Conditional Planck-Triad Readouts and a Mass-Modulus Branch Ledger in the CHC Framework

The standard unreduced Planck length, Planck time, and Planck mass are

\ell_{P}=\sqrt{\frac{\hbar G}{c^3}},\qquad
t_P=\sqrt{\frac{\hbar G}{c^5}},\qquad
m_P=\sqrt{\frac{\hbar c}{G}}.

In the revised SI, the exact numerical values of the defining constants, including $c$TeXc and $h$TeXh, define the units, while $G$TeXG remains a measured constant with standard uncertainty; CODATA therefore supplies the standard numerical readout of the Planck triad [citation]. Equation reference is not questioned here. The question is narrower: once a declared CHC calibration class supplies length and time calibration factors together with the associated dimensionless length-to-time calibration factor, what is the lawful dimensional readout of reference, and what remains open?

The input is the CCL Solar-System calibration class. On that class, a path-harmonic Earth-unit normalization readout and a local clock-rate calibration share one curvature-layered calibration map. The imported readouts are

\frac{V_{\rm read}}{c_\oplus}=\pi,
\qquad
D_0=\tau_0\pi^2F_{3/4},

Here $V_{\rm read}$TeXV_{\rm read} names only the CCL path-normalization readout; it is not a local, group, signal, or universal propagation speed. The factor $F_{3/4}$TeXF_{3/4} is the corresponding Solar-window path factor [citation]. The CCL object is not used here as a universal propagation law, a universal clock law, a horizon theorem, or a cosmology result. It is used only as a declared calibration class.

The first role of the present paper is negative. It prevents the shortcut $c_\oplus\mapsto\pi c_\oplus$TeXc_\oplus\mapsto\pi c_\oplus in reference from being mistaken for a CHC Planck-triad lift. Such a shortcut moves a speed symbol while leaving $\hbar$TeX\hbar and $G$TeXG in their local readout layer. It is therefore a mixed-layer diagnostic rather than a lawful dimensional calibration.

The second role is to keep the mass slot explicit. The imported CCL class fixes length and time factors but does not own a mass-sector law. Hence the Planck-mass component remains $m_{P,\Hcal}=\mu m_{P,\oplus}$TeXm_{P,\Hcal}=\mu m_{P,\oplus} until a mass-bearing invariant is separately declared. The phase-rigidity tension branch displayed below is a declared conditional branch whose formula is available only under its branch-local invariant premise. It is not a branch-independent prediction and is not evidence that the construction derives a universal mass scale.

The construction below uses one declared CCL calibration class as an input and develops only a conditional dimensional readout with an explicit mass-modulus branch ledger. It adds no root-level standing assumption and does not promote the calibration readout into a universal propagation, clock, SI, Planck-scale, mass-origin, or quantum-gravity theorem. All statements below are restricted to the declared calibration class and to explicitly declared mass-modulus branches.

Companion certificate note..

The CCL/PTM-VP0 metrology-reference result supports only PTM-CONDITIONAL-TRIAD-CHECK-SATISFIED on the conditional Planck-triad reference-check lane, with the shared umbrella label CCL-PTM-VP0-REFERENCE-CHECK-SATISFIED on the declared BIPM--NIST/CODATA--IAU reference surface. It does not alter the theorem or equation status of this construction, does not select the mass modulus $\mu$TeX\mu, does not validate any empirical CCL witness outside the declared metrology-reference lane, and does not supply an SI redefinition, universal Planck-mass prediction, or quantum-gravity scale theorem.

Figure or table content is omitted from the web reader; use the canonical manuscript for the exact object.

The guiding reading rule is branch-local: a recovery, compatibility, benchmark, or exact status may be read only on its declared branch, family, window, map, comparator datum, or observation domain. Restricted recovery is not universal identity. The construction therefore supplies a declared-class calibration readout and a conditional branch ledger only.

Earth/SI Planck triad

The Earth/SI readout of the standard Planck triad is

\ell_{P,\oplus}=\sqrt{\frac{\hbar_\oplus G_\oplus}{c_\oplus^3}},\qquad
 t_{P,\oplus}=\sqrt{\frac{\hbar_\oplus G_\oplus}{c_\oplus^5}},\qquad
 m_{P,\oplus}=\sqrt{\frac{\hbar_\oplus c_\oplus}{G_\oplus}} .

These are imported standard dimensional definitions. Numerical values below use SI/CODATA readouts and are not new measurements [citation].

Declared CCL calibration factors

The imported CCL calibration class uses a path-harmonic Earth-unit normalization readout

\frac{V_{\rm read}}{c_\oplus}=\kpath^{-1},
 \qquad
 \kpath=\frac{L}{\displaystyle\int_{r_\odot}^{r_\oplus}dr/\kappa(r)},
 \qquad
 L=r_\oplus-r_\odot,

This is the same dimensionless calibration relation as the parent CCL statement, written without treating $V_{\rm read}$TeXV_{\rm read} as a physical speed. The scale-free radial profile is

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

For $q\ne1$TeXq\ne1, define

x\equiv\frac{\RsunN}{\au},
 \qquad
 F(q)=\frac{(1-q)(1-x)}{1-x^{1-q}}.

The imported CCL witness selects $q=3/4$TeXq=3/4 on its declared Solar-System window [citation]. The astronomical unit is exactly $149\,597\,870\,700$TeX149\,597\,870\,700 m by IAU Resolution B2, and the IAU nominal solar radius $R_{\odot}^{\mathrm N}=695\,700\,000$TeXR_{\odot}^{\mathrm N}=695\,700\,000 m is an exact nominal conversion constant, not the true time-varying solar radius [citation].

The imported accepted CCL readout is

\frac{V_{\rm read}}{c_\oplus}=\pi,
\qquad
R\equiv \frac{D_0}{\tau_0}=\pi^2F_{3/4},
\qquad
F_{3/4}\equiv F(3/4).

Equivalently,

\kpath=\frac1\pi,
\qquad
\kop=\frac{1}{\pi F_{3/4}}.

The path-harmonic path-normalization factor $\kpath$TeX\kpath and the point-local Earth factor $\kop$TeX\kop are distinct imported objects. They are not collapsed in this paper.

definition: Declared calibration factors. On the imported class reference, define

\Tcal\equiv \pi F_{3/4},\qquad
\Lcal\equiv \pi^2F_{3/4},\qquad
\Ccal\equiv \frac{\Lcal}{\Tcal}=\pi .

Here $\Tcal$TeX\Tcal is the time calibration factor, $\Lcal$TeX\Lcal is the length calibration factor, and $\Ccal$TeX\Ccal is the associated dimensionless length-to-time calibration factor on the declared readout.

remark: No validation of the imported class. The paper does not prove the CCL class and does not use it as a general clock or distance law. If the imported class fails under its own admissibility and range conditions, the factors reference are unavailable.

definition: Dimension vector. A dimensional quantity $Q$TeXQ has dimension vector $(a,b,c)$TeX(a,b,c) when

[Q]=M^aL^bT^c.

For the constants used below,

[c]=LT^{-1},\qquad [\hbar]=ML^2T^{-1},\qquad [G]=M^{-1}L^3T^{-2}.

definition: Declared-class dimensional lift. Let

\mu\equiv\frac{M_{\Hcal}}{M_\oplus}>0

be the independent mass-calibration modulus. For any dimensional quantity $Q$TeXQ with dimension vector $(a,b,c)$TeX(a,b,c), define its declared-class readout by

Q_{\Hcal}=\mu^a\Lcal^b\Tcal^cQ_\oplus .

This is a calibration readout rule on one declared class, not a redefinition of SI units and not a derivation of the underlying constants.

proposition: Lifted dimensional readout components. Under reference,

c_{\Hcal}=\frac{\Lcal}{\Tcal}c_\oplus=\pi c_\oplus,

 \hbar_{\Hcal}=\mu\frac{\Lcal^2}{\Tcal}\hbar_\oplus=\mu\pi^3F_{3/4}\hbar_\oplus,

 G_{\Hcal}=\frac{\Lcal^3}{\mu\Tcal^2}G_\oplus=\frac{\pi^4F_{3/4}}{\mu}G_\oplus.

remark: Reference-component status of $c_{\Hcal}$TeXc_{\Hcal}. The symbol $c_{\Hcal}$TeXc_{\Hcal} in reference is a branch-reference component of the declared dimensional lift. It is not an observed local light speed, not a new SI value of $c$TeXc, and not a propagation-speed prediction.

proof. Substitute the dimension vectors of $c$TeXc, $\hbar$TeX\hbar, and $G$TeXG into reference. The simplified forms follow from reference.

remark: Readout-vector status. The symbols $c_{\Hcal}$TeXc_{\Hcal}, $\hbar_{\Hcal}$TeX\hbar_{\Hcal}, and $G_{\Hcal}$TeXG_{\Hcal} are components of the declared dimensional readout vector associated with reference. They are not new SI definitions, not new measurements of the corresponding constants, and not assertions that the Earth/SI constants have changed. In particular, a displayed factor multiplying $G_\oplus$TeXG_\oplus is a branch-local bookkeeping component of the lifted readout, not a measured rescaling of Newtonian gravity.

remark: Why $\mu$TeX\mu is explicit. The imported calibration class supplies length, time, and dimensionless length-to-time calibration readouts. It does not supply a mass-rigidity law. Therefore $\mu$TeX\mu cannot be set by the lift alone. Keeping $\mu$TeX\mu visible prevents a hidden mass-origin theorem.

definition: Declared-class Planck-triad readout. For a fixed $\mu>0$TeX\mu>0, define

\ell_{P,\Hcal}=\sqrt{\frac{\hbar_{\Hcal}G_{\Hcal}}{c_{\Hcal}^3}},
\qquad
 t_{P,\Hcal}=\sqrt{\frac{\hbar_{\Hcal}G_{\Hcal}}{c_{\Hcal}^5}},
\qquad
 m_{P,\Hcal}=\sqrt{\frac{\hbar_{\Hcal}c_{\Hcal}}{G_{\Hcal}}}.

theorem: Declared-class Planck-triad calibration with open mass modulus. On the imported calibration class, the readout reference satisfies

\ell_{P,\Hcal}=\Lcal\ell_{P,\oplus}=\pi^2F_{3/4}\ell_{P,\oplus},

 t_{P,\Hcal}=\Tcal t_{P,\oplus}=\pi F_{3/4}t_{P,\oplus},

 m_{P,\Hcal}=\mu m_{P,\oplus}.

proof. Using reference,

\frac{\ell_{P,\Hcal}}{\ell_{P,\oplus}}
=\left[\frac{(\mu\Lcal^2\Tcal^{-1})(\Lcal^3\mu^{-1}\Tcal^{-2})}{(\Lcal\Tcal^{-1})^3}\right]^{1/2}=\Lcal,

\frac{t_{P,\Hcal}}{t_{P,\oplus}}
=\left[\frac{(\mu\Lcal^2\Tcal^{-1})(\Lcal^3\mu^{-1}\Tcal^{-2})}{(\Lcal\Tcal^{-1})^5}\right]^{1/2}=\Tcal,

\frac{m_{P,\Hcal}}{m_{P,\oplus}}
=\left[\frac{(\mu\Lcal^2\Tcal^{-1})(\Lcal\Tcal^{-1})}{\Lcal^3\mu^{-1}\Tcal^{-2}}\right]^{1/2}=\mu.

All factors are positive, so the positive square-root branch is selected.

corollary: Speed-only substitution is not the lift. If only $c_\oplus$TeXc_\oplus is replaced by $\pi c_\oplus$TeX\pi c_\oplus in reference while $\hbar_\oplus$TeX\hbar_\oplus and $G_\oplus$TeXG_\oplus remain in the local layer, the mixed diagnostic is

\ell_P^{\rm mix}=\pi^{-3/2}\ell_{P,\oplus},
\qquad
 t_P^{\rm mix}=\pi^{-5/2}t_{P,\oplus},
\qquad
 m_P^{\rm mix}=\pi^{1/2}m_{P,\oplus}.

It is not the declared-class calibration reference--reference.

proof. The powers of $c$TeXc in reference give reference. The calculation violates reference because it moves only one dimensional constant.

theorem: No branch closure without a mass-bearing invariant. The declared length-time calibration class and reference do not select a branch value of $\mu$TeX\mu. A branch value of $\mu$TeX\mu is lawful only if it is induced by a positive dimensional invariant with nonzero mass dimension on a declared branch. Marked diagnostic values of $\mu$TeX\mu may be recorded as diagnostic rows, but they are not branch closures and cannot be used to infer a Planck-mass prediction.

proof. The factors $\Lcal$TeX\Lcal and $\Tcal$TeX\Tcal are fixed by reference, while reference contains an independent positive factor $\mu$TeX\mu for mass dimension. The proof of reference cancels $\mu$TeX\mu in the length and time components and leaves it exactly in the mass component. Therefore no value of $\mu$TeX\mu follows from the imported calibration class alone.

theorem: Mass-modulus selection by a declared invariant. Let $Q$TeXQ be a positive dimensional quantity with $[Q]=M^aL^bT^c$TeX[Q]=M^aL^bT^c and $a\ne0$TeXa\ne0. If a branch declares

Q_{\Hcal}=Q_\oplus,

then the mass modulus is fixed uniquely on the positive branch as

\mu_Q=\Lcal^{-b/a}\Tcal^{-c/a}.

proof. By reference, the invariant condition is $\mu^a\Lcal^b\Tcal^c=1$TeX\mu^a\Lcal^b\Tcal^c=1. Since $a\ne0$TeXa\ne0 and $\mu>0$TeX\mu>0, the positive branch gives reference.

Figure or table content is omitted from the web reader; use the canonical manuscript for the exact object.

remark: No hidden preference. reference is not a ranking of physical truth. It records how different invariant choices would close $\mu$TeX\mu on the declared lift. The phase-rigidity tension row is adopted only as a declared conditional branch in reference; the remaining rows are diagnostics and are not adopted.

proposition: Compton quantities are consistency relations. Let $m_{\Hcal}=\mu m_\oplus$TeXm_{\Hcal}=\mu m_\oplus. With

\lambda_C=\frac{\hbar}{mc},
\qquad
\omega_C=\frac{mc^2}{\hbar},

the lift gives

\lambda_{C,\Hcal}=\Lcal\lambda_{C,\oplus},
\qquad
\omega_{C,\Hcal}=\Tcal^{-1}\omega_{C,\oplus}.

Both relations are independent of $\mu$TeX\mu.

proof. Substitute reference and reference together with $m_{\Hcal}=\mu m_\oplus$TeXm_{\Hcal}=\mu m_\oplus. The factors of $\mu$TeX\mu cancel.

proposition: Dimensionless gravitational coupling is neutral. For

\alpha_G(m)=\frac{Gm^2}{\hbar c},

one has

\alpha_{G,\Hcal}(m_{\Hcal})=\alpha_{G,\oplus}(m_\oplus).

Thus the dimensionless coupling cannot select $\mu$TeX\mu.

proof. The numerator lifts as $(\Lcal^3\mu^{-1}\Tcal^{-2}G_\oplus)(\mu^2m_\oplus^2)=\mu\Lcal^3\Tcal^{-2}G_\oplus m_\oplus^2$TeX(\Lcal^3\mu^{-1}\Tcal^{-2}G_\oplus)(\mu^2m_\oplus^2)=\mu\Lcal^3\Tcal^{-2}G_\oplus m_\oplus^2. The denominator lifts as $(\mu\Lcal^2\Tcal^{-1}\hbar_\oplus)(\Lcal\Tcal^{-1}c_\oplus)=\mu\Lcal^3\Tcal^{-2}\hbar_\oplus c_\oplus$TeX(\mu\Lcal^2\Tcal^{-1}\hbar_\oplus)(\Lcal\Tcal^{-1}c_\oplus)=\mu\Lcal^3\Tcal^{-2}\hbar_\oplus c_\oplus. The factors cancel.

proposition: Schwarzschild radius scales as a length. For $r_s=2Gm/c^2$TeXr_s=2Gm/c^2,

r_{s,\Hcal}=\Lcal r_{s,\oplus}.

Again $\mu$TeX\mu cancels.

proof. Substitution gives

\frac{2(\Lcal^3\mu^{-1}\Tcal^{-2}G_\oplus)(\mu m_\oplus)}{\Lcal^2\Tcal^{-2}c_\oplus^2}=\Lcal\frac{2G_\oplus m_\oplus}{c_\oplus^2}.

A clock supplies a time relation, a wavelength supplies a length relation, and a dimensionless coupling supplies a consistency relation. None of these quantities fixes the mass unit unless a mass-bearing invariant has first been declared.

definition: Phase-rigidity tension. A phase-rigidity tension $\Tph$TeX\Tph is a declared positive branch variable used here as the mass-bearing invariant for one conditional calibration branch. Its use is an additional branch-local mass-sector assumption, not an output of the length--time calibration class and not a paper-wide mass-modulus closure. It has dimension

[\Tph]=\frac{E}{L}=MLT^{-2}.

It is not identified in this paper with a Standard Model Higgs parameter, a Yukawa map, a QCD trace-anomaly contribution, a QCD string tension, a flux-tube tension, a Regge slope, a Nambu--Goto tension, a worldsheet coupling, or a fundamental string tension. Those structures remain non-identity comparison or boundary objects.

definition: Phase-tension branch. The declared conditional phase-tension branch is available only when the branch-local invariant premise

\mathfrak T_{\mathrm{ph},\Hcal}=\mathfrak T_{\mathrm{ph},\oplus}

is supplied. This definition does not rank the branch above other possible mass-bearing invariants, does not adopt it as a paper-wide branch assumption, and does not select it empirically.

proposition: Declared conditional tension-branch value of the mass modulus. On the phase-tension branch reference alone,

\mu=\frac{\Tcal^2}{\Lcal}=F_{3/4}.

Consequently,

m_{P,\Hcal}^{(\Tph)}=F_{3/4}m_{P,\oplus}.

proof. Because $[\Tph]=MLT^{-2}$TeX[\Tph]=MLT^{-2}, the lift gives

\frac{\mathfrak T_{\mathrm{ph},\Hcal}}{\mathfrak T_{\mathrm{ph},\oplus}}=\mu\frac{\Lcal}{\Tcal^2}.

The declared invariance condition sets this ratio equal to one, so $\mu=\Tcal^2/\Lcal$TeX\mu=\Tcal^2/\Lcal. Substituting $\Tcal=\pi F_{3/4}$TeX\Tcal=\pi F_{3/4} and $\Lcal=\pi^2F_{3/4}$TeX\Lcal=\pi^2F_{3/4} gives reference. Equation reference follows from reference.

corollary: Branch-local dimensional readout components. Substituting $\mu=F_{3/4}$TeX\mu=F_{3/4} into reference--reference gives the declared conditional tension-branch readout components

c_{\Hcal}^{(\Tph)}=\pi c_\oplus,
\qquad
 \hbar_{\Hcal}^{(\Tph)}=\pi^3F_{3/4}^2\hbar_\oplus,
\qquad
 G_{\Hcal}^{(\Tph)}=\pi^4G_\oplus.

These are branch-local readout-vector components inside the declared lift. The last relation is not a new measured Newtonian constant, not a replacement of SI/CODATA $G$TeXG, and not a universal rescaling of gravity.

This section records non-identity boundaries only. The following standard structures are not used to prove, motivate uniquely, or empirically select the phase-tension branch.

QCD boundary..

QCD mass decomposition uses the QCD energy--momentum tensor, quark and gluon contributions, trace-anomaly terms, and renormalization conventions to analyze hadronic mass [citation]. The present branch does not model those contributions, does not compute a hadron spectrum, and does not identify $\Tph$TeX\Tph with a QCD trace-anomaly term, a QCD string tension, or a flux-tube tension. The QCD references are cited only to mark a comparator boundary.

Electroweak boundary..

The Brout--Englert--Higgs mechanism and related gauge-symmetry-breaking formulations provide the standard electroweak mass-loading language [citation]. In the present construction, the global phase scalar is not identified with the Standard Model Higgs doublet; no Yukawa matrix or Standard-Model-complete mass spectrum is constructed. This is the electroweak non-identity boundary used here [citation].

String-like boundary..

String and dual-resonance constructions provide standard tension and slope languages for extended-object spectra [citation]. The present branch does not identify $\Tph$TeX\Tph with a fundamental string tension, a Regge slope, a Nambu--Goto tension, or a worldsheet coupling. The comparison is dimensional and lexical only. Universal string/M equivalence, unrestricted holography, all-compactification claims, and unrestricted ultraviolet completion remain outside this construction [citation].

Using

\au=149\,597\,870\,700~{\rm m},
\qquad
\RsunN=695\,700\,000~{\rm m},

one obtains

x=0.004650467260962157,
F_{3/4}=0.3367857708787376,

\Tcal=1.058043703626217,
\Lcal=3.323942326489060.\nonumber

The point-local Earth factor and the associated bounded proxy are

\kop=0.9451405424678726,
\qquad
1-\kop^2=0.1067093549835355.

The action-anchor diagnostic and its inverse are

\frac{1}{\pi^3F_{3/4}}=0.09576275847120606,
\qquad
\pi^3F_{3/4}=10.442472793854199.

The gravity-anchor diagnostic is

\pi^4F_{3/4}=32.805995814483634.

Using CODATA 2022 central readouts,

\ell_{P,\oplus}=1.6162550244\times10^{-35}~{\rm m},

t_{P,\oplus}=5.3912464483\times10^{-44}~{\rm s},

m_{P,\oplus}=2.1764343427\times10^{-8}~{\rm kg},

the declared-class readout gives

\ell_{P,\Hcal}=5.3723384861\times10^{-35}~{\rm m},
\qquad
 t_{P,\Hcal}=5.7041743593\times10^{-44}~{\rm s}.

The mass readout remains

m_{P,\Hcal}=\mu\,(2.1764343427\times10^{-8}~{\rm kg}).

On the declared conditional phase-tension branch alone,

m_{P,\Hcal}^{(\Tph)}=7.3299211788\times10^{-9}~{\rm kg}.

These are central Earth/SI numerical readouts of the declared calibration algebra. Uncertainty is inherited from the standard Planck readout where applicable; no new constant measurement is introduced.

Figure or table content is omitted from the web reader; use the canonical manuscript for the exact object.

The construction is unavailable, rejected, or overread under any of the following conditions.

F0. Imported calibration-class failure..

If the CCL Solar-System calibration class fails its own reconstruction, witness, loop-exactness, range/Doppler admissibility, or boundary-stability conditions, then the factors $\Lcal$TeX\Lcal and $\Tcal$TeX\Tcal are unavailable.

F1. Mixed-layer substitution..

If only $c_\oplus$TeXc_\oplus is replaced by $\pi c_\oplus$TeX\pi c_\oplus while $\hbar_\oplus$TeX\hbar_\oplus and $G_\oplus$TeXG_\oplus remain in the local layer, the result is reference, not the declared-class calibration reference--reference.

F2. Hidden mass closure..

If a value of $\mu$TeX\mu is used as a branch closure without a declared mass-bearing invariant, then the mass part is overclaimed. If a diagnostic value of $\mu$TeX\mu is displayed without being explicitly marked as diagnostic, or if it is used beyond its diagnostic row, then the diagnostic reading is also overclaimed.

F3. Branch hardening..

If the phase-rigidity tension branch is read as forced, unique, empirical, or branch-independent, then the reading exceeds reference.

F4. Standard-sector collapse..

If $\Tph$TeX\Tph is identified with a Higgs/Yukawa object, a QCD trace-anomaly term, a QCD string tension, a flux-tube tension, a Regge slope, a fundamental string tension, or any Standard-Model mass mechanism, then the construction collapses distinct lanes.

F5. Universalization..

The paper does not provide an SI replacement, a new measurement or redefinition of $c$TeXc, $h$TeXh, $\hbar$TeX\hbar, or $G$TeXG, a derivation of those constants, a minimum-length or minimum-time theorem, a universal mass-origin theorem, a complete Standard Model spectrum, a QCD replacement, a string/M equivalence, a universal compactification theorem, or a UV completion.

The formal core is positive-real dimension-vector algebra with explicitly declared constants.

definition: Formal variables. Assume positive symbols

\pi>0,
\qquad F>0,
\qquad \mu>0,
\qquad \ell_0>0,
\qquad t_0>0,
\qquad m_0>0.

Define

T=\pi F,
\qquad
L=\pi^2F,
\qquad
C=L/T,

H_\hbar=\mu L^2/T,
\qquad
H_G=L^3/(\mu T^2).

proposition: Symbolic verification target. In the ordered field of positive real numbers,

C=\pi,

\sqrt{\frac{H_\hbar H_G}{C^3}}=L,
\qquad
\sqrt{\frac{H_\hbar H_G}{C^5}}=T,
\qquad
\sqrt{\frac{H_\hbar C}{H_G}}=\mu,

and for any positive invariant $Q\sim M^aL^bT^c$TeXQ\sim M^aL^bT^c with $a\ne0$TeXa\ne0,

\mu_Q=L^{-b/a}T^{-c/a}.

proof. The first identity follows from $C=(\pi^2F)/(\pi F)=\pi$TeXC=(\pi^2F)/(\pi F)=\pi. The three square-root identities reduce to $\sqrt{L^2}$TeX\sqrt{L^2}, $\sqrt{T^2}$TeX\sqrt{T^2}, and $\sqrt{\mu^2}$TeX\sqrt{\mu^2}; positivity fixes the positive branch. Equation reference follows by solving $\mu^aL^bT^c=1$TeX\mu^aL^bT^c=1 on the positive branch.

Figure or table content is omitted from the web reader; use the canonical manuscript for the exact object.

A declared CCL Solar-System calibration class with $V_{\rm read}/c_\oplus=\pi$TeXV_{\rm read}/c_\oplus=\pi and $D_0=\tau_0\pi^2F_{3/4}$TeXD_0=\tau_0\pi^2F_{3/4} induces a lawful Planck-triad readout only when all dimensional quantities are lifted together. The declared lift fixes

\ell_{P,\Hcal}=\pi^2F_{3/4}\ell_{P,\oplus},
\qquad
 t_{P,\Hcal}=\pi F_{3/4}t_{P,\oplus},

while leaving

m_{P,\Hcal}=\mu m_{P,\oplus}

open. That open status is the central guardrail: it prevents a length-time calibration note from becoming a hidden mass-origin theorem.

The branch ledger shows how $\mu$TeX\mu can be closed only after a mass-bearing invariant is explicitly declared. Compton quantities, Schwarzschild radii, and dimensionless couplings do not select $\mu$TeX\mu by themselves. A phase-rigidity tension $\Tph\sim E/L$TeX\Tph\sim E/L is recorded as a declared conditional branch, not a selected, forced, empirical, or paper-wide closure. On that branch, and only under its declared branch-local invariant premise, it yields $\mu=F_{3/4}$TeX\mu=F_{3/4} and $m_{P,\Hcal}^{(\Tph)}=F_{3/4}m_{P,\oplus}$TeXm_{P,\Hcal}^{(\Tph)}=F_{3/4}m_{P,\oplus}. This is a branch-conditioned calibration readout, not a universal Planck-mass prediction and not an identity with QCD, electroweak, string, or ultraviolet-completion structures.

For $q=3/4$TeXq=3/4,

F_{3/4}=\frac14\frac{1-x}{1-x^{1/4}},
\qquad
x=\frac{695700000}{149597870700}.

The exact symbolic calibration factors are

\Tcal=\pi F_{3/4},
\qquad
\Lcal=\pi^2F_{3/4},
\qquad
\Ccal=\pi.

The imported path and point-local factors are

\kpath=\frac1\pi,
\qquad
\kop=\frac{1}{\pi F_{3/4}},
\qquad
1-\kop^2=1-\frac{1}{\pi^2F_{3/4}^2}.

On the declared conditional phase-tension branch alone,

\mu_{\Tph}=F_{3/4},
\qquad
 c_{\Hcal}^{(\Tph)}=\pi c_\oplus,
\qquad
 \hbar_{\Hcal}^{(\Tph)}=\pi^3F_{3/4}^2\hbar_\oplus,
\qquad
 G_{\Hcal}^{(\Tph)}=\pi^4G_\oplus.

The CHC references below are cited only for the scientific objects imported or bounded by the present construction: the root global-phase-field branch, the declared Solar-System calibration class, mass-rigidity and bound-state grammar, electroweak non-identity firewall, strong-sector tension boundary, and typed string/M and large-$N$TeXN comparator ledgers [citation]. No internal reference is used as a substitute for the local derivations above.

Funding and competing interests..

No external funding was received for this work. The author declares no competing interests.