Record Codebook Size Bounds Proper-Time Quantum Erasure

1. Proper Time and Embodied Counts

Proper time appears in three established laws. Stationary clocks at different gravitational potentials accumulate different durations. In a path-integral or WKB description, a free massive branch $\gamma$ carries phase $\theta \left[\gamma \right]=-m{c}^2\tau \left[\gamma \right]/\hslash$. A timelike geodesic locally maximises the same duration between fixed endpoints [1,2]. Optical clocks resolve gravitational redshift across laboratory height differences [3,4]. I ask how a finite embodied record constrains the phase correction inferred from such a duration.

I use the Stack Theory definitions needed for that question [5,6]. Let $\Phi$ be a nonempty set of mutually exclusive states. A program is a set $p\subseteq \Phi$. A finite vocabulary $\mathfrak{v}\subseteq 2^{\Phi }$ encodes $\varphi \in \Phi$ by

$${Enc}_{\mathfrak{v}}\left(\varphi \right)=\{p\in \mathfrak{v}\mid \varphi \in p\}.$$

(1)

For a finite timeline fragment $h=({\varphi }_0,\dots ,{\varphi }_q)$, define

$$\begin{array}{cc}\hfill C_{\mathfrak{v}}\left(h\right)& =\left\{j\in \{1,\dots ,q\}|{Enc}_{\mathfrak{v}}\left({\varphi }_j\right)\ne {Enc}_{\mathfrak{v}}\left({\varphi }_{j-1}\right)\right\},\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill {\kappa }_{\mathfrak{v}}\left(h\right)& =|C_{\mathfrak{v}}\left(h\right)|.\hfill \end{array}$$

(3)

Layer space and encoding capacity are

$$\begin{array}{cc}\hfill S_{\mathfrak{v}}\left(\varphi \right)& =|{Enc}_{\mathfrak{v}}\left(\varphi \right)|,\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill \mathsf{I}\left(\mathfrak{v}\right)& ={log}_2\left|Im\left({Enc}_{\mathfrak{v}}\right)\right|.\hfill \end{array}$$

(5)

Layer space counts programs true together at one state. Encoding capacity counts the distinct encodings reached across states. The two come apart, and the distinction carries the result below. Capacity is blind to whether a layer can hold many distinctions at once or only a few, and it is that simultaneity which physical constraints act on.

For the record result, let ${\Phi }_{\mathfrak{r}}\subseteq \Phi$ be the nonempty set of states reachable when the correction is selected. Fix an operational record vocabulary

$$\mathfrak{r}=\{p_1,\dots ,p_n\},$$

(6)

and its correction-state codebook

$${\mathcal{C}}_{\mathfrak{r}}:=\{{Enc}_{\mathfrak{r}}\left(\varphi \right)\mid \varphi \in {\Phi }_{\mathfrak{r}}\}.$$

(7)

Program $p_i$ means that binary input line i of the correction interface is asserted at the correction event. A low line is represented by absence from the encoding, rather than by an added complementary program. Under this convention, $S_{\mathfrak{r}}\left(\varphi \right)$ is the Hamming weight of the interface word. The codebook size $M_{\mathfrak{r}}:=\left|{\mathcal{C}}_{\mathfrak{r}}\right|$ is the number of interface words reachable when correction is selected. The theorem is relative to this fixed interface. A different vocabulary describes a different interface.

Lemma 1

(Bounded-weight record count). Let $\left|\mathfrak{r}\right|=n$ and let $S_{\star }\in \{0,\dots ,n\}$. If $S_{\mathfrak{r}}\left(\varphi \right)\le S_{\star }$ for every $\varphi \in {\Phi }_{\mathfrak{r}}$, then

$$M_{\mathfrak{r}}=\left|{\mathcal{C}}_{\mathfrak{r}}\right|\le V(n,S_{\star }):={\displaystyle \sum _{j=0}^{S_{\star }}}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{j}}\right).$$

(8)

Equality requires every binary word of weight at most $S_{\star }$ to be reachable.

Proof. 

Every member of ${\mathcal{C}}_{\mathfrak{r}}$ is a subset of $\mathfrak{r}$ with cardinality at most $S_{\star }$. There are $V(n,S_{\star })$ such subsets. □

Interpretation. If at most $S_{\star }$ of n interface lines can be asserted together, the record cannot use arbitrary n-bit words. It can use only words inside the corresponding Hamming ball [7]. The counting is standard. The physical content arises when the record layer fixes $S_{\star }$. Section 4 gives two such cases.

Fix a physical map from each timelike path segment $\gamma$ in a family $\Gamma$ to a timeline fragment $h_{\gamma }$.

Definition 1

(Calibrated clock). A clock vocabulary $\mathfrak{c}$ has frequency $\nu >0$ and error $B<\infty$ on Γ when

$${\kappa }_{\mathfrak{c}}\left(h_{\gamma }\right)=\nu \tau \left(\gamma \right)+e\left(\gamma \right),\left|e\left(\gamma \right)\right|\le B$$

(9)

for every $\gamma \in \Gamma$.

Interpretation. The count tracks proper duration at rate $\nu$, with at most B ticks of error. Stack Theory defines the count of changes visible to the clock vocabulary, which means the number of transitions between distinct clock-layer encodings along the fragment. Physical calibration determines whether the chosen vocabulary functions as a clock.

2. Gravity Changes the Count Rate

Consider a connected static region with zero cosmological constant and

$$\mathrm{d}{s}^2=-N^2\left(\mathbf{x}\right)c^2\mathrm{d}{t}^2+h_{ij}\left(\mathbf{x}\right)\mathrm{d}{x}^i\mathrm{d}{x}^j,$$

(10)

where $N>0$ and t is normalised to a reference clock. A stationary path obeys $\mathrm{d}\tau =N\mathrm{d}t$. If stationary segments of arbitrary coordinate duration T lie in the calibrated family, Eq. (9) gives

$$r\left(\mathbf{x}\right):=\underset{T\to \infty }{lim}{\displaystyle \frac{{\kappa }_{\mathfrak{c}}\left(h_{\mathbf{x}}\left(T\right)\right)}{T}}=\nu N\left(\mathbf{x}\right).$$

(11)

Let $n^{\mu }$ be the future unit normal. Define $\mathcal{E}=T_{\mu \nu }{n}^{\mu }{n}^{\nu }$ and $\mathcal{S}=h^{ij}{h_i}^{\mu }{h_j}^{\nu }{T}_{\mu \nu }$. The standard static lapse equation becomes

$$D_i{D}^{i}r={\displaystyle \frac{4\pi G}{c^4}}(\mathcal{E}+\mathcal{S})r,$$

(12)

where $D_i$ is the covariant derivative of $h_{ij}$ [8]. For a perfect fluid at rest relative to $n^{\mu }$, $\mathcal{E}+\mathcal{S}=ϵ+3p$.

A Newtonian monotonicity statement for added density under common boundary data follows from Eq. (12) and is given in the Supplemental Material. It is not used below.

Scope of this section. Equations (11) and (12) establish that the calibrated count is an operational stand-in for proper duration and that gravity acts on its rate. The visibility bound uses the gravitational phase-window width $\Theta$ of Eq. (28). The dynamic $3+1$ dictionary, the count-maximiser theorem, and the jointly resolved stationary-phase result are likewise placed in the Supplemental Material, because they fix the shared variable rather than the ceiling.

For two calibrated branches, write $u=\tau \left({\gamma }_1\right)-\tau \left({\gamma }_0\right)$ and $\Delta \kappa ={\kappa }_{\mathfrak{c}}\left(h_{{\gamma }_1}\right)-{\kappa }_{\mathfrak{c}}\left(h_{{\gamma }_0}\right)$. Equation (9) gives

$$\left|u-{\displaystyle \frac{\Delta \kappa }{\nu }}\right|\le {\displaystyle \frac{2B}{\nu }}.$$

(13)

A relative phase $-\omega u$ is therefore a calibrated layer-time difference with a bounded error. The record theorem concerns the finite interface used to infer and correct that phase.

3. Finite Outcomes Limit Quantum Erasure

Proper-duration differences can correlate an interferometer path with internal clock or environmental degrees of freedom and reduce visibility [9,10,11]. Here quantum erasure means measuring a separate correlated record and applying an outcome-conditioned phase correction to the path system. This conditions on side information without retaining the quantum record coherently [12,13,14]. Finite reference frames and restricted environmental access impose related coherence limits [15,16]. Miatto and colleagues restrict the accessible Hilbert-space dimension and obtain full recovery for some finite-dimensional conditional states [17,18]. Here the unresolved phase has an absolutely continuous distribution and only M classical values are retained. A finite value set cannot resolve almost every phase exactly.

Let $\omega >0$. Let u be uniformly distributed on an interval J of width $\Delta$. Conditional on u, take

$$|{\psi }_u\rangle ={\displaystyle \frac{|0\rangle +e^{-i\omega u}|1\rangle }{\sqrt{2}}},\Theta =\omega \Delta ,0<\Theta \le 2\pi .$$

(14)

The uniform law is the maximum-entropy density on a fixed bounded interval. The Supplemental Material treats arbitrary densities and a Gaussian example.

Let R be a separate record with conditional state ${\rho }_R\left(u\right)$,

$${\rho }_{PR}={\displaystyle \frac{1}{\Delta }}{\int }_J|{\psi }_u\rangle \langle {\psi }_u|\otimes {\rho }_R\left(u\right)\mathrm{d}u.$$

(15)

A measurement of R is followed by storage of one classical encoding in $\mathfrak{r}$. Let $\mathcal{R}\subseteq {\mathcal{C}}_{\mathfrak{r}}$ be the retained encodings and let $M=\left|\mathcal{R}\right|\ge 1$. An arbitrary measurement induces $p\left(y\right|u)$. After outcome y, an optimal y-dependent phase shift aligns the conditional coherence. The average recovered visibility is

$${\mathcal{V}}_{\mathfrak{r}}={\displaystyle \frac{1}{\Delta }}\sum _{y\in \mathcal{R}}\left|{\int }_{J}p\left(y\right|u)e^{-i\omega u}\mathrm{d}u\right|.$$

(16)

This includes a classical side record and any positive-operator-valued measurement on a separate quantum record when only the classical outcome remains available for path correction.

I use the unnormalised convention $sincx=sinx/x$ and $sinc\left(0\right)=1$.

Theorem 1

(Finite-outcome ceiling). Under Eqs. (14)–(16),

$${\mathcal{V}}_{\mathfrak{r}}\le F(M,\Theta ):={\displaystyle \frac{2M}{\Theta }}sin\left({\displaystyle \frac{\Theta }{2M}}\right)=sinc\left({\displaystyle \frac{\Theta }{2M}}\right)<1.$$

(17)

The function F is strictly increasing in M. Equality is attained when the record channel perfectly distinguishes M equal contiguous phase bins and the correction interface retains their labels.

Proof. 

Set $z_y={\int }_{J}p\left(y\right|u)e^{-i\omega u}\mathrm{d}u$. Choose ${\beta }_y$ so that $e^{i{\beta }_y}{z}_y=\left|z_y\right|$, with arbitrary ${\beta }_y$ when $z_y=0$. Assign each u to the smallest index that maximises $cos({\beta }_y-\omega u)$. Let $\tilde{\mathcal{V}}$ be the visibility of the resulting measurable deterministic partition, so ${\mathcal{V}}_{\mathfrak{r}}\le \tilde{\mathcal{V}}$. Under the change of variable $\vartheta =\omega u$, let $E_y$ be its phase cells and let ${\ell }_y$ be their arc lengths. Then

$$\tilde{\mathcal{V}}={\displaystyle \frac{1}{\Theta }}\sum _y\left|{\int }_{E_y}{e}^{-i\vartheta }\mathrm{d}\vartheta \right|.$$

(18)

Circular rearrangement gives

$$\left|{\int }_{E_y}{e}^{-i\vartheta }\mathrm{d}\vartheta \right|\le 2sin({\ell }_y/2),$$

(19)

with equality for a contiguous arc. The cell lengths obey ${\sum }_y{\ell }_y=\Theta$. Jensen’s inequality for the concave function $x\mapsto sin(x/2)$ gives Eq. (17). Strict monotonicity follows because $sincx$ decreases on $(0,\pi ]$ and $\Theta /\left(2M\right)$ decreases with M. The Supplemental Material gives each step. □

Interpretation. Each retained value groups a range of unresolved phases and applies one correction to the whole group. Equal phase bins lose the least coherence, but every finite partition leaves some phase spread inside each bin. No finite record can therefore recover full coherence for this absolutely continuous phase law.

4. Layer Space Fixes the Codebook

At the correction event, one of M distinguishable record encodings is retained. Across possible runs, the interface must admit each encoding that triggers a distinct correction. Therefore

$$M\le M_{\mathfrak{r}}=\left|{\mathcal{C}}_{\mathfrak{r}}\right|\le V(n,S_{\star }).$$

(20)

The first inequality is the immediate capacity requirement. The second is the layer-space constraint for the fixed interface vocabulary.

Theorem 2

(Record codebook ceiling). Let the operational record vocabulary contain n binary programs, let $S_{\star }\in \{0,\dots ,n\}$, and assume $S_{\mathfrak{r}}\left(\varphi \right)\le S_{\star }$ for every $\varphi \in {\Phi }_{\mathfrak{r}}$. Then

$$\begin{array}{|c|}\hline {\mathcal{V}}_{\mathfrak{r}}\le {\overline{\mathcal{V}}}_{n,S_{\star }}(\Theta ):=sinc\left({\displaystyle \frac{\Theta }{2{\sum }_{j=0}^{S_{\star }}\left(\genfrac{}{}{0pt}{}{n}{j}\right)}}\right).\\ \hline\end{array}$$

(21)

Equality requires every bounded-weight encoding to be reachable, perfect discrimination of the corresponding equal phase bins, and a distinct correction for each retained label.

Proof. 

Combine Eq. (17), monotonicity in M, and Eq. (20). □

Equation (21) is only as interesting as the reason for $S_{\star }$. Read as a comparison between codebooks it says little, since a designer who wants more correction labels will simply choose a denser code, and standard capacity accounting tracks that choice perfectly well. The physical content appears where $S_{\star }$ is not available to the designer. Two such cases follow.

4.1. A Complete Record Channel

The record must correlate with u without disturbing the interfering particle, and it must be read and fed forward before recombination. Position detection of the interfering particle itself would destroy the superposition and does not realise Eqs. (15)–(16).

The residual arm-height difference z is a property of the guiding potential rather than of the interfering particle. Let a probe system R couple to the apparatus coordinate that sets z, for example by scattering from the guide or mirror element whose displacement fixes the arm offset, and let its outcome be resolved by n position-sensitive detectors. Writing y for the detector index gives a channel $p\left(y\right|u)$ directly, since $u=gTz/c^2$. The protocol then retains y, applies the phase $e^{i{\chi }_y}$ to one arm, and recombines.

Three conditions are required and are experimentally separable. The probe must couple to the apparatus coordinate and not to the branch index, so that it carries no which-path information. Probe backaction on z during the dwell must stay below the bin width, or the effective channel is broadened. The feed-forward must complete inside the coherence window.

4.2. Particle Number Fixes the Weight

Condition on one detected probe quantum under ideal number-resolving detection. Exactly one detector then fires. The reachable codebook is the set of n singletons, so $M\le n$ and

$${\mathcal{V}}_{\mathfrak{r}}\le sinc\left({\displaystyle \frac{\Theta }{2n}}\right),{\mathcal{V}}_{\mathfrak{r}}\le sinc\left({\displaystyle \frac{\pi }{n}}\right)\mathrm{at}\Theta =2\pi .$$

(22)

For $n=8$ this gives ${\mathcal{V}}_{\mathfrak{r}}\le 0.9745$, and reaching $0.999$ at one full cycle requires $n\ge 41$ detectors. This is a design constraint on the readout rather than a correction to a naive estimate: a correct capacity accounting also assigns an ideal one-hot array $M=n$, hence ${log}_{2}n$ bits. What Eq. (22) adds is the conversion of that count into an attainable interferometric ceiling, and the observation that the weight is set by particle number and is therefore not available to the designer.

4.3. Peak Assertion and One Energy Realisation

More generally, suppose the interface presents n readable lines but the layer can assert at most $S_{\star }$ of them together at the correction event. One realisation is a budget E for simultaneous assertion at cost $\epsilon$ per asserted line, giving $S_{\star }=\lfloor E/\epsilon \rfloor$ and

$${\mathcal{V}}_{\mathfrak{r}}\le sinc\left({\displaystyle \frac{\Theta }{2{\sum }_{j=0}^{\lfloor E/\epsilon \rfloor }\left(\genfrac{}{}{0pt}{}{n}{j}\right)}}\right).$$

(23)

This is a peak-weight constraint, not a thermodynamic law. Equation (23) assigns zero cost to unasserted lines and omits the decoder, wiring, reset, discrimination, and correction-actuator budgets, each of which is a separate and possibly dominant cost. The substantive content is that peak simultaneity, not total capacity, is what the constraint acts on, and that it changes an exponent. For fixed $S_{\star }\ge 1$ and large n,

$$1-{\overline{\mathcal{V}}}_{n,S_{\star }}(\Theta )={\displaystyle \frac{{\Theta }^2{(S_{\star }!)}^2}{24n^{2S_{\star }}}}\left[1+O\left(n^{-1}\right)\right],$$

(24)

against ${\Theta }^2/(24·4^n)+O\left({16}^{-n}\right)$ for an unrestricted n-line codebook. Figure 1 shows the resulting slopes. Equation (24) is asymptotic in n and should not replace the exact entries of Table 1 and Table 2; at $n=8$ and $S_{\star }=2$ it sits about a third above the exact ceiling.

Table 1 fixes eight readable binary lines and one full phase cycle. The exact one-hot row is the physically forced case of Eq. (22); the remaining rows are Hamming balls, which include the all-zero word. Table 2 fixes sixteen lines and varies the energy budget alone.

Doubling the budget from one line to two improves the ceiling by nearly two orders of magnitude in deficit at fixed interface size. Read the other way, fix a target ${\mathcal{V}}_{\star }\in (0,1)$ and let $x_{\star }\in (0,\pi )$ be the unique solution of $sinc{x}_{\star }={\mathcal{V}}_{\star }$. Any protocol reaching the target must obey

$$V(n,S_{\star })\ge M_{\star }:=⌈{\displaystyle \frac{\Theta }{2x_{\star }}}⌉.$$

(25)

This condition is necessary. It becomes sufficient only when the corresponding codewords and equal-bin record channel are physically available. For fixed $S_{\star }$ and large $M_{\star }$, inversion of the Hamming-ball asymptotic gives $n\gtrsim {(S_{\star }!M_{\star })}^{1/S_{\star }}$. At $\Theta =2\pi$ the target ${\mathcal{V}}_{\star }=0.999$ requires $M_{\star }=41$, which a budget of one line meets only at $n=40$, a budget of two lines meets at $n=9$, and a budget of three or more meets at $n=6$. Energy and interface size trade against each other at fixed coherence target.

5. Gravitational Phase Window

Let $\eta \ll 1$ bound both $\left|\phi \right|/c^2$ and ${\left|\mathbf{v}\right|}^2/c^2$ on the branches. In a weak static field,

$${\displaystyle \frac{\mathrm{d}\tau }{\mathrm{d}t}}=1+{\displaystyle \frac{\phi }{c^2}}-{\displaystyle \frac{{\left|\mathbf{v}\right|}^2}{2c^2}}+O\left({\eta }^2\right).$$

(26)

Consider held or guided branches with the same velocity history in $\phi \left(z\right)=gz$. Let their height difference be $d_0+z$, where $d_0$ is the nominal arm separation and z is the residual arm-height difference after the known mean phase has been removed. A common vertical displacement of both arms does not contribute. For coordinate dwell time T,

$$u\left(z\right)={\displaystyle \frac{gTz}{c^2}}+O\left(T{\eta }^2\right).$$

(27)

If z spans a window of width $\Delta z$, then

$${\Theta }_C={\displaystyle \frac{mgT\Delta z}{\hslash }},{\Theta }_0={\displaystyle \frac{{\omega }_{0}gT\Delta z}{c^2}}.$$

(28)

The first expression is the proper-time contribution from the point-particle action $-m{c}^2\tau$ [1]. It does not treat one internal energy eigenstate as an operational Compton clock. The second is the internal-clock scale [9,11]. A closed freely falling light-pulse interferometer can cancel the uniform-field redshift term. Equation (28) therefore concerns held or guided branches, or an equivalent clock-initialisation scheme [11,19]. Differential trapping, guide, laser, separation, and endpoint phases must be common-mode, calibrated, or independently measured.

For ⁸⁷Rb, $T=1\mathrm{ms}$, and $g=9.81\mathrm{m}{\mathrm{s}}^{-2}$, one full rest-mass phase cycle corresponds to

$$\Delta z={\displaystyle \frac{2\pi \hslash }{mgT}}\simeq 0.47\mu \mathrm{m}.$$

(29)

Combining Eqs. (25) and (28) gives

$$\begin{array}{|c|}\hline V(n,S_{\star })\ge ⌈{\displaystyle \frac{mgT\Delta z}{2\hslash x_{\star }}}⌉.\\ \hline\end{array}$$

(30)

Mass, field, dwell time, residual arm-height width, record architecture, and target visibility now occur in one relation. With a single-particle position record, Eq. (22) replaces the left side by the detector count n.

For Gaussian residual separation $z\sim \mathcal{N}(0,{\sigma }_z^2)$, let $s=mgT{\sigma }_z/\hslash$. With no retained record,

$${\mathcal{V}}_{\mathrm{none}}=e^{-s^2/2}.$$

(31)

A one-bit sign record attains

$${\mathcal{V}}_{\mathrm{sign}}=e^{-s^2/2}\sqrt{1+{erfi}^2(s/\sqrt{2})}.$$

(32)

For ⁸⁷Rb, $T=1\mathrm{ms}$, $g=9.81\mathrm{m}{\mathrm{s}}^{-2}$, and ${\sigma }_z=0.10\mu \mathrm{m}$, these values are $0.406$ and $0.733$. The sign partition is an attainable benchmark rather than a global optimum.

A direct protocol prepares a balanced path superposition, varies the residual arm-height difference according to a known law, retains only a chosen codeword, removes access to the finer controller setting, applies the codeword-dependent phase shift, and recombines the paths. Shot-to-shot randomisation tests the resource bound on the retained protocol record. It does not assert irreversible decoherence. The decisive comparison holds the interface size n fixed and varies $S_{\star }$ through the record physics, either by detecting a single-quantum probe of the apparatus coordinate or by restricting the peak assertion weight of Eq. (23). Optical clocks establish laboratory access to gravitational frequency gradients [3,4]. The coherent two-branch protocol and its phase controls remain additional experimental requirements.

6. Discussion

The quantum theorem depends directly on M, the number of retained correction values. The Stack variables constrain the observable through

$$(n,S_{\star })⟶M_{\mathfrak{r}}⟶M⟶\mathcal{V}.$$

(33)

The first arrow is the bounded-weight record count over correction states. The final arrow is the finite-outcome theorem.

Equation (8) is Hamming-ball enumeration and agrees with standard coding theory. Capacity accounting determines ${log}_{2}M$ once the reachable codebook is known. The physical record determines that codebook. Particle number fixes the one-hot codebook of Eq. (22), while a peak-assertion budget fixes the bounded-weight codebook of Eq. (23). Combining those embodied constraints with Eq. (17) yields exact coherence ceilings and their attainability conditions.

Layer space defines peak weight as a property of the encoding layer and places it in the same vocabulary as the layer-time count of Eq. () and the calibrated clock of Eq. (9). The chain from $r=\nu N$ through $\Theta$ to $\mathcal{V}$ then becomes one accounting of clock rate, phase, record architecture, and recovered coherence.

The result is vocabulary-relative. I fix the vocabulary by the correction interface, where each program is one directly readable asserted line. A different interface changes n, $S_{\star }$, and the reachable encoding image. The theorem does not identify a vocabulary-independent property of the substrate.

The static gravity equation is the standard lapse equation after $r=\nu N$. The dynamic $3+1$ dictionary, count-to-phase error bound, count-maximiser theorem, and jointly resolved stationary-phase result are given in the Supplemental Material. Those results establish a shared operational variable. They do not derive Einstein’s equation, Hilbert space, the Born rule, the point-particle action, or clock calibration.

The visibility ceiling assumes a uniform phase window with $0<\Theta \le 2\pi$, an absolutely continuous mixture, a retained classical record, correction conditioned only on that record, and no postselection. Coherent retention of the quantum record, joint path-record operations, filtering, extra side information, wrapped windows, and other phase laws define different protocols. The Gaussian benchmark illustrates one such law without claiming the uniform closed form. Equation (23) assumes a fixed per-line assertion cost with none for unasserted lines, and neglects decoder, wiring, reset, discrimination, and actuator budgets.