Exact organized-source state modulo a global child exchange¶
Conditional construction. The two exchangeable children need not have equal form or phase. Their averaged coordinates together with one joint rank-at-most-one Gram matrix retain the complete primary state up to the single exchange of those children. On the specified organized-source law, these coordinates close both nonlinear continuous rows, energy and loss, the fixed energy guard and the admitted marked Reception–Coherence–Silence maps. Separate pair summaries can lose the relations required for this prediction.
This extends the represented domain of the fixed-sector chart, whose 72 primary coordinates require equal children. It does not replace its separate finite-error results. The present generic primary dimension is still 90. Removing a redundant global label is different from reducing the number of continuous degrees of freedom. Section 7 separately proves a quantitative approach to that sector in the admitted acute energy domain, with finite tolerance deadlines and decaying instantaneous prediction defects. Section 8 bounds the accumulated error of a separately evolved symmetric core, for unit capacities without events.
The construction applies the existing support-symmetry criterion and joint Gram method to the actual 45-node source, reciprocal phase row and marked ideal word. Its new obligation is to identify which relations across the nine paired slots this support preserves. It introduces no new constitutive law.
1. Actual symmetry and retained state¶
Use the source support and law: nodes 9*c+j,
five nine-cycles and the matching child edges
(0,1),(2,3),(0,2),(1,3),(0,4). All ninety edges have unit conductance.
The global permutation J exchanges whole children one and two and fixes
children zero, three and four. It fixes the marked targets zero and one.
The cycles couple adjacent slots, so swapping just one corresponding
pair of nodes is generally not a support automorphism.
Keep the ordered child orbits A={0}, B={1,2}, C={3}, D={4}, with nine
slots each. Their masses are 1,2,1,1; write M_mass for the diagonal
36-coordinate mass matrix. P copies projected coordinates to their fine
nodes and R=M_mass^-1 P^T averages the fibers. Thus Pi=PR=(I+J)/2.
The inherited weighted Laplacian is L_w, as derived in the fixed-sector
owner; its 63 edges have total weight ninety.
Declare the raw real carrier lift psi_(9c+j)=alpha*(j-4),
alpha=2*pi/9. Retain 36 projected forms u and 36 projected real phase
offsets d. Write vartheta=bar_psi+d. The unequal children have
All other children are copied from their singleton orbit. Positive held
capacities must satisfy nu_(1j)=nu_(2j). Their 36 retained coordinates
are separate law parameters, not part of the 90-primary-coordinate count.
They may vary between slots and between the four child orbits.
Set v=(a_0,...,a_8,b_0,...,b_8) and retain
The domain requires one jointly symmetric positive-semidefinite matrix of rank at most one. Its 171 distinct entries are redundant coordinates. They are products of one deterministic hidden vector, not an ensemble covariance, and cannot be replaced by unrelated positive local blocks. The joint-realizability owner explains the general distinction between pairwise admissibility and a common underlying state.
If G is nonzero, choose its first positive diagonal entry G_kk and set
v_k=sqrt(G_kk)>0, v_i=G_ik/v_k. Rank-one realizability proves that
these values reconstruct G, and its only real lifts are v and -v. At
G=0 the only lift is zero. Hence (u,d,G,nu) reconstructs the fine
primary state up to exactly J. Separate signs for different slots or for
the form and phase blocks do not describe the same equivalence relation.
The generic dimension is 72+18=90, with a singular zero-hidden stratum.
The canonical sign rule is an algebraic representative, not a globally
smooth choice of labeled state. At v=0 the differential of v -> vv^T
vanishes; this coordinate singularity does not make the fine flow singular.
Labeled nonsymmetric observations need a choice between the two lifts.
Invariant observations and fixed-target values do not.
These coordinates retain declared real phase lifts. They do not quotient independent turns, define a global chart on circular phases, or infer a preferred lift from wrapped observations.
2. Closed projected nonlinear law¶
The complete unforced law in structural time tau is
where S_f(Theta)_i=sum_(k~i) sin(Theta_k-Theta_i) and Gamma is absent.
Every consumed capacity is held between events; the same capacity
multiplies both rows. Define q=M_mass^-1 L_w u and let D be the
diagonal retained capacity matrix. Hidden form cancels in the projected
Laplacian, so R L_f x=q even away from the fixed sector.
For nonnegative r, put
Their power series in r remove any division at zero. For a B-cycle edge jk define
The projected sine current s_G=R S_f(Theta) is obtained by these exact
contributions; each displayed term is added to the indicated row:
| Row and neighbor | Contribution |
|---|---|
| B slot j, adjacent B slot k | sin(vartheta_Bk-vartheta_Bj)*C(r_jk) |
| B slot j, U slot j, U=A or C | sin(vartheta_Uj-vartheta_Bj)*C(r_j) |
| U slot j, B slot j, U=A or C | 2*sin(vartheta_Bj-vartheta_Uj)*C(r_j) |
| Every unaffected cycle or A–D edge | Its original fine sine term |
These follow from sin(t+b)+sin(t-b)=2*sin(t)*cos(b) and the
inherited neighbor counts. Consequently both projected rows close:
The phase row does not depend directly on hidden coordinates, but the form row consumes cross-slot entries of G_bb. Their evolution will also consume the mixed block G_ab. Keeping just the phase correction at one instant does not establish future closure.
3. Joint Gram evolution and exact balance¶
Let H=L_C9+2I, with diagonal four and adjacent entries minus one, and
V=diag(nu_Bj). Define the symmetric matrix K as a weighted cycle
Laplacian with edge weight
plus the diagonal entry
The half-difference of the two fine sine currents is exactly -K b.
Subtracting the complete fine rows therefore gives
The entries of K depend on (d,G_bb), not on the lift's overall sign.
The exact joint equation is thus
These equations are smooth on the realized domain, including G=0; the power series also define a smooth extension to ambient coordinates. Starting from a realized G, the fine lifted solution proves preservation of rank-one realizability and supplies the unique projected solution. The two signs give the same retained trajectory. This is a constructive closure proof, not a prescription to integrate redundant unconstrained entries or repair a matrix that has drifted out of its domain.
Let bar E be the fixed-sector excess at (u,d), and
H_*=45*(1-cos(alpha)). Exact edge grouping yields
Each unoriented cycle edge appears once. This identity does not require acute angles; the correction need not be nonnegative outside an acute projected domain. For arbitrary admitted symmetric positive capacities,
Indeed the two hidden fine gradients are q_B+Ha and q_B-Ha, whose
weighted squares cancel the cross terms. Slot-dependent V must remain
between the two H factors. The fine inverse-capacity-weighted form and
phase-offset totals reduce to sum(mass/nu*u) and sum(mass/nu*d);
hidden terms cancel because the paired capacities agree.
These are exact identities for the supplied law. They neither select its constitutive coefficients nor identify these quantities with laboratory energy, time or measured information.
4. Guard and marked hybrid maps¶
Use the same supplied fixed energy level and mandatory first downward crossing from the guard owner. Since the full energy is reconstructed exactly, its crossing set is the same for both fine lifts and the retained trajectory. Existence, positive first occurrence and transversality remain the guard owner's separate premises; this representation does not establish them anew.
For the complete ideal word, intersect with the
section 23 word domain:
initial unit capacities, its acute lift cell, E<1/100,
abs(initial form mean)<1/30, and complete supplied admission context.
The continuous identity above allows more general held capacities; it
does not widen this particular first-word admission theorem.
Let beta, lambda and kappa be the already admitted Reception, Coherence and Silence factors. Each stage uses one common fine snapshot at the fixed targets i=0,1. Reception retains the same inherited mean:
The hidden forms cancel in the two exchanged neighbors. Only these two singleton projected forms change; d, G and capacities are held.
For Coherence, write j=i for the target's slot. Its relative resultant is
This is the exact five-neighbor fine sum. On the existing regular argument
branch, d_i^+=d_i+lambda*Arg(z_i); every other projected phase, all
forms, G and capacities are held. Silence then sends the two target
capacities to kappa*nu_i, holding u, d and G. It preserves equality
between the capacities of children one and two, so the subsequent
held-capacity law remains in this domain.
Each primary stage commutes with J. The represented state therefore reconstructs the marked ideal word and its changed-capacity continuation up to that same global exchange. The hidden vector itself is unchanged by these marked maps; a capacity jump does not reset G.
Histories, ordinals, debt, prior Coherence, policies and any labeled selector inputs remain separately retained and admitted. Arbitrary runtime contexts are not functions of G. Full labeled execution needs the matching fine context and lift; no quotient of arbitrary context or new occurrence law follows. Stored-pressure jumps and fresh-law association retain their existing contracts. See the operator descent and grammar descent conditions. The proof concerns the primary ideal maps under those premises, not a new runtime executor.
5. Frozen information-loss controls¶
Both controls were fixed at h=1/128, unit capacity and carrier
projected phase before evaluation. They are static state and derivative
identities, not samples from a trajectory or unknown crossing states.
They disprove the named omissions; they do not prove that all 171
entries are a minimal collection of stored numbers.
Cross-slot phase association is visible immediately¶
Set projected form h*(e0-e1) on child A, every other projected form
zero, and a=0. In the first hidden phase vector all b_j are +h except
b_2=-h; in the second, only b_4=-h. Every local pair Gram is
(a_j^2,a_j*b_j,b_j^2)=(0,0,h^2). Their hidden means agree, so the
means of each labeled child also agree. Both states have
Their marked targets and every incident initial form, phase and capacity
agree. Thus initial fresh target pressure and direct ideal word outputs
at the targets agree as well, conditional on equal admitted context.
The largest cycle gap is at most alpha+2h<pi/2, matching gaps have
magnitude h and the global form mean is zero.
Nevertheless their projected B-slot-1 form rates differ, first minus second, by
The missing entry G_bb,1,2 is respectively -h^2 and +h^2.
Adjacent phase associations matter even when local pair Grams, child
means, energy, instantaneous loss and initial target data all agree.
Separate complete form and phase Grams still lose mixed information¶
Now set every projected form to zero, a=h*e0, and compare
b=+h*e1 with b=-h*e1. All local pair Grams agree because the two
supports are disjoint. Even the full separate matrices G_aa and G_bb
agree, as do both projected first derivative rows. Both preparations
are acute, have zero global form mean, and satisfy
But G_ab,0,1 is respectively +h^2 and -h^2. Since
(Ha)_1=-h, the squared hidden phase has
This distinction reaches a projected observable at the next derivative:
To verify the coefficient, the initial projected phase rate is zero and
the two projected form rates coincide, so their q derivatives coincide.
Only the derivative of the cycle phase correction differs. Here
dot b_1-dot b_0=-5*gamma*h/4; differentiating the two cosines gives
a projected sine-rate contrast 5*gamma*h*sin(alpha)*sin(h)/2.
The additional form-row factor gamma/4 yields the stated curvature.
The mixed block records a relative association between hidden form and
phase that neither separate full Gram retains.
6. Detached implementation and boundary¶
The private orbit reader encodes fine primitives, decodes admitted invariant coordinates to a canonical fine lift and observes the same complete law through the shared fine capacity reader. Projected rows are actual fiber averages; selecting one representative off the fixed sector would be incorrect. The joint Gram derivative is built from the fine hidden derivatives. No second sine kernel, numerical integrator or event executor is introduced.
The shared exact rank-one lift normalizes every matrix entry before checking symmetry and realizability. It also serves the existing P5 invariant decoder. The implementation retains exact rational or exactly represented real input values and rational canonical lifts. A realizable rational Gram whose canonical square root is irrational is explicitly unsupported by this decoder, though it is a valid point of the real mathematical domain. It is not rounded to a nearby lift or declared physically impossible.
The encoded fine state and every consumed primitive are readmitted; cached reports, Boolean scalars, nonfinite values, unequal paired capacities or arbitrary positive-semidefinite higher-rank matrices do not replace these premises. The fine state in a decoded report is the canonical representative, not a claim to recover the original labeled preparation without its missing overall sign.
Independent controls compare exact reconstruction, the shared-law observation and the prospectively fixed state/derivative distinctions. Algebraic closure and the conditional word proof establish sufficiency under this selected law. These reconstruction results alone establish neither a minimal state, faster simulation, an attractor, autonomous hierarchy formation, nor an empirical mapping.
7. Quantitative approach to the symmetric sector¶
The exact joint state explains what unequal children retain. A different question is when their difference becomes small enough for a declared approximation. The existing full-state capture already implies eventual equality on its admitted source. The result here adds a uniform decay rate and a finite tolerance deadline while the projected environment continues to move. It uses the same complete law; no feedback, noise removal or equality reset is added.
This is a common quadratic argument for a time-varying hidden stiffness. The weak-contact residual theorem provides a related method on a different support. The formed-class maintenance theorem uses a fixed-target potential. Neither its constants nor a fixed-potential derivative can be imported into the present moving hidden subsystem.
Domain, phase collar and capacities¶
Admit the actual fixed winding-one acute lift cell and E<1/100 from
the full fine primitives. Keep positive held capacities, with equal
values on paired children as in section 1. The
source word-domain argument
gives, on this unrestricted acute cell,
where u_e is each edge's deviation from its carrier angle. This follows
from the cycle Taylor bound with cancellation of the carrier's first
variation, together with nonnegative form and matching storage. Thus
every cycle angle has magnitude below alpha+1/5<pi/3, and every
matching angle has magnitude below 1/5. These inequalities concern the
complete actual state, rather than only its projected phase.
The straight lifted segment from a state to its global child exchange has the same bounds: each edge angle interpolates two angles inside the same interval. Each weight in the secant matrix K from section 3 is an average of cosine along this segment. Therefore, throughout the admitted flow,
The shared acute boundary barrier exceeds 1/100, and the actual positive-capacity storage balance is nonpositive. Initial membership therefore retains this domain. Capacities of the other three children may be nonuniform; they affect the moving projected environment but do not enter the paired matrix V. This proof does not differentiate K or require it to commute with H or V.
A common quadratic metric for both hidden rows¶
Define the auxiliary quantity
W is a positive proof metric in the declared normalized coordinates, not the physical storage or a new primitive parameter. It is a linear functional of the diagonal and same-slot mixed entries of G. Keeping this scalar supplies an envelope; section 5 still excludes those local entries as a complete predictive state.
With nu_min=min_j V_jj and nu_max=max_j V_jj, the metric sandwich is
Differentiate using both actual hidden rows, including the reciprocal phase row. The capacity factors cancel exactly:
From 0<=K<=H, congruence gives K H^-1 K<=K; no simultaneous
diagonalization is used. Young's inequality then gives
Since 0<gamma<1/2, the remaining form coefficient
1/4-gamma^2/2 is at least gamma^2/4. Using the two matrix lower
bounds proves
Consequently, for every elapsed structural time tau in the declared law,
The rate is not uniform when paired capacity tends to zero; zero paired capacity is outside this certificate. A common clock rescaling must rescale the rate and elapsed time consistently. The represented norm does not identify form and phase with the same laboratory unit.
Passage through the marked word and prediction scope¶
The separately admitted ideal Reception–Coherence–Silence word at
targets zero and one holds a, b and V exactly. Thus it has Delta W=0
and Delta N=0, including Silence's change of the A-target capacities.
The existing passivity theorem retains the same acute energy domain.
Before and after this word, the same rate applies, without restarting
the bound or requiring the environment to have equilibrated first.
Elapsed tau is the sum of the continuous segments; the ideal jumps
have no invented finite duration. Word admission, history, policy,
selection and any represented map error remain separate premises.
This proves acquisition of any positive declared tolerance, not entry into exact equality at a finite time. A nonzero hidden vector cannot reach zero under its smooth homogeneous differential equation in finite time: uniqueness applied backwards from zero would contradict its nonzero initial state. The marked word also cannot make it zero because it holds that vector. Exact symmetry remains an asymptotic statement.
At one actual state compare its projected field with the fixed-sector field evaluated at the same projected coordinates and capacities. The earlier quadratic sine defect gives
The global capacity maximum is held on a continuous segment; the
admitted Silence factor is at most one, so its initial value also
bounds the segment after this marked word. At a target, the actual
and projected Coherence resultants differ by at most b_j^2.
Both have real part greater than four in the admitted energy domain,
so their argument difference is at most N/4, and their ideal phase
reset difference is at most lambda*N/4 for 0<=lambda<=1.
Reception commutes with this projection, and Silence holds the primary
coordinates. These instantaneous defects inherit the displayed decay.
This section does not follow a separately evolved reduced trajectory. Its accumulated error has the separate coupled stability proof in section 8, under narrower capacity and event premises. Nor does a small N give a uniform late-time energy-guard timing error: that question still needs a strictly positive crossing-loss margin.
Prospectively fixed static controls¶
Both controls fix unit capacities, h=1/128, and zero projected form
and phase offsets. They evaluate algebra and bounds, not a trajectory
or a performed operator event.
First take a=h*e0, b=h*e1. The section 5 mixed-information control
already establishes its acute energy admission; its initial metric
is exactly W(0)=2h^2. For outward gamma bounds g_lo<=gamma<=g_hi,
define the rational certified rate and elapsed time
Since exp(-10)<2^-10 and g_hi<1/2, the certificate implies
Initially N(0)=2h^2; the declared approximation tolerance is therefore
absent initially and guaranteed by this conservative structural deadline.
This finite-tolerance acquisition does not assert formation of a new
NFR identity, spontaneous support selection or a physical timescale.
Second take a=epsilon*h*e0, b=h*e0, with epsilon=1/65536.
Here K_00=2*(1+cos(alpha))*sinc(h), so the complete fine law gives
Indeed cos(alpha)<4/5, sinc(h)<1 and gamma>1/4096 imply
gamma*(4-K_00)>1/10240>4*epsilon. Its excess is
4*epsilon^2*h^2+4*(1+cos(alpha))*(1-cos(h))<1/100, and all fine
edge angles are acute. Meanwhile the proved inequality gives W'<0.
Thus exponential control by W does not assert monotonically decreasing
Euclidean hidden distance. Both rows and their mixed term matter.
Primitive certificate and numerical boundary¶
The relaxation reader rebuilds the canonical fine lift, shared observation, raw offset angles, energy and boundary margins from primitives. It supplies a conditional metric and squared-distance allowance, with explicit unavailability when the geometric premises or exponential work budget are not certified. It admits no event context and performs no integration or reset.
The report-free quadratic kernel keeps this common-metric theorem separate from its existing fixed-potential helpers. The reader uses the existing exact rational negative-exponential evaluator, whose nonnegative exponent is limited to 4096. That limit is a numerical work boundary; it does not limit the analytical theorem. Exact rational scalar bounds preserve nonzero allowances below the trigonometric interval grid rather than materializing them as zero.
Independent controls check the full fine derivative, the metric sandwich, nonuniform capacities, the frozen deadline and transient-growth distinction, event-map invariance and unavailable domains. The result supplies a justified tolerance for discarding this asymmetry in instantaneous predictions under the selected law. It neither establishes a unique canonical constitutive law nor makes the lossy reduced state exactly sufficient at finite time.
For the frozen first control, the rational certificate gives
r>2.4196e-8 and T<4.133e8 in structural tau units, with a joint
squared-norm allowance below 5.544e-9, compared with the declared
h^2/256>2.384e-7. The instantaneous form-field defect is bounded by
2.157e-12 and the target argument defect by 1.386e-9. These are
conservative analytical allowances at the prescribed deadline, not
observed errors or a laboratory settling time.
8. An independently evolved symmetric approximation¶
Small instantaneous defects do not by themselves bound an accumulated
prediction error. Here the complete state z evolves under the same fine
law while a symmetric state w evolves independently from w(0)=Pi z(0).
All capacities are exactly one and remain held; Gamma, external input,
events and changes of support are absent. Time is the same structural
tau in both rows. The initial fine state must lie in section 7's actual
winding-one acute cell with E<1/100.
The prediction is the error between Pi z(tau) and w(tau), including both form and lifted phase. The result is uniform for all nonnegative tau and decays to zero. It strengthens the earlier finite comparison on this narrower invariant domain; it does not replace that comparison's global-sine or first-hit scope. Neither calculation evaluates a trajectory.
Two retained domains and the correct error space¶
In the acute lifted cell the storage is convex and invariant under the whole-child exchange J. Consequently
The projection stays in the same convex lift cell. The initial core
therefore satisfies E(w(0))<=E(z(0))<1/100. Each independent solution
retains its acute sublevel by the actual storage loss and boundary barrier.
The core remains symmetric because the complete law commutes with J.
At every time, Pi z and w both have the section 7 phase collar. The
segment joining their lifted phases stays in that collar, so its sine
secant Hessian obeys L/2 <= K_e(tau) <= L as quadratic forms.
Write p=(Pi x)-w_x and q=(Pi delta)-w_delta in the fine 45-node lift.
At unit capacity both complete rows conserve their ordinary sums, and
Pi preserves those sums. Matched initialization therefore gives
sum(p)=sum(q)=0 for every time. The initial source means may be nonzero.
The actual full-support spectral bound
is p^T Lp >= (4/15)||p||^2 on this mean-zero space, and likewise for q.
The hidden nine-slot gap two is not the gap for these errors.
All norms below use the 45-node lift. Equivalently, the 36-coordinate
norm weights the child orbits by their masses 1,2,1,1. An unweighted
36-coordinate norm does not give the displayed identities.
Actual edge forcing and a forced common metric¶
Orient each edge and let B have +1 at its tail and -1 at its head, so
L=B B^T and the sine field is B sin(theta_head-theta_tail).
The difference between the projected sine field and the sine field at
the projected state is B j. On each of the two copies of a B-child
cycle edge from slot l to k, put
On each of the four matching edges adjacent to the exchanged children
at slot l, use j_e=sin(bar theta_e)[cos(b_l)-1], with its own oriented
projected angle. Other entries vanish. Pairing the two child currents
proves this identity at singleton as well as paired nodes; the linear
form row and reciprocal phase row commute with Pi exactly.
Since |1-cos v|<=v^2/2, the actual edge multiplicities give
For the last inequality, each squared edge difference is at most
2||b||^2, their sum is at most 4||b||^2, and the fourth-power slot
sum is at most ||b||^4. No nodal maximum-degree approximation replaces
this fine edge count. The error obeys both complete rows:
Define W_e=||p||^2+||q||^2+2 gamma p^T q. It lies between
(1-gamma)(||p||^2+||q||^2) and
(1+gamma)(||p||^2+||q||^2). The common-metric argument of section 7,
now on the full mean-zero space, bounds the homogeneous derivative by
-p^T Lp/8-gamma^2 q^T Lq/8. The additional term is
(gamma/2)(p+gamma q)^T B j. Apply Young's inequality separately:
Using 0<gamma<=1/2, the full spectral gap and the metric sandwich yields
Both stiffnesses may change with the moving environment. This proof neither differentiates them nor assumes simultaneous diagonalization.
Uniform accumulated error and its limits¶
For unit capacities let C_0=W_h(0)/(1-gamma), with W_h the hidden
metric of section 7. Then ||b(tau)||^2<=C_0 exp(-r_h tau) and
r_h=15 r_e. Integrating the forced scalar inequality from W_e(0)=0
gives
The bracket is nonnegative, bounded by one and converges to zero. Since C_0 is quadratic in the initial hidden coordinates, the projected error norm has a uniform quadratic dependence on their amplitude. This is a controlled lossy approximation; it does not make the symmetric state exactly sufficient for a nonzero hidden preparation at finite time.
For comparison with the unprojected fine state, orthogonality gives
||z-w||^2=2N+||p||^2+||q||^2. Its hidden contribution has the factor
two from the two copies of each slot. Thus the quadratic projected-norm
claim must not be attributed to the full reconstructed error, whose
leading norm contribution can be linear in hidden amplitude.
Matched initialization is essential: a reference with a different common phase offset has a constant error even at zero hidden state. This result does not extend through the marked word, whose Silence changes capacities and whose resets require a separate error budget. It supplies no guard-time accuracy without crossing transversality, new NFR identity, physical identification or unique selection of the constitutive law.
Detached certificate and prospectively fixed controls¶
The tracking reader
admits all primitives and unit capacities, rebuilds the section 7 domain
at time zero, and constructs the matched reference with the existing exact
quotient lift. No supplied report or proposed reference replaces these
premises. From outward g_lo<=gamma<=g_hi, use
The evaluated metric allowance is
F integral_0^T exp[-r(T-s)] exp[-30r s] ds; the uniform allowance is
F/(29r). Divide each by 1-g_hi for the lifted joint squared error.
The shared convolution kernel
uses the smaller of a simple positive bound and the exponential difference
quotient, keeping exact zero at T=0 and avoiding small-time cancellation.
The existing cycle relaxation consumer delegates to that same kernel.
The shared negative-exponential evaluator admits each argument up to
4096; this is a numerical work cap, not a bound on the analytical time
domain. If only this cap fails, the uniform bound remains available and
the requested pointwise evaluation is explicitly unavailable.
The prospective preparation fixes unit capacities, h=1/128, zero
projected form and offsets, a=h e0, and b=h e1. Its reduced reference
is the carrier equilibrium. At time zero the comparison error is exactly
zero, yet its projected form-rate defect at child-one slot zero is
(gamma/4) sin(alpha) [cos(h)-1]<0. The full and reduced evolutions
therefore need not agree even from matched projected initialization.
The frozen acceptance thresholds are a uniform joint squared-error
allowance below h^2/128 and, at T=10/r, a pointwise allowance below
h^2/2^20. A companion preparation with h/2 must reduce both allowances
by exactly 1/16. The time and thresholds were fixed before evaluation.
Primitive and numerical controls, independent edge and forced-metric controls and convolution controls test these contracts, including means, raw lifts, tiny rational scaling and unavailable domains. They evaluate static derivatives and analytical allowances, not independent trajectories, measured errors or solver speed.
The frozen certificate gives a uniform joint squared-error allowance below
3.087e-7, against h^2/128>4.768e-7. Its certified rate exceeds
1.6131e-9, and the prescribed structural time is below 6.200e9.
At that time the allowance is below 1.402e-11, against
h^2/2^20>5.820e-11. The companion has exactly 1/16 of both allowances;
the initial projected form-rate defect is strictly negative. These numbers
are conservative proof bounds in structural units, not measured settling
times. No historical continuation supplies evidence for this comparison.