish a full-interval contraction.
For a starting endpoint of norm at most 2^13, its uniterated source term
has norm below 2^14. Twenty Jacobian insertions leave a functional-series
tail below 2^14*(1/32)^21/(1-1/32)<2^-90. This establishes a candidate
route, not an implemented quadrature or a polynomial residual certificate.
Every approximated integral and coefficient must still pass (97)--(98).
A fixed grid of length sixteen, except the last shortened panel, uses at
most 2^15 panels. The validator stores O(Nd) coefficients and performs
O(Nd) interval bookkeeping and O(N) packed integer products per panel;
the bit cost of each product depends on the integer backend and coefficient
magnitudes. No dense spatial matrix or Python quadratic coefficient-product
loop is required. This reduces the prospective number of panels and makes
signed polynomial validation executable. It does not establish the cost of
building 97 canonical input samples or generating a candidate on each panel.
The polynomial controls independently check signed products, interval corners, carrying, derivatives and work limits. The panel controls check whole-polynomial cancellation, time scaling, uncertainty and invalid domains. Their full-size synthetic countercontrol deliberately omits generated degree-97 through degree-192 source terms and requires their nonzero residual to survive. This is software evidence, not a canonical variational correction. Section 17 evaluates the first canonical panel; a resource-admitted full interval remains pending.
17. Canonical first-panel construction and result¶
The bounded evaluation fixes the first normalized-clock panel [0,16],
the same C1024, m=1/2 reference, degree 96, and zero initial correction.
The input builder
prepares its own authoritative finite-input closure and encloses all 97
Lobatto samples. The shared transform uses ordinary Chebyshev coefficients,
including the full constant term:
Dyadic integer center/radius dot products enclose every signed sum before one final interval projection per coefficient. Sampling uncertainty and transform roundoff remain in the coefficient boxes; the analytic tails (96) remain separate. A finite sampled polynomial alone supplies no such between-node tail for an arbitrary function.
A proposal method with independent posterior validation¶
For a degree-d row f, define B_d f as its antiderivative projected to
degree d and re-centered to vanish at s=-1. Its retained coefficients are
b_1=f_0-f_2/2, b_j=(f_(j-1)-f_(j+1))/(2j) for 2<=j<=d, and
b_0=-sum_(j=1)^d (-1)^j b_j, with unavailable coefficients zero.
For d=1, the first formula gives b_1=f_0. The discarded exact-integral term
is f_d*(T_(d+1)-(-1)^(d+1)T_0)/(2(d+1)). Its derivative is part of the
posterior equation defect; projecting the proposal does not delete it
from validation.
Let q=sqrt(1-sigma^2/4), alpha=sigma/2+iq and
mu=-sigma/2+iq. The complex encoding psi=x+alpha*theta gives
psi'=mu*A*psi for the constant tangent, since mu*alpha=-1.
The inverse is theta=Im(psi)/q, x=Re(psi)-(sigma/2)*theta.
A full complex spatial FFT retains all 2N real coordinates; no same-mode
conjugacy or half-spectrum omission is imposed. For cycle eigenvalue
lambda_j=1-cos(2*pi*j/N), factor the temporal matrix
There are at most N/2 distinct nonzero factors on the even cycle. These small temporal matrices are separate from a dense full-state generator. Pivoted factorization avoids dividing by a frequency difference. Reuse the factors for the signed source solve and twenty fixed feedback updates. Each feedback product is evaluated with 2d+1 Lobatto points, retaining its degree-2d polynomial before projecting the next proposal to degree d.
The candidate owner uses this finite iteration count as a proposal policy. The exact-Volterra contraction and insertion tail in section 16 do not certify the projected iteration or its floating arithmetic. Binary64 produces only candidate coefficients; re-admit their represented values as exact rationals and set their constant coefficients so the proposed initial state is exactly zero. The independent outward validator then reconstructs both signed equation rows, all degree-192 products and the real full-state norm. Its acceptance depends on that rebuilt defect, not a solver tolerance or iteration delta.
The prospective local policy is P<=2^35, R<=2^-54 and initial endpoint
enclosure norm at most 2^-70, with the separate analytic input errors in
(96). Record input, proposal and validation costs separately. Keep the
first outcome, including failure, without changing the profile, degree,
iteration count, precision or panel after observing it. This local gate
does not evaluate the rest of T, its joins, a nonlinear bootstrap or a
source-to-composite response. A full-interval cost estimate is a resource
assessment, not a prediction that later panels will pass.
Retained local result and cost boundary¶
The frozen protocol declares this single evaluation before its response. The freeze, source manifest and source archive associate the evaluator, runtime and pre-evaluation proof bytes. They do not authenticate chronology or execution. The first attempt is preserved with its primitive response and result.
The numerical policy passes on this panel. Rounded-up decimal bounds are:
| Whole-panel quantity | Upper bound |
|---|---|
| Candidate full-state norm P | 9.913e-6 |
| Complete polynomial defect norm R | 3.005e-21 |
| Integrated defect, including analytic input tails | 4.808e-20 |
| Initial enclosure norm | 4.145e-36 |
Finite-input correction error, 512*(initial + integrated defect) |
2.462e-17 |
The candidate has exactly zero initial state as rational coefficients;
the small enclosure width is nevertheless charged to the error allowance.
All degree-192 terms remain in the residual. The last row bounds error
to the finite-input linear correction on [0,16], conditional on the
retained input enclosures and section 16's analytic tails. It does not
include the earlier exact-model input truncation allowances in (88),
and is not an error bound for a nonlinear trajectory. The source and edge
coefficient-radius summaries are respectively below 8.858e-33 and
1.527e-35; these are arithmetic widths, not the analytic input tails.
The resource policy fails its input threshold: preparation took 666.814 s against 600 s. Of that cost, sampling took 648.298 s and interpolation 16.842 s. The proposal took 5.521 s and interval validation 37.433 s, within their respective 180 s and 120 s thresholds. Total recorded time, including artifact I/O, was 713.873 s. The thresholds assess the completed attempt; they are not abort timers or mathematical restrictions on the law. The length-sixteen grid, with a shortened last panel, has 8,988 panels over the exact horizon. Repeating this observed total cost would take about 74.3 sequential days; this is a conditional cost extrapolation, not a runtime bound or evidence that later panels satisfy the numerical policy.
The input controls check canonical preparation, clocks, interpolation and separate error channels. The candidate controls compare independent dense small-cycle equations, retain all spatial modes and reject equating a projected fixed point with zero continuous defect. The evidence audit checks archive associations and re-admits primitive coefficients to rebuild the signed defect and verdict without running any canonical producer. It does not independently regenerate the retained sample enclosures.
This closes one local construction and its numerical assessment. The retained first attempt starts at zero; repeating that preparation would reset the correction and would not solve the full-interval problem. The maintained input reuse and carried-state construction below are separate implementation improvements. Initial mismatches, join defects and every panel's signed residual must share (91)'s budget. The deferred-dependency contract in the sole execution plan owns reactivation and further admission. No whole-horizon correction, nonlinear maintenance, composite acquisition or physical identification follows from this panel.
Exact endpoint carry without resetting accumulated error¶
The maintained candidate owner
also accepts two incoming scalar rows. It admits their exact or represented
values before numerical work; an uncertain interval is not an incoming
point. Add the full spatial Fourier transform of
psi_0=x_0+alpha*theta_0 to the constant temporal coefficient on the
right-hand side of (100). The same modal factors and twenty feedback
updates then propose the carried panel. The zero-initial entry point
delegates to this constructor and preserves its declared preparation.
After admitting the proposed tail coefficients as rationals, set
a_0=y_0-sum_(k=1)^d (-1)^k*a_k separately in each row. Thus the left
endpoint is exactly the supplied y_0 even when its rational representation
is not binary64. Carrying the preceding polynomial's exact rational right
endpoint sum_k a_k makes the mathematical join exactly zero. An interval
midpoint or a rounded endpoint is a different preparation whose mismatch
must be charged; overlapping endpoint boxes do not prove a zero join.
This does not make the incoming approximation equal to the true solution. Equation (91) retains its existing global error and applies gain 512 once to initial error plus all integrated defects and join norms. Neither a restart nor repeated multiplication of that gain is justified. Independent small-system controls check both real rows, nonzero means, homogeneous damped transport and two-panel joins. They do not evaluate a second canonical panel or admit its precision.
Input-reuse cost pilot and result¶
The implementation pilot keeps the first panel, all 97 sampling locations, degree 96 and the analytic input errors fixed. It builds only the optimized source and edge interpolants, then revalidates the exact candidate already stored in the first response. No candidate generator, old frozen producer or nonlinear trajectory is executed. Reusing those coefficients tests changed input arithmetic without generating a new scientific response or rewriting the first outcome.
The numerical policy remains P<=2^35, R<=2^-54, exact zero initial
state and initial enclosure norm at most 2^-70. The prospective local
resource thresholds are 180 s for inputs and 120 s for validation. Retain
the first outcome without automatic retries. Report cold preparation,
sampling and transformation separately. An explicit full-grid resource
scenario uses at most 24 sequential hours and 8 GiB of retained artifacts
as engineering policies, not dynamical hypotheses. Its time extrapolation
combines the new input/validation measurements with the old proposal cost;
its storage extrapolation uses the original per-panel response size.
Neither is a bound or a full-run admission. Repeated all-panel snapshots
and simultaneous allocation of the whole coefficient history are not
required by (91); a future storage design must retain auditable primitives
and checkpoints within a declared budget.
The separate protocol,
freeze,
source manifest
and source archive
retain this implementation and associate all eight previous evidence files.
Its first response
and report
pass every local numerical and resource check. The same retained candidate
still has finite-input correction error below 2.462e-17 on [0,16].
This is a revalidation of its existing polynomial, not a newly generated
variational response or an improvement to the physical model.
| Measured phase | Original attempt (s) | Input-reuse pilot (s) |
|---|---|---|
| Complete input construction | 666.814 | 109.466 |
| Input sampling | 648.298 | 90.324 |
| Input interpolation | 16.842 | 15.548 |
| Complete interval validation | 37.433 | 36.908 |
Complete input construction is about 6.09 times faster in these recorded runs. The new cold factory cost is 2.042 s, including preparation of the 100,352 encoded spatial entries; the cached basis uses three integers per entry. Its memory tradeoff is explicit. Different process conditions may change these timings; the single retained comparison is not a universal speed guarantee. The pilot's 150.687 s total excludes candidate generation, so it is not directly comparable to the original complete-attempt total.
The mixed-cost full-grid scenario still gives about 15.80 sequential days before artifact I/O and 59.16 GiB at the original response size. Both exceed their stated 24-hour and 8-GiB engineering policies. The new archive omits the reused candidate and must not be used as a complete per-panel storage estimate. Input sampling remains material; transformation, validation and evidence storage also need explicit treatment before a full run. The read-only audit checks the two evidence associations and rebuilds the complete signed defect and policy assessments from retained primitive coefficients. It does not regenerate inputs, candidates or old responses.
18. Admission result and execution boundary¶
The original symmetry and separated carrier energies do not supply a
discriminating formation mechanism. Positive curvature at an exact lower
carrier supplies a concrete phase-sensitive envelope balance, generated
harmonics and a conditional complete-law validation method. This identifies
a candidate nonlinear retention relation needing a separate preparation.
The coarse acute-chart method is excluded on a full nonlinear phase time,
so a different proved stability or residual treatment is needed before a
quantitative retention certificate. Equations (25)--(35) now give a
complete-law relative-storage comparison with explicit means, metric
positivity, moving coefficients, residuals and positive loss. The carrier
resonance and small-divisor calculation require retaining a near-resonant
block; exact-resonance-only cancellation is not uniformly a small positive
correction. This resolves a method-admission question, not its quantitative
use for a localized reference. Equations (40)--(44) now supply uniform
profile, retained-block and moving-metric bounds. The specified scalar
defect allowance prevents a full phase-time certificate even if all other
growth is suppressed; sharpening resonance constants or source precision
cannot resolve this particular failure. Equations (49)--(63) provide a
different treatment: signed two-row harmonic transport, explicit aliases
and tails, a response-corrected reference and a full Euclidean comparison
without spectral-gap amplification. Its transported-defect and nonlinear
budgets still need a quantitative closure before any maintenance claim.
The canonical-profile preflight (64)--(72) now excludes the specified
positive coefficient-norm implementation of that comparison before a
coefficient or trajectory campaign. Signed phase transport and nonlinear
work correlations remain unassessed by this obstruction.
The full variational comparison now has a finite phase-resolved
representation with explicit profile, Jacobian, source and chronological
tails, whose combined analytical error is below 2^-48 on the fixed
phase interval. Its finite inputs and sparse action have a
shared private implementation.
The Chebyshev panel validator now retains the
signed continuous defect and all generated degrees, with explicit uniform
input interpolation bounds. Raising the general comparison solver's
dimension cap would lose the cancellations, and the admitted direct Taylor
realization has excessive work for this route. The
first canonical panel now passes its
numerical policy, while its input construction exceeds the local cost
threshold. Shared signed input arithmetic and exact nonzero endpoint carry
now have maintained implementations. The separate reuse pilot passes its
local policies, while its full-grid time/storage scenario still fails.
Full-interval resource admission and a canonical joined correction remain
unresolved. Their reactivation conditions
belong to the sole execution plan; the calculation is deferred, with its
shared implementations and frozen evidence retained. The whole-horizon
correction and nonlinear maintenance bound remain unevaluated.
No result here proves that two patterns acquire the relation.
The static controls derive the local balance, cubic harmonics, correctors and envelope identities independently. The relative-storage controls check the two-row balance, missing means, coercivity, resonant production and small-divisor obstruction. The periodic-bound controls check derivative budgets and the scalar-defect exclusion. The phase-defect controls check actual-loss modal transport, two-row projection, aliases, signed convolutions, correction residuals and full-state metric identities. The defect-budget controls check the independent profile averages, source-moment condition, resolvent floor and strict rational obstruction, including its distinction from an actual oscillatory response. The variational controls check chronological order, coupled-mode structure, resonance limits, factorial and coefficient tails, propagation gains and separated input errors. Their arithmetic controls reuse the existing positive-series tail owner; they do not instantiate the large finite variational sum. The finite-input controls check independent reconstruction derivatives, sparse actions, scalar and clock admission, the declared three-point arithmetic preflight and the distinction between samples and a continuous residual certificate. They execute neither a trajectory nor frozen evidence. The actual rate implementation remains the shared sine owner; transport and nonlinear budgets retain their shared comparison owner. No envelope runtime or replacement nodal law is installed. Quantitative maintenance, a source-to-composite experiment and physical identification remain separate obligations in the sole execution plan.