# Protein-Clock Check: mathematical contract

99 Small Problems: Useful models for assumptions with expensive ambitions. No. 04. v0.6.0-alpha.

## Scope and state

Input: measured or explicitly synthetic functional RNA fraction \(m(t)=M(t)/M_0\). Output: normalized protein \(p(t)=P(t)/P_0\). Initial protein is one. Constant translation coefficient and first-order loss are assumptions, not fitted facts.

\[
\frac{dP}{dt}=k_{\mathrm{tr}}M-\lambda P,\qquad
k_{\mathrm{tr}}M_0=\lambda P_0.
\]

Consequently:

\[
\frac{dp}{dt}=\lambda[m(t)-p(t)],\qquad p(0)=1.
\]

The baseline relation requires steady state and compatible normalizations. The RNA driver must represent the RNA pool relevant to translation. Unchanged RNA abundance does not establish unchanged translation, as synthesis and degradation can both change during dynamic responses ([Ross et al., 2021](https://pmc.ncbi.nlm.nih.gov/articles/PMC7950106/)).

## Loss convention

Effective mode:

\[
\lambda=\frac{\ln2}{H_{\mathrm{eff}}}.
\]

Intrinsic-degradation mode:

\[
\lambda=\frac{\ln2}{H_{\mathrm{deg}}}+\frac{\ln2}{T_d}.
\]

The second term is omitted when doubling time is blank. Division dilution must not be counted again when already incorporated in the supplied effective half-life. The distinction between true degradation and division dilution is established turnover accounting ([Ross et al., 2021](https://pmc.ncbi.nlm.nih.gov/articles/PMC7950106/)).

For population protein amount \(A\), cell number \(N\), per-cell RNA \(M\), and balanced growth \(\dot N=\mu N\), assuming \(\dot A=Nk_{\mathrm{tr}}M-k_{\mathrm{deg}}A\) gives

\[
\frac{d(A/N)}{dt}=k_{\mathrm{tr}}M-(k_{\mathrm{deg}}+\mu)(A/N).
\]

This derivation concerns mean protein per cell, not total protein per well. Stable growth and baseline normalization must hold. Treatment-dependent growth requires a different time-dependent model and normalization.

## Solution and complete-shutoff bound

\[
p(t)=e^{-\lambda t}+
\lambda\int_0^t e^{-\lambda(t-s)}m(s)\,ds.
\]

For nonnegative RNA:

\[
p(t)\geq e^{-\lambda t}.
\]

Across a declared range of rates, the fastest rate provides the lowest common shutoff floor. A protein value below that floor is incompatible with this model and rate range, absent measurement error. It is not proof of a unique alternative mechanism.

## Sustained step

For \(m(t)=r\):

\[
p(t)=r+(1-r)2^{-t/H}.
\]

For \(r<q<1\), the first downward crossing of protein fraction \(q\) is

\[
t_q=H\log_2\left(\frac{1-r}{q-r}\right).
\]

At \(r=0.1,H=48\) h, \(p(24)=0.7363961031\), and \(t_{0.5}=56.15640007\) h. Synthetic, not a target-specific measurement. Protein half-life effects on silencing kinetics were explored in the more extensive modeling of [Bartlett and Davis (2006)](https://pmc.ncbi.nlm.nih.gov/articles/PMC1331994/); this is not a reproduction of their full model.

## Exact linear-segment propagation

For \(m(t_0+u)=m_0+su\), define \(x=\lambda\Delta\), \(Q=1-e^{-x}\):

\[
p(t_0+\Delta)=p(t_0)+(m_0-p(t_0))Q+
s\Delta\left(1-\frac{Q}{x}\right).
\]

Implementation uses the equivalent form \(p(t_0)e^{-x}+m_0Q+s\Delta(1-Q/x)\) to retain small surviving protein amounts rather than subtracting nearly equal numbers. \(Q=-\operatorname{expm1}(-x)\). For \(|x|<0.001\), the ramp factor uses

\[
1-\frac{1-e^{-x}}{x}
=\frac{x}{2}-\frac{x^2}{6}+\frac{x^3}{24}
-\frac{x^4}{120}+\frac{x^5}{720}+O(x^6).
\]

For zero rate or zero duration, return the initial protein. Validation restricts UI rates to positive finite values; the zero-rate branch is a numerical limiting case.

## Extrema and thresholds

The sign of the derivative is the sign of \(m-p\). On a linear segment, \(d=m-p\) satisfies \(\dot d=s-\lambda d\); its solution is monotone or constant, so there is at most one zero in its interior. Bisection identifies that zero when endpoint signs differ. Evaluate boundaries and extrema to find the finite-window minimum; the earliest minimum wins ties.

Threshold roots are bisected on the resulting monotone pieces. Below-threshold intervals are merged. Isolated zero-duration tangencies do not count as time below threshold. No threshold crossing outside the declared observation window is inferred.

## Sensitivity and display grid

Use 31 logarithmically spaced half-lives from lower to upper, plus the central value. Deduplicate equal values. Compute the pointwise minimum/maximum across these models. This sampled envelope is neither a rigorous continuous-parameter bound nor a statistical interval. In recovery cases, intermediate half-lives may control a pointwise extreme, so comparing only endpoints is unsafe.

Display/export points: 481 uniform times, all input knots within the window, central extrema, and inspection time. Solve inspection and roots directly from segments; do not interpolate on the display grid.

## Old protein and continued synthesis are different contributions

The integrating-factor solution decomposes the predicted protein pool:

\[
p_{\mathrm{old}}(t)=e^{-\lambda t},\qquad
p_{\mathrm{new}}(t)=\lambda\int_0^t e^{-\lambda(t-s)}m(s)\,ds
=p(t)-p_{\mathrm{old}}(t).
\]

Both quantities are relative to baseline protein, not percentages of the remaining pool. The old share of the remaining pool is \(p_{\mathrm{old}}/p\), when \(p>0\). This is a mathematical decomposition under constant loss, not a molecular-age measurement. Total protein can fall while new protein continues to be synthesized.

The initial-pool contribution equals the complete-shutoff bound. This equality does not imply that subsequent synthesis is zero in the actual scenario. Nonnegative synthesis and positive initial protein also mean that exact zero cannot be reached at finite time; zero-threshold crossings are suppressed even if floating-point underflow gives a numerically zero value.

## Explicit hold counterfactual

At inspection time \(t_a\), let \(p_a=p(t_a)\) and \(r_a=m(t_a)\). If RNA were held at that value for an additional \(\tau\), independently of the supplied future RNA:

\[
p_{\mathrm{hold}}(t_a+\tau)
=r_a+(p_a-r_a)e^{-\lambda\tau}.
\]

The conditional plateau is \(r_a\), and the distance to it decreases by \(e^{-\lambda\tau}\). This is an explicitly declared counterfactual, not extrapolation or a claim that one RNA point establishes a plateau.

For \(p_a>q\) and \(r_a<q\), a downward target \(q\) is reached after

\[
\tau_q=\frac{1}{\lambda}\ln\left(\frac{p_a-r_a}{q-r_a}\right).
\]

If \(p_a>q=r_a\), the target is approached asymptotically. If \(p_a>q\) and \(r_a>q\), it cannot be reached by waiting under this hold. If the target is already met, the audit says so; RNA recovery may subsequently reverse that result. Both delay and residual synthesis can contribute at the same time.

## Matched-endpoint comparison

This laboratory is separate from the measured-RNA model. Assume each hypothesis imposes an instantaneous and sustained RNA fraction \(r_i\) from time zero, with effective half-life \(H_i\), \(p_i(0)=1\), and a shared protein anchor \(p_i(t_a)=y\).

\[
\lambda_i=\ln2/H_i,\qquad
r_i=\frac{y-e^{-\lambda_i t_a}}{1-e^{-\lambda_i t_a}}.
\]

Feasibility requires \(e^{-\lambda_i t_a}\le y<1\), with \(0\le r_i<1\). A negative inferred residual fraction is rejected, not clamped. This is algebraic construction from an anchor, not an empirical parameter fit.

With \(H_A\ge H_B\), \(r_A\le r_B\): a slower-turnover explanation of the same endpoint requires deeper sustained suppression. Their different RNA histories must not be portrayed as both fitting the same reliably measured functional RNA course. Even a perfectly matched protein endpoint does not erase that distinction.

Default synthetic example: \(t_a=24\) h, \(y=0.75\), \(H_A=48\) h, \(H_B=12\) h.

| Derived quantity | A: slower turnover | B: faster turnover |
|---|---:|---:|
| Residual RNA and eventual protein | 0.1464466094 | 0.6666666667 |
| Protein at 72 h | 0.4482233047 | 0.671875 |
| Time to 50% protein | 61.03455855 h | Unreachable |

The instantaneous decline at the matched anchor is \(\lambda_i(y-r_i)\). A longer half-life need not imply a smaller decline at that anchor because the residual synthesis levels also differ. No hypothesis is inferred to be a particular delivery failure, resistant subpopulation, or therapeutic mechanism.

## Follow-up separation rule

A user-supplied allowance \(\varepsilon\) is placed around each fixed prediction. On the post-anchor interval, disjoint allowances require

\[
D(t)=p_B(t)-p_A(t)>2\varepsilon.
\]

Allowances are clipped to 0–1 for display. They are deterministic tolerances, not estimated confidence intervals, measurement-error distributions, or a power calculation. No parameter uncertainty is included in this companion module.

For the ordered, matched sustained steps, \(D(t)\) is nondecreasing after \(t_a\). To see this, write

\[
-\dot p_i(t)=(1-y)
\frac{\lambda_i e^{-\lambda_i t}}{1-e^{-\lambda_i t_a}}.
\]

The factor on the right decreases with \(\lambda_i\) when \(t\ge t_a\). Its log-derivative is \(1/\lambda_i-t-t_a/(e^{\lambda_i t_a}-1)<0\). Thus the slower-rate hypothesis A declines at least as fast after the common anchor, so \(D'(t)\ge0\). Bisection locates the touching boundary \(D=2\varepsilon\); the intervals are disjoint only after it.

The default \(\varepsilon=0.05\) gives a boundary near 45.13519410 h. At the boundary the intervals touch. If \(r_B-r_A\le2\varepsilon\), no arbitrarily long follow-up separates these fixed hypotheses by this rule. If they could separate eventually but not within the chosen horizon, that distinction is reported.

This experiment-selection aid does not establish statistical distinguishability. Reliable RNA measurements or independently measured turnover may already reject either hypothesis, and experimental precision, baseline uncertainty, parameter uncertainty, and sampling cost must be considered separately.

## Export separation and compatibility

Main scenarios use `protein-clock-check/1`; comparison scenarios use `protein-clock-comparison/1`. The two formats are not interchangeable. Importing the main v0.1.0-alpha format adds the explicit 72-hour hold-duration default, without changing its RNA input or main forward trajectory. Comparison imports validate feasibility and recalculate residual RNA and all outputs.

## Bounded RNA and turnover uncertainty

The uncertainty audit extends, rather than silently changes, the fixed-hypothesis reference. It propagates bounded sets of sustained RNA fractions and effective half-lives, conditioning each set on a declared protein-anchor interval. It is set membership, not Bayesian conditioning, a likelihood fit, or a confidence interval. Defaults are synthetic stress-test values; no universal assay precision is implied.

RNA is bounded multiplicatively, using a user-supplied log-base-two half-width \(w\):

\[
r\in[r_c2^{-w},\min(1,r_c2^w)].
\]

This is a bound on the final normalized RNA quantity. Exponential conversion from quantification cycles, reference-gene normalization, amplification efficiency, and inter-run calibration all contribute to uncertainty in relative qPCR quantities; a raw technical-replicate Cq standard deviation is not a substitute for the final bound ([Hellemans et al., 2007](https://genomebiology.biomedcentral.com/articles/10.1186/gb-2007-8-2-r19)). The tool does not reproduce qBase or calculate uncertainty from raw qPCR data.

The half-life interval is \(H\in[H_c/f,H_cf]\), where \(f\ge1\). The corresponding rate interval is reversed: \(\lambda\in[\ln2/(H_cf),\ln2/(H_c/f)]\). These are uncertain parameter values, not a population of cells with different turnover rates. Measurement limitations and biological context both matter in turnover estimation ([Ross et al., 2021](https://pmc.ncbi.nlm.nih.gov/articles/PMC7950106/)); the model does not estimate a target-specific error distribution.

The protein anchor has tolerance \(\eta\):

\[
y_L=\max(0,y-\eta),\qquad y_U=\min(1,y+\eta).
\]

Within each hypothesis, retain the joint set

\[
\mathcal F=\{(H,r):H_L\le H\le H_U,\ r_L\le r\le r_U,\
y_L\le r+(1-r)2^{-t_a/H}\le y_U\}.
\]

This constraint prevents the common error of independently perturbing RNA and turnover after fitting an exact protein anchor and then pretending that every new trajectory still matches it. No frequency, likelihood, or posterior weight is assigned within \(\mathcal F\). An empty set means inconsistency of these bounds with this model, not proof of an alternative mechanism.

### Independent RNA evidence versus hypothetical RNA

In hypothesis-specific mode, A and B have separate user-entered RNA centers. Initially they match the synthetic nominal construction; the copy-centers button updates them explicitly. These centers are assumptions, not independent measurements. In shared mode, the same entered RNA interval constrains both parameter sets. An independently justified sustained functional RNA interval may eliminate either or both families.

A single qPCR point cannot establish a constant RNA history or equivalence to functional RNA. The RNA interval is one constant unknown input throughout each trajectory, not independent noise redrawn at every time point. Values are restricted to suppression (0–1); this companion does not model RNA overshoot. A zero center remains zero under multiplicative uncertainty, so a below-quantification-limit observation must not be entered as proven zero. Detection-limit censoring requires a different input model.

### Continuous parameter extrema without Monte Carlo sampling

For fixed \(H\), define \(e_a=2^{-t_a/H}\). The admissible RNA interval is

\[
\ell(H)=\max\left(r_L,\frac{y_L-e_a}{1-e_a}\right),\qquad
u(H)=\min\left(r_U,\frac{y_U-e_a}{1-e_a}\right).
\]

The implementation first restricts the half-life interval using the monotonicity of anchor protein in both \(H\) and \(r\). A fixed-\(r\) anchor intersection, when \(r<y<1\), has

\[
H^*(r,y)=\frac{t_a\ln2}{\ln[(1-r)/(y-r)]}.
\]

Lower and upper predictions at time \(t\) are \(P(t;H,\ell(H))\) and \(P(t;H,u(H))\). On a piece where the active constraint is constant RNA, prediction is monotone in \(H\). Where the active constraint is an anchor boundary \(y_b\), it is

\[
P(t;H,y_b)=1-(1-y_b)
\frac{1-2^{-t/H}}{1-2^{-t_a/H}}.
\]

For fixed \(t\), this expression is monotone in \(H\), with direction depending on which side of \(t_a\) is inspected; at \(t=t_a\) it equals \(y_b\). Therefore extrema occur at feasible half-life endpoints or intersections of an RNA bound with an anchor bound. The implementation evaluates those candidates directly, retaining the parameters generating each extreme. This is a continuous-parameter envelope, to floating-point precision, not a quantile or a sampled half-life envelope. Numerical boundary tolerance is \(2\times10^{-12}\) in the feasibility arithmetic.

The CSV exports the lower/upper predictions and their generating \(H,r\) pairs. Pointwise extrema can use different parameter pairs at different times; the plotted envelope boundary need not itself be a physical trajectory. Displayed marginal RNA and half-life ranges are not a Cartesian product of admissible pairs.

### Robustness of the follow-up

At the inspection time, denote the parameter envelopes by \([L_A,U_A]\) and \([L_B,U_B]\). The possible B-minus-A difference interval is conservatively

\[
[L_B-U_A,\ U_B-L_A].
\]

With prospective protein-readout allowance \(\varepsilon\) on each prediction, define the separation margin

\[
S(t)=\max(L_B-U_A,\ L_A-U_B)-2\varepsilon.
\]

A positive margin means all expanded predictions are disjoint under the declared bounds; zero means touching. The numerical implementation requires \(S>10^{-12}\) to avoid reporting round-off as separation. A nonpositive margin means some combinations remain unresolved, not that every combination is indistinguishable.

The parameter-envelope plot does not include prospective readout allowance; the decision text applies it explicitly. Anchor tolerance and prospective readout allowance belong to different measurements and should not be populated twice with the same already-propagated error. All combinations across bounds are retained, with no probabilistic independence assumption. Known shared-baseline correlations could tighten this conservative box treatment; unmodeled bias, changed RNA history, translation, or turnover could invalidate it.

For follow-up search, evaluate 481 equally spaced times from anchor to horizon, plus the exact inspection time. Report the first checked disjoint time, not an exact earliest boundary or guaranteed persistent separation. Fixed-hypothesis monotonic separation is not assumed for the envelope comparison.

RNA-only and turnover-only displays keep the same anchor tolerance while fixing the other input at its center. Their widths are not additive variance components. An empty one-factor set is reported rather than replaced by the joint set.

### Default uncertainty example

Retain the nominal 75% protein anchor at 24 h, half-lives 48/12 h, and their nominal residual RNA centers. Add \(w=0.5\), \(f_A=f_B=1.25\), and anchor tolerance \(\eta=0.03\). At 72 h, A spans 39.51–50.62% protein and B spans 59.72–73.42%. The minimum B-minus-A gap is 9.10 percentage points, less than the 10 points required for nonoverlap after ±5-point readout allowances. The nominal curves separate, but this stronger robustness criterion does not.

The first checked disjoint follow-up is 75 h on the default 0.3-hour grid. These calculations are synthetic consequences of selected bounds, not recommended assay precision, a sampling prescription, or published biological measurements.

The separate `protein-clock-uncertainty/1` JSON includes both the fixed-comparison configuration and uncertainty inputs. Import validates and recomputes everything, updates the fixed comparison, and leaves the main measured-RNA driver unchanged. Original comparison JSON files contain no uncertainty settings; v0.2.0-alpha comparison files and v0.1.0/v0.2.0-alpha main files remain readable.

## Shared protein baseline and correlated measurement errors

This separate measurement audit inherits the sustained-step biological families and their RNA/turnover bounds. It does not change the measured-RNA forward model. The prior rectangular anchor tolerance and prospective allowance are replaced, not added. Those inputs remain active in the earlier uncertainty panel for comparison. This panel does not turn bounded input knowledge into a probability distribution.

### A single denominator across a candidate trajectory

Let \(b\) be the measured protein reference divided by the true baseline reference, and let numerator errors be expressed relative to true baseline protein:

\[
Y_a=\frac{P(t_a;H,r)+e_a}{b}=y,\qquad
Y_t=\frac{P(t;H,r)+e_t}{b},\qquad
b\in[b_c/f_b,b_cf_b].
\]

The same \(b\) applies to both measurements within each candidate trajectory. Values above one depress normalized readouts at a fixed numerator. An uncertain systematic baseline factor is not fresh noise at each time and is not divided by the square root of a replicate count. Each competing explanation is profiled over its own feasible nuisance-factor values; the audit does not assume both biological hypotheses are simultaneously true with different physical denominators.

Protein-reference uncertainty does not rescale the RNA bounds. RNA and protein assays need not share a reference. The residual error budget below must exclude the denominator error already assigned to \(b\). All combinations of the declared baseline interval and residual-error region are permitted; dependency between those two sources is not separately modeled.

Shared input quantities can induce correlation among derived measurements, and dependency belongs in the measurement equations rather than being discarded during propagation ([Kessel and Kacker, 2009](https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=902877)). The implementation below is an original bounded-set construction, not a reproduction of that paper's uncertainty-estimation procedures.

### Correlated residuals without invented confidence coverage

For positive anchor scale \(\sigma_a\), nonnegative follow-up scale \(\sigma_t\), radius \(k\), and residual correlation parameter \(\rho\), define \(z=e_a/\sigma_a\). When \(|\rho|<1\) and \(\sigma_t>0\), the permitted joint errors satisfy

\[
z^2+\frac{(e_t/\sigma_t-\rho z)^2}{1-\rho^2}\le k^2.
\]

Equivalently, at a given \(z\in[-k,k]\),

\[
e_t\in \sigma_t\left[\rho z-\sqrt{1-\rho^2}\sqrt{k^2-z^2},\
\rho z+\sqrt{1-\rho^2}\sqrt{k^2-z^2}\right].
\]

At \(\rho=\pm1\), the region becomes the line \(e_t=\sigma_t\rho z\). At \(\sigma_t=0\), follow-up numerator error is exactly zero. The software requires \(0.0001\le\sigma_a\le0.25\), \(0\le\sigma_t\le0.25\), \(-1\le\rho\le1\), and \(0.1\le k\le4\). These are software bounds, not estimates of assay performance.

This is a covariance-shaped error region. Its scales can be calibrated standard deviations only if justified externally; by default they are shape parameters for a bounded sensitivity audit. No density, likelihood, posterior, confidence level, or coverage probability is assigned. In particular, \(k=2\) in two dimensions is not automatically a 95% contour. At \(\rho=0\), the region is still an ellipse, not the earlier rectangular allowance model.

Correlation is between anchor and follow-up numerator residuals, not between A and B or between RNA and protein. The audit specifies a pairwise error geometry for each follow-up, not a full multi-time stochastic covariance process. At the anchor time, the follow-up represents a new replicate, not the identical anchor observation. Perfect positive correlation with equal error scales recovers the same normalized anchor exactly.

### Conditioning and exact inner extrema

For fixed \(H,b\), let \(E_a=2^{-t_a/H}\), \(Q_a=1-E_a\). The anchor enforces

\[
r(z)=\frac{by-E_a-\sigma_a z}{Q_a}.
\]

Intersect the resulting RNA constraint with the joint-error radius:

\[
z_L=\max\left[-k,\frac{by-E_a-Q_ar_U}{\sigma_a}\right],\qquad
z_U=\min\left[k,\frac{by-E_a-Q_ar_L}{\sigma_a}\right].
\]

A node is feasible only when \(z_L\le z_U\). Feasibility of the full family is first screened continuously using monotonicity of protein in \(H,r\), then feasible half-life boundaries are calculated for each baseline node.

At follow-up, write \(E_t=2^{-t/H}\), \(Q_t=1-E_t\), and

\[
C=E_t+\frac{Q_t}{Q_a}(by-E_a),\quad
A=\sigma_t\rho-\sigma_a\frac{Q_t}{Q_a},\quad
B=\sigma_t\sqrt{1-\rho^2}.
\]

The two conditional readout boundaries are

\[
Y_t^\pm(z)=\frac{C+Az\pm B\sqrt{k^2-z^2}}{b}.
\]

The lower minimum and upper maximum occur at interval endpoints or the admissible stationary points

\[
z_\pm=\pm\frac{kA}{\sqrt{A^2+B^2}}.
\]

The implementation evaluates these candidates directly, including the degenerate linear cases. Consequently, RNA/error extrema are exact at each sampled \(H,b\) node to floating-point precision. Exported witnesses retain \(H,r,b,e_a,e_t\); these reconstruct the anchor, the displayed readout extreme, and the error-region constraint.

### Outer profiling and numerical limits

Unlike the preceding continuous two-parameter envelope, this extension samples the outer \(H,b\) domain. Standard resolution uses up to 65 logarithmic half-life nodes for each of 33 baseline nodes; fine resolution uses 129 by 65. Feasible endpoints, central values, and additional RNA/anchor intersections are included where applicable. Equal bounds collapse to one node.

The plot evaluates 61 post-anchor times plus the exact inspection time. Its boundaries can come from different feasible trajectories at different times. It is not a simulated noise trace, and the audit does not search for an exact earliest separating follow-up.

An inspection diagnostic compares extrema with a lower-resolution sweep (33 by 17, or 65 by 33). The maximum change is reported in percentage points. It is not a guaranteed error bound, and zero change does not prove that the global extrema have been found.

For observed ranges \([L_A,U_A]\), \([L_B,U_B]\), define the sampled signed gap

\[
G=\max(L_B-U_A,\ L_A-U_B).
\]

Positive \(G>10^{-10}\) means disjoint sampled ranges only. Missed outer extrema can shrink or remove that gap, so sampled disjointness is not a continuous-parameter separation certificate. These ranges already include their declared measurement errors; do not add the old prospective allowance again. Negative or above-one observed values are retained rather than clipped. The biological protein and RNA model remains suppression-only.

### Baseline cancellation is useful, not a mechanism identifier

With exactly the same denominator,

\[
\frac{Y_t}{Y_a}
=\frac{P_t+e_t}{P_a+e_a}.
\]

The denominator cancels algebraically from this ratio. Numerator errors do not cancel, nor does the ratio uniquely determine RNA suppression or turnover. Inference conditioned on an absolute normalized anchor still depends on its unknown baseline factor. Separate denominators, reference drift, treatment-dependent translation or loss, and an incorrect RNA history are outside this observation model.

Positive correlation can offset part of the anchor-conditioned error in the coefficient \(A\), but it need not monotonically improve biological discrimination. The interface therefore compares the selected correlation with \(\rho=0\) at the same bounds, and compares profiling the baseline with fixing it at the selected center. A fixed center may be incompatible even when the full baseline interval contains feasible trajectories.

### Synthetic default case and reproducibility

Retain the earlier 24 h/75% anchor, 48/12 h half-life centers, RNA centers, and RNA/turnover bounds. Set \(b_c=1\), \(f_b=1.1\), \(\sigma_a=0.015\), \(\sigma_t=0.025\), \(\rho=0.6\), \(k=2\). At 72 h, the standard numerical profile gives A 33.57–55.04% and B 55.01–75.37% observed normalized protein. The sampled gap is −0.027 percentage points: a slight overlap that should not be rounded into separation.

With baseline fixed at one, the corresponding ranges are A 39.66–50.37% and B 59.82–73.85%. With baseline profiled and zero residual correlation, they are A 32.46–56.35% and B 52.38–77.30%. These are synthetic consequences of assumptions, not published assay data or general claims about typical bias/correlation.

Measurement JSON uses `protein-clock-measurement/1`, includes the fixed comparison, RNA/turnover bounds, and all measurement inputs, and recomputes results on import. Import updates those comparison panels but not the main measured-RNA model. CSV exports profiles and extremum witnesses; annotated PNG exports state the assumptions and numerical limitations. Earlier v0.3.0-alpha uncertainty files and earlier main/comparison files remain readable. Editing either upstream inputs or measurement assumptions disables stale measurement exports.

## Prospective design comparison, sensitivity map, and anchor-relative diagnostic

The v0.5 companion implements three practical refinements: an explicit comparison of experimental-assumption changes; a baseline-width/correlation map; and a re-expression of observed predictions relative to the measured anchor. These are original computations built on the observation and biological equations above. No empirical calibration or new biological mechanism is claimed.

### Change the assumption, not the historical datum

Let \(a_b,a_e\in[0,1]\) be fractions retained after hypothetical improvements and let \(\Delta t\ge0\). Define

\[
f_b'= \exp(a_b\ln f_b),\qquad
\sigma_t'=a_e\sigma_t,\qquad
t'=t+\Delta t.
\]

Tighter calibration changes the log half-width of the baseline interval around the same \(b_c\), not the denominator's central value. At \(a_b=0\), it fixes the baseline at the chosen center; at \(a_b=1\), nothing changes. Independent calibration evidence is needed to justify any narrowed bound. A real calibration result may move the center, which this comparison deliberately holds fixed.

Improved precision scales only the prospective numerator-error parameter \(\sigma_t\). It does not retroactively improve the historical anchor, remove shared baseline bias, narrow the RNA/turnover ranges, or change the joint radius or residual correlation. It is not converted into a replicate count using an independence assumption. If a new protocol changes correlation or introduces bias, edit the measurement model explicitly.

The later-readout strategy evaluates an anchor-conditioned prediction at \(t'\), retaining sustained RNA, constant turnover, and the same measurement-error geometry. It does not assimilate an additional observed value and does not quantify the joint information from two follow-ups. Joint inference would require the additional observations and a valid multi-time error model. Times outside the declared horizon are unavailable, not clipped or extrapolated.

The table shows current design, each one-factor change, and all three changes together. For each strategy \(s\), report

\[
G_s=\max(L_{B,s}-U_{A,s},L_{A,s}-U_{B,s}),\qquad
\Delta G_s=G_s-G_0.
\]

Every row includes its coarse/fine extremum-change diagnostic. Changes have unspecified and generally unequal costs, so the table is not cost-adjusted optimization, expected information gain, statistical power, or a recommended experiment. Empty families yield no comparable gap rather than artificial success. All numerical-profile caveats from the measurement audit apply.

### Why reducing one error scale need not monotonically increase this gap

At each \(H,b\), recall the anchor-conditioned coefficient

\[
A=\sigma_t\rho-\sigma_a Q_t/Q_a.
\]

Reducing \(\sigma_t\) also changes how the follow-up error covaries with the uncertain anchor error. The conditional prediction region is not necessarily a nested shrinkage of the old one: cancellation in \(A\) can be weakened even as the marginal follow-up error scale falls. This is a feature of the stated observation model, not evidence that intrinsically better assays are harmful.

For a synthetic counterexample, retain the default biology and error scales, fix \(b=1\), and set \(\rho=0.9\). At 72 h, halving \(\sigma_t\) from 0.025 to 0.0125 increases A's sampled prediction width from 6.58 to 7.12 percentage points. The sampled separation gap falls from 13.48 to 13.10 points. The covariance structure, not simply the marginal error size, determines this conditional sensitivity. This case is covered by a regression test.

### Baseline-width versus residual-correlation map

For a user-selected maximum fold width \(F\in[1,2]\), the axes are

\[
f_{b,i}=F^{i/10},\quad i=0,\ldots,10;\qquad
\rho_j=-1+j/5,\quad j=0,\ldots,10.
\]

Each of 121 cells recomputes the measurement profile at the current inspection time, current error scales, current baseline center, and current RNA/turnover bounds, changing only \(f_b,\rho\). Proposed strategy improvements do not silently modify the map. Feasible parameter families are reusable across correlations because anchor feasibility does not depend on \(\rho\) or \(\sigma_t\); inner extrema are still recomputed for each cell.

The cell exposes observed ranges, sampled signed gap, refinement change, and four generating witnesses. Teal plus signs indicate positive sampled gaps, amber minus signs indicate overlap/touch, and a cross indicates incompatibility. Color magnitude is scaled to the largest absolute gap within that map and is not directly comparable between separate maps. No statistical or universal mechanism classification is encoded.

The map is a discrete assumption grid with no interpolated guarantee between cells. The outer profile within each cell is itself numerical. Refinement changes are not error bounds, and positive cells are not continuous-parameter separation certificates. Setting \(F=1\) intentionally collapses the baseline dimension to repeated fixed-center calculations.

### Anchor-relative re-expression cannot create discrimination

The observed anchor is fixed at the positive value \(y\) while its compatible biological states remain uncertain. Re-express each observed follow-up range as

\[
R_i=Y_{t,i}/y,\quad
R_i\in[L_i/y,U_i/y],\quad
D_i=1-R_i\in[1-U_i/y,1-L_i/y].
\]

Here \(D_i\) is apparent decline relative to the observed anchor, not true-protein depletion. Negative apparent decline and ratios above one are retained. Algebraically, the same physical denominator cancels within each candidate ratio:

\[
R_i=\frac{P_{t,i}+e_{t,i}}{P_{a,i}+e_{a,i}}.
\]

Yet baseline uncertainty still affects which biological trajectories can explain the absolute anchor. The tool does not refit the model while discarding that information. Since both prediction intervals are divided by the same positive observed constant,

\[
G_R=G_Y/y,
\]

and overlap status is unchanged. The interface inherits the original numerical separation classification rather than introducing a second floating-point threshold in different units. This diagnostic is a re-expression, not an independent observation, and must not be double-counted as evidence. Distinct denominators or reference drift break the simple cancellation.

### Default comparison and reproducibility

With default inputs, retain half the baseline log-width, halve the follow-up error scale, and delay the readout by 24 h. The baseline fold width becomes \(\sqrt{1.1}=1.04881\), follow-up scale becomes 0.0125, and the later time is 96 h. Standard-resolution sampled gaps are:

- **Current design:** −0.027 percentage points at 72 h.
- **Tighter baseline calibration:** 3.984 points at 72 h.
- **Better follow-up precision:** 2.645 points at 72 h.
- **Later readout:** 6.366 points at 96 h.
- **All three changes:** 12.102 points at 96 h.

These are synthetic conditional results, not a ranking of real experiments or a guarantee that waiting is best. Strategy CSV records ranges, gaps, changes, assumptions, and refinement diagnostics. Map CSV records all 121 cells and their witnesses. Annotated PNG preserves map assumptions and the selected cell; it is a map export, not a complete experimental-design report.

The `protein-clock-design/1` JSON preserves the fixed comparison, RNA/turnover uncertainty, measurement assumptions, and design controls. Import validates and recomputes all comparison panels, leaving the main measured-RNA model unchanged. An explicit “use this cell” action applies its baseline width and correlation to the measurement audit; it invalidates prior design results until recalculation. Older v0.4.0-alpha measurement JSON remains readable but contains no design controls. Upstream or local edits disable stale exports and cell application.

## Worked example: calibration against another day

Two synthetic cases are computed in `web/example.js` by the same `designPoint` used for the design comparison, so the numbers on `worked-example.html` cannot drift from the workbench.

Plateau case. Anchor y = 0.75 at 24 h; hypothesis A has H = 16 h with matched sustained RNA r_A = (y − 2^{−24/16}) / (1 − 2^{−24/16}) ≈ 0.61327, hypothesis B has H = 4 h with r_B ≈ 0.74603; half-life bounds ×/÷1.08 each; RNA log2 half-width 0.25; shared baseline center 1 with range ×/÷1.08; anchor and follow-up error scales 0.015 and 0.025; ρ = 0.6; radius 2; follow-up at 96 h; horizon 288 h. The central trajectories are p_A(t) = r_A + (1 − r_A) 2^{−t/16} and p_B(t) = r_B + (1 − r_B) 2^{−t/4}, so at 96 h both lie within 0.7 pp and 0.0001 pp of their plateaus. At standard resolution the current design gives sampled gap ≈ −0.7541 pp; retaining half the baseline log-width (×/÷1.08^{0.5} ≈ 1.0392) gives ≈ +1.0199 pp; halving the follow-up error scale gives ≈ +2.1682 pp; waiting 24 h to 120 h gives ≈ −0.5647 pp; the sampled gap over 96–288 h is non-decreasing and converges to ≈ −0.4733 pp at the horizon. Every readout inside the horizon therefore overlaps under these bounds while the single calibration change does not.

Contrast case. The shipped defaults (H = 48 h and 12 h, follow-up 72 h, baseline ×/÷1.10) give current ≈ −0.0273 pp, calibration ≈ +3.9841 pp, later readout at 96 h ≈ +6.3663 pp. The ranking reverses because the slower trajectory is still falling steeply at 72 h.

Threshold. For fraction f of retained log-width the baseline range is ×/÷F^f at fixed center. The page bisects for the largest f with positive sampled gap at the current follow-up time and reports the gap immediately on both sides of the crossing (plateau case f ≈ 0.7833, i.e. ×/÷1.0621; contrast case f ≈ 0.9942). Monotonicity of the gap in f is assumed by the bisection and checked in tests on an eleven-point grid, not proven in general. Edge cases are reported explicitly: f = 1 when the current design already separates, and no threshold when even an exactly known baseline leaves overlap.

Separation budget. The sampled gap admits an exact decomposition rather than only a sign. Let the two central predictions at time t be a(t) and b(t) in observed units, a(t) = (E_t + (Q_t/Q)(β y − E)) / β with E = 2^{−anchor/H}, Q = 1 − E, E_t = 2^{−t/H}, Q_t = 1 − E_t and β the declared baseline center; call the hypothesis with the larger central value the upper one. Write D = central separation = upper central − lower central, R_up = upper central − min of the upper sampled range, R_down = max of the lower sampled range − lower central. Then

    sampled gap = D − R_up − R_down.

This is an accounting identity over the same enumerated profile family, not an approximation: `separationBudget` recomputes the sampled gap independently and reports the residual, which is exactly 0 in double precision for every strategy of both shipped cases. It is what makes the two levers distinguishable rather than merely rankable. Tighter baseline calibration leaves D untouched and can only shrink R_up + R_down, because the shared baseline enters both predictions through the same factor. A later readout adds to D but simultaneously widens both reach terms, since the anchor-conditioned ranges are extrapolated further from the measured anchor. In the plateau case calibration converts 1.774 pp of reach into gap with D fixed at 12.672 pp, while a 24 h delay adds 0.391 pp to D and gives 0.201 pp of it back as extra reach. The identity also explains why the central prediction always lies inside its own sampled range: `centralInside` is true for every strategy and every cell of the regime grid.

Plateau timing. The ceiling on D follows from the sustained step in closed form. With matched sustained-RNA centers r_A and r_B, the central protein distance is d(t) = [r_B + (1 − r_B) 2^{−t/H_B}] − [r_A + (1 − r_A) 2^{−t/H_A}], which increases to the plateau difference L = r_B − r_A as t → ∞. `plateauDiagnostics` reports L, the realized share d(inspect)/L, the residue L − d(inspect) available to all future time, the part of that residue lying inside the declared horizon, d(inspect + delay) − d(inspect) for one delay step, the follow-up expressed in half-lives of each hypothesis, and the times at which 95% and 99% of L are realized, obtained by 80-step bisection of d on [anchor, max(horizon, 128 H_slow)] with explicit reporting when the target is not reached in that bracket or when the plateaus coincide. For the plateau case L = 13.276 pp, of which 95.4% is already realized at 96 h (6.00 slower half-lives), with t95 = 93.8 h and t99 = 131.0 h; all remaining time to infinity can add at most 0.604 pp. For the contrast case L = 52.022 pp, only 43.0% is realized at 72 h (1.50 slower half-lives), t95 = 241.7 h lies beyond the 168 h horizon, and the next 24 h are worth 8.448 pp. These quantities are properties of the central trajectories alone and carry no uncertainty statement.

Exchange rates. Three one-factor equivalences are solved against the sampled gap, each holding every other declared value fixed. `equivalentWait` bisects the readout time on [inspect, horizon] for the earliest time matching a target gap, and refuses with the gap at the horizon when no in-horizon readout reaches it — the plateau case refuses with a horizon gap of −0.4733 pp, while the contrast case needs 14.7 h past the current follow-up to match calibration. `equivalentCalibration` bisects the retained log-width fraction f on [0, 1], where the baseline range is ×/÷F^f at fixed center, and refuses with the gap at f = 0 when even an exactly known baseline cannot match the target; the plateau case needs f = 0.942 (×/÷1.075) to match one delay step, the contrast case f = 0.291 (×/÷1.028). Both searches assume the gap is monotone in their argument over the scanned interval, which tests check on a grid rather than prove. `replicateEquivalent` converts a fraction into n = n₀/f² under an assumed 1/√n scaling of the baseline log half-width and rounds up: from three replicates, the plateau case's f = 0.942 implies 4, its positive-gap threshold f = 0.783 implies 5, and the contrast case's f = 0.291 implies 36. This is deterministic width arithmetic on a declared bound. It is not a power calculation, carries no error rate or coverage, and says nothing about how many replicates a real baseline needs.

Regime map. A single case can only show that the ranking is contingent. `regimeMap` reruns the same one-factor comparison over a grid of slower half-lives × follow-up times with H_fast = H_slow / ratio, rebuilding the matched sustained-RNA centers in every cell so both hypotheses continue to reproduce the declared anchor exactly, and recording per cell the winner by larger sampled gap, the margin between the two changes, whether either change separates, and the follow-up in elapsed slower half-lives. Infeasible kinetics — those for which 2^{−anchor/H} > y, i.e. H > anchor·ln2 / ln(1/y) — are marked unavailable with a reason rather than approximated; for the shipped anchor (y = 0.75 at 24 h) that ceiling is 57.8 h. The family cache is rebuilt per half-life row, because `designPoint` keys its cache only on measurement and grid fields and is therefore valid only while the kinetic and RNA bounds are held fixed. On the shipped 7 × 9 grid the winner flips in six rows, and every crossover is bracketed between 3.0 and 5.0 elapsed slower half-lives: the boundary is dimensionless in t/H_slow rather than fixed in hours. `regimeSummary` reports these brackets from adjacent columns only and never interpolates between them; the resolution of the claim is the spacing of the grid.

Editable case. `buildCase` accepts any subset of the declared inputs, rebuilds both matched sustained-RNA centers from the anchor, and validates through the same scenario, uncertainty, measurement, and design constructors as the workbench, so an infeasible hypothesis is refused with the kinetic reason rather than silently approximated. The page states the implied half-life ceiling for the declared anchor when that is the cause.

What the example establishes. The ranking of calibration against time is determined by how much separation the two central trajectories still have left to generate before the follow-up, relative to the shared-baseline term that calibration shrinks. The budget identity localizes the difference — one lever acts on D, the other only on R_up + R_down — and the plateau diagnostics say how much of D remains purchasable at all. It is a statement about declared bounds, not a general rule, a power calculation, a confidence statement, or a recommendation, and the costs of the two changes are not modeled.

## Boundary conditions and excluded inferences

Piecewise-linear RNA driver; no extrapolation; strictly increasing times beginning at zero; 2–300 points. RNA fraction 0–2; horizon 0.1–720 h; declared half-lives and any doubling time 0.1–10,000 h. These are software guardrails, not biologically endorsed parameter ranges.

No target-specific kinetics, parameter estimation, efficacy, cell killing, delivery/RISC dynamics, secretion, trafficking, feedback, changed translation coefficient, time-varying loss, non-exponential decay, or population-mixture resolution. An independent target/assay validation remains necessary.

## References

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