C6 response, contraction and represented rounding¶

Parked C6 response domains, numerical reserves and rounding-cell restrictions.

Part of Cycle winding under Coupling: scope and retained evidence. Section numbers are stable across the collection; hypotheses and model changes remain local to each result.

Archived research record. This preserves conditional derivations and source-bound evidence. Its local next-step language is historical and does not schedule work.

8. One shared local response for circular means¶

B2.d.15 compares the previous antipodal obstruction with winding under the default bidirectional all-target UM stage. It does not change operator coefficients or disable functional links. Sections 1-7 retain their distinct target-only hypotheses.

For a nonzero mean phasor S_s=sum_{j in N_s} exp(i*theta_j), differentiation in a regular local chart gives

R_sj = 1[j in N_s]*Re(exp(i*theta_j)/S_s)
     = 1[j in N_s]*sum_{k in N_s} G_jk / sum_{l,k in N_s} G_lk,
G_jk = cos(theta_j-theta_k).

Rows sum to one by rotation covariance, but may have negative entries. The antipodal bridge's (1,1,-1) row therefore cannot be treated as convex diffusion. For one common phase factor t, receiver i averages the proposals from its declared source targets T_i, giving the derivative J_ij=(1-t)*delta_ij+t*mean_{s in T_i} R_sj. Pointwise IL uses only source i; bidirectional UM includes every source that proposes a write to i.

phase_response.py is the shared exact owner for these two operations. It validates an exact unit planar cosine Gram: symmetry, diagonal one, positive semidefiniteness and rank at most two. A Schur complement about one unit vector must be a positive rank-at-most-one matrix; this avoids numerical angle reconstruction. Zero mean resultants are rejected. A rounded cosine table that fails these identities is not repaired silently. Gram data do not identify oriented winding, live graph support, phase branches or binary64 derivatives. Both the antipodal reference and the C6 reference now derive their matrices through this common implementation.

9. Default all-target C6 response and a phase neighborhood¶

Consider the mathematical winding base theta_i=i*pi/3, i=0,...,5, on the unit simple cycle, with the canonical gate pi/2. Adjacent phases differ by g=pi/3; non-neighbors have circular separation at least 2*pi/3. Thus the base has strict gate margin pi/6 both for existing edges and against new compatible nonedges. This base is declared preparation, not a derived birth.

Let A denote cycle adjacency. Each closed UM neighborhood has phasor resultant two, whereas each open IL neighborhood has resultant one. Hence

R_closed = I/2 + A/4,                  R_IL = A/2,
W = (I+A)*R_closed/3 = (2*I+3*A+A^2)/12,
U = (1-t)*I+t*W,                      M = (1-alpha)*I+alpha*A/2,
P = M*U.

Every UM receiver averages three sources: itself and its two neighbors. The individual base displacements +t*g,0,-t*g cancel in the merge; they do not vanish individually. This distinguishes the default simultaneous stage from both one bidirectional target and the earlier target-only stage.

The Fourier multipliers of W, in mode order k=0,...,5, are (1,1/2,0,0,0,1/2). Those of P are

(1,lambda1,lambda2,lambda3,lambda2,lambda1),
lambda1 = (1-t/2)*(1-alpha/2),
lambda2 = (1-t)*(1-3*alpha/2),
lambda3 = (1-t)*(1-2*alpha).

For centered tangent energy Q(e)=sum_i(e_i-mean(e))^2/2, symmetry gives the optimal linear bound Q(Pe)<=q*Q(e), q=max(lambda1^2,lambda2^2,lambda3^2). It is strictly below one for 0<t<=1, 0<=alpha<=1. The global rotation multiplier remains one, so full uncentered contraction is not claimed. The algebraic boundary t=0 remains a detached control, not canonical UM admission. This exact reference uses the rational cosine Gram of C6; mathematical pi is not replaced by math.pi.

A sufficient nonlinear phase-map invariant box¶

An exact phase neighborhood can also be retained without linearizing. Write theta_i=i*g+e_i in consistent periodic lifts, with all e_i in [m-r,m+r] and 0<=r<=pi/24. The center m is a common rotation, not a new interaction parameter. The radius is a sufficient analytical bound.

Translate m to zero and examine one closed UM sum relative to its base center. Its unperturbed value is two, and |exp(i*e)-1|<=|e| gives |S-2|<=3*r. Since r<=pi/24<1/6, Re(S)>=3/2 and |Arg(S)|<=2*r. Every included phase is then at angular distance at most g+3*r<=11*pi/24<pi/2 from the mean. All its mean derivatives are positive. Monotonicity between the all-lower and all-upper preparations consequently sharpens the mean error c_s to [m-r,m+r].

The source-to-receiver shortest arcs stay on their fixed branches. After averaging the three baseline displacements, the exact UM error is

e_i' = (1-t)*e_i + t*(c_(i-1)+c_i+c_(i+1))/3.

This is a convex combination in the same interval. For IL, the two neighbor phases have lifted separation at most 2*g+2*r<=3*pi/4<pi. Their circular mean is the midpoint, so

e_i'' = (1-alpha)*e_i' + alpha*(e_(i-1)'+e_(i+1)')/2.

It preserves the interval as well. Throughout these exact phase maps, adjacent separation is at most g+2*r<=5*pi/12, and nonedge separation is at least 2*g-2*r>=7*pi/12. Both retain margin pi/12 from the gate; functional-link creation therefore has no compatible candidates, and winding stays one. This proves invariance for finite repetitions of these restricted exact phase maps. It does not assert the linear factor q globally throughout the box, nonlinear asymptotic convergence, positive-EPI admission, a repeated complete engine policy or a binary64 invariant class. The fixed interval center m is not asserted to be the conserved arithmetic error mean. Nonlinear UM generally has mean(e')=(1-t)*mean(e)+t*mean(c), which can differ from mean(e). IL preserves that mean inside the midpoint chamber; both local linear Jacobians preserve it at the winding base.

Connection to nodal pressure¶

In this same midpoint neighborhood, the phase-pressure channel satisfies the exact identity pi*g_phase=(A/2-I)*e. IL's phase jump and the pressure readout therefore use the same structural local discrepancy, with their distinct configured coefficients. Nodal EPI evolves through dEPI/dt=nu_f*(w_epi*g_epi+w_phase*g_phase+...); a phase update is not an independent continuous EPI law. The implementation retains the pi-scaled tangent as a rational vector. Finite captures separately measure represented phase pressure and its difference from the midpoint and tangent expressions.

10. Finite default C6 controls and remaining scope¶

c6_winding_phase_response.py predeclares five preparations: the nominal winding, plus positive and negative epsilon=2^-12 in directions

k1 = (1,1/2,-1/2,-1,-1/2,1/2),
k3 = (1,-1,1,-1,1,-1).

EPI is initially 0.5, capacity is one, all six conductances/lengths are one, and the seed is 17. The single stored base tuple i*math.pi/3 is used both for preparation and wrapped phase comparisons. The default bidirectional UM, capacity mixing and functional links remain enabled. Each case executes one all-target UM, refresh, IL, refresh, one shared h=0.25 Euler interval with four held-pressure substeps, then SHA after measurement.

The shared stage observer retains actual admission, an independent strict IL readiness check, pure-kernel predictions versus actual writes, histories, metrics and signed reference/flow budgets. Its phase readout is now supplied explicitly, so the same mechanics serve the earlier antipodal controls without imposing their phase chart on this cycle. The existing certify_phase_winding observes the actual oriented cycle with the resolved UM gate. Rotation drift, centered phase deviation, new-link exclusion margins and numerical residues are recorded separately. A rational tangent sum never supplies the winding integer.

The finite measured centered-energy ratios after UM/IL are:

Preparation	Observed Q_after/Q_before	Linear prediction
`+2^-12*k1`	`0.558580560057`	`0.558580559487`
`-2^-12*k1`	`0.558580560058`	`0.558580559487`
`+2^-12*k3`	`0.092062966493`	`0.092062966493`
`-2^-12*k3`	`0.092062966493`	`0.092062966493`

The small k1 excess above the tangent bound is retained, not rounded into a finite nonlinear certificate. The k3 scalar response has the exact local finite multiplier (1-t)*(1-2*alpha) in its alternating chart, with recorded binary64 residuals. Every observed stage and flow endpoint retains winding one and excludes all nine nonedges; the minimum observed nonedge margin is about 0.523325 radians. Phase stays fixed during the nodal interval while EPI responds to its pressure; for positive k1, EPI changes from uniform 0.5 to approximately (0.499994493,0.499997246,0.500002754,0.500005507,0.500002754,0.499997246). This is a finite forced response, not EPI localization or self-restoration. The null preparation retains a phase residual around 1.30e-17; its initial centered phase energy is zero, so the ratio is undefined. Its stored EPI remains uniform after the interval despite small represented pressure residuals. Neither coordinate stasis nor a tiny residual proves a full fixed runtime state.

The exact matrix, mode and cache-revalidation controls extend coupling_winding.py and live in test_c6_winding_phase_response.py. test_phase_response.py supplies independent signed-response, Gram and receiver-average controls. The finite runtime tests also compare independent nonlinear phasor formulas, preserve the null case's roundoff evidence, and verify winding, actual source refresh and nodal flow.

Generate the independently manifested local artifact with:

.venv313\Scripts\python.exe -X utf8 benchmarks/c6_winding_phase_response.py

This block supplies a stronger prepared phase candidate than the antipodal UM/IL configuration. A retained winding and phase neighborhood do not by themselves preserve a localized EPI shape. The joint exact-model result below addresses pressure refresh and nodal integration. Formation of the winding still needs a declared canonical mechanism. Autonomous physical structures and laboratory correspondence remain open.

11. Nonlinear contraction and a joint phase/capacity/EPI domain¶

Keep the prepared unit C6, all-target bidirectional UM and midpoint IL from section 9, with fixed factors 0<t<=1 and 0<=alpha<=1. Write phase error in units of mathematical pi, z_i=e_i/pi, and let v=osc(z)=max(z)-min(z)<=1/12. This normalization is an analysis coordinate, not a new nodal field or an approximation of pi. The interval midpoint supplies the common rotation m; the phase radius is at most pi/24.

A uniform nonlinear oscillation bound¶

The sharpened closed-mean error lies between the smallest and largest input errors. Every participating phase is therefore within g+2*r<=5*pi/12 of that mean. For its nonzero phasor sum S,

d Arg(S)/d theta_j = cos(theta_j-Arg(S))/|S|,
cos(5*pi/12)>1/4,       |S|<=2+3*r<=2+pi/8<5/2.

Each of the three active derivatives exceeds 1/10. This rational lower bound follows from the geometry; it does not replace an operator coefficient. Receiver averaging in UM consequently supplies derivative at least t/30 for each of the five columns at cycle distance at most two. Every two such rows share at least four columns. Their overlap is at least 2*t/15, so the row-stochastic Jacobian contracts oscillation by at most

rho = 1-2*t/15 < 1.

Integrate the Jacobian along the segment from a common rotation to the input inside the same invariant box. The same row overlap survives integration. The midpoint IL map is row stochastic and cannot increase oscillation. Thus this bound applies to the nonlinear phase map T, not just its tangent: osc(T(z))<=rho*osc(z). It is a conservative range bound, distinct from the optimal local squared-energy factor in section 9. No assertion that the local quadratic factor holds throughout the nonlinear box is needed.

Both phase maps keep each error inside its previous min/max interval. Nested intervals and v_n<=rho^n*v_0 imply convergence to a common error z_infinity: a rotated uniform winding. That rotation need not equal the initial arithmetic mean. This is an exact phase-map convergence theorem on the declared domain, not a binary64 or complete-runtime limit theorem.

From phase pressure to a preserved EPI reserve¶

Assume uniform fixed positive capacity nu, fixed normalized channel weights w_epi>0,w_phase>=0, and a held Euler interval of duration h>=0 after each UM/IL phase update. Uniform capacity is unchanged by these UM and IL maps. The unit regular support gives zero capacity and topology gradients. The midpoint pressure identity and the nodal equation yield

z_plus = T(z),
DeltaNFR = -w_epi*L_rw*x - w_phase*L_rw*z_plus,
x_plus = (I-s*L_rw)*x - b*L_rw*z_plus,
s = h*nu*w_epi,                 b = h*nu*w_phase.

These are the refreshed inputs at the interval boundary. The default solver's four held-pressure substeps do not refresh diffusion inside the interval. In exact arithmetic they trace the same straight segment and have the displayed endpoint. Require 0<=s<=1, so I-s*L_rw is row stochastic. Since ||L_rw*z_plus||_infinity<=v_plus<=rho*v,

min(x_plus) >= min(x)-b*v_plus,
max(x_plus) <= max(x)+b*v_plus.

Define the derived future-forcing reserve coefficient

B = b*rho/(1-rho).

The identity (b+B)*rho=B proves that both inequalities

min(x)-B*v >= ell,             max(x)+B*v <= U

are preserved at cycle endpoints. For declared 0<ell<=U<=1, this is a joint phase/EPI domain with constant positive capacity. Every held internal substep is between the old and new EPI endpoints, so it also stays in [ell,U]; hard clipping in the canonical interval [-1,1] is unnecessary. At an internal fraction u of the interval, the same estimate uses (u*b+B)*rho<=B, so the reserve inequalities hold there as well. This does not make every internal substep a new pressure-refreshed cycle. The phase box, winding and exclusion of new links persist throughout the held interval.

The same coefficient gives the explicit accumulated forcing bound sum_(n>=0) b*||L_rw*z_(n+1)||_infinity<=B*v_0. The configured timestep and channel factors remain declared inputs. The bound derives their allowed effect; it neither chooses an action nor adds a restoring pressure law.

The positive lower bound is not, by itself, every application condition. The runtime must also match the actual UM minimum EPI, capacity and phase gate, independent strict IL readiness, hard clipping, pressure realization and word/history context. SHA is a terminal closure and is excluded from this constant-positive-capacity domain.

What tends to persist in this restricted model¶

The exact EPI arithmetic mean is conserved because both Laplacian terms sum to zero. For fixed 0<s<1, its nonuniform-mode factor is

gamma = max(|1-s/2|, |1-3*s/2|, |1-2*s|) < 1.

Writing y=x-mean(x)*1, the norm estimate ||y_(n+1)||_2<=gamma*||y_n||_2+sqrt(6)*b*v_0*rho^(n+1) tends to zero by the geometric convolution. Therefore EPI tends to its initial mean while phase tends to a rotated winding. This class supports prepared phase organization; it does not maintain an asymptotically localized EPI shape. At s=1, zero phase error and an alternating EPI perturbation give multiplier -1, so positivity and boundedness do not imply convergence. At h=0, EPI does not evolve. Neither boundary is silently promoted to the strict theorem.

Executable algebra and its boundary¶

The joint-domain reference and observer extend coupling_winding.py, reusing the cycle Laplacian and exact input readers. The reference computes rho, B, the Euler matrix and the separate EPI mode bound. The observer receives six declared pre/post phase-error coordinates in pi units and the EPI state. It checks interval nesting and contraction, evaluates the exact nodal endpoint and reports the input/output reserves. Public reference caches are rebuilt. Supplying two vectors that pass these checks does not authenticate a production phase update, its phase chart or its operator admission.

12. Two-cycle admission and the represented null boundary¶

c6_winding_joint_domain.py predeclares exactly two UM/IL/flow cycles for three inherited preparations: the nominal winding and the positive k1 and k3 probes at 2^-12. Each cycle uses one h=0.25 interval; a single terminal SHA follows both measurements. The graph, default operator factors, seed and source preparation are unchanged from section 10. No horizon or coefficient search is performed.

The EPI band is read from actual application and clipping settings. On the positive chart, default UM requires EPI>=0.05 and nu>=0.01, whereas independent strict IL requires EPI>0 and nu>0. The chosen band therefore has lower endpoint equal to the represented UM floor and upper endpoint one. The benchmark calls the independent UM/IL/SHA precondition validators in addition to both whole-word validators and actual per-stage admission. It does not enable a different precondition policy to obtain acceptance.

The full word UM IL UM IL SHA passes both word validators. A separate read-only UM UM SHA control records string-validator rejection alongside ValidatedSequence acceptance and live-candidate acceptance after the first UM. It is not executed. These interfaces answer different checks and their results must remain visible; a string-word refusal is not a universal live kernel refusal. This preserves the earlier P2 distinction rather than changing the transition policy during a research comparison.

For the executed defaults, the conservative exact-model coefficients are

rho  approximately 0.967806265733,
B    approximately 5.700849477232,
gamma approximately 0.977105818448.

The initial positive k1 and k3 probes have represented pi-scaled phase diameter about 0.000155425, well inside the prepared box and the joint reserve band. The entire maximum-width phase box at uniform EPI 0.5 would only give a conservative lower reserve about 0.0249292, below UM's floor. Positivity or phase-box membership alone therefore does not satisfy this sufficient joint admission condition. Failure of a sufficient reserve does not prove that an actual trajectory must cross the floor.

Both nonzero preparations pass the conditional phase-transition checks in each cycle. The benchmark separates the exact nodal endpoint on the supplied coordinates from pressure realization and actual integrator endpoint defects; their signed sum closes exactly. All three preparations retain winding one, unit capacity before closure and admissible EPI in the observed words. For k1, the minimum stored EPI changes from 0.5 to approximately 0.499994493 and 0.499990503; the corresponding lower future-forcing reserves are approximately 0.499332272 and 0.499495571. These are finite measured reserves, not a runtime invariant-class theorem.

The null preparation exposes the next boundary. Its initial represented phase error has zero diameter, but the first UM/IL stage produces a small nonzero diameter. The homogeneous bound v_plus<=rho*v consequently fails on that represented chart. The second cycle fails the conditional transition check as well. Positive contraction residuals are approximately 4.12e-18 and 2.90e-18 in represented pi units. The actual runtime words still succeed. The report retains both outcomes rather than replacing the observed phase by exact zero or weakening the test tolerance.

Captured wrapped errors are divided by represented binary64 math.pi for these readouts; they are not silently identified with division by mathematical pi. The null result refutes a defect-free transfer of the homogeneous bound to this represented chart. It does not refute the exact phase theorem or every possible numerical invariant class. Any such extension needs explicit phase, pressure and EPI execution-defect bounds and the actual admission band.

The exact-domain tests, independent boundary tests and runtime tests separate the geometric budget, Euler boundary, application floors, public cache revalidation, finite word evidence and represented null obstruction. Generate only this block's artifact with:

.venv313\Scripts\python.exe -X utf8 benchmarks/c6_winding_joint_domain.py

The additive extension below accounts for these retained defects. The exact uniform-EPI limit must still guide action selection and formation: retaining winding alone does not explain sustained EPI localization or physical particle formation.

13. Additive defects, finite reserves and the neutral EPI mean¶

Reuse the same joint reference: T=I-s*L_rw, b=h*nu*w_phase, rho=1-2*t/15, B=b*rho/(1-rho). For supplied ordered endpoints define

v = osc(z),        v_plus = osc(z_plus),
epsilon = max(0, v_plus-rho*v),
delta = x_plus - (T*x-b*L_rw*z_plus),
mu = mean(delta),                 delta_c = delta-mu*1.

These are arithmetic residuals of the supplied coordinates, not new driving terms added to the nodal law. The additive observer retains them even when the defect-free transition inequalities fail. Interval expansion and membership in the original phase box remain separate readouts. A bound on oscillation alone does not establish that a phase tuple was generated by UM/IL or that a particular wrapped chart remains valid in the future.

The uniform EPI mode is neutral, so the mean identity is exactly mean(x_plus)-mean(x)=mu. For the earlier reserves ell_x=min(x)-B*v and U_x=max(x)+B*v, let C=B+b=b/(1-rho). The stochastic Euler matrix and ||L_rw*z_plus||_infinity<=v_plus give

ell_x_plus >= ell_x - [C*epsilon-min(delta)],
U_x_plus   <= U_x   + [C*epsilon+max(delta)].

The signed lower/upper loss bounds can be negative under favorable common drift. Equivalently, separate mu and use nonnegative centered costs C*epsilon-min(delta_c) and C*epsilon+max(delta_c). Both signed losses are bounded above by the conservative absolute cost C*epsilon+||delta||_infinity. This retains cancellation in the actual endpoint defect instead of summing the magnitudes of its constituent errors.

For a nonempty adjacent finite endpoint chain, summing these signed losses gives lower/upper reserve bounds at every recorded prefix. The phase envelope obeys v_bound_next=rho*v_bound+epsilon, and the EPI mean follows the exact signed prefix mean(x_n)=mean(x_0)+sum_(j<n)mu_j. The implementation rebuilds all public observation caches and verifies endpoint adjacency; this supplies a detached finite arithmetic telescope, not a new causal execution seal.

14. A conditional invariant bound with persistent numerical defects¶

The finite reserve cost need not be summable over an unbounded sequence. Use the existing EPI diffusion to control persistent spatial error separately from the neutral mean. For 0<s<1, the first row of T^2 is

(a0,a1,a2,0,a2,a1),
a0=(1-s)^2+s^2/2,        a1=s*(1-s),        a2=s^2/4.

Its optimal oscillation overlap is k=min(s^2,4*s*(1-s))>0, hence osc(T^2*x)<=(1-k)*osc(x). Opposite rows attain the minimum overlap 4*min(a1,a2); the other distinct row shifts have overlaps 2*a1+2*min(a1,a2) and a1+3*a2, both no smaller. The coefficient changes branch at s=4/5. This is a two-step matrix identity; it does not require executing two additional runtime intervals.

Define the translation-invariant range functional

V(x)=osc(x)+osc(T*x),           r=1-k/2 < 1.

Since osc(T*x)<=osc(x), V(T*x)<=V(x)-k*osc(x)<=r*V(x). This gives a rational one-step bound without treating the phase tangent factor as a nonlinear or EPI gain.

Now assume independently for every step and prefix being claimed:

epsilon_n <= epsilon_bar,
osc(delta_n)=osc(delta_c_n) <= E_bar,
abs(sum_(j<n)mu_j) <= M.

The last condition concerns signed cumulative mean error. It is not implied by a uniform per-step error bound. Choose the derived envelopes

D = max(v_0, epsilon_bar/(1-rho)),
F = 2*b*D+E_bar,
R = max(V(x_0), 2*F/(1-r)).

The phase recursion preserves v<=D. If D<=1/12, it stays in the earlier oscillation class, subject to a consistent winding chart. The complete additive EPI input -b*L_rw*z_plus+delta has oscillation at most F. Subadditivity gives V(x_plus)<=r*V(x)+2*F, so V<=R is preserved. The sharp six-coordinate mean/range inequality yields

mean(x_0)-M-5*R/6 <= x_i <= mean(x_0)+M+5*R/6.

Requiring this interval inside the declared application band supplies a conditional joint phase/range/mean guarantee. Persistent nonzero errors may maintain a nonzero spatial discrepancy; strict contraction of homogeneous diffusion is not a claim of convergence under those errors. At s=0 or s=1, k=0; this bound rejects nonzero forcing. With zero forcing it retains the initial range functional and makes no strict contraction claim.

The mean hypothesis is essential. In an abstract additive-error sequence, take zero phase error, x_0=(1/2)*1 and delta_n=-(1/1024)*1. Spatial range remains zero, but EPI decreases uniformly: at step 460 it is 52/1024, above the default UM floor, and at step 461 it is 51/1024, below that floor while still positive and far from hard clipping. This refutes the sufficiency of uniformly small per-step errors alone. It does not assert that production arithmetic generates this particular error sequence at a zero-pressure state.

The executable finite and conditional-uniform bounds extend coupling_winding.py, with shared joint-reference validation, nodal pressure evaluation and exact matrix powers. The finite-budget tests and independent uniform-bound controls separate exact telescope identities, optimal overlap, persistent error, neutral drift, excluded Euler boundaries and public-cache revalidation.

15. Offline audit of the retained numerical boundary¶

c6_winding_defect_budget.py reads the existing three two-cycle records from section 12. It does not execute another operator or integrate a longer trajectory. The input file is bound by SHA-256; its producer manifest and source scope remain historical, while this analysis receives its own source manifest. The input and output paths must differ, and both input bytes and analysis source are checked across the audit.

The adapter rebuilds ordered unit-C6 snapshots and their exact gradient caches; checks fixed capacity, channel weights, stage EPI preservation, recorded phase charts, event/flow endpoint continuity and held Euler evidence; and recomputes the signed pressure/endpoint identities. Stored admission records remain declarations from that producer, not newly executed checks. The retained represented-pi chart is explicit. Both null cycles now enter the additive budget while keeping their failed defect-free status.

The EPI defect is separated exactly into:

phase contribution = h*nu*w_phase*(g_phase_production+L_rw*z_plus),
assembly contribution = h*nu*(kernel_pressure_defect+stored_pressure_residual),
integration contribution = x_after-(x_before+h*nu*stored_pressure).

Their signed sum equals delta. The assembly term includes rounding in the combined pressure channels; it is not mislabeled as a pure phase error. In the null case these components substantially cancel. The observed two-cycle conservative costs and exact signed mean changes are:

Preparation	Sum of absolute defect costs	Total EPI mean change
Null	`4.347315188882e-17`	`0`
Positive k1	`3.167528703228e-16`	`7.401486830834e-17`
Positive k3	`4.477724788020e-16`	`7.401486830834e-17`

All finite prefix reserve bounds remain within the declared band. The retained extrema also produce algebraically admissible examples of the conditional uniform envelope: approximately [0.406278,0.593722] for the nonzero probes and a very small interval around 0.5 for the null. These are deliberately labeled illustrations with future hypotheses unverified. Two measured cycles cannot establish uniform error or cumulative-mean bounds.

The offline report tests check the retained arithmetic and reject inconsistent captures and bindings. To reproduce the detached report from the retained input:

.venv313\Scripts\python.exe -X utf8 benchmarks/c6_winding_defect_budget.py

The local nodal arithmetic result below resolves one part of the numerical task. Pressure realization, phase/backend accuracy and a horizon-independent signed-mean-prefix bound still require their own declared chart/domain arguments. Additional samples alone cannot supply them. The exact and finite results remain available for the separate canonical action-selection and formation question; no numerical defect is promoted to a new physical source.

16. Held nodal rounding cells and the remaining phase/mean boundary¶

The actual quarter-flow arithmetic¶

Keep the retained interval's uniform capacity nu=1, duration h=1/4, four equal substeps and zero Gamma contribution. Write RN for binary64 rounding to nearest with ties to even. For one represented stored pressure p, the unprojected scalar arithmetic is

q = RN(p/16),
x_(j+1) = RN(x_j+q),       j=0,1,2,3.

Capacity multiplication and addition of zero do not change the finite pressure. The production helper dynamics/_euler_kernel.py retains the separate multiplication and addition; replacing them by a fused multiply-add or one full-interval update would change the numerical map. The reference in physics/binary64_nodal_flow.py describes these arithmetic operations, not an alternative pressure law. It does not certify how a live graph produced p.

At x=1/2, the neighboring represented values are 1/2-2^-54 and 1/2+2^-53. The significand of 1/2 is even, so both midpoint ties belong to its rounding cell:

RN(1/2+q)=1/2  iff  -2^-55 <= q <= 2^-54.

For represented finite pressure in this quarter-flow map, this is equivalent to

-2^-51 <= p <= 2^-50.

The endpoints are included. The interval is asymmetric because 1/2 lies at a binary exponent boundary. A pressure in this interval can be nonzero while all four stored EPI updates remain exactly 1/2. The instantaneous nodal product nu*p then remains nonzero: arithmetic stasis of the represented EPI is not the zero-pressure equilibrium condition. This statement holds for the fixed held input; it does not make the complete UM/IL/refresh word stationary.

A local error bound and a zero-sum counterexample¶

Assume finite inputs, nearest-even binary64 arithmetic and no clipping. The initial EPI and every unprojected rounded substep endpoint must lie in the declared positive band [ell,U], with 0<ell<=U<=1. Let u_min=2^-1074 be the least positive subnormal. Scaling by 1/16 is exact when representable; otherwise its absolute rounding error is at most u_min/2. Each addition returning a positive value at most one has absolute rounding error at most 2^-53, including the wider upper cell at 1. Therefore

eta = x_4-(x_0+p/4),
|eta| <= 4*(u_min/2) + 4*2^-53 = 2^-51 + 2^-1073.

This is an a priori local bound under the stated arithmetic and endpoint conditions. It includes subnormal scaling; an execution that clips or leaves the declared positive band is outside this claim. A platform probe checks format and selected rounding behavior, but is not a proof of every native kernel's conformance. Individually captured arithmetic can be compared with the exact map without promoting those checks to a future execution theorem.

The bound cannot be replaced by zero, even for small normal pressure. For x_0=1/2 and p=2^-50, each increment is the upper midpoint tie 2^-54. All four additions return 1/2, giving eta=-2^-52. One direct full-step addition instead returns 1/2+2^-52; the two numerical schedules differ.

For six initial values x_i=1/2, take the held pressure tuple (+2^-50,-2^-50,+2^-50,-2^-50,+2^-50,-2^-50). Its sum is zero. The positive-pressure coordinates remain at 1/2, whereas each negative one ends exactly at 1/2-2^-52. Consequently the represented EPI mean changes by -2^-53, despite the zero-sum input. This is a numerical-kernel counterexample to automatic mean preservation, not a claim that canonical C6 phase refresh generates that pressure tuple. No additional graph trajectory is required to establish the rounding-cell calculation.

Across N otherwise admissible intervals, the absolute local integration bound supplies only a generic cumulative bound proportional to N*(2^-51+2^-1073). It does not give a horizon-independent bound on signed mean error. Pressure and phase errors also remain separate contributions; the local integration estimate alone cannot discharge section 14's all-prefix mean hypothesis.

Retained observations and their provenance¶

c6_winding_rounding_cells.py audits the same six retained flows as section 15, through its existing capture, pressure and chronology validation. It compares scalar and NumPy evaluations of the shared arithmetic helper with those endpoints and records per-coordinate cells and local error bounds. This is detached arithmetic replay of stored inputs, not another operator invocation or a longer research trajectory. The input bytes, producer manifest and SHA-256 remain distinct from the new analysis manifest and output artifacts/research/c6_winding_rounding_cells.json.

The exact stored-EPI mean changes in these retained flows are:

Preparation	First flow	Second flow
Null	`0`	`0`
Positive k1	`1/(3*2^52)`	`0`
Positive k3	`2^-53`	`-1/(3*2^53)`

The nonzero stored pressure in the null control is approximately -2.14453350193743e-16, inside the 1/2 pressure cell above. Its stored EPI remains uniform. Node zero's phase nevertheless changes in both cycles, and the earlier defect-free phase-transition checks still fail. Neither the EPI stasis nor the retained mean increments establish a fixed full state, an eventual numerical attractor or a uniform future error class.

The rounding-report tests compare the retained endpoint and defect identities with each of the four scalar and NumPy arithmetic states. Reproduce the detached audit with:

.venv313\Scripts\python.exe -X utf8 benchmarks/c6_winding_rounding_cells.py

Phase evaluation has additional, distinct numerical contracts¶

The phase computations do not all use the same numerical path:

Readout or update	Current operations
UM closed phasor mean	`math.sin/cos`, `math.fsum`, `math.atan2` in the coupling proposal kernel
IL open neighborhood mean	`math.sin/cos`, then NumPy mean/arctangent when available; compensated scalar fallback in trigonometric helpers
Refreshed pressure for the retained C6	NumPy sine/cosine, `np.add.at` neighborhood sums, NumPy arctangent, represented modulo and division by represented pi in the fused pressure kernel
Alternate pressure mean path	Cached trigonometric values, averaged sums and a resultant check in pressure evaluation; separate scalar fallback
Scalar phase difference	Another sine/cosine/arctangent evaluation through unified numerical operations, including UM proposal and receiver merging
Array phase difference	Subtraction, addition of represented pi, remainder modulo represented tau and subtraction of represented pi in numeric utilities

The default C6 pressure dispatch uses compute_fused_gradients_symmetric. Its twelve outgoing edge entries stay below the n_edges>100 JIT threshold, so the NumPy branch applies. That branch selects nodes with neighbors but does not replace a zero resultant by the node's original phase. The resultant fallback in the alternate neighbor-mean path is not part of these retained C6 evaluations.

The shared binary64 probe explicitly does not certify arbitrary native numerical kernels. Python documents platform C library dependencies and possible double rounding in fsum; it provides no blanket correctly-rounded sine/cosine/arctangent contract here. See the Python math documentation. NumPy describes its arctangent through the underlying C implementation, while reduction precision can depend on the reduction path; see NumPy arctangent and NumPy summation. These dependencies do not show that the installed functions are inaccurate. They prevent inferring a universal ULP allowance from the binary64 format probe alone.

A conditional propagation bound identifies what a future numerical proof must supply. If an exact complex phasor sum obeys |S|>=m>0, its evaluated sum has complex error at most e<m, and arctangent evaluation has angular error at most a, then its wrapped angle error is at most

e/(m-e)+a.

For example, rotate S to the positive real axis: the perturbed real part is at least m-e, and the imaginary displacement is at most e; use atan(u)<=u. In the prepared pi/24 box, UM's closed sums permit m=3/2. The open two-neighbor sums instead permit m=3/4, since their resultant is at least 2*cos(3*pi/8)>3/4; an averaged sum uses m=3/8. The full stage must also account for scalar phase-difference evaluation, arithmetic, modulo operations and a consistent lifted chart. These error allowances are numerical proof premises, not new TNFR parameters or forces.

The next proof obligation is therefore narrower: establish useful bounds for the actual phase/pressure implementation and a separate cumulative-mean mechanism on a declared represented domain. The held arithmetic theorem, finite reports and conditional envelopes do not establish nonlinear formation, autonomous action selection or a global TNFR stability theorem.

17. Opposite-pair closure and a restricted mean-preserving arithmetic class¶

An exact partial observable of the nonlinear phase map¶

Use the same ordered C6, winding chamber and simultaneous all-target UM/IL maps. For normalized phase errors z=e/pi, define three centered opposite pair sums:

(P*z)_i = z_i+z_(i+3)-2*mean(z),       i=0,1,2.

Their sum is zero. This observable retains two independent coordinates; it neither reconstructs the six phases nor fixes their common rotation. Let A be cycle adjacency, L=I-A/2, K=(I+A)/3, and let c contain the six closed-neighborhood phasor-mean errors in the same normalized chart. The exact finite phase maps in the established chamber are

z_U=(1-t)*z+t*K*c,
z_plus=M*z_U,           M=(1-alpha)*I+alpha*A/2.

Opposite receiver triples partition all six sources. Consequently

P*K=0,
P*M=(1-3*alpha/2)*P,
P*L=(3/2)*P.

These are exact matrix identities, independent of the nonlinear function that produced c. They give the finite, nonlinear observable law

h_phi_plus=lambda_pair*h_phi,
lambda_pair=(1-t)*(1-3*alpha/2),      h_phi=P*z.

The multiplier equals the earlier k=2 tangent coefficient, but its use here has a separate finite-map proof. The full tangent energy bound is not promoted to a nonlinear bound. For 0<t<=1 and 0<=alpha<=1, abs(lambda_pair)<1; opposite-pair error decays geometrically for exact repetition in this chamber. The subspace h_phi=0 is preserved.

This is pairing about the current mean. Indeed mean(z_U)=(1-t)*mean(z)+t*mean(c), so a nonlinear common rotation can change while centered pairing remains exact. Literal z_(i+3)=-z_i would additionally require the mean to remain zero. Geometric reflection with phase conjugation is a different symmetry and is not inferred here.

The executable observe_c6_winding_pairing rebuilds the existing joint reference and applies these identities to supplied closed-mean coordinates. Their membership in the initial phase interval is a necessary chamber property, not authentication as actual phasor means. It keeps the previously derived oscillation-contraction and reserve checks separate. The observer does not call trigonometric kernels, admit an operator word or identify an executed graph.

The nodal equation closes the corresponding EPI observable¶

For h_x=P*x, s=h*nu*w_epi and b=h*nu*w_phase, the same exact held nodal map gives

phase-pressure opposite sums = -(3/2)*h_phi_plus,
total-pressure opposite sums = -(3/2)*(w_epi*h_x+w_phase*h_phi_plus),
h_x_plus=(1-3*s/2)*h_x-(3*b/2)*h_phi_plus.

Thus each of the three dependent pair coordinates follows one common two-component recurrence:

                  [ lambda_pair                 0        ]
Q_pair =          [ -(3*b/2)*lambda_pair        1-3*s/2    ].

This closes a partial observable of the phase-first nonlinear model. Both sectors retain their zero-sum relation over the three pair indices. For 0<s<=1, both diagonal eigenvalues have modulus below one, so these pair defects tend to zero, including the coincident-eigenvalue case of a triangular matrix. At s=0, EPI pairing can remain unchanged. This result does not control the opposite-odd spatial modes or the common phase mean; in particular it does not remove the alternating EPI boundary in section 11.

When both initial pair defects vanish, total pressures are opposite in each pair. The real-valued nodal evolution then preserves x_i+x_(i+3). The arithmetic counterexamples in section 16 explain why this real-model identity alone does not preserve represented pair sums.

A closed binary64 class for the isolated EPI channel¶

There is a useful restricted arithmetic class, without choosing a new physical coefficient. Keep unit capacity, zero Gamma, fixed unit C6, four held substeps of 1/16, and a fixed represented 0<=w_epi<=1. Pressure is refreshed from the EPI channel before each quarter interval. The class is specified by the represented state:

All six EPI values lie in the positive application band [float(0.05),1] and strictly inside one common normal binade (B,2*B).
Its spacing is Delta=B*2^-52; the exact pair sums are all 2*c.
The center lies on that lattice: c/Delta is an integer.

B, Delta and c are derived from the input representation. They are arithmetic premises, not TNFR forces or adjustable model parameters.

On this domain, subtracting two EPI values and averaging the two neighbor differences are exact. The production scalar and vector difference reducers therefore both materialize

g_i=(x_(i-1)+x_(i+1))/2-x_i,
p_i=RN(w_epi*g_i).

The statement covers their mixed-sign exact fallback too. The observer checks both reducers against the rational pressure and its final rounding. Opposite-pair reflection commutes with cycle adjacency, so g_(i+3)=-g_i. Nearest-even rounding is sign-odd numerically; hence the represented pressures and their scaled increments remain opposite.

First, the initial EPI interval [m,M] is preserved. For nonnegative pressure, write M-x_i=n*Delta. Monotonic rounding gives 0<=q_i=RN(p_i/16)<=(M-x_i)/16; the upper bound is representable on this positive band. A rounded lattice update advances at most floor(n/16+1/2) spacings. For every integer n>=0,

4*floor(n/16+1/2)<=n.

For n<=7 the left side is zero; for n>=8 it is at most n/4+2<=n. An induction bounds all four partial updates inside the original interval. Negative pressure has the corresponding lower-bound argument. Thus every result cell stays strictly inside the same binade.

Second, reflection x -> 2*c-x preserves this uniform lattice and its even-significand tie decisions because c/Delta is integral. The reflected addition cells from section 16 coincide, so each opposite pair keeps sum 2*c at every substep. The common mean stays exactly c. The interval, lattice-center and pressure-pair premises therefore hold again after refreshing this pure-channel map. This proves forward invariance and zero signed mean drift under its arbitrary finite repetition. It permits nonuniform rounding plateaus; it is not a binary64 consensus theorem.

The executable observe_binary64_paired_c6_diffusion reuses the B18 flow and its exact cells. It checks the declared class and one supplied scalar/vector kernel application, keeping repeated pure-channel scope separate from a live executor or a complete canonical UM/IL word. The exact pairing tests and binary64 class tests exercise the identities, class boundaries and arithmetic obstructions.

Two exclusions are substantive. At a half-lattice center, even ties can break pair sums: x=(0.75,0.75+2^-53) and opposite pressures +/-2^-50 make both first updates equal 0.75. At the inherited center 0.5, the two sides have different spacings; there is no nontrivial symmetric common-binade neighborhood. The earlier quarter-flow counterexample continues to apply. Moreover, phase forcing can move uniform EPI outside its initial min/max interval. The pure-channel interval proof does not replace the joint phase-source budget.

What the retained default execution actually preserves¶

The detached c6_winding_pairing.py audits the same B16 records through the B18 continuity and arithmetic validation. It computes centered phase pairs, EPI pairs and pressure pairs at the initial, raw-operator, refreshed and flow boundaries. It does not execute another graph, change factors or extend the horizon.

Let sigma=g_phase_production+L*z be the phase-realization defect. At any retained capture the pressure-pair accounting is

p_i+p_(i+3) = -(3/2)*w_epi*(P*x)_i
              -(3/2)*w_phase*(P*z)_i
              +w_phase*(sigma_i+sigma_(i+3))
              +kernel_assembly_defect_i+kernel_assembly_defect_(i+3)
              +stored_pressure_residual_i+stored_pressure_residual_(i+3).

The last term matters at raw UM/IL boundaries, where stored pressure need not equal a refreshed kernel. This identity separates loss of geometric pairing from pressure arithmetic and from the operator's stored-pressure writes. All represented-pi chart assumptions remain explicit.

All three preparations have opposite centered errors relative to the stored winding base, yet their initial represented pressures do not pair. For example the base has theta_1+theta_2-Fraction(math.pi)=-2^-52. This is a defect of that represented reflection identity, not a claim about an exact physical conjugation or a universal impossibility result. The first UM produces nonzero centered pair residuals in all three cases, so the discrepancy cannot be removed by subtracting a common rotation.

One existing boundary is especially discriminating: after the first UM refresh in the k3 record, all opposite pressure sums are exactly zero. After the following IL and pressure refresh,

g_phase[0]+g_phase[3] = -5215/2^65,
p[0]+p[3] = -15823/2^67.

Thus the recorded B16 IL/refresh does not preserve that observed paired pressure class. The nonzero preparations' EPI endpoints also straddle the 0.5 binade boundary, outside the restricted arithmetic theorem. Historical producer declarations and input hashes remain distinct from this analysis's source manifest and derived output.

Reproduce this detached stage audit from the retained B16 input with:

.venv313\Scripts\python.exe -X utf8 benchmarks/c6_winding_pairing.py

The next joint-runtime obligation is now specific: a preserved relational contract for total pressure pairs and compatible EPI rounding cells, or an explicit summable budget for their leakage. A bound on each phase error separately still does not give a horizon-independent signed-mean bound. The exact partial quotient and restricted pure-channel class are useful positive results; complete winding-runtime preservation, canonical formation and laboratory correspondence remain open.