Task-Fixed pi Readout within a Declared Solar-System Calibration Window

A present-day comoving horizon scale is commonly quoted as a radius of order $46$TeX46--$47$TeX47 billion light-years, while the Earth-inferred cosmic age is commonly taken as $\tau_0\simeq 13.8~\mathrm{Gyr}$TeX\tau_0\simeq 13.8~\mathrm{Gyr}. These quantities are operationally distinct: $\tau_0$TeX\tau_0 is tied to local clocks and calibration conventions, whereas a present-day horizon radius $D_0$TeXD_0 depends on a chosen horizon definition and a chosen parameter set. Reference cosmological parameter sets are quoted in primary cosmological analyses [citation].

Scope..

No FLRW dynamics or standard horizon integrals are re-derived. The problem addressed here is narrower: whether a declared Solar-System calibration-layer hypothesis with shared curvature layering in propagation and clock rate can yield a declared-class relation between a present horizon-scale quantity $D_0$TeXD_0 and the Earth-inferred age $\tau_0$TeX\tau_0 once the admissible window, witness package, and completion task are fixed. The exact symbolic chain is the paper-internal core, whereas any wider DSN-to-Earth-unit calibration claim remains conditional on a witness-backed set of delay, angular, and clock data that is stated but not certified here. The shared map below is used only as a local calibration-layer hypothesis on that declared Solar-System window, and no general propagation law, general clock law, or cosmological horizon theorem is claimed outside that declared class.

Target quantity..

The quantity $D_0$TeXD_0 is treated as a present-day horizon radius expressed in Earth light-year units (Gly). It is not a directly measured local observable; it is a derived scale whose meaning depends on stated conventions. The question is whether the adopted calibration class yields a declared-class relation between $D_0$TeXD_0 and $\tau_0$TeX\tau_0 after the Solar-System window, the admissibility conditions, and the completion task have been fixed.

Calibration identity..

Let

\rho\equiv\Phi(\gamma_{\rm ref})=\frac{V_{\rm read}}{c_\oplus}

be estimated from an independently specified reference-path data set. If (i) a scale-free Solar-System layering profile with exponent $q$TeXq is admissible, (ii) the effective refractive geometry $n(r)\propto 1/\kappa(r)$TeXn(r)\propto 1/\kappa(r) satisfies the solar-window Herglotz monotonicity condition introduced below, and (iii) the same boundary-normalized $\kappa(r)$TeX\kappa(r) survives delay, angular, and clock consistency tests, then the exact algebraic comparison is

D_0=\tau_0\,\rho^2\,F(q).

Here $F$TeXF is the closed-form path factor derived in Sec. reference. The relation is a comparison identity, not a prediction of $D_0$TeXD_0, unless $q$TeXq and $\rho$TeX\rho have been determined without using $D_0$TeXD_0 or $\tau_0$TeX\tau_0 and the horizon definition has been fixed in advance. The point hypothesis $H_\pi:\rho=\pi$TeXH_\pi:\rho=\pi must be tested against the fitted alternative; it is not supplied by the phase-discriminator threshold.

Logical separation of the calibration steps..

Delay/range reconstruction, exponent estimation, inverse-profile testing, detector phase closure, and cosmological comparison are distinct operations. The exponent $q=3/4$TeXq=3/4 follows only from the specified shell-content hypothesis. The completion constant $\mathcal C=\pi$TeX\mathcal C=\pi is a property of a binary first-harmonic discriminator. It does not fix $\rho$TeX\rho, which remains a path observable.

Companion metrology-reference note..

The CCL/PTM-VP0 metrology-reference result supports only CCL-METROLOGY-REFERENCE-CHECK-SATISFIED on the declared BIPM--NIST/CODATA--IAU reference-check surface, with the shared umbrella label CCL-PTM-VP0-REFERENCE-CHECK-SATISFIED. It confirms the declared symbolic/metrological chain for the reference constants used by the CCL/PTM readout; it does not certify any wider delay--angular--clock empirical witness outside the declared certificate surface, does not assign the Earth-unit calibration outside the declared class, and does not convert the construction into a general propagation, clock, horizon, or SI-redefinition theorem.

Logical dependency of the comparison..

Lemma reference fixes the harmonic path functional; Proposition reference gives $q=3/4$TeXq=3/4 only under its shell-extensivity premise; Theorem reference fixes the detector threshold; Theorem reference proves that this threshold cannot determine $\rho$TeX\rho; and Proposition reference gives $R=F(q)\rho^2$TeXR=F(q)\rho^2. The empirical value of $\rho$TeX\rho must be supplied by an independent likelihood.

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Symbolic dependency boundary..

Table reference separates mathematical identities from parameters that require data. Delay/range data estimate the path response; angular/VLBI and clock-transfer data test the same Solar-System map.

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Shared distortion factor Let $T$TeXT denote global time and $\tau$TeX\tau denote local (Earth) clock time. We introduce a bounded Solar-System curvature-layering proxy $\Xiloc(r)\in[0,1)$TeX\Xiloc(r)\in[0,1) on the declared propagation window (Definition reference).

definition: Bounded Solar-System curvature-layering proxy. The symbol $\Xiloc(r)$TeX\Xiloc(r) denotes an operational curvature-layering measure used only through the shared distortion map developed here. It is bounded by construction on the declared Solar-System window, and it is not identified here with the unbounded CHC phase-gradient measure $\Xi$TeX\Xi used in dynamical CHC papers. Any analysis that would require values outside $[0,1)$TeX[0,1) lies outside the admissible scope of the present calibration map and is not interpreted within the framework developed here.

Define

\kappa(r)\equiv \sqrt{1-\Xiloc(r)}\in(0,1].

definition: Declared shared calibration-map hypothesis. A path-normalization calibration variable and a local clock-rate factor share the same curvature-layering factor:

v(r)=V\,\kappa(r),\qquad d\tau(r)=\kappa(r)\,dT,

where $v(r)$TeXv(r) and $V$TeXV are internal calibration-layer variables used only inside the declared travel-time functional. They are not modified metric light speeds, local group velocities, signal velocities, or universal propagation speeds.

remark: Single-geometry admissibility of the shared map. Equation reference is used here only as a local calibration-layer hypothesis on the declared Solar-System window. It is not promoted in this paper to a general CHC propagation law, a general clock law, or a cosmological distance law. Within that declared window, propagation observables and clock observables are read through one common geometric structure with no second-clock effect. In the Ehlers--Pirani--Schild viewpoint, light propagation fixes a conformal structure, free fall fixes a projective structure, and the absence of a second-clock effect is what permits a common Lorentzian metric representative [citation]. This motivates treating reference as a hypothesis to be tested by cross-channel equality rather than as an unconditional identification.

Earth-unit path-normalization readout as a path-harmonic mean Solar-System path-normalization readouts are induced by additive travel time along a declared path. We fix the Solar-System propagation window by adopting external conventions for its boundary constants.

definition: Declared Solar-System window and fixed boundary constants. We take $r_\oplus\equiv \mathrm{au}$TeXr_\oplus\equiv \mathrm{au} to denote the heliocentric radius scale of Earth's orbit, where $\mathrm{au}$TeX\mathrm{au} is defined exactly by IAU 2012 Resolution B2 [citation]. We fix the inner boundary as the nominal solar radius,

r_\odot \equiv R_{\odot}^{\mathrm{N}},

recommended as a standard conversion constant by IAU 2015 Resolution B3 [citation]. The associated dimensionless window ratio is

x\equiv \frac{r_\odot}{r_\oplus}=\frac{R_{\odot}^{\mathrm{N}}}{\mathrm{au}},

consistent with reference. The symbol $r_\odot$TeXr_\odot marks the domain boundary of the Solar-System propagation segment; it is not a fit parameter and is not adjusted in the inference below.

definition: Admissible solar-window convexity (Herglotz monotonicity). We call the declared Solar-System profile admissible on the window $[r_\odot,r_\oplus]$TeX[r_\odot,r_\oplus] if

\frac{d}{dr}\left(\frac{r}{\kappa(r)}\right)>0
\qquad\text{for all } r\in[r_\odot,r_\oplus].

Equivalently, for the effective refractive index $n(r)\propto 1/\kappa(r)$TeXn(r)\propto 1/\kappa(r), one has $\frac{d}{dr}(r n(r))>0$TeX\frac{d}{dr}(r n(r))>0, the radial Herglotz condition used in travel-time tomography [citation]. Within the declared radial-isotropic class, this is the monotonicity condition under which the travel-time profile is Abel-invertible.

lemma: No-new-scale lemma for the inner boundary. Within the scale-free Solar-System premise adopted in Sec. reference, allowing the inner boundary $r_\odot$TeXr_\odot to vary introduces an additional adjustable scale into $F(q)$TeXF(q). Therefore $r_\odot$TeXr_\odot is fixed by external convention as in Definition reference.

proof. The closed-form path factor $F(q)$TeXF(q) depends on the dimensionless ratio $x=r_\odot/r_\oplus$TeXx=r_\odot/r_\oplus through reference. Freedom in $x$TeXx would shift the comparison continuously and constitutes an extra scale beyond the premise used to fix $q$TeXq (Proposition reference). Fixing $r_\odot$TeXr_\odot as a standard nominal constant removes that nuisance degree of freedom.

For a radial path from $r_\odot$TeXr_\odot to $r_\oplus$TeXr_\oplus,

t_{\mathrm{prop}}
=\int_{r_\odot}^{r_\oplus}\frac{dr}{v(r)}
=\frac{1}{V}\int_{r_\odot}^{r_\oplus}\frac{dr}{\kappa(r)}.

definition: Path-harmonic Earth-unit path-normalization readout. Define the Earth-unit path-normalization readout

c_\oplus \equiv \frac{L}{t_{\mathrm{prop}}}=V\,\kappa_{\mathrm{path}}, 

\quad \kappa_{\mathrm{path}}\equiv\frac{L}{\displaystyle\int_{r_\odot}^{r_\oplus} dr/\kappa(r)}, 

\quad L\equiv r_\oplus-r_\odot.

lemma: Uniqueness of the harmonic mean for additive travel time. If a travel time accumulates additively along a path, $t_{\mathrm{prop}}=\int_0^{L} d\ell/v(\ell)$TeXt_{\mathrm{prop}}=\int_0^{L} d\ell/v(\ell), then the unique path-average calibration-rate variable consistent with $L=\bar v\, t_{\mathrm{prop}}$TeXL=\bar v\, t_{\mathrm{prop}} is the harmonic mean:

\bar v \;=\;\frac{L}{t_{\mathrm{prop}}}
\;=\;\left(\frac{1}{L}\int_0^{L}\frac{d\ell}{v(\ell)}\right)^{-1}.

proof. The relation $L=\bar v\,t_{\mathrm{prop}}$TeXL=\bar v\,t_{\mathrm{prop}} determines $\bar v=L/t_{\mathrm{prop}}$TeX\bar v=L/t_{\mathrm{prop}}. With additive travel time

t_{\mathrm{prop}}=\int_0^L \frac{d\ell}{v(\ell)},

substitution gives

\bar v
=
\frac{L}{t_{\mathrm{prop}}}
=
\left(\frac{1}{L}\int_0^L\frac{d\ell}{v(\ell)}\right)^{-1}.

No other path-average calibration-rate variable is compatible with both additive travel time and the defining relation $L=\bar v\,t_{\mathrm{prop}}$TeXL=\bar v\,t_{\mathrm{prop}}.

definition: DSN phase-count and Doppler observables. Fix a declared DSN tracking link (one-way, two-way, or three-way; coherent or non-coherent as declared) and a user specified count time $T_c$TeXT_c. Let $P_{\mathrm{obs}}(T_c)$TeXP_{\mathrm{obs}}(T_c) denote the total-count phase observable, i.e.\ the cumulative number of counted cycles over $T_c$TeXT_c in the corresponding DSN Doppler/phase-count data type. The DSN Doppler observable is the corresponding average cycle-rate

D_{\mathrm{obs}}\equiv \frac{P_{\mathrm{obs}}(T_c)}{T_c}\quad (\mathrm{cycles/s}),

and is recorded together with range and ancillary identification metadata (station IDs, band, count time, ramp and turnaround information) in standard navigation data products [citation].

definition: Observed/computed pairing and minimal computed model. Navigation data analysis is performed on paired observed and computed values for the same DSN data type and the same count interval [citation]. For a declared datum, let $P_{\mathrm{cmp}}[\mathcal{M}](T_c)$TeXP_{\mathrm{cmp}}[\mathcal{M}](T_c) denote the computed total-count phase (cycles) produced by a declared model $\mathcal{M}$TeX\mathcal{M} (trajectory, station geometry, media corrections, and RF-system information), and define the computed reference as

P_{0}(T_c)\equiv P_{\mathrm{cmp}}[\mathcal{M}_{\min}](T_c)\quad (\mathrm{cycles}).

Here $\mathcal{M}_{\min}$TeX\mathcal{M}_{\min} is a minimal computed model specification: the smallest declared model for which the computed phase-count is unambiguous and comparable to $P_{\mathrm{obs}}(T_c)$TeXP_{\mathrm{obs}}(T_c). Concretely, $\mathcal{M}_{\min}$TeX\mathcal{M}_{\min} must specify, at minimum,

- the DSN data type and link multiplicity (one-way/two-way/three-way) and whether the link is coherent or non-coherent; - the count interval $T_c$TeXT_c (and the time tags defining its endpoints); - the frequency plan used by the DSN for that interval, including any ramp table and the spacecraft transponder turnaround ratio [citation]; - the reference trajectory/geometry used to compute light-time along the signal path (including the participating stations for three-way links) together with the declared light-time model terms used in the computed observable; - any additional hardware/time-scale delays and calibration terms that are part of the DSN data type definition for the chosen observable (e.g.\ transponder delay, station delays, time-scale conversions) [citation]; - the declared set of propagation and media corrections used in the computed observable (e.g.\ troposphere, ionosphere, solar plasma), or an explicit declaration that a given correction is not applied.

If any required element is missing or ambiguous for a given pass, then $P_{0}$TeXP_{0} is not well-defined for that pass and the pass is excluded from $\Phi$TeX\Phi-based inference.

definition: Admissible phase-count segment. A tracking pass (or sub-arc) is admissible for estimating $\Phi$TeX\Phi and testing the scale-free signature if it satisfies both:

- Model specification: the minimal computed model $\mathcal{M}_{\min}$TeX\mathcal{M}_{\min} in Definition reference is satisfied for the pass, so that $(P_{\mathrm{obs}},P_{0})$TeX(P_{\mathrm{obs}},P_{0}) is an observed/computed pair for the same DSN data type and count interval; - Solar-plasma gate: the Sun--Probe--Earth (SPE) angle is sufficiently large that unmodeled solar-plasma effects cannot masquerade as a curvature-layering signature on the phase count. The DSN Services Catalog notes that tracking accuracy degrades for SPE angles below $10^\circ$TeX10^\circ, and that S-band data are generally unusable below $5^\circ$TeX5^\circ [citation]. Accordingly, a conservative admissibility rule is adopted: we exclude the near-conjunction regime $\mathrm{SPE}<5^\circ$TeX\mathrm{SPE}<5^\circ for all inferences, and unless an explicit dual-frequency or validated solar-plasma calibration is declared as part of $\mathcal{M}_{\min}$TeX\mathcal{M}_{\min} we require $\mathrm{SPE}\ge 10^\circ$TeX\mathrm{SPE}\ge 10^\circ [citation]. If a calibration is available, an extended analysis may relax the $\mathrm{SPE}\ge 10^\circ$TeX\mathrm{SPE}\ge 10^\circ rule while keeping the $\mathrm{SPE}<5^\circ$TeX\mathrm{SPE}<5^\circ exclusion [citation].

This gate prevents the Solar-System domain boundary (Definition reference) from being effectively replaced by a variable cutoff set by conjunction-dependent plasma contamination.

lemma: Functorial count structure of coherent total-count phase. Fix a declared coherent DSN data type and a phase-continuous tracking pass. Let $I\mapsto P_{\mathrm{obs}}(I)$TeXI\mapsto P_{\mathrm{obs}}(I) denote the observed total-count phase on declared reception intervals $I$TeXI within that pass. Then:

- if $I=I_1\sqcup I_2$TeXI=I_1\sqcup I_2 is a disjoint union of contiguous subintervals within the same phase-continuous pass, then

P_{\mathrm{obs}}(I)=P_{\mathrm{obs}}(I_1)+P_{\mathrm{obs}}(I_2);

- the same additivity holds for the computed reference $P_{0}(I)$TeXP_{0}(I) when the observed/computed pairing is declared for the same DSN data type, count interval, frequency plan, and coherent turnaround rule; - repeated subdivision and recombination of a declared interval leaves the total count unchanged.

Hence coherent total-count phase defines a finitely additive, refinement-stable interval functional on declared count intervals.

proof. By Definition reference, total-count phase is a cumulative cycle count on a user-specified count interval. In standard DSN processing, such counts are meaningful only across subintervals on which the counted phase is continuous, and observed/computed values are paired for the same DSN data type and count interval under a fixed frequency plan and, for coherent links, a fixed transponder turnaround ratio [citation]. Concatenating contiguous phase-continuous subintervals therefore concatenates their counted cycles, giving additivity for both observed and computed counts. Since the count on a union is the sum on its pieces, further subdivision changes only the bookkeeping, not the total accumulated count.

lemma: Cycle count and range change over a count interval. A DSN Doppler/phase-count observable is a counted carrier phase-difference measurement; each counted cycle corresponds to one effective carrier wavelength of accumulated phase-path change in the declared observable [citation]. Therefore the total-count phase $P_{\mathrm{obs}}(T_c)$TeXP_{\mathrm{obs}}(T_c) provides a direct measure of propagation-distance change over $T_c$TeXT_c:

\Delta \mathcal{L}(T_c)=\lambda_{\mathrm{eff}}\,P_{\mathrm{obs}}(T_c),

where $\lambda_{\mathrm{eff}}$TeX\lambda_{\mathrm{eff}} is the effective carrier wavelength determined by the declared frequency plan (including turnaround and ramp information) already contained in $\mathcal{M}_{\min}$TeX\mathcal{M}_{\min} (Definition reference) [citation]. The corresponding one-way range change satisfies

\Delta R(T_c)=\frac{\lambda_{\mathrm{eff}}}{\chi}\,P_{\mathrm{obs}}(T_c),

where $\chi$TeX\chi is a fixed link multiplicity factor ($\chi=1$TeX\chi=1 for one-way, $\chi=2$TeX\chi=2 for two-way and three-way coherent links) [citation]. Since $\chi$TeX\chi is fixed by the declared DSN data type, it cannot be tuned to absorb an $r$TeXr-scaling signature used in Sec. reference.

proof. The counted phase change is $\Delta\varphi=2\pi P_{\rm obs}$TeX\Delta\varphi=2\pi P_{\rm obs}. For the fixed effective wavelength in the declared frequency plan, the same phase change is $\Delta\varphi=(2\pi/\lambda_{\rm eff})\Delta\mathcal L$TeX\Delta\varphi=(2\pi/\lambda_{\rm eff})\Delta\mathcal L. Equating the two expressions gives reference. A coherent two- or three-way observable contains two one-way geometric legs, whereas a one-way observable contains one; division by the fixed multiplicity $\chi$TeX\chi gives reference.

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definition: Frequency-free dimensionless layered path functional. For a declared propagation segment $\gamma$TeX\gamma of Euclidean length $L_\gamma\equiv\int_\gamma d\ell$TeXL_\gamma\equiv\int_\gamma d\ell, define the propagation time

t_{\mathrm{prop}}(\gamma)=\int_{\gamma}\frac{d\ell}{v(\ell)}
 =\frac{1}{V}\int_{\gamma}\frac{d\ell}{\kappa(\ell)}.

Define the unlayered reference travel time for the same segment by the model restriction $\kappa(\ell)\equiv 1$TeX\kappa(\ell)\equiv 1:

t_{0}(\gamma)\equiv t_{\mathrm{prop}}(\gamma)\big|_{\kappa\equiv 1}=\frac{L_\gamma}{V}.

For an admissible DSN phase-count pass (Definition reference) let $P_{\mathrm{obs}}(\gamma)$TeXP_{\mathrm{obs}}(\gamma) denote the observed total-count phase (cycles) on $\gamma$TeX\gamma and let $P_{0}(\gamma)$TeXP_{0}(\gamma) denote the corresponding computed reference count produced by $\mathcal{M}_{\min}$TeX\mathcal{M}_{\min} (Definition reference) for the same DSN data type and count interval. The frequency-free, dimensionless phase-ratio functional is

\Phi(\gamma)\equiv \frac{P_{\mathrm{obs}}(\gamma)}{P_{0}(\gamma)}.

Under the matched observed/computed pairing and count-interval contract fixed by Definitions reference and reference, Lemma reference shows that on admissible coherent, non-dispersive segments the operational ratio $\Phi$TeX\Phi is the quotient of two coherent count measures over the same declared interval algebra and, absent any extra admissible channel, is forced to factor through the elapsed-time quotient of the declared segment. In particular, for the Solar-System reference path in reference it yields the dimensionless path-normalization readout $V_{\rm read}/c_\oplus=\Phi$TeXV_{\rm read}/c_\oplus=\Phi by Lemma reference. The observed-minus-computed residual in cycles is $P_{\mathrm{obs}}-P_{0}=(\Phi-1)\,P_{0}$TeXP_{\mathrm{obs}}-P_{0}=(\Phi-1)\,P_{0}, so $\Phi-1$TeX\Phi-1 is the natural dimensionless residual used to test the layered signature against DSN computed observables [citation].

definition: Declared operational interpretation package. We say that a DSN-calibrated inference instance satisfies the declared operational interpretation package if all of the following hold:

- Declared data type and count interval: the link definition, data type, band, and count interval are declared as in Definition reference; - Observed/computed pairing: the computed reference $P_{0}$TeXP_{0} is generated as $P_{0}=P_{\mathrm{cmp}}[\mathcal{M}_{\min}]$TeXP_{0}=P_{\mathrm{cmp}}[\mathcal{M}_{\min}] under the minimal computed model contract in Definition reference, so that observed and computed values refer to the same DSN data type and count interval [citation]; - Admissibility condition: the pass satisfies the admissibility criteria of Definition reference; - Primary inference channel and control comparison: primary inference is performed on coherent two-way/three-way links (F2/F3), while one-way non-coherent links (F1) are retained as a control comparison for oscillator/clock contamination [citation]; - Dispersive-media classification when multi-band is available: if the same declared segment is observed at two or more downlink carrier frequencies, apply the dispersive scaling test in Definition reference; passes classified as dispersive are excluded from curvature-layering inference unless a validated dispersive calibration is explicitly declared as part of $\mathcal{M}_{\min}$TeX\mathcal{M}_{\min} (Definition reference). - Cross-channel consistency when external witnesses are available: if admissible angular/lens data or admissible clock-transfer data exist for the same boundary-normalized Solar-System window, they are used as external witness channels and must be consistent with the $\kappa(r)$TeX\kappa(r) reconstructed from the delay channel in the sense formalized in Definition reference. If the clock witness is available on a closed comparison network, the loop-exactness condition in Definition reference is imposed in addition. Otherwise the shared-map identification reference is rejected for that window. - Declared $\pi$TeX\pi-normalization class: for the present class-bound readout, the normalization is performed on the coherent, non-dispersive admissible reparameterization class of the radial Solar-System segment from $r_\odot$TeXr_\odot to $r_\oplus$TeXr_\oplus underlying reference. We denote this class by $\Gamma_{\pi}$TeX\Gamma_{\pi} and its reference representative by $\gamma_{\mathrm{ref}}$TeX\gamma_{\mathrm{ref}}. This clause introduces no additional geometric scale beyond the segment already fixed by reference; the class-bound readout property used below is established in Lemma reference.

definition: Three-channel acceptance criterion for the shared map. Fix the same Solar-System window, the same boundary normalization, and the same admissibility rules. Let $\kappa_{\mathrm{delay}}(r)$TeX\kappa_{\mathrm{delay}}(r) denote the profile reconstructed from admissible line-of-sight delay/range data. If admissible angular/lens data can be reduced to a boundary-normalized witness profile $\kappa_{\mathrm{angle}}(r)$TeX\kappa_{\mathrm{angle}}(r) on that window, then it must agree with $\kappa_{\mathrm{delay}}(r)$TeX\kappa_{\mathrm{delay}}(r). If admissible clock-rate transfer or redshift data can be reduced to a boundary-normalized witness profile $\kappa_{\mathrm{clock}}(r)$TeX\kappa_{\mathrm{clock}}(r), then it too must agree with $\kappa_{\mathrm{delay}}(r)$TeX\kappa_{\mathrm{delay}}(r). In the fully witnessed case, where both witness profiles are available, the acceptance criterion becomes

\kappa_{\mathrm{delay}}(r)=\kappa_{\mathrm{angle}}(r)=\kappa_{\mathrm{clock}}(r)
\qquad \text{for } r\in[r_\odot,r_\oplus].

When the clock witness is available on a closed comparison network, this agreement requirement is supplemented by the loop-exactness condition in Definition reference. Any failure of reference, of the corresponding pairwise agreement when only one witness channel is available, or of the applicable loop-exactness condition falsifies the shared-map identification rather than redefining $\kappa$TeX\kappa.

definition: Clock-loop exactness on closed comparison networks. Suppose admissible clock-transfer or redshift data are available on a closed comparison network for the same stationary or quasi-stationary Solar-System window. Let $R_{ij}$TeXR_{ij} denote the boundary-normalized frequency ratio assigned to the oriented edge $i\to j$TeXi\to j under the declared transfer reduction. The clock witness is called loop-exact if for every closed loop $\mathcal L=(i_1\to i_2\to\cdots\to i_m\to i_1)$TeX\mathcal L=(i_1\to i_2\to\cdots\to i_m\to i_1),

\prod_{a=1}^{m} R_{i_a i_{a+1}} =1,\qquad i_{m+1}\equiv i_1.

Equivalently, $\sum_{\mathcal L}\ln R_{ij}=0$TeX\sum_{\mathcal L}\ln R_{ij}=0. Under the shared-map identification this is the operational no-holonomy condition $\oint_{\mathcal L} d\ln\kappa =0$TeX\oint_{\mathcal L} d\ln\kappa =0. Failure of reference rejects the shared map on that window.

remark: Logical status of the three-channel criterion. Definition reference and Definition reference are admissibility rules. Here $\kappa_{\mathrm{delay}}$TeX\kappa_{\mathrm{delay}} is constructed operationally from DSN-style delay/range observables, $\kappa_{\mathrm{angle}}$TeX\kappa_{\mathrm{angle}} enters as an external uniqueness witness within the declared radial-isotropic class, and $\kappa_{\mathrm{clock}}$TeX\kappa_{\mathrm{clock}} enters as a single-geometry equality test and falsifier. On stationary or quasi-stationary windows, the clock witness may additionally be checked by loop exactness of reduced frequency ratios, reflecting the exact redshift-potential picture expected for standard clocks [citation]. The angular and clock channels therefore remain external witnesses of unequal status relative to the internal delay derivation.

definition: Multi-band dispersive scaling test (plasma classification). Let $\gamma$TeX\gamma be an admissible segment and let $\rho(f)\equiv \Phi(\gamma;f)-1$TeX\rho(f)\equiv \Phi(\gamma;f)-1 denote the dimensionless observed-minus-computed phase ratio residual obtained from the same declared DSN data type and count interval at downlink carrier frequency $f$TeXf. If two or more bands are available for the same geometry and the residual family is consistent (within measurement uncertainty and within the declared computed model) with the dispersive scaling law

\rho(f)\ \propto\ \frac{1}{f^{2}},

equivalently $\rho(f_i)f_i^{2}$TeX\rho(f_i)f_i^{2} is constant across bands, then the residual is classified as dispersive (media-dominated) rather than curvature-layering dominated. Such $1/f^{2}$TeX1/f^{2} scaling is a characteristic signature of plasma dispersion in deep-space radiometric observables and appears explicitly in standard error models [citation]. Under Contract $\mathsf{C}$TeX\mathsf{C}, dispersive-classified passes are excluded from curvature-layering inference unless a validated dispersive calibration is explicitly declared in $\mathcal{M}_{\min}$TeX\mathcal{M}_{\min} (Definition reference) [citation].

lemma: Coherent count-measure quotient representation and no-extra-channel theorem under Contract $\mathsf{C}$TeX\mathsf{C}. Assume Contract $\mathsf{C}$TeX\mathsf{C} (Definition reference) and the CHC shared map reference. Let $\gamma$TeX\gamma be an admissible coherent DSN phase-count segment for which $P_{\mathrm{obs}}(\gamma)$TeXP_{\mathrm{obs}}(\gamma) and $P_{0}(\gamma)$TeXP_{0}(\gamma) are paired observed/computed total-count phase observables of the same DSN data type, count interval, frequency plan, and turnaround ratio. Let $\mathfrak I_\gamma$TeX\mathfrak I_\gamma denote the algebra of declared reception subintervals of a phase-continuous pass realizing $\gamma$TeX\gamma, and define

\mu_{\mathrm{obs}}(I)\equiv P_{\mathrm{obs}}(I),\qquad
\mu_{0}(I)\equiv P_{0}(I),\qquad I\in\mathfrak I_\gamma.

Assume further that: (i) $\mu_{\mathrm{obs}}$TeX\mu_{\mathrm{obs}} and $\mu_0$TeX\mu_0 satisfy the functoriality of Lemma reference; (ii) both are locally absolutely continuous with respect to reception time on phase-continuous passes; and (iii) after the dispersive rejection of Definition reference, no residual band label remains. Then the coherent observable quotient is representable only as a quotient of two count measures over the same declared interval algebra, and any curvature-layering contribution consistent with the matched observed/computed contract must factor through the same elapsed-time measure of the declared segment. Consequently

\Phi(\gamma)=\frac{P_{\mathrm{obs}}(\gamma)}{P_{0}(\gamma)}
=\frac{\mu_{\mathrm{obs}}(\gamma)}{\mu_0(\gamma)}
=\frac{t_{\mathrm{prop}}(\gamma)}{t_{0}(\gamma)}
=\frac{1}{L_\gamma}\int_{\gamma}\frac{d\ell}{\kappa(\ell)}.

Moreover, any additional channel affecting $\Phi$TeX\Phi without appearing in the same elapsed-time functional would violate at least one of: finite count additivity, coherent observed/computed pairing, band-independence after dispersive rejection, or refinement stability on declared intervals.

proof. By Lemma reference, the coherent total-count phase defines a finitely additive, refinement-stable interval functional on the declared interval algebra $\mathfrak I_\gamma$TeX\mathfrak I_\gamma. By local absolute continuity with respect to reception time, there exist local count densities such that

d\mu_{\mathrm{obs}}=\lambda_{\mathrm{obs}}(t)\,dt,
\qquad
d\mu_{0}=\lambda_{0}(t)\,dt.

Power-law profile and closed-form path factor Assume a scale-free Solar-System window ($r_\odot\ll r\lesssim r_\oplus$TeXr_\odot\ll r\lesssim r_\oplus) in the local self-similarity sense: there exists an open interval $\mathcal{B}\subset(0,\infty)$TeX\mathcal{B}\subset(0,\infty) containing $1$TeX1 such that for any scale factor $b\in\mathcal{B}$TeXb\in\mathcal{B} for which both $r$TeXr and $br$TeXbr remain within the window, the layering profile obeys

\kappa(br)=G(b)\,\kappa(r),

for some positive function $G$TeXG that depends only on the scale factor. Under mild regularity (differentiability in $r$TeXr and differentiability of $G$TeXG at $b=1$TeXb=1), this local condition is enough to force a power-law profile [citation]:

lemma: Local self-similarity implies a power-law profile. Assume $\kappa:(0,\infty)\to(0,\infty)$TeX\kappa:(0,\infty)\to(0,\infty) is differentiable on the scale-free window and satisfies reference for all $b\in\mathcal{B}$TeXb\in\mathcal{B} with $1\in\mathcal{B}$TeX1\in\mathcal{B}, where $G$TeXG is differentiable at $b=1$TeXb=1. Then there exists an exponent $q\in\mathbb{R}$TeXq\in\mathbb{R} such that $G(1)=1$TeXG(1)=1, $q=G'(1)$TeXq=G'(1), and

\kappa(r)=\kappa(r_\oplus)\left(\frac{r}{r_\oplus}\right)^q

on the window.

proof. Setting $b=1$TeXb=1 in reference gives $G(1)=1$TeXG(1)=1. Taking natural logs yields

\ln\kappa(br)-\ln\kappa(r)=\ln G(b).

Differentiate with respect to $b$TeXb at $b=1$TeXb=1 (for fixed $r$TeXr). Since $\kappa$TeX\kappa is differentiable,

\left.\frac{d}{db}\ln\kappa(br)\right|_{b=1}=\frac{r\,\kappa'(r)}{\kappa(r)},

and since $G(1)=1$TeXG(1)=1 and $G$TeXG is differentiable at $b=1$TeXb=1,

\left.\frac{d}{db}\ln G(b)\right|_{b=1}=G'(1)=q.

Therefore $r\,\kappa'(r)/\kappa(r)=q$TeXr\,\kappa'(r)/\kappa(r)=q, i.e.\ $d(\ln\kappa)/d(\ln r)=q$TeXd(\ln\kappa)/d(\ln r)=q on the window. Integrating gives $\ln\kappa(r)=q\ln r + C$TeX\ln\kappa(r)=q\ln r + C, hence $\kappa(r)=C' r^q$TeX\kappa(r)=C' r^q. Evaluating at $r=r_\oplus$TeXr=r_\oplus fixes $C'=\kappa(r_\oplus)r_\oplus^{-q}$TeXC'=\kappa(r_\oplus)r_\oplus^{-q}, yielding reference.

Thus the minimal family consistent with scale-freeness is the power-law profile

\kappa(r)=\kappa_\oplus\left(\frac{r}{r_\oplus}\right)^q,\qquad q\ge 0,\qquad \kappa_\oplus\equiv \kappa(r_\oplus).

Let $\mathrm{au}$TeX\mathrm{au} denote the astronomical unit, defined as exactly $149\,597\,870\,700$TeX149\,597\,870\,700 m by IAU 2012 Resolution B2 [citation], and let $R_{\odot}^{\mathrm{N}}\equiv 695\,700\,000$TeXR_{\odot}^{\mathrm{N}}\equiv 695\,700\,000 m denote the nominal solar radius recommended by IAU 2015 Resolution B3 [citation]. Define the dimensionless window ratio

x\equiv \frac{R_{\odot}^{\mathrm{N}}}{\mathrm{au}}.

lemma: Closed form for the path factor. For $q\neq 1$TeXq\neq 1, inserting reference into reference yields

\kappa_{\mathrm{path}}=\kappa_\oplus\,F(q),
\qquad
F(q)\equiv\frac{(1-q)(1-x)}{1-x^{1-q}}.

proof. With $L=r_\oplus-r_\odot=r_\oplus(1-x)$TeXL=r_\oplus-r_\odot=r_\oplus(1-x), the harmonic definition gives

\frac{1}{\kappa_{\rm path}}=
\frac{1}{L\kappa_\oplus}\int_{r_\odot}^{r_\oplus}
\left(\frac{r}{r_\oplus}\right)^{-q}\mathrm dr
=\frac{1-x^{1-q}}{\kappa_\oplus(1-q)(1-x)}.

Taking the reciprocal proves reference.

Fixing $q=3/4$TeXq=3/4 from a scale-free invariant (and a no-go)Fixing q=3/4 from a scale-free invariant (and a no-go)

proposition: Premise-fixed 3+1 shell-extensivity selects $q=3/4$TeXq=3/4 on the declared scale-free window. Assume the shared distortion map reference so that both spatial propagation scales and local clock scales are reduced by the same factor $\kappa(r)$TeX\kappa(r). In a scale-free Solar-System window, require that no additional dimensionful scale beyond $r$TeXr enters the layering profile. Let $\alpha$TeX\alpha denote the a priori unknown self-similarity exponent of the local scaling action in Lemma reference. Require further that the orbit label for adjacent radial shells be extensive under shell concatenation. For one shared time dimension and $d$TeXd spatial dimensions, the corresponding shell-content candidate is

\mathcal{I}(r)\equiv \frac{\kappa(r)^{d+1}}{(r/r_\oplus)^{d}}.

Require this shell-content candidate to be invariant across the window. Then

(d+1)\alpha-d=0.

Under the power-law ansatz reference, the self-similarity exponent equals $q$TeXq, and therefore

q=\frac{d}{d+1}.

For $d=3$TeXd=3 this yields

q=\frac{3}{4}.

proof. By the stated shell-extensivity premise, the orbit label must compose consistently across adjacent radial shells. Because the shared map carries one time dimension and $d$TeXd spatial dimensions through the same scalar $\kappa$TeX\kappa, the shell-content candidate reference is the monomial orbit label compatible with that 3+1 shell valuation. By Lemma reference, this candidate is invariant under the general scaling action reference if and only if $(d+1)\alpha-d=0$TeX(d+1)\alpha-d=0, which gives reference. Under the power-law ansatz, $\kappa(br)=b^{q}\kappa(r)$TeX\kappa(br)=b^{q}\kappa(r), so the self-similarity exponent equals $q=\alpha$TeXq=\alpha. Hence reference and reference.

remark: Why the invariant uses $\kappa^{d+1}$TeX\kappa^{d+1} and $(r/r_\oplus)^d$TeX(r/r_\oplus)^d. Under the shared distortion map reference, the same dimensionless profile $\kappa(r)$TeX\kappa(r) rescales the local time unit ($d\tau=\kappa\,dT$TeXd\tau=\kappa\,dT) and the calibration-layer path-normalization variable ($v=V\kappa$TeXv=V\kappa). Hence a single $\kappa$TeX\kappa acts as a common scale factor on one time dimension and on the $d$TeXd spatial dimensions of an isotropic organizer, so the natural isotropic (time+space) monomial scale factor is $\kappa^{d+1}$TeX\kappa^{d+1}. In a scale-free Solar-System window, $r$TeXr (measured in Earth units $r/r_\oplus$TeXr/r_\oplus) is the only available length scale. If, in addition, the orbit label for adjacent radial shells is required to compose extensively under shell concatenation, then the monomial shell-content candidate is $(r/r_\oplus)^{-d}\kappa^{d+1}$TeX(r/r_\oplus)^{-d}\kappa^{d+1}. Requiring the ratio $\mathcal{I}(r)=\kappa(r)^{d+1}/(r/r_\oplus)^d$TeX\mathcal{I}(r)=\kappa(r)^{d+1}/(r/r_\oplus)^d to be constant is therefore the strongest shell-extensive statement available within the stated 3+1 shared-map premises. Equivalently, the local self-similarity relation reference defines a scaling group acting on profiles $(r,\kappa(r))$TeX(r,\kappa(r)). In this setting, $\mathcal{I}(r)$TeX\mathcal{I}(r) acts as a maximal invariant for the induced action on the pair $(r,\kappa)$TeX(r,\kappa): any scale-free invariant scalar factors through $\mathcal{I}$TeX\mathcal{I} [citation]. In particular, once one common distortion scalar $\kappa$TeX\kappa carries one time dimension and $d$TeXd spatial dimensions, the shell-content index $\mathcal{I}(r)=\kappa(r)^{d+1}/(r/r_\oplus)^d$TeX\mathcal{I}(r)=\kappa(r)^{d+1}/(r/r_\oplus)^d is the shell-content candidate selected by the stated 3+1 extensivity premise; Lemma reference then shows that, if this candidate is required to be invariant under the general action reference, the self-similarity exponent must satisfy $(d+1)\alpha-d=0$TeX(d+1)\alpha-d=0, and under the power-law ansatz this yields $q=\alpha=d/(d+1)$TeXq=\alpha=d/(d+1).

lemma: General-$\alpha$TeX\alpha maximal invariant and orbit rigidity of the shell-content index. Assume the shared distortion map reference and the local self-similarity premise reference on a scale-free Solar-System window. Consider the induced one-parameter scaling action on the pair $(r,\kappa)$TeX(r,\kappa) defined (in Earth units) by

(r,\kappa)\ \mapsto\ (br,\ b^{\alpha}\kappa),\qquad b>0,

where $\alpha$TeX\alpha is a priori unknown. Then

M_{\alpha}(r,\kappa)\equiv \frac{\kappa}{(r/r_\oplus)^{\alpha}}

is a maximal invariant for reference: it is invariant under the action, and two points $(r_1,\kappa_1)$TeX(r_1,\kappa_1) and $(r_2,\kappa_2)$TeX(r_2,\kappa_2) lie on the same orbit if and only if $M_{\alpha}(r_1,\kappa_1)=M_{\alpha}(r_2,\kappa_2)$TeXM_{\alpha}(r_1,\kappa_1)=M_{\alpha}(r_2,\kappa_2). Consequently any dimensionless scalar invariant of the scaling action is a function of $M_\alpha$TeXM_\alpha [citation]. In particular, the shared-map shell-content index

\mathcal{I}(r)=\frac{\kappa(r)^{d+1}}{(r/r_\oplus)^d}

is invariant under reference if and only if

(d+1)\alpha-d=0.

proof. First, reference leaves $M_\alpha$TeXM_\alpha invariant:

\frac{b^\alpha\kappa}{(br/r_\oplus)^\alpha}=\frac{\kappa}{(r/r_\oplus)^\alpha}.

For orbit separation, suppose $(r_2,\kappa_2)=(br_1,b^\alpha\kappa_1)$TeX(r_2,\kappa_2)=(br_1,b^\alpha\kappa_1) for some $b>0$TeXb>0; then $M_\alpha(r_2,\kappa_2)=M_\alpha(r_1,\kappa_1)$TeXM_\alpha(r_2,\kappa_2)=M_\alpha(r_1,\kappa_1). Conversely, if

\frac{\kappa_2}{(r_2/r_\oplus)^\alpha}=\frac{\kappa_1}{(r_1/r_\oplus)^\alpha},

then $\kappa_2=(r_2/r_1)^\alpha\kappa_1$TeX\kappa_2=(r_2/r_1)^\alpha\kappa_1. Taking $b=r_2/r_1$TeXb=r_2/r_1 yields $(r_2,\kappa_2)=(br_1,b^\alpha\kappa_1)$TeX(r_2,\kappa_2)=(br_1,b^\alpha\kappa_1), so the points lie on the same orbit. Hence $M_\alpha$TeXM_\alpha indexes the orbits and is maximal invariant.

Now apply the same action to the shell-content index:

\mathcal{I}\mapsto
\frac{(b^\alpha\kappa)^{d+1}}{(br/r_\oplus)^d}
=
b^{(d+1)\alpha-d}\,
\frac{\kappa^{d+1}}{(r/r_\oplus)^d}.

Therefore $\mathcal{I}$TeX\mathcal{I} is invariant if and only if $(d+1)\alpha-d=0$TeX(d+1)\alpha-d=0. In particular, once the shell-content candidate reference is selected by the stated 3+1 extensivity premise, the general-$\alpha$TeX\alpha action leaves no remaining freedom in the exponent, so the fixed point is forced rather than chosen.

remark: Status of the exponent fixing. The value $q=3/4$TeXq=3/4 is fixed only within the combined premises of (i) local scale-freeness on the Solar-System window, (ii) one shared distortion scalar acting on one time and three spatial dimensions, and (iii) shell-extensivity under radial concatenation. It is not extracted here from DSN delay/range data alone. The observational role of Solar-System time-transfer is narrower: it tests whether an admissible scale-free window remains compatible with the exponent selected by those premises.

remark: No-go for $q\neq d/(d+1)$TeXq\neq d/(d+1) under the stated scale-free premise. If $\kappa(r)=\kappa_\oplus (r/r_\oplus)^q$TeX\kappa(r)=\kappa_\oplus (r/r_\oplus)^q with $q\neq d/(d+1)$TeXq\neq d/(d+1), then $\mathcal{I}(r)\propto r^{(d+1)q-d}$TeX\mathcal{I}(r)\propto r^{(d+1)q-d} is not constant, so the window is not scale-free in the sense of an invariant: an additional scale is implicitly introduced. The stated scale-free premise is therefore equivalent to the exponent condition.

remark: Primary falsifier for $q$TeXq. Solar-System time-transfer constraints act directly on $\int dr/\kappa(r)$TeX\int dr/\kappa(r) (Sec. reference); if those constraints exclude a scale-free exponent near $q=3/4$TeXq=3/4 over the relevant window, the choice reference is falsified.

proposition: Declared-class Earth-ratio identity on the adopted Solar-System map. Under reference and reference,

R=\frac{1}{\kappa_\oplus\,\kappa_{\mathrm{path}}}
=\frac{1}{\kappa_\oplus^2\,F(q)}.

proof. By definition, $\kappa_{\rm path}=\kappa_\oplus F(q)$TeX\kappa_{\rm path}=\kappa_\oplus F(q). The shared-map ratio is $R=(\kappa_\oplus\kappa_{\rm path})^{-1}$TeXR=(\kappa_\oplus\kappa_{\rm path})^{-1}. Substitution gives $R=[\kappa_\oplus^2F(q)]^{-1}$TeXR=[\kappa_\oplus^2F(q)]^{-1}.

proposition: Earth-read reconstructions on the declared Solar-System window. From reference,

\kappa_\oplus(q)=\sqrt{\frac{1}{R\,F(q)}},\qquad
\Xi_\oplus(q)=1-\kappa_\oplus(q)^2,

and since $d\tau=\kappa_\oplus dT$TeXd\tau=\kappa_\oplus dT at $r_\oplus$TeXr_\oplus,

T_0(q)=\frac{\tau_0}{\kappa_\oplus(q)}.

Moreover, define the Earth-unit path-normalization readout by

\frac{V_{\rm read}}{c_\oplus}=\frac{1}{\kappa_{\mathrm{path}}}
=\frac{1}{\kappa_\oplus(q)\,F(q)}=\sqrt{\frac{R}{F(q)}}.

proof. Solving $R=[\kappa_\oplus^2F(q)]^{-1}$TeXR=[\kappa_\oplus^2F(q)]^{-1} for the positive branch gives $\kappa_\oplus=[RF(q)]^{-1/2}$TeX\kappa_\oplus=[RF(q)]^{-1/2} and $\Xi_\oplus=1-\kappa_\oplus^2$TeX\Xi_\oplus=1-\kappa_\oplus^2. The local clock equation gives $T_0=\tau_0/\kappa_\oplus$TeXT_0=\tau_0/\kappa_\oplus. Finally $\kappa_{\rm path}=\kappa_\oplus F(q)$TeX\kappa_{\rm path}=\kappa_\oplus F(q) implies $V_{\rm read}/c_\oplus=\kappa_{\rm path}^{-1}=\sqrt{R/F(q)}$TeXV_{\rm read}/c_\oplus=\kappa_{\rm path}^{-1}=\sqrt{R/F(q)}.

A closure-based normalization is only as strong as the operational closure that implements it. In radiometric tracking, coherent two-way/three-way DSN links realize a phase-coherent uplink/downlink loop: the spacecraft transponder locks the downlink to the received uplink (up to a fixed turnaround ratio), and the ground receiver counts accumulated carrier phase over a declared count interval [citation]. We remove ambiguity by fixing (i) a single operational decision variable $S$TeXS and (ii) a single completion functional $\mathcal{C}$TeX\mathcal{C} that is defined as the minimal phase accumulation required to flip $S$TeXS within the same declared pipeline.

Operational decision variable

definition: Sign-distinguishing completion variable. Fix a declared phase-discriminator pipeline that outputs a real-valued discriminator signal $y(t)$TeXy(t), and fix a measurement window of duration $T_w$TeXT_w. Assume that, within the declared pipeline, $y(t)$TeXy(t) is (up to a fixed gain and offset) a sinusoid of a dimensionless phase argument, e.g.\ $y(t)\propto \cos\theta(t)$TeXy(t)\propto \cos\theta(t) or $y(t)\propto \sin\theta(t)$TeXy(t)\propto \sin\theta(t). This covers an interferometric difference channel, coherent I/Q discrimination (Lemma reference), and multiplier phase detection (Lemma reference). Define the binary decision variable

S \equiv \operatorname{sign}\!\left(\int_{t_0}^{t_0+T_w} y(t)\,dt\right)\in\{+1,-1\}.

A completion is an operational transformation that flips $S$TeXS while remaining within the same declared measurement pipeline.

remark: Interference/discriminator readout and the pipeline phase argument. The detector outputs used here depend sinusoidally on a phase-error coordinate; the elementary trigonometric forms are written explicitly in Lemma reference and Lemma reference. The sign decision in reference is therefore a fixed functional of a dimensionless phase-like pipeline coordinate.

In the task class used below, the phase argument is not identified with the raw carrier phase $\omega t$TeX\omega t. Instead, the declared DSN phase-count pipeline constructs a frequency-free coordinate from radiometric observables by pairing an observed total-count phase $P_{\mathrm{obs}}$TeXP_{\mathrm{obs}} with its computed reference $P_{0}$TeXP_{0} under the minimal computed-model conditions of Definition reference and forming the ratio $\Delta\Theta(\gamma)=\Phi(\gamma)=P_{\mathrm{obs}}(\gamma)/P_0(\gamma)$TeX\Delta\Theta(\gamma)=\Phi(\gamma)=P_{\mathrm{obs}}(\gamma)/P_0(\gamma) (Definition reference) [citation]. The admissibility gate in Definition reference prevents conjunction-dependent solar-plasma contamination from reintroducing an effective variable cutoff on the declared Solar-System window. Because $\Phi$TeX\Phi is a ratio of like-formally specified observed/computed cycle counts, the cycles-to-radians conversion cancels identically (Lemma reference), so no extra $2\pi$TeX2\pi factor remains at the level of $\Delta\Theta$TeX\Delta\Theta.

The relevant closure is operational: coherent two-way/three-way DSN links realize a phase-coherent uplink/downlink loop with a fixed transponder turnaround ratio [citation]. One-way non-coherent links depend on an independent spacecraft oscillator and therefore break this closure, providing an internal control in Sec. reference. Accordingly, any nondegenerate binary detector formed within the declared coherent pipeline belongs, up to fixed gain and phase offset, to the first-harmonic family $D(\Theta)=\operatorname{sign}(a\cos\Theta+b\sin\Theta)$TeXD(\Theta)=\operatorname{sign}(a\cos\Theta+b\sin\Theta).

lemma: Coherent I/Q phase discriminator yields sinusoidal outputs. Consider a coherent in-phase/quadrature-phase detector for a received carrier of the form $s(t)=A_c m(t)\cos(2\pi f_c t)$TeXs(t)=A_c m(t)\cos(2\pi f_c t) mixed with local oscillators $\cos(2\pi f_c t+\varphi)$TeX\cos(2\pi f_c t+\varphi) and $\sin(2\pi f_c t+\varphi)$TeX\sin(2\pi f_c t+\varphi), followed by ideal low-pass filtering. Then the baseband components satisfy

v_I(t)=\frac{A_c}{2}\,m(t)\cos\varphi,\qquad v_Q(t)=\frac{A_c}{2}\,m(t)\sin\varphi,

up to an overall amplitude normalization. Thus coherent phase discrimination naturally produces $\cos\varphi$TeX\cos\varphi and $\sin\varphi$TeX\sin\varphi terms of a phase-error coordinate $\varphi$TeX\varphi.

proof. Multiply $s(t)$TeXs(t) by $\cos(2\pi f_c t+\varphi)$TeX\cos(2\pi f_c t+\varphi) and use $\cos a\cos b=\tfrac{1}{2}[\cos(a-b)+\cos(a+b)]$TeX\cos a\cos b=\tfrac{1}{2}[\cos(a-b)+\cos(a+b)]; after low-pass filtering the high-frequency term, one obtains $v_I(t)=(A_c/2)m(t)\cos\varphi$TeXv_I(t)=(A_c/2)m(t)\cos\varphi. The quadrature branch follows similarly using $\cos a\sin b=\tfrac{1}{2}[\sin(b-a)+\sin(b+a)]$TeX\cos a\sin b=\tfrac{1}{2}[\sin(b-a)+\sin(b+a)], yielding $v_Q(t)=(A_c/2)m(t)\sin\varphi$TeXv_Q(t)=(A_c/2)m(t)\sin\varphi after low-pass filtering. This is the standard coherent I/Q low-pass reduction.

In coherent carrier tracking, these sinusoidal discriminator outputs are used to form a phase-error signal for a phase-locked loop. This is the operational setting of coherent Doppler tracking receivers, including DSN carrier tracking loops used for radiometric Doppler measurement [citation].

lemma: Multiplier phase detector output is proportional to $\cos\varphi$TeX\cos\varphi. Let $x(t)=A\cos(\omega t)$TeXx(t)=A\cos(\omega t) and $y(t)=B\cos(\omega t+\varphi)$TeXy(t)=B\cos(\omega t+\varphi) be two sinusoids with phase difference $\varphi$TeX\varphi. The low-pass filtered product satisfies

\mathrm{LPF}\{x(t)y(t)\}=\frac{AB}{2}\cos\varphi,

up to a fixed gain, so a multiplier phase detector naturally produces a $\cos(\text{phase error})$TeX\cos(\text{phase error}) output.

proof. Using $\cos a\cos b=\tfrac{1}{2}[\cos(a-b)+\cos(a+b)]$TeX\cos a\cos b=\tfrac{1}{2}[\cos(a-b)+\cos(a+b)] gives

x(t)y(t)=\frac{AB}{2}\cos\varphi+\frac{AB}{2}\cos(2\omega t+\varphi).

Low-pass filtering removes the $2\omega$TeX2\omega term, leaving reference.

lemma: First-harmonic detector universality forces the antipodal completion. After centering each coherent detector to zero threshold and absorbing fixed gains and phase offsets into the coefficients, every nondegenerate binary detector produced within the declared coherent pipeline has the form

D(\Theta)=\operatorname{sign}\!\big(a\cos\Theta+b\sin\Theta\big),\qquad (a,b)\neq(0,0).

Equivalently,

D(\Theta)=\operatorname{sign}\!\big(R\cos(\Theta-\phi)\big),\qquad R>0.

Its zero set is an antipodal pair on $S^1$TeXS^1, and the unique minimal global phase shift that flips the decision on every coherent phase class is