Relational coefficient and capacity identification¶
Spectral and finite-amplitude discrimination, temporal coefficient acquisition and capacity-intervention records with their separate calibration and observation scopes.
Part of Native relational exchange admission. Section numbers are stable across the collection; hypotheses and model changes remain local to each result.
5. A prospective spectral discriminator¶
Take common positive capacity nu, zero form and phase consensus. Here
H=pi D, Dg=-L/pi and Bx=0. A nonzero Laplacian mode with eigenvalue
lambda has the exact linearization
All nonzero spatial modes share the same damped/critical/oscillatory class:
the discriminant has the sign of e^2-4w^2/(beta*pi^2). Their rates scale
linearly with lambda. For the existing regular-support cotangent model,
the determinant instead scales with lambda at held degree, not lambda^2.
Two distinct nonzero modes can therefore distinguish these complete laws
without tuning a coefficient to a trajectory or repeating the C8 experiment.
For example, the C6 eigenvalues 1/2 and 3/2 give determinant ratio 9
for this exchange premise and 3 for the cotangent premise. These ratios
cancel the respective unknown storage/exchange scale and common clock.
This is a conditional linear prediction. Section 9 establishes a separate
finite-amplitude projected identity and a prepared-state implementation
control; neither identifies physical matter.
Coefficient audit: selection, covariance and identifiable response¶
The dimensions and unit changes in section 4 give the invariant
chi=beta*(e/w)^2. Here chi names only this coefficient combination, not a
field diagnostic. It characterizes the two-channel family, not the complete
four-channel pressure, an arbitrary phase closure or a physical constant.
For a fixed positive reference capacity nu_ref, set
u=x/sqrt(beta), tau=nu_ref*w*t/sqrt(beta) and R=N/nu_ref. The full
nonlinear rows, on the same regular phase chart, become
Thus one dimensionless coefficient remains after these form/time scale choices
within this family. Support, relative capacities, initial u, phase geometry
and the observation remain independent data. This does not reduce four pressure
weights to one parameter or identify a physical clock.
Balance and composition leave a family. The section 3 loss identity holds
for every e>=0, w,beta>0. Capacity separability/additivity in section 11
selects the phase-row form for each such choice, not its numerical coefficient.
For instance, e=w=1/2 with either beta=1 or beta=1/4 satisfies the same
admission premises, while chi=1 and chi=1/4 have different consensus pole
classes. At the same state their form rates agree and their phase rates differ
by four. Complete synchronized replicas in section 4 preserve beta rather than
selecting it. These are countermodels to selection by the stated premises,
not a proof that an additional independently justified premise cannot select.
A clock cannot choose the invariant. Section 4's normalized representation
e_new=e/k, w_new=a*w/k, beta_new=a^2*beta, N_new=k*N/b, with k=e+a*w,
leaves chi unchanged. Omitting the capacity compensation is a different law.
A regular positive state-dependent clock change scales the Jacobian at an
equilibrium by a common positive factor; it cannot remove its pole class.
Away from equilibrium every row and memory argument must transform according
to the whole-law clock contract.
The numeric 4/pi^2 boundary uses the declared normalization g=Arg(Z)/pi.
For any constant r>0, re-express the same unnormalized mathematical law as
g_new=r*g, H_new=H/r, w_new=w/r. Both rows and storage are unchanged,
but chi_new=r^2*chi and its numeric boundary is 4*r^2/pi^2. The squared
damping ratio zeta^2=pi^2*chi/4 in the original normalization is unchanged
when expressed consistently in the new one. Circular phase does not select an
absolute coefficient from that normalization. No engine convention is changed.
Capacity dependence does not supply a hidden selection rule. If the same
two rows are composed with a smooth positive function N(x,theta) on fixed
support, then at a joint equilibrium Bx=g=0 all capacity-derivative terms
multiply zero. Its nodal Jacobian is the held-capacity one evaluated at N_*.
For an independent law nu_dot=A(x,theta,nu) at a full equilibrium the full
Jacobian has blocks [[J_*,0],[A_(x,theta),A_nu]]. Capacity modes may affect
stability, but no equation fixing chi follows from this block structure.
Since storage has no capacity argument, its instantaneous loss identity also
holds along such a composition while capacities are nonnegative. Positivity,
continuation and long-time recovery require their own capacity-law hypotheses.
The executor still holds capacity; no continuous capacity law is installed.
Restoring the capacity-pressure channel changes these premises. At a state
with g!=0, even uniform form does not cancel the capacity-derivative terms.
A common capacity factor nu(t)>0 instead changes accumulated activity
s(t)=integral_0^t nu(u) du. It does not choose chi, and finite total activity
need not realize the entire constant-capacity asymptotic response. This reuses
the capacity-clock boundary,
not an assertion that arbitrary heterogeneous capacity changes are a clock.
Variational structure is a distinct premise. Power cancellation permits
the skew tensor (w/beta)*[[0,-NH^-1],[NH^-1,0]], not automatically a Poisson
tensor. Already on P2, for delta=theta_1-theta_0 and
h=pi*sinc(delta), its Jacobiator on (x_1,x_0,theta_0) is
-(w/beta)^2*nu_0*nu_1*h'(delta)/h(delta)^3, nonzero for
0<delta<pi/2 and positive capacities. Choosing a positive beta cannot repair
that defect. The Jacobi and cotangent owner
retains alternative geometric premises; they cannot select a coefficient of
this unchanged tensor merely by being called variational. Selecting critical
damping as an optimum likewise supplies an objective and constraints, rather
than deriving a law from storage balance.
The same invariant in hidden-phase memory and temporal poles¶
Return to the fixed tangent at uniform form and phase consensus. Allow
heterogeneous strictly positive held capacities and put T=ND^-1 B.
With M=ND^-1, T=M B is similar to sqrt(M)*B*sqrt(M), so its nonzero
eigenvalues mu are positive. The complete tangent is
For common capacity, mu=nu*lambda; the spatial eigenvalue does not multiply
the P2 block a second time. At other phase equilibria the actual Hessian and
phase metric are needed. Section 15's uniform cycle twist gives the boundary
chi*cos(kappa)=4/pi^2, not the consensus value at every winding.
Eliminate only the phase perturbations, keeping all form coordinates and their
initial hidden phase. With k=w^2/(beta*pi^2), exact elimination gives
The second-order equation inherits
xdot(0)=-eT*x(0)-(w/pi)*T*theta_0; arbitrary initial velocity would add
solutions not belonging to the original state, including uniform-form drift.
The hidden block is zero, so its kernel is constant K(t)=-kT^2, not a decaying
exponential. In the metric M^-1, M^-1 K=-k B M B is negative semidefinite;
this is not an entrywise sign assertion. Stable joint dynamics can therefore
have nondecaying projected memory. No reservoir or forcing is added by this
exact rewrite. Writing A_x=-eT exposes the common coefficient identity
The coordinate-memory owner already derives these blocks and the hidden-initial-state source. Only this phase-only elimination of the fixed consensus tangent has the constant kernel; eliminating fine nodes, nonlinear evolution or varying capacities generally gives a different kernel and retains its own closure obligations.
For one resolved nonzero spatial mode, the two temporal poles satisfy
sigma_1+sigma_2=-e*mu and sigma_1*sigma_2=k*mu^2. Consequently
This is a conditional identification formula: spatial rate and a constant clock scale cancel. Both poles must belong to the same mode and be present in the observation and preparation. An eigenline preparation can excite only one pole despite a two-dimensional all-state observation; mixing poles from different spatial modes can give a false value, or accidental agreement without valid identification. Critical repeated poles require their generalized response to be resolved. Common modes, partial observation, noise, finite sampling and time-varying clocks need separate admission. Even an identified chi neither determines every physical scale nor selects one universal microscopic coefficient.
Real poles do not imply monotone form. On the P2 consensus tangent with
e=w=1/2, beta=nu=1, the roots sigma_+>sigma_- are both negative.
Initial form contrast u_0>0 and zero phase contrast give
Its slow residue is negative; the signal crosses zero and approaches zero from below. A sum of pure-diffusion modes can also be nonmonotone at one node, although each diffusion mode has a single real decay rate. Pole class, prepared response and nonlinear trajectory are different statements.
The coefficient audit controls reuse the native field/tangent and exact observation/memory owners. Rational references use represented pi explicitly; numerical comparisons do not certify mathematical pi or a continuous trajectory. The existing symbolic exchange, replica and Jacobi controls retain their independent proofs. Separate calibration could estimate identifiable combinations under an admitted measurement model; only subsequently frozen, reserved responses would test them. This audit supplies no physical dataset, fit, selected coefficient or new runtime law.
Full-state consensus derivative and a bounded linear response¶
The consensus derivative does not require uniform initial form. Retain the
same fixed connected unit support, strictly positive held capacities and
unforced native law. Write L for the Laplacian denoted B above, and put
K=ND^-1, a=w/pi, b=w/(beta*pi). At any state with all phases equal,
the relative resultant is z_i=d_i. For a phase variation v,
Dz_i[v]=i*sum_j(v_j-v_i) is purely imaginary. Therefore
Here H_i=pi*|z_i|*sinc(Arg z_i): the modulus derivative vanishes at a
positive real resultant, and sinc'(0)=0. Consequently the phase derivative
of H^-1 Lx vanishes even when Lx!=0. The actual full Jacobian at any such
phase-consensus source is exactly
This extends the derivative's admission, not its time interval of validity. A nonuniform-form source is not an equilibrium. Once its phases separate, the native metric and argument derivative generally change, and the full Jacobian must include their new state dependence.
For the constant linear comparison generated by J_0, use
xi=K^(-1/2)x, eta=K^(-1/2)theta after removing the common origins, and
The matrix mathcal B is positive definite on mathcal Q, and each of its
eigenvalues lambda>0 has temporal roots
In particular, e=w=1/2,beta=1 gives
e^2-4ab=1/4-1/pi^2>0: every relative mode has distinct negative real
poles on any such support and with any strictly positive held capacities.
Neither an isolated-ring approximation nor a discarded hidden mode is used.
This does not exclude the separately defined work-port frequency maximum,
and it does not classify finite nonlinear formation.
There is also a full-form transient bound, rather than only a pole test.
Assume e^2>4ab and prepare arbitrary relative form xi_0 with eta_0=0.
Define
The exact linear comparison satisfies eta(t)=h(t*mathcal B)xi_0.
For s>=0, h(s)>=0, with one positive maximum at
s_*=log(r_+/r_-)/delta. To bound it without fitting a response, consider
the scalar system u'=-e*u-a*v, v'=b*u, u(0)=1,v(0)=0.
It has v=h, and u(s_*)=0. Integrating its first row to that time gives
The spectral theorem, including commutation with mathcal B, then yields
This is the quadratic phase storage of the constant linear comparison, not
the native nonlinear cosine storage along its actual trajectory. For the
current beta=chi=1 and F_0<=E_ref<9, every original lifted edge gap obeys
Thus no arbitrary form preparation in this budget creates a winding in the constant consensus linear comparison. The zero source is stationary and is included by the nonstrict storage inequality. This statement neither extends the copied-reflected nonlinear obstruction to every full form nor shows that the actual native phase gaps satisfy the linear bound. A full-state nonlinear estimate must control the evolving argument pressure and reciprocal phase metric before using that conclusion. The separate full nonlinear bound now excludes the same maintained target through actual pressure work and loss. It neither imports this tangent bound nor proves acute confinement of the nonlinear trajectory. Searching unobserved linear modes is unnecessary for that completed target-exclusion question.
The shared native derivative owner
exposes evaluate_relational_consensus_tangent for the actual initial
phase-consensus Jacobian. That detached observation does not integrate the
constant comparison or certify a nonlinear remainder. The
observation contract
separates its admitted source from the evolving nonlinear field. The
resonance foundation supplies
the compatible full-network spectral framework and its input/output limits.
A prepared mode identifies chi from its initial response jet¶
This specifies a sufficient preparation/observation for the preceding
identification relation. Keep the fixed connected simple reciprocal unit
support, common held capacity nu>0, fixed e>=0, w,beta>0, and the
unforced two-channel law. Choose a real nonzero mode from the support alone,
Lv=lambda*v, lambda>0, and prepare
No coefficient value is needed to select v. A constant common capacity does
not change its spatial shape. Declare the observation y=ell^T x with
ell=Dv/(v^TDv). Reversibility gives ell^T L=lambda*ell^T,
ell^T v=1 and ell^T 1=0. On irregular graphs the unweighted projection
v/(v^Tv) need not remove the common form offset. For example, on P3 choose
v=(1,-1,1), lambda=2 and y=(x_0-2*x_1+x_2)/4.
The full node order, support and phase preparation remain recorded; a scalar
signal alone does not prove those premises.
Put mu=nu*lambda and m_j=y^(j)(0). Consensus gives g=0,
Dg=-L/pi, H=pi D. Differentiating the actual form row once, using the
actual phase row rather than a reconstructed pressure, yields
These initial jets are exact for finite A within the stated model: the form row is linear in x and the source derivative is evaluated at prepared phase consensus. This reuses section 9's instantaneous response identity; it does not assert that the nonlinear trajectory stays in one Fourier mode. Consequently
The ratio cancels amplitude and a constant clock scale. All three jets must
be expressed in the same measurement units: an unknown constant nonzero
gain multiplying y also cancels, but replacing measured m0 by a differently
scaled latent A does not. A residual additive sensor offset does not cancel.
For e>=0, the preparation further requires m_0*m_1<=0; an incompatible
sign cannot be repaired by squaring m1. The e=0 limit gives chi zero without
singular division. A single pure-diffusion mode instead has Delta zero.
No unexcited-pole exception in the tangent preparation. Write the modal
tangent as J_m=[[-a,-b],[c,0]], with a=e*mu, b=w*mu/pi>0 and
c=w*mu/(beta*pi)>0. For z_0=(A,0) and O=(1,0),
The initial direction excites, and the form observation retains, both tangent state dimensions. This includes the critical repeated pole and its generalized response. The general cancellation counterexamples still apply to different preparations or observations. A nonzero initial phase component changes m1, so the ratio cannot silently assume that omitted phase was zero. The phase is prepared once, not held fixed during the subsequent joint evolution.
Clock boundary. If tau=T(t), alpha=T'>0, then in that clock
An affine time conversion cancels; a general changing clock can bias the ratio. This is a measurement-clock condition, not a new capacity law.
Finite samples require a declared error budget and abstention¶
Suppose the noiseless observed y is C3 on one regular interval [0,2h],
h>0, and an independently justified bound abs(y''')<=M_3 holds there.
Uniform samples z_j=y(j*h)+eta_j, abs(eta_j)<=epsilon, give estimators
Taylor's integral remainder gives one-signed kernels with absolute integrals
h^2/3 and h, respectively; the noise constants are the absolute sums of
the stencil weights. These are bounds for the full noiseless observation,
not automatically for a tangent approximation or an Euler sample sequence.
M3, error bounds, timing accuracy and the regular horizon must be justified
in the measurement units before using the bound. Three samples cannot infer
their own third-derivative ceiling. Numerical solver error, if relevant,
needs its own enclosure; a work residual alone is insufficient.
Use I0=[z_0-epsilon,z_0+epsilon] and the two derivative-error intervals
I1,I2. With outward interval arithmetic set S1=I1^2 and
D1=S1-I0*I2. A sufficient arithmetic admission requires 0 notin I0,
inf(D1)>0 and no strictly positive I0*I1 interval. Then
Shared samples create interval dependence and can widen this enclosure; they
do not justify treating correlated quantities as measured independently.
If signal or restoring gap is unresolved, no point estimate is certified.
Denominator positivity also does not imply useful precision: freeze a maximum
permitted width separately. An interval crossing 4/pi^2 does not establish
criticality. At fixed noise, decreasing h increases derivative noise and can
turn an available bound into abstention.
The shared observer
bound_relational_coefficient_from_jet accepts declared jet intervals and
uses the existing outward dyadic and mathematical-pi enclosure. Its immutable
report retains the widened inputs, restoring gap, coefficient bounds or
unavailable reasons. It does not estimate derivatives, inspect a graph, or
authenticate phase preparation, modal isolation, timing or physical data.
Its API contract
and shared JSON exporter keep that boundary explicit.
Native P2/P3 controls use the shared pressure-derivative owner with the actual
phase rates; symbolic controls check excitation, criticality and Taylor-error
constants. A separate synthetic C3 sample control exercises uncertainty, not a
physical acquisition or a TNFR trajectory. A local mixture
3*exp(-t)-exp(-2t) has jets (2,-1,-1) and positive Delta despite containing
only diffusion modes: passing arithmetic cannot establish modal provenance.
The research controls
retain this and the hidden-phase, irregular-projection and nonaffine-clock
counterexamples; API controls
cover malformed intervals, unresolved data and exact export.
This completes a conditional preparation/observation admission, not a selection of chi or the nonlinear law: sections 10–12 allow different completions with the same consensus jets. Physical use still needs the independent observation, preparation, clock and uncertainty bridge. Calibration may estimate chi; a separately reserved response under frozen conditions must test the result.
A bounded computational temporal acquisition on P2¶
This admits one known-source software acquisition, not physical data,
blind calibration or a comparison selecting the constitutive law. Use the
unchanged unit-P2 model with nu=1, e=w=1/2, held support/capacity and no
Gamma, events or clipping. Before evaluating the response, declare the prior
beta in [1/2,2], source beta=1, preparation x=(1/16,-1/16),
theta=(0,0), and observation u=x_0-x_1. Its gain is one and baseline zero
by construction; this is twice the dual-mode amplitude, an admissible constant
gain in the preceding identification. Primitive phase is prepared at consensus
only initially. The clock is independent structural time, not inferred phase.
Write phi=theta_0-theta_1, the opposite orientation to the P2 pulse subsection
below. The exact nonlinear contrast dynamics, including the removable limit,
are
These are not tangent rows. The ideal arithmetic means are constant on P2;
the numerical acquisition retains both nodal coordinates without imposing
mean conservation or reflection. The error proof concerns the two contrasts,
not individual-node or common-mode errors. Use a fixed normalized form chart,
radians and the declared structural clock. The sup norm on (u,phi) below
uses those fixed coordinate scales, not an untyped sum of physical units.
Whole-window bounds before sampling. Set U=1/8, r=1/256,
h=2^-10, and H=2h=1/512. On abs(phi)<=r, sinc's integral representation
gives sinc(phi)>=1-r^2/6 and abs(sinc'(phi))<=r/3. Thus
Using mathematical pi>3 and the prior, define rational upper bounds
Until a first phase exit, nonincreasing storage gives abs(u)<=U, then
abs(phi_dot)<=R. Since H*R=64/393215<r, that exit cannot occur before H.
The smooth field continues throughout the window on this compact regular
rectangle. Differentiation gives abs(u_ddot)<=U2,
abs(phi_ddot)<=Phi2, and u'''=-u''-phi''/pi, establishing the displayed
M2 and M3 for the full nonlinear trajectory uniformly over the prior.
The field is Lipschitz on the convex rectangle with
L=max(4/3,(2/3)*(q0+U*q1))<2. No measured sample is used to fit these bounds.
Native acquisition and continuous error. The
instrument advances
only through step_relational_exchange, with s=2^-21, 4096 steps and
sample indices (0,2048,4096). Every actual numerical contrast point must
remain in the same rectangle. Every actual graph/report clock must equal
its exact dyadic grid value; a clock defect rejects the acquisition.
Native pressure and both rows are refreshed at every step. The predeclared
precision gate is an available chi interval of width at most 1/8 containing
the source truth 1. No refinement or retuning follows an evaluated failure.
At each numerical point z_k, bound the ideal field F(z_k) independently
with the shared mathematical-pi interval and the alternating-series sandwich
This is positive and zero-safe on the rectangle. Form both state and rate
contrasts by exact subtraction of rationalized nodal values, not by
rationalizing an already rounded difference. If the actual contrast rate is
fhat_k, retain
The first bound includes native transcendental, pressure and rate arithmetic;
the second is the actual numerical update defect, not physical Gamma. For
e_k=||z_k-z(k*s)||_infinity, Taylor's integral remainder and Lipschitzness give
The initial represented preparation is exact, hence e_0=0. Summing and taking
a prefix maximum, using k*s<=H<1/2, yields the conservative enclosure
The separately bootstrapped ideal trajectory and every checked numerical
point lie in the same convex rectangle, so the comparison needs no assumed
error tube. It neither treats a small work residual as a solver bound nor
uses empirical grid convergence. Set epsilon=max(E_0,E_2048,E_4096) and
use the preceding three-sample theorem with M3. The shared
bound_relational_coefficient_from_samples retains exact stencils and error
terms and delegates coefficient arithmetic to the existing jet observer.
Evidence boundary. The truth-side field residual uses the predeclared
source beta; the estimator receives samples and their justified error budget,
not beta as its answer. This is not parameter-blind experimental uncertainty.
The derivative bound uses the full prior, not the estimated value. A separate
CLI invocation freezes the protocol and an archive of all project Python
sources under src/tnfr, the producer and pyproject.toml before execution.
Imported TNFR modules must resolve to that checkout. All accepted states,
rates, clock checks, separate error contributions and the original verdict
are retained; an analysis failure does not discard the acquired frames.
Unresolved bounds give unavailable conclusions, not a passing precision flag.
The controls check nonlinear derivative identities, independent high-precision interval references, short-step wiring and protocol/source rejection. The shared read-only auditor binds the embedded protocol to the frozen original before reconstructing all saved errors and the verdict. Its admission controls separate record consistency, original success and unavailable evidence without invoking the producer or evolving a graph. The API contract owns the supported record and numerical work limits. Source hashes do not authenticate chronology, physical provenance, or installed dependency binaries. No claim of universal chi or laboratory identification follows from this computational admission.
First retained evaluation, 2026-10-02. The separate prepare and evaluate
invocations produced artifacts/research/relational_coefficient_acquisition/
with result.protocol.json, result.source.zip and result.json. The source
archive contains 608 project files. All 4096 steps completed on the frozen
grid, stayed inside the contrast rectangle and used fused_canonical native
pressure. No source, preparation, budget or precision rule was revised after
the response. The exact rational chi enclosure lies inside the outward
decimal display
Its width is approximately 0.0748844172, below the predeclared 1/8, and
contains the known value 1. This is a deterministic conditional enclosure,
not a statistical confidence interval. Approximate recorded error quantities
in the declared coordinates and clock are:
| Quantity | Recorded value, rounded for display |
|---|---|
| Uniform sample-error ceiling epsilon | 7.2044724626e-11 |
| Prior whole-window third-form-derivative ceiling M3 | 0.1821501469 |
| Initial-rate error ceiling | 3.5299916433e-7 |
| Initial-acceleration error ceiling | 4.8005847948e-4 |
| Accumulated field-evaluation contribution before growth factor | 8.7884015660e-21 |
| Accumulated update-rounding contribution before growth factor | 1.4240542810e-14 |
| Accumulated Euler-truncation contribution before growth factor | 7.1749059369e-11 |
The record retains the exact rational quantities rather than these rounded displays. The read-only audit reconstructed every step's contrast defects, accumulated bounds, three sample stencils and original verdict without regenerating the trajectory. Independent ideal initial-jet values also lie inside the recorded jet bounds. SHA-256 identifiers are:
- Protocol:
02cbaa7de7ebb9a6e10ef71a0095b7de72fbdb441eded286ed6b9de2cdfbdd55. - Response:
68e3e8f99c78a7d8f3ec2d5d6733e51f324d7976c1bf8bd9f7d9a9121aa59879. - Source archive:
6a0213a8c0ec43aa7811d69d6d56576ea30d8d09d8deeb9fa1abf59c8b630ee4.
This closes the computational temporal-acquisition gate. It does not test which same-jet nonlinear completion is correct, nor provide a physical preparation or parameter-blind sample-error model. The local artifact audit is unavailable when these retained files are absent; it never substitutes a new producer run. Subsequent calibration and reserved discrimination remain separate obligations in the sole execution plan.
Pulse, recurrence and the nondissipative boundary¶
Use the complete law of sections 1–3 on fixed connected unit support, with
held strictly positive capacities, w,beta>0 and a regular chamber where
every H_i>0. There are no inputs, events, clipping or extra common phase
velocity. These hypotheses distinguish this model from a generic runtime
or the auxiliary graph-wave spectrum reported by Network.rhythm().
Positive dissipation excludes a nonstationary full-state cycle. Suppose
e>0 and a solution returns after T>0 modulo common form and phase offsets:
x(T)=x(0)+a*1, theta(T)=theta(0)+b*1 on the phase circles. Joint storage
is invariant under both offsets. Its exact balance therefore gives
Continuity and positive capacities force q=0 throughout. Connectivity
then gives x=c(t)*1, while the selected phase row gives theta_dot=0.
The form row requires c'=w*nu_i*g_i at every node. Multiplying by
H_i/nu_i and summing yields c'*sum(H_i/nu_i)=w*sum(H_i*g_i)=0 because
Hg=-grad(V_phi) and common phase rotation leaves V_phi unchanged.
Thus c'=0 and g=0: the entire solution is stationary. This excludes
nonstationary periodic and relative-periodic full-state solutions inside
the admitted chamber. It does not establish global attraction or classify
every lossy observable; the existing local recovery theorem
supplies its separate neighborhood and convergence hypotheses.
The selected phase representative also obeys, at every admitted state,
Consequently a common rigid phase velocity theta_dot=Omega*1 requires
Omega=0, even when e=0. This does not freeze the arithmetic mean during
general nonuniform phase motion. Adding a common phase velocity would be a
different full-state law, although storage alone would not detect it.
P2 admits a precise oscillation comparison. With common capacity nu>0,
write u=x_0-x_1, delta=theta_1-theta_0, h(delta)=pi*sinc(delta).
On |delta|<pi/2, both arithmetic means are constant and the exact rows are
At consensus the roots are
-e*nu +/- nu*sqrt(e^2-4*w^2/(beta*pi^2)), the lambda=2 case above.
Oscillatory roots at e>0 describe damped small perturbations, not a
maintained pulse. In particular, e=w=1/2, beta=1 is overdamped on P2;
the cycle-mode calculation
explains why the same coefficients can give damped oscillations around a
nonuniform winding geometry.
At e=0, every level 0<E<beta is a compact smooth closed curve wholly
inside the acute chamber. The vector field has no zero on that curve, so
it gives a genuine nonlinear periodic form/phase exchange. Its period is
The turning-point singularity is integrable, and
T(E) -> pi^2*sqrt(beta)/(w*nu) as E -> 0. Conservation of storage
prevents attraction to one such orbit from an open neighborhood of other
energy levels; amplitude, relative timing and an attracting synchronization
law are not selected. This consolidates the exact P2 sector equivalence in
section 8 and the
existing reversible cotangent sector,
rather than introducing a new oscillator mechanism. The engine already
admits epi_weight=0, with effective normalized phase_weight=1; that
constitutive boundary is not inferred from a desire for sustained pulses.
These continuous statements neither certify finite Euler periodicity nor
exclude pulses in other justified TNFR completions, driven regions or
declared hybrid systems.
9. Exact finite-amplitude discriminator and prepared-state control¶
Use one unit regular cycle of degree d=2, unit held capacity, no forcing or
events, and the same fixed e,w>0. Prepare each mode separately:
x(0)=a*v, a!=0, L*v=lambda*v, 1^T*v=0, and primitive phase consensus.
For the cotangent comparison use its complete general-scale law
At consensus, H=pi*d*I, DH=0, K=0, Dg=-L/pi, and both form
velocities are -e*lambda*a*v. Let A=v^T*x/(v^T*v). Although differentiating
the cotangent correction can give a nonzero cubic vector, its projection is
exactly zero: v^T*dot(K)*x=a*v^T*dot(K)*v=0. Consequently
These are exact instantaneous finite-amplitude identities, with no
small-amplitude remainder. On C6 choose (lambda_low,lambda_high)=(1/2,3/2).
The phase-gain ratio is 3 versus 1; the restoring ratio is 9 versus
3. The respective unknown storage scale and a common clock cancel.
Under a clock conversion, the diffusion rate subtracted in Q must transform
too. Different per-mode clocks or refitted coefficients are not admitted.
Boundary controls. This does not make a nonlinear Fourier mode invariant.
For v=(1,1/2,-1/2,-1,-1/2,1/2) and eta=1, the cotangent contribution
dot(K)*x is a^3*(-1,1,-1,1,-1,1)/(16*pi^2): a nonzero orthogonal harmonic.
Adding a common form offset mu instead changes its projected acceleration
divided by a by -5*mu^2/(24*pi^2). A mixed preparation can transfer
connection work between the observed modes. The exact controls differentiate
the complete connection before projection and retain these exclusions.
Finite constitutive probe, distinct from temporal evolution. Initialize two nearby states on the tangent line of each declared law,
Keep their lifted phase range below pi/2. On this degree-two acute chart,
the neighbor resultant has the neighbor midpoint angle, hence g=-L*theta/pi.
At both endpoints x remains collinear with v, so the additional cotangent
v^T*K*x vanishes. Therefore, writing f for the complete model form field,
This finite prepared-state identity has no time-discretization remainder.
The endpoints are not evolved trajectory samples. The two candidates
supply their own phase directions; the same native pressure implementation
evaluates the projected responses. This verifies constitutive implementation
consistency, not an independent experiment selecting an autonomous law.
The full cotangent vector is not computed by dropping Kx; only its proven
zero projection is omitted.
The maintained producer
fixes a=1/16, h=1/8, e=w=1/2, eta=1, beta=1/2, and integer modes
(2,1,-1,-2,-1,1) and (2,-1,-1,2,-1,-1). The scales match the low mode
a priori through beta=eta*d*w*lambda_low; neither is refitted on the high
mode. Both preparations also lie inside their respective conditional
regular-domain storage bounds. The probe phases stay in their separately
checked acute chart.
Before pressure execution, preparation freezes every materialized state,
exact-rational midpoint references using represented binary64 pi, source
fingerprints, runtime versions and a restoring allowance epsilon=1e-10.
Given reference restoring values q_l>epsilon,q_h, its decision interval is
[(q_h-epsilon)/(q_l+epsilon),(q_h+epsilon)/(q_l-epsilon)]. Both prospective
intervals exclude the fixed separator 6. Evaluation records actual native
pressure and the propagated bound
and rejects if this exceeds the frozen allowance. Fraction arithmetic makes this an exact defect calculation on materialized rates. It is not an interval certificate for transcendental real arithmetic or a temporal solver estimate. The independently derived symbolic identities and represented-input checks have distinct roles. No random input is used.
Retained first evaluation (2026-09-26). The
prediction
was written before the separate
response.
Observed ratios were 9.000000000000036 (relational) and
3.000000000000035 (cotangent). Their largest propagated restoring defects
were respectively 5.63e-17 and 2.76e-16, below the predeclared 1e-10.
All eight probes used the checked fused_canonical NumPy path, binary64,
and the retained node order; there were no random inputs or temporal steps.
The JSON records contain exact represented rates, fractions and fingerprints.
Their SHA-256 hashes are, respectively,
34fa669cb5a1c494f5df3e38d71d1d6670fd3e02f31c3511143e1a6186591486
and 1246771b7d0f6cd4ec78251f4ecdc31b027fe7aa805a1d94b9fb439c2029dd48.
The producer fingerprint is
ea19b78874c2dc8cac9a9b0869417fcc5bec0a3aa9a6b942ed17b99f649c50e7.
The listed file hashes are not a complete transitive dependency archive.
Controls: focused comparison tests cover full projected acceleration, harmonic leakage, excluded preparations, finite tangent-line algebra, actual pressure and pre-execution protocol rejection. The benchmark guide owns invocation and immutable-record handling.
A scale-independent reserved capacity test¶
K3 relative-rate protocol v1 (2026-10-03). This section specifies the
conditional analytic prediction and uncertainty budget; the separate
bounded software response tests
that protocol. Compare exactly the two complete laws above: the native form row with phase row
F_rel, versus the same form row with F_rel+Z. Both hold the unit K3 support
and capacities fixed, use the same e>=0, w>0, beta>0 and structural clock,
have no forcing or events, and evolve both form and phase in the regular chart.
The capacity change defines two matched initial preparations, not two
successive times on a freely evolving trajectory or an installed capacity law.
Use the exact form/phase witness above, with capacities (1,1,1) and
(1,1/2,1), respectively. No nodal readout is replaced by tetrad telemetry.
The prepared geometry gives
Write
For the relative initial phase rate R=F_0-F_2, the exact predictions are
| Candidate | Before preparation | After preparation |
|---|---|---|
| Capacity-separable | k*alpha |
k*alpha |
| Named capacity-mediated alternative | k*(alpha+c) |
k*(alpha+2*c/3) |
Normalize each candidate by its own before rate:
Both denominators are positive. An unknown common nonzero observation gain
and positive constant affine-clock scale multiply both rates and cancel,
as does k. Relative phase removes a common angular reference. Distinct
gains/clock rates between preparations, additive rate offsets or a changed
preparation do not cancel. A derivative readout must use a consistent local
phase lift; differencing wrapped angles across a branch cut is not admitted.
The coordinates and intervention are prescribed in the same form/capacity
units for both laws; normalization does not authenticate that preparation.
This removes the need to identify k or tighten the previous chi estimate
for this discriminator. Coefficient calibration remains a separate result:
its uncertainty is not set to zero, but neither chi nor k is consumed by
this normalized prediction. The known-source P2 record is therefore not
reused as blind calibration. No response is used to fit either law.
Prospective error admission. Choose the common readout orientation so
the ideal before rate B is positive. Supply measured rate estimates m_b,
m_a and independently justified total errors epsilon_b, epsilon_a.
Freeze the following sufficient signal-resolution gate before acquisition:
The errors must enclose the difference from the specified ideal rates, not
just sensor noise. Exact preparation is the present mathematical premise;
otherwise form, phase, capacity, support, timing and gain discrepancies need
their own justified propagation into this budget. Since L<=B, each error
is at most B/200. For the two measured intervals
I_b=[m_b-epsilon_b,m_b+epsilon_b] and
I_a=[m_a-epsilon_a,m_a+epsilon_a], each endpoint can deviate by at most
B/100 from the corresponding true rate: centering on a noisy measurement
and then padding it are two separate contributions.
With eta=1/100, monotonicity of the positive-denominator ratio gives the
exact-arithmetic report envelopes
They are disjoint whenever eta<d/(4-d), which this budget satisfies.
Equivalently, every exact quotient corner differs from its ideal Q by at
most 2/99. Reserve a further arithmetic allowance 1/1584 on each side:
the final report must then lie within Q +/- 1/48, since
2/99+1/1584=1/48. These two enlarged envelopes are still disjoint because
d>1/24. The constants are a declared precision policy for this preparation,
not physical constants or universal TNFR thresholds.
The shared rate-contrast observer
computes I_a/I_b-1, retaining the denominator dependency that naive
(I_a-I_b)/I_b loses. Its outward dyadic bounds do not authenticate errors.
For acquisition, compare the returned endpoints with the exact rational
corner hull of the supplied intervals and require outward inflation at most
1/1584 per side. Mathematical prediction constants also require certified
enclosures. Unresolved denominators or excess arithmetic/measurement error
mean unavailable precision, not a passing observation. Fixed absolute-grid
arithmetic and finite sensor errors do not guarantee resolution as k->0.
Keep both candidate predictions frozen. A report incompatible with one
candidate can exclude it only under the independently admitted premises;
overlap remains inconclusive, and incompatibility with both challenges the
two-candidate/preparation/error conjunction. No nearest-model classification
is forced. A chosen-law simulation can check this algebra, not independently
select the true constitutive law. Only the named alternatives are separated:
the larger family F_rel+gamma*Z has
Q_gamma=-gamma*c/[3*(alpha+gamma*c)], tending to zero as positive gamma
tends to zero. There is no uniform separation from all mediated laws.
Boundary of the closed gate. The specified initial-rate contrast and
error budget are mathematically separated without a new trajectory. Temporal
sampling additionally needs the admission developed below. For a relative-phase observation
y=theta_0-theta_2, the shared three-point first-derivative stencil has error
4*epsilon/h + M3*h^2/3 only with independently justified sample error and
|y'''|<=M3 throughout each window. Both candidates and every retained
parameter value need regular-chart, horizon and derivative bounds. The P2
form-response bound cannot be transferred to this phase response. Unknown
scale cancellation at the initial jet does not bound finite-time derivatives.
No physical bridge, autonomous connection birth or new engine law follows.
The selection controls derive this ratio from the full symbolic fields and check the bounded static uncertainty comparison. The observer controls exercise interval admission, scale/sign covariance, abstention and exact export. Neither set acquires a reserved temporal response.
Prior-uniform temporal admission for the normalized test¶
K3 phase-sampling admission v1 (2026-10-03). The initial-rate test can
be approached by temporal samples without knowing which candidate generated
them or substituting an estimated coefficient into the truncation bound.
Both candidate laws are used to derive the common bound. This is a conditional
acquisition specification; no reserved samples are evaluated here. Supply
e=w=1/2, beta in [3/4,5/4], the two exact preparations above, gain one,
radians and the same structural clock. These are a finite computational
reference prior and coordinate choices, not derived universal coefficients
or an admitted physical measurement map. Both named laws and both capacity
arms must satisfy the same bounds.
Geometry and continuation. On K3, let j,k be the other two nodes and
write
The relative phasor resultant is exactly 2*cos(b_i)*exp(i*a_i). In the
acute lifted chart its argument is a_i, so
This is the full nonlinear metric and pressure source, not a linearization.
Put a closed box of radius r=1/64 in every one of the six normalized
coordinates about the exact initial state. Its edge gaps have magnitude at
most pi/3+2*r<pi/2, and |a_i|<=pi/4+2*r<1. The cosine and sinc factors
are strictly positive. The interval certificate uses mathematical-pi
enclosures, the shared trigonometric bounds and the shared sinc derivative
series, including each derivative's remainder.
For all states in these boxes and all beta in the prior, the
static certificate
encloses all four complete vector fields and proves ||F||_infinity<2/5.
The norm refers to the declared normalized coordinates. Set T=1/32.
Since T*(2/5)=1/80<r, a first-exit argument keeps the ideal trajectories
inside their boxes throughout [0,T]. Smoothness and continuation follow
from the positive metric on that compact chart. This proves the domain
before reading samples; a simulated path staying acute would not suffice.
Third derivative on the whole window. At an arbitrary interval-enclosed
point of each box, recursively evaluate the normalized Taylor coefficients
of the complete field. If z_j encloses z^(j)/j!, the recurrence is
z_(j+1)=[F(sum z_l t^l)]_j/(j+1). Only orders through three are needed.
The recurrence is evaluated on the entire box, not advanced as a trajectory.
The shared interval-Taylor owner bounds the elementary functions and their
derivatives separately. Applied to y=theta_0-theta_2, all four cases certify
This bound covers both alternatives and the whole declared beta prior. It does not reuse the earlier P2 form bound, assume phase consensus or evaluate an error against the known source beta. Static native-field and independent derivative controls check the K3 field representation; the certificate retains exact outward bounds, domain margins and the admission verdict.
Frozen sampling and error budget. Take samples at 0,h,2h with
h=1/64, so 2h=T. Require each supplied sample of the lifted phase
difference to have total error at most epsilon=2^-24 relative to the ideal
prepared trajectory. The initial-rate stencil then gives
Both ideal before rates exceed 1/8 uniformly in the prior. In particular,
k>=2/5 and alpha>1/3 give R_sep>2/15; the mediated baseline is larger.
Consequently even the worst permitted measurement has
m_b-delta>1/8-2*delta>200*delta. The previous operational rate-resolution
gate therefore holds for both arms. The quotient's separate outward
arithmetic allowance remains required. The sample-error ceiling is an
acquisition requirement, not an observed accuracy or a quantity fitted from
the two-law discrepancy.
The shared sample-rate observer implements the stencil and its exact error arithmetic without assuming coefficient identification. The coefficient-sample helper delegates that same calculation; phase samples do not acquire pure-form/consensus premises merely by reusing it. Each rate interval feeds the existing contrast observer.
What remains before evaluation. Actual samples still need independent
error evidence. A numerical implementation must enclose the irrational
initial phase and non-binary form coordinates, evolution error, both phase
coordinate errors and any final subtraction error within epsilon.
The exact 1/3 and pi/6 preparation cannot be treated as exactly binary64.
Uncertainty from different beta trajectories is model uncertainty, not
solver error. A validated interval-Taylor calculation is a possible bounded
software source; it must freeze source, parameter, order/grid and work budget
before response evaluation. Its candidate-specific solution enclosure would
verify that supplied simulator, not establish the unknown physical law.
Real measurements require independent preparation, gain, clock and noise
admission. The mathematical temporal budget is now feasible; no acquired
response, physical identification or connection-birth result follows.
Retained static admission. The local record
artifacts/research/relational_capacity_discriminator/sampling-admission-v1.json
retains the exact certificate, source inventory/digests and interpreter
version. Its sibling sampling-admission-v1.source.zip preserves 611 project
Python/configuration/control files. These files record a static admission,
not a response or a completed numerical sampling experiment:
- Record SHA-256:
c4d0889441e2e970b7f553bd64059c34116153ad75d5413e5da9232bc8e60806. - Source archive SHA-256:
51f057a0044b8d9d4dcfb088f39bfe362916f8e33e8f9b843a686676baf91fce.
The artifact directory is ignored by Git; a fresh clone can recompute the static certificate but cannot audit these retained bytes without the bundle. Source hashes do not authenticate chronology or dependency binaries. Preserve the record; a future response requires its own prospectively frozen source and numerical method, rather than appending an evaluation to this admission.
Bounded software verification of the phase-sampling protocol¶
The sampling instrument
separates preparation from evaluation. It freezes beta=1 as a supplied
software source inside the admitted prior, both named laws, both capacity
arms, the exact six-coordinate preparation, and the existing times
0,1/64,1/32. Gain, clock, epsilon=2^-24 and the common M3=1/2 are
unchanged. This is a verification of those supplied mathematical models and
the observation procedure, not an acquisition from an unknown physical law.
Fixed numerical method. Use a single Taylor expansion from the initial
state, polynomial order six and whole-box remainder order seven. For each
coordinate, the shared flow-jet recurrence gives normalized coefficients
c_j=z^(j)/j!. Mathematical-pi and rational-form intervals enclose the exact
initial point. If C7 encloses the seventh flow coefficient at every point
of the previously certified box, the integral remainder proves
The prior first-exit proof keeps the actual solution inside B. This finite
Taylor theorem needs smoothness on that domain, not convergence of an
infinite power series. There is no stepping, recentering, adaptive order or
grid refinement. Whole-box derivative/remainder admission precedes response
evaluation; initial polynomials are evaluated only after the source/protocol
freeze. The interval field is a verification representation of the complete
candidate, not a new engine law or operator schedule.
Subtract the two enclosed phase coordinates to obtain each y interval;
retain polynomial, signed remainder, final enclosure, its exact rational
midpoint and radius. The radius includes both coordinate errors, initial
representation and outward arithmetic and must not exceed the frozen
2^-24. The midpoint is the sample supplied to the neutral observer.
Converting it to binary64 would need an additional error bound. All six
initial-coefficient rows and remainder coefficients remain in the record,
including unsuccessful or partial cases.
Prospective decision. Analyze the three midpoint samples in each arm
with the common frozen sample-error ceiling and prior-wide C3 bound,
rather than tightening the noise policy using the observed smaller radii.
Then form the normalized contrast with the shared observer. Require all
sample radii within budget, resolved baseline and operational rate-error
gate, outward ratio inflation at most 1/1584 per side against exact
rational corners, containment of the supplied candidate's ideal prediction
enclosure, exclusion of the other candidate, containment inside its declared
Q +/- 1/48 envelope, and disjoint final contrast intervals. These conditions
are software acceptance rules; labels are supplied to the generator and
comparison, not discovered microscopic identities.
The source bundle, protocol and original response are exclusive files. Evaluation first checks current source/runtime and archive bytes against the separate frozen declaration. Any precision failure is retained without changing order, time, law or budget; partial evidence and analysis errors remain distinct. Project-source hashes authenticate bytes, not chronology, dependency binaries or physical provenance. The initial static-admission record remains a separate immutable bundle.
Original retained result, 2026-10-03. The first frozen evaluation completed
all four law/arm cases and passed every declared check without changing the
protocol. The twelve phase-contrast samples have maximum enclosure radius
less than 1.701e-13, below 2^-24; the observer still uses the original
sample-error ceiling and derivative bound. The following decimals are rounded
outward from the exact rational intervals in the retained response:
| Supplied law | Enclosed normalized response Q |
Frozen decision |
|---|---|---|
| Capacity-separable | [-0.000610621, 0.000609731] |
Contains zero; excludes the named alternative |
| Named capacity-mediated | [-0.080862868, -0.079974160] |
Contains the predicted approximately -0.080418444; excludes zero |
Both intervals also lie inside their respective prospective Q +/- 1/48
envelopes. The saved-record audit
reconstructs all twelve polynomial/remainder enclosures, sample radii,
observations and original decision from retained data without executing the
response producer. Static and synthetic controls remain separate from that
audit. These checks verify the specified software experiment; the supplied
source labels do not become an independently selected constitutive law.
The local bundle is under
artifacts/research/relational_capacity_discriminator/:
| File | SHA-256 |
|---|---|
response-v1.protocol.json |
6414312fb044c91712e339deb6f7ce98745ac56aeb9dc973d31a77f618fa8f49 |
response-v1.source.zip |
8c760e41b89283632a47a61bbcffeb80e3efc67f386ea95503680692b04237f4 |
response-v1.json |
6aa12cecc0bddd8eab3fb42072cfd3e5cad6248ee88ba323bbe2b861a918613c |
The source archive contains 610 files. These ignored local artifacts are not included in a clean clone; an external retained-record audit requires the original bundle. Missing files remain unavailable rather than triggering a replay. Preserve the original verdict and the separate static-admission bundle. This closes the bounded sampling verification, not physical identification, uniqueness among all nonlinear completions or primitive connection formation.
14. Finite capacity-intervention response¶
This F4 control evolves the admitted reference and the section 12
countermodel after the same capacity intervention. The latter remains a
benchmark-only comparison, not a new production policy. Both use the native
pressure and the same supplied unit K3, form and phase. Set e=w=1/2,
beta=1; compare held capacities (1,1,1) and (1,1/2,1), with
x=(1/3,0,-1/3) and theta=(0,pi/6,-pi/6) in node order (0,1,2).
The support, initial state, clock and storage scale are independent inputs.
A relative observation tied directly to pressure¶
Use the common-rotation-invariant contrast
Use the common real lift fixed by the supplied preparation and retained by
the admitted Euler segments. On this chart, the exact neighbor-midpoint identity gives
g_0=-chi/pi. Thus the proposed phase observation directly determines the
receiver's phase-pressure contribution; no new diagnostic or fitted mapping
is required. For each law M define the intervention response
R_M[y](t)=y_{M,intervened}(t)-y_{M,baseline}(t) and the difference between
those responses D_y=R_alt[y]-R_rel[y].
The contrast is invariant under common phase rotation. It is not a global
real-valued function of arbitrary independently shifted 2*pi phase
representatives; those representatives require a consistent lift first.
The initial form gradient is Bx=(1,0,-1). Changing only capacity 1 therefore
leaves the entire initial reference phase row unchanged. It does not
leave subsequent trajectories unchanged: node 1 already has nonzero native
form pressure, and its changed form rate feeds back into other nodes later.
In particular R_rel[chi]'(0)=0 is an instantaneous statement, not an
identically zero finite response.
For the alternative, the active pair mobility changes from 1/2 to 1/3.
Its complete correction vector at this preparation is therefore multiplied
by 2/3. Section 12 then gives the exact initial relative response
The two laws have identical initial form rates for each capacity preparation.
Differentiating the native receiver row, with nu_0=1 in both cases, cancels
their equal initial diffusion contribution and gives
Smoothness on the admitted chart implies a first-order phase difference and a second-order form difference for sufficiently small positive time. This derivation identifies how a constitutive capacity assumption reaches the form row through native pressure. It does not specify an error enclosure at an arbitrary finite time or choose which capacity premise is physically true.
The complete receiver identity and what the leading orders mean¶
The connection is valid along each admitted continuous trajectory, not only at its initial state. On the same common acute lift, every receiver row has
The unchanged receiver capacity nu_0=1 and common pressure coefficients are
essential. Taking the same four-way linear difference gives the exact identity
For the equal initial states, integrating the receiver row yields
Thus the phase difference contributes through the existing pressure law,
while neighboring form differences remain part of the response. Omitting
those terms would manufacture an autonomous one-node description that this
comparison has not derived. The sign of D_chi alone does not determine
the form response at every later time. Changing the receiver's own capacity
would add product terms and would require a different intervention formula.
The leading D_chi=O(t) and D_x0=O(t^2) orders describe smooth response
near the initial state. They do not assert a finite propagation delay: both
can be nonzero at every sufficiently small positive time. In an explicit
Euler execution the two laws first separate their phase endpoints, then
their form endpoints at a following step. That discrete ordering is not a
new physical clock or signal-speed law.
At the three retained checkpoints, the same identity can be checked on actual stored state and native rates, with a represented-arithmetic residual. The continuous integral and all intermediate numerical states cannot be recovered from those checkpoints alone. Exact conditional identities, snapshot checks and three-grid trajectory evidence are therefore separate levels of support.
Prospective numerical protocol¶
The producer separates preparation from evaluation. Before executing any trajectory, freeze:
- Structural horizon
T=1/32, with128,256and512Euler steps; all four law/capacity combinations start from the same supplied form and phase. - Signed finite hypotheses
D_chi(T)/(T*D_chi'(0)) in [3/4,5/4]andD_x0(T)/(T^2*D_x0''(0)/2) in [3/4,5/4]. These intervals are prospective numerical hypotheses motivated by the initial Taylor coefficients, not proved Taylor-remainder bounds. - Each trajectory stays inside the selected chart with minimum acute-edge
margin greater than
pi/8; actual represented work residual is at most1e-12in the declared normalization. The cutoff is an arithmetic acceptance policy, not a new structural constant. - Successive full-state endpoint differences and both discriminating signals
decrease under step refinement: the fine-to-middle difference is at most
3/4of the middle-to-coarse difference plus1e-12. Each finest signal exceeds32times its fine-to-middle difference plus the same numerical floor inside that factor. These checks supply empirical numerical separation, not a rigorous ODE error bound. - Source/runtime fingerprints, scalar precision, effective coefficients, native pressure path and preparation order accompany the prediction. Initial, intermediate and final state evidence, phase margins and actual work/Euler defects accompany each execution. The reference uses the shared production step; the comparison reuses its field and Euler arithmetic.
There is no fitting, trajectory-based pressure reconstruction or independent physical measurement in this control. Failure of a sign, interval, chart or refinement condition must remain a failed or unresolved frozen result; it does not authorize adjusting the response afterward. Earlier constitutive and source-relative records remain immutable.
First retained evaluation: 2026-09-27¶
The frozen prediction
preceded all twelve trajectories. The
response record
retains their complete initial, midpoint and final states and per-trace
evidence. All prospective gates passed without changing the inputs or
acceptance rules. Source hashes and Python/NumPy/NetworkX versions are in the
prediction; the observed native pressure path was fused_canonical throughout.
Observation at T=1/32, finest grid |
Difference of intervention responses | Ratio to the leading Taylor prediction | Change from the middle grid |
|---|---|---|---|
Relative phase chi |
-6.004469361879845e-4 |
0.9885036850 |
1.3625854652e-8 |
Receiver form x_0 |
1.4825332026457971e-6 |
0.9814494374 |
2.8041313160e-9 |
The refinement changes approximately halved. Full-state midpoint and endpoint
refinements passed separately for every law/capacity pair. The smallest chart
margin was 0.5207904177 radians, above the predeclared pi/8; the smallest
phase metric was 5.4369609494. Maximum actual represented work residual was
1.377e-16. The report separately retains nonzero Euler storage defects;
that small instantaneous residual is not a bound on time-integration error.
The reference's own finite phase response to the capacity change was
-6.909006713886213e-6, despite its zero initial response slope. This confirms
the need to distinguish local instantaneous dependence from later network
feedback. The alternative's response was -6.073559429018707e-4. Their
difference supplies the first row of the table rather than assuming the
reference response remains zero.
Disposition. This closes the named finite-response F4 obligation in its numerical scope. The two supplied capacity laws produce the predicted distinct phase-to-pressure-to-form responses while retaining the same ideal total storage balance. This demonstrates a discriminating internal model prediction, not evidence preferring one law in nature, a certified ODE enclosure, formation of a new NFR or indefinite persistence. The controls independently check initial identities, a comparison step, fail-closed protocol handling and the retained endpoint/refinement arithmetic without regenerating the campaign.
Here "same balance" means the same functional loss identity evaluated on each law's own state. Once the trajectories differ, their numerical storage and loss values need not be identical. Agreement with that identity therefore cannot replace an intervention-sensitive response test.
Retained-evidence consolidation¶
The read-only audit is the single reconstruction owner for the retained F4 record. Tests reuse it rather than maintaining another copy of the numerical decision logic. It does not advance a state, refit a parameter, replace the prediction or rerun the twelve simulations. Its checks separate the following obligations:
| Check | Evidence available | Boundary |
|---|---|---|
| Preparation | All twelve initial checkpoints agree with the frozen support, form, phase frame and capacity choice | Internal record binding, not proof of physical acquisition |
| Checkpoint state and arithmetic | Thirty-six initial/midpoint/final states admit their stored field/rate/balance data; the K3 midpoint pressure and receiver identity are checked independently | A matching nu*p product alone does not validate pressure |
| Frozen numerical verdict | Intervention responses, Taylor ratios and refinement decisions are reconstructed from retained coordinates | Rechecking the evaluated prediction is not a new reserved prediction |
| Current source/runtime | Listed byte fingerprints and runtime versions determine availability of current-engine snapshot replay; numerical replay agreement is reported separately | The first manifest does not identify the operating system, libm or every dependency, so listed agreement alone does not guarantee bitwise reproduction |
| Unretained steps | Reported extrema must be compatible with the saved checkpoints | Their exact values over omitted steps and the complete nodal time integral cannot be independently recovered |
The receiver check retains its actual represented defect, rather than assigning
zero. With rho=x_0-(x_1+x_2)/2 and represented pi64, it decomposes as
These terms separate nodal arithmetic, the native pressure split and the
midpoint representation. Their four-way difference checks the complete receiver
identity at the retained times. The audit's 1e-12 comparison allowance is
a read-only arithmetic policy; it neither changes the original frozen gates
nor becomes a physical constant or an exact-flow error bound.
Current-source replay is reported as unavailable if the installed source or runtime differs. That does not change the historical verdict reconstructed from the record. If the listed source/runtime matches but exact numerical replay disagrees, the audit retains that disagreement separately from its independent record-consistency checks. Conversely, source agreement cannot make contradictory state, pressure, storage or decision fields acceptable. The original producer and prediction/response bytes remain unchanged by this consolidation.