Complete prefix estimates for the compensated writer

D Complete prefix estimates for the compensated writer

This appendix closes the clock and pointer derivative estimates used before $t_a=0.0366\,\mathrm s$. All norms are full spinor Hilbert norms, including both internal sectors. Write $t=0$ at $t_h=-0.002\,\mathrm s$, and set

Clock derivatives are physical $S$ derivatives; pointer derivatives are with respect to the dimensionless $Z$. All scalar numbers below use SI units for the clock. Removing the same constant Galilean carrier from both exact waves leaves the clock generator $vp_S+p_S^2/(2M)$. The preparation estimates are those of Sections B.5 and B.6; the old-wave bounds are obtained from Sections C.6 and C.5. Thus neither wave nor actual law is replaced at the handoff.

D.1 Finite Gaussian coefficient calculation

Let $w=0.003$, $b(s)=24B_9(s/w)$ with its constant extensions, and

Its first logarithmic derivative is

The scalar phase in $d$ is retained. Successive logarithmic derivatives are $cy+d$, $c'y-cb'+d'$, $c''y-2c'b'-cb''+d''$, and $c'''y-3c''b'-3c'b''-cb'''+d'''$. The following rational derivative ceilings specify every coefficient used below:

Here $B_1=24(315/128)/w$ follows from $B_9'=630x^4(1-x)^4$. Differentiating this identity, using $x(1-x)\le1/4$ and $|1-2x|\le1$, gives $B_2=24(2520/64)/w^2$ and $B_3=24(7560/16+5040/64)/w^3$. For $j=4,5$ the sufficient coefficient bound is

The polynomial and constant extensions are globally $C^4$; their fifth weak derivative is bounded. No endpoint distribution enters these calculations.

Define $C_j=B_j+\mu B_{j+1}$ for $0\le j\le4$ and $q_0=C_0+1$. The $C_j$ bound the derivatives of $b+\mathrm i\mu b'$. Define four positive affine polynomials in a formal variable $Y$:

For a positive polynomial $P(Y)=\sum p_jY^j$, put

This uses the exact Gaussian moment $\|y^j\gamma\|^2=(2j-1)!!/2^j$. Complete, evaluable formulas for the four pointer jets are

so that $\|\partial_s^j\gamma\|\le g_j$. In particular,

The defining expressions, rather than these shortened displays, are used in the numerical bounds below.

For $A=\partial_Z+Z-\Pi_1(b+\mathrm i\mu b')$ and $\mathsf Q(f)=(\|Zf\|^2+\|p_Zf\|^2)^{1/2}$, the oscillator form identity and $[A,Z]=1$ give

They hold fibrewise and then for the Hilbert direct sum over all other coordinates. Differentiating $A\gamma=0$ gives the weighted jet bounds

Finally, for the writer coefficients $F=b/\mu+\mu b''$ and $C_b=b^2/(2\mu)+\mu bb''/2$, define

These bound $|F|,|F'|,|F''|,|C_b'|,|C_b''|$, respectively.

D.2 Exact intertwiner and pulse integrals

Let $G_*=V(S)(\psi_o g_0)$, where $\psi_o$ is the full exact old wave, and $V$ applies the above Weyl displacement in sector one and the identity in sector zero. Since $\mathrm i\hbar v\gamma_S=H_Z\gamma$, product differentiation gives, up to an irrelevant overall sign in the forced error equation,

This identity holds separately in the orthogonal sectors and hence for any normalized qubit input. Its support is exactly the open writer pulse; the stationary displaced plateau contributes no source. Write $\delta=\Psi-G_*$ and $I_j=\int_0^T\|\mathbf1_{\rm pulse}\partial_S^j\psi_o\|\,dt$. The old characteristic helper has clock centre $t_{\rm lab}+0.104$ and half-width $0.0010000001$. Its earliest pulse contact is $0.0364499999\,\mathrm s$, leaving $\Delta=0.0001500001\,\mathrm s$. Throughout those late slices the initial offset lies in an endpoint strip of relative width $d=0.1500002$. With $\sigma=0.001$ and $m_\phi=1.000001$, define the rational bounds

The inequality $\cos(\pi x/(2\sigma))\le \pi(\sigma-x)/(2\sigma)$, positive convolution and Jensen's inequality prove that $\beta$ bounds the strip mass and $\eta$ its amplitude first-derivative norm. The mollification radius is included in $d$. On these slices $\|u_S\|\le N_1/a$, $N_1=34359738369$. The old-wave estimates, evaluated through elapsed time $0.0404$, give $\|\psi_o-G_o\|<3.1\times10^{-11}$ and $\|\partial_S(\psi_o-G_o)\|<36000$. Consequently

The last bound uses the full old Hessian, including its error. The preparation transfer and old right collar imply $\|\delta(0)\|\le3\times10^{-30}+1.2\times10^{-37}$; the latter term bounds $(V-I)\psi_og_0$ without discarding the tail. Duhamel therefore proves the uniform ceiling

where $\widehat\kappa=1.055\times10^{-35}>\kappa$. All subsequent sharp bounds use the defining expression for $D_*$.

D.3 Closing the simultaneous pointer hierarchy

For this first, deliberately coarse closure take $h_-=1.054\times10^{-34}<\hbar$, and use $\widehat\kappa$ in positive numerators and $h_-$ in denominators. The old physical force ceilings are $F_o=8.00000000001\times10^{-6}$ and $F_{o,2}=1$. The handoff bounds proved in the preparation appendix are

These already include the remote Weyl change. Differentiating the residual once gives the coefficients $1,3/2,1/2$. Define its integrated bounds

Oscillator rotation and the differentiated error equation give the successive bounds

In particular $Q_d<6.847\times10^{-16}$, $e_S<1.482\times10^6$, and $A_1<6.192373945249072\times10^{-23}$. The last row follows from the exact identity

For the second power the exact equation is

The last source is essential and comes from $[A,\partial_S]=\Pi_1c'/a$.

For candidate constants $(X,P,U,B)$ bounding $(\mathsf Q(\delta_S),\|\Psi_{SS}\|,\|\partial_S(A\Psi)\|, \|A^2\Psi\|)$, put

A simultaneous constant supersolution consists of the following four strict inequalities:

These follow from the twice differentiated Schrödinger equation, (97), and (93). In particular the force on $A\Psi$ costs $B+q_0A_1$, not merely $A_1$. For $(X,P,U,B)=(10^{14},10^{40},2\times10^{11},2\times10^{-17})$ the respective right sides are strictly below

Every dependence on the other three unknowns has been retained. The positive integral inequalities and a simultaneous first-crossing argument prove the four bounds. The large $10^{40}$ bound is used only inside this hierarchy; the next subsection supplies the much sharper error Hessian needed for the clock current.

D.4 Old third clock derivative with complete coefficients

Only an ordinary third spatial clock derivative of the old wave is needed; no fourth radial energy graph is invoked. Sections B.3 and B.4 give the following outward bounds for the gauged old preparation error $E_g$:

where $\|p_S^jE_g\|\le P_j$ and $n_2$ bounds the second left collar. For a cutoff equal to one through $0.10005$, zero from $0.10015$, and $g=0.0001$, the localized interpolation identities give

This cutoff covers every preparation-phase derivative, whose support ends at $0.10001$, and lies inside the full-one region of the old second collar, which ends at $0.100249999873$. For the preparation-rate jets use

and put $r_j=\mu_r^+F_pa_j/2$ for $j=1,2,3$. Expanding $p_S^3(e^{\mathrm i\theta}E_g)$, with $\hbar|\partial_S^j\theta|\le r_j$, proves the handoff bound

The last term bounds the original packet's third derivative. The helper's radial state is $S$ independent at handoff and its preparation phase is zero there. Thus $\|\psi_{o,SSS}(0)\|\le E_{3,h}<2.250395507380083\times10^{42}$.

Here are the old-wave input ceilings used on the enlarged horizon $T_o=0.0404$, computed with the formulas of Sections C.6 and C.5:

For clarity the enlargement keeps the same ramp interval $0.0033$ and uses static duration $T_o-0.0033=0.0371<0.0377$. The helper age remains below $4/3<\pi/2$. Its packet ceilings are

so that, with $N_2=1.001(2^{35}+1)^2$,

The three error entries are the sums of the transported preparation, short ramp, static, common-to-clipped and clipped-to-static component bounds evaluated at these durations; no new propagation assumption is introduced by extending the time argument.

The remote third force, including the preparation phase-square term, has the complete upper expression

The last summand is the third quiet-force derivative obtained from the finite product-rule calculation in Section C.4. Thus $F_{c,3}<68.48117358490407$. Put

The remote common-minus-clipped force is supported on $S\le0.10001$. The helper and all its $S$ derivatives vanish there for the entire prefix, since its left support starts at $0.1009999999$ and moves right. The third differentiated unitary equation therefore gives

In particular the large helper Hessian is not charged on the remote force support. Integrating the linear envelope once more yields

D.5 Localized first derivative and refined mixed moment

Take a fixed decreasing Lipschitz cutoff $\chi=1$ on $S\le0.126$, $\chi=0$ on $S\ge0.136$, with $|\chi'|\le1/g_L$, $g_L=0.01$. Every old clock-force derivative is supported in its full-one region ($S\le0.106$), while the residual starts at $S=0.14145$. The positive constant drift favors departure from this left region. The forced continuity identity, with no source on the support of $\chi$, gives

Since $AG_*=0$, improve the oscillator bound to $\widetilde Q=A_1+q_0D_*$. Let $\widetilde R_S=(\kappa/\widehat\kappa)R_S$ and similarly $\widetilde R_{SQ}=(\kappa/\widehat\kappa)R_{SQ}$. For the candidate $e=0.1$, define

The positive two-variable comparison closes because $\mathcal E<0.07327295767972046<e$. Consequently $\|\delta_S\|\le0.1$ and $n_L\le n_*<1.197180\times10^{-35}$ throughout the prefix. The improvement retains the old force but multiplies it by its actual localized error norm.

On that force support $A=\partial_Z+Z$ and $AG_*=0$. Applying the oscillator identity on each $S$ slice, without differentiating a sharp indicator, gives $\mathsf Q(\mathbf1_{\rm old\ force}\delta) \le\sqrt{A_1^2+n_*^2}$. Set $B=2\times10^{-17}$ in $Q_2$ above. The differentiated oscillator rotation equation now proves the much sharper uniform bound

This is the mixed moment required in the clock estimate; the coarse $10^{14}$ bound from (98) is not substituted here.

D.6 Second clock derivative and the two absolute currents

Two residual derivatives give exactly

Its integrated norm is bounded by the explicit expression

The refined new-minus-old handoff Hessian from Section B.6, plus both derivatives of the remote Weyl change, gives the following expression. Here $n_{2,R}=5.970369147764257548\times10^{-38}$ is the right second-collar ceiling from Section B.5. It equals the numerical left-collar ceiling $n_2$ because reflecting the moving-cutoff construction gives the same positive momentum inequality: the right boundary translates at $v$ plus the drift ceiling and the left boundary at $v$ minus that ceiling. The right collar is one throughout the writer support; its handoff full-one edge is below $0.103750000127$, while $S_{\rm on}=0.14145$. Thus the required right-tail amplitude is bounded independently of the left-tail localization used in (99). Explicitly,

For $H(t)=\|\delta_{SS}(t)\|$, integration by parts with $\chi$ gives

The cutoff cost is retained. Twice differentiating the error equation therefore yields the differential comparison $H^\prime\le 2\alpha\sqrt H+f(t)$, where $\alpha=F_o\sqrt{n_*}/\hbar$, $f\ge0$, and $H(0)+\int_0^T f(t)\,dt\le\mathcal R$ with

Indeed, writing $F(t)=H_h+\int_0^t f(u)\,du$, the function $[\alpha t+\sqrt{F(t)}]^2$ has derivative at least $2\alpha[\alpha t+\sqrt{F(t)}]+f(t)$ and dominates the initial value. Scalar differential comparison therefore proves

Regularization at zero justifies this comparison when a norm vanishes. Smooth-domain approximation and these uniform finite estimates justify the oscillator and weak clock commutations without an oscillator cutoff.

On $S<S_{\rm on}$, $G_*=\psi_og_0$ and $A=\partial_Z+Z$. The oscillator form and graph identities imply, after restriction to that region,

For example $\|p_Z^2f\|^2\le\|(p_Z^2+Z^2)f\|^2+2\|f\|^2$ and $\|(\mathcal B^\dagger\mathcal B+1)f\|^2 =\|\mathcal B^2f\|^2+4\|\mathcal Bf\|^2+\|f\|^2$ with $\mathcal B=\partial_Z+Z$ prove the second bound. The conditional-rank stability functional then gives

Here the comparison's radial direction is independent of $Z$; $\|g_0'\|=1/\sqrt2$, $\|g_0''\|=\sqrt3/2$ and $\|g_0'^2/g_0\|=\sqrt3/2$ supply the displayed coefficients. For the clock functional use the old characteristic helper instead. Expanding its positive stability majorant with the additional error $\delta$ gives

This is an increment of an absolute-current upper bound, not a subtraction of signed currents. All hidden-coordinate absolute values precede integration. Both estimates apply on the common whole-prefix good-clock event; its complement is charged separately in the event composition. The current components must be added before optimizing the single rank collar.

Reproducible outward arithmetic.

Equations (92)–(109), together with the outward input tables, form a finite coefficient specification. Every displayed decimal input is an exact rational ceiling. One fully rational evaluation uses a bracket for $\pi$ of width $2\times10^{-69}$ about $3.141592653589793238462643383279502884197169399375105820974944592307817$ and $\hbar=6.62607015\times10^{-34}/(2\pi)$. For a nonnegative rational $x=n/d$, replace each square root by

All these coefficient expressions have positive inputs; use upper numerators and lower denominators. This procedure gives $X_*<616734.119066647397144$, $H_*<1.117214708915152634\times10^{21}$, and the strict current bounds (108)–(109). It specifies the calculations inside the manuscript and requires no external coefficient file.