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 from the CCL inverse problem is the dimensionless path functional $\rho>0$TeX\rho>0 and the profile factor $F(q)>0$TeXF(q)>0. The corresponding length and time ratios are

\frac{V_{\rm read}}{c_\oplus}=\rho,
\qquad
D_0=\tau_0\rho^2F(q).

Here $V_{\rm read}$TeXV_{\rm read} is a path-normalization functional, not a local, group, signal, or universal propagation speed. The parameters $(\rho,q)$TeX(\rho,q) must be estimated from calibration data under the CCL rank condition [citation]. The specialization $H_\pi:(\rho,q)=(\pi,3/4)$TeXH_\pi:(\rho,q)=(\pi,3/4) is a testable submodel and not a consequence of detector phase completion.

Replacing only $c_\oplus$TeXc_\oplus by $\rho c_\oplus$TeX\rho c_\oplus in reference changes one dimensional component while holding the others fixed. Such a calculation is inconsistent with a common change of dimensional calibration. The transformation must act on every dimensional quantity according to its mass--length--time exponents.

The mass factor remains explicit because the CCL inverse problem contains no mass-sector observable. Thus $m_{P,\Hcal}=\mu m_{P,\oplus}$TeXm_{P,\Hcal}=\mu m_{P,\oplus} until an additional mass-bearing condition is imposed. More generally, dimensional covariance implies that no dimensionless observable changes under the calibration map. This observation supplies a precise criterion for physical content: a proposed extension must alter at least one independently measurable dimensionless relation.

The analysis concerns a dimensional calibration map with parameters $(\rho,q,\mu)$TeX(\rho,q,\mu). It neither redefines SI units nor derives $c$TeXc, $h$TeXh, $\hbar$TeX\hbar, $G$TeXG, a minimum length, or a quantum-gravity scale. General theorems are stated for arbitrary positive $(\rho,F(q),\mu)$TeX(\rho,F(q),\mu); $H_\pi$TeXH_\pi is separated as a numerical submodel.

The companion metrology calculation checks the use of BIPM, CODATA, NIST, and IAU reference values. Such a reference check does not identify $(\rho,q,\mu)$TeX(\rho,q,\mu) and supplies no physical evidence for $H_\pi$TeXH_\pi.

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

All conditional statements below display their additional hypotheses explicitly. In particular, numerical agreement within $H_\pi$TeXH_\pi cannot establish $H_\pi$TeXH_\pi because the same quantities were used to define that specialization.

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].

CCL calibration factors

Let the path-normalization readout be

\rho\equiv\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 pair $(\rho,q)$TeX(\rho,q) is an empirical parameterization. It is locally identifiable only when the profiled CCL sensitivity matrix has rank two. 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 resulting relations are

\frac{V_{\rm read}}{c_\oplus}=\rho,
\qquad
R\equiv \frac{D_0}{\tau_0}=\rho^2F(q).

Equivalently,

\kpath=\frac1\rho,
\qquad
\kop=\frac{1}{\rho F(q)}.

The path-harmonic normalization $\kpath$TeX\kpath and the point-local factor $\kop$TeX\kop are distinct quantities. No detector boundary condition fixes either one.

definition: Calibration factors. For positive $\rho$TeX\rho and $F(q)$TeXF(q), define

\Tcal\equiv \rho F(q),\qquad
\Lcal\equiv \rho^2F(q),\qquad
\Ccal\equiv \frac{\Lcal}{\Tcal}=\rho .

Here $\Tcal$TeX\Tcal and $\Lcal$TeX\Lcal are time and length calibration factors, while $\Ccal$TeX\Ccal is their ratio.

remark: Identifiability requirement. Equations reference--reference propagate estimates; they do not estimate $(\rho,q)$TeX(\rho,q). If the CCL inverse problem fails its admissibility or rank condition, the numerical factors are not determined.

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: Dimensional calibration map. Let

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

be an independent mass-calibration factor. For any dimensional quantity $Q$TeXQ with dimension vector $(a,b,c)$TeX(a,b,c), define

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

This map is a change of dimensional representation. It acquires physical content only if the theory supplies a change in a dimensionless observable.

proposition: Transformation of dimensional constants. Under reference,

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

 \hbar_{\Hcal}=\mu\frac{\Lcal^2}{\Tcal}\hbar_\oplus=\mu\rho^3F(q)\hbar_\oplus,

 G_{\Hcal}=\frac{\Lcal^3}{\mu\Tcal^2}G_\oplus=\frac{\rho^4F(q)}{\mu}G_\oplus.

remark: Status of $c_{\Hcal}$TeXc_{\Hcal}. The symbol $c_{\Hcal}$TeXc_{\Hcal} in reference is a component of the dimensional transformation. It is not an observed local light speed, a new SI value of $c$TeXc, or 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: Interpretation. The symbols $c_{\Hcal}$TeXc_{\Hcal}, $\hbar_{\Hcal}$TeX\hbar_{\Hcal}, and $G_{\Hcal}$TeXG_{\Hcal} are components of the same dimensional map. They are neither new measurements nor assertions that the SI constants have changed.

remark: Why $\mu$TeX\mu is explicit. The CCL observables supply length and time factors but no mass-sector equation. Therefore $\mu$TeX\mu cannot be inferred from them.

definition: Transformed Planck triad. 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: Covariance of the Planck triad. Under the dimensional calibration map, reference satisfies

\ell_{P,\Hcal}=\Lcal\ell_{P,\oplus}=\rho^2F(q)\ell_{P,\oplus},

 t_{P,\Hcal}=\Tcal t_{P,\oplus}=\rho F(q)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: Inconsistency of a speed-only substitution. If only $c_\oplus$TeXc_\oplus is replaced by $\rho c_\oplus$TeX\rho c_\oplus in reference while $\hbar_\oplus$TeX\hbar_\oplus and $G_\oplus$TeXG_\oplus are held fixed, the result is

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

It is not the common dimensional transformation reference--reference.

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

theorem: Non-identifiability without a mass-bearing observable. Length--time calibration and reference do not identify $\mu$TeX\mu. A value can be obtained only after an independent positive quantity of nonzero mass dimension is constrained. Hence the transformed Planck mass is not predicted by the CCL calibration data.

proof. The factors $\Lcal$TeX\Lcal and $\Tcal$TeX\Tcal may be estimated from $(\rho,q)$TeX(\rho,q), whereas reference contains the independent factor $\mu$TeX\mu. For every $\mu>0$TeX\mu>0 the same length and time calibration is obtained. The forward map from $(\rho,q,\mu)$TeX(\rho,q,\mu) to length--time observables is therefore constant along the $\mu$TeX\mu direction and its Jacobian has a zero $\mu$TeX\mu column. Thus $\mu$TeX\mu is not locally or globally identifiable from those observables.

theorem: Mass-factor determination by an independent 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 an independent physical hypothesis imposes

Q_{\Hcal}=Q_\oplus,

then the mass factor is fixed uniquely 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: Mutually independent hypotheses. reference lists algebraic consequences of different conditions; it does not rank or establish them. Simultaneously imposing two rows generally overconstrains $(\rho,q,\mu)$TeX(\rho,q,\mu) and must therefore be tested as a separate model.

theorem: Dimensionless-observable invariance. Let $Q_1,\ldots,Q_n$TeXQ_1,\ldots,Q_n have dimension vectors $\boldsymbol d_i=(a_i,b_i,c_i)$TeX\boldsymbol d_i=(a_i,b_i,c_i), and let

I=\prod_{i=1}^{n}Q_i^{s_i}

be dimensionless, so that $\sum_i s_i\boldsymbol d_i=\boldsymbol 0$TeX\sum_i s_i\boldsymbol d_i=\boldsymbol 0. Under reference,

I_{\Hcal}=I_\oplus .

Consequently, a multiplicative recalibration of base dimensions cannot by itself predict a change in any dimensionless observable.

proof. Applying reference to every factor gives

\frac{I_{\Hcal}}{I_\oplus}
=\mu^{\sum_i s_i a_i}\Lcal^{\sum_i s_i b_i}\Tcal^{\sum_i s_i c_i}.

All three exponents vanish because $I$TeXI is dimensionless, proving reference. Since experimental comparisons reduce to dimensionless ratios of readings to standards, a calibration map alone has no observable effect. A physical theory must additionally change a dimensionless constitutive relation, coupling, spectrum ratio, or correlation function.

corollary: Planck-triad non-predictivity. The ratios $\ell_{P,\Hcal}/\Lcal$TeX\ell_{P,\Hcal}/\Lcal, $t_{P,\Hcal}/\Tcal$TeXt_{P,\Hcal}/\Tcal, and $m_{P,\Hcal}/\mu$TeXm_{P,\Hcal}/\mu reproduce their reference values, and every dimensionless combination of the transformed Planck triad and consistently transformed standards is unchanged. Thus reference expresses dimensional covariance, not new Planck-scale phenomenology.

proof. The three equalities follow from reference; the general statement follows from reference.

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 positive variable of dimension

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

Its invariance is an additional mass-sector hypothesis, not an output of length--time calibration. It is not identified with a Standard Model Higgs parameter, a Yukawa coupling, a QCD trace-anomaly contribution, a QCD string tension, or a fundamental string tension.

definition: Phase-tension invariance hypothesis. Consider the additional hypothesis

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

No result in the preceding sections implies this equality; it requires independent empirical or microscopic justification.

proposition: Consequence of phase-tension invariance. Under reference,

\mu=\frac{\Tcal^2}{\Lcal}=F(q).

Consequently,

m_{P,\Hcal}^{(\Tph)}=F(q)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 invariance condition sets this ratio equal to one, so $\mu=\Tcal^2/\Lcal$TeX\mu=\Tcal^2/\Lcal. Substituting $\Tcal=\rho F(q)$TeX\Tcal=\rho F(q) and $\Lcal=\rho^2F(q)$TeX\Lcal=\rho^2F(q) gives reference. Equation reference follows from reference.

corollary: Conditional transformed constants. Substituting $\mu=F(q)$TeX\mu=F(q) into reference--reference gives

c_{\Hcal}^{(\Tph)}=\rho c_\oplus,
\qquad
 \hbar_{\Hcal}^{(\Tph)}=\rho^3F(q)^2\hbar_\oplus,
\qquad
 G_{\Hcal}^{(\Tph)}=\rho^4G_\oplus.

These are components of the dimensional map under an added hypothesis. They do not represent new measurements of the corresponding constants.

proof. Insert $\mu=F(q)$TeX\mu=F(q) from reference into $c_{\Hcal}=\rho c_\oplus$TeXc_{\Hcal}=\rho c_\oplus, $\hbar_{\Hcal}=\mu\rho^3F(q)\hbar_\oplus$TeX\hbar_{\Hcal}=\mu\rho^3F(q)\hbar_\oplus, and $G_{\Hcal}=\rho^4F(q)G_\oplus/\mu$TeXG_{\Hcal}=\rho^4F(q)G_\oplus/\mu. Cancellation gives reference. Since the calculation uses only a dimensional lift and the additional phase-tension invariance postulate, it has no independent empirical content.

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].

For illustration only, define

H_\pi:\qquad q=\frac34,\qquad \rho=\pi.

Using

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

one then obtains

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

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

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 dimensional transformation 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}).

If phase-tension invariance is additionally imposed,

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

These numbers are conditional evaluations of reference; they are not predictions of the theory. A statistical test of $H_\pi$TeXH_\pi must estimate $(\rho,q)$TeX(\rho,q) from calibration data not used to define the hypothesis, report the covariance or profile likelihood, and evaluate predictive performance on held-out observations. Without those steps, the numerical table has no evidential force.

The interpretation fails under any of the following conditions.

F0. Imported calibration-class failure..

If the CCL inverse problem fails its reconstruction, admissibility, or rank conditions, then numerical values of $\Lcal$TeX\Lcal and $\Tcal$TeX\Tcal are unavailable.

F1. Mixed-layer substitution..

If only $c_\oplus$TeXc_\oplus is replaced by $\rho c_\oplus$TeX\rho c_\oplus while $\hbar_\oplus$TeX\hbar_\oplus and $G_\oplus$TeXG_\oplus remain fixed, the result is reference, not the common transformation reference--reference.

F2. Hidden mass closure..

Inferring $\mu$TeX\mu without an independent mass-bearing observable contradicts reference.

F3. Branch hardening..

Treating phase-tension invariance as derived, unique, or empirical exceeds reference.

F4. Standard-sector collapse..

Identifying $\Tph$TeX\Tph with a Higgs/Yukawa object, a QCD trace-anomaly term, a flux-tube tension, or a fundamental string tension requires an explicit dynamical map absent here.

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

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

Define

T=\rho F,
\qquad
L=\rho^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=\rho,

\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 $C=(\rho^2F)/(\rho F)=\rho$TeXC=(\rho^2F)/(\rho F)=\rho. 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.

The compact scalar branch adds a coordinate-invariant target circumference $L_\chi=\oint M_{\rm eff}(\mathcal H)\,d\mathcal H$TeXL_\chi=\oint M_{\rm eff}(\mathcal H)\,d\mathcal H, but it does not select its value in Planck units. A field redefinition changes coordinate periods while preserving $L_\chi$TeXL_\chi; a change of metrological units changes numerical representations while preserving all dimensionless observables. Neither operation supplies a mass modulus, a Planck-triad identity, or an empirical calibration.

Predictive dimensional linkage requires a common action that fixes the conversion functions before data are inspected. If $q$TeXq such parameters control $m>q$TeXm>q dimensionless readouts, their image has codimension at least $m-q$TeXm-q and obeys corresponding compatibility equations. Assigning an independent modulus to each readout saturates the observable rank and reproduces the non-predictivity theorem of this paper. Compact topology is therefore genuine global structure, but not a cure for scale underdetermination.

The closure test for the conditional Planck-triad and mass-modulus readout is applied to a dimensionless observable vector $y\in\mathbb R^m$TeXy\in\mathbb R^m formed from fixed reference scales and the declared basket of dimensionless Planck-triad ratios, mass ratios, and metrology invariants. Let $a$TeXa range over the independent constitutive inputs comprising unit convention, compact circumference, modulus response, and mass calibration.

proposition: Functional saturation, finite closure, and sector admissibility. Suppose the unrestricted prediction map $F:a\mapsto y$TeXF:a\mapsto y is continuously differentiable on a Banach space of constitutive inputs. If $D_aF$TeXD_aF is surjective and has a bounded right inverse at the calibration point, the unrestricted family is locally open in observable space and supplies no nonzero local equality restriction on $y$TeXy. Suppose instead that a single microscopic closure replaces $a$TeXa by finite parameters $\theta\in\mathbb R^p$TeX\theta\in\mathbb R^p, with profiled nuisance coordinates $\eta\in\mathbb R^q$TeX\eta\in\mathbb R^q. If

J=D_\eta F_{\rm cl}D_\theta F_{\rm cl},
 \qquad \operatorname{rank}J=r<m,

then there are $m-r$TeXm-r independent first-order restrictions

w^{\mathsf T}\delta y=0,
 \qquad w\in\ker J^{\mathsf T}.

If the rank is constant locally, these restrictions are tangent to a compatibility manifold of codimension $m-r$TeXm-r. For this sector, the finite closure is admissible only if only dimensionless invariants produced by one finite modulus model are compared, with unit rescalings quotiented out before rank evaluation.

proof. Split surjectivity gives a bounded right inverse $R$TeXR with $D_aF\,R=I_m$TeXD_aF\,R=I_m. The Banach-space submersion theorem then makes $F$TeXF locally onto a neighborhood of the calibrated observable vector. Any smooth equality holding throughout that image must therefore vanish on an open set and contributes no model-specific local restriction. Under finite closure, the attainable first-order variations are exactly the column space of $J$TeXJ. Its orthogonal complement is $\ker J^{\mathsf T}$TeX\ker J^{\mathsf T}, whose dimension is $m-r$TeXm-r by rank--nullity, which proves reference. The constant-rank theorem supplies the stated local manifold. The sector condition is necessary because dimensionful coincidences can be created by changing units and do not close the mass-modulus branch. Failure of that condition therefore rejects the proposed microscopic closure before parameter estimation can be counted as evidence for it.

For an identifiable CCL calibration $(\rho,q)$TeX(\rho,q), a consistent dimensional map gives

\ell_{P,\Hcal}=\rho^2F(q)\ell_{P,\oplus},
\qquad
 t_{P,\Hcal}=\rho F(q)t_{P,\oplus},

while leaving

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

with $\mu$TeX\mu unidentified by length--time data.

The dimensionless-observable invariance theorem shows that these rescalings have no physical content by themselves. A genuine extension must change an independently testable dimensionless relation. Phase-tension invariance would imply $\mu=F(q)$TeX\mu=F(q), but it is an additional hypothesis. The specialization $H_\pi$TeXH_\pi supplies conditional numbers only and requires an independent likelihood analysis before it can be treated as evidence.

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

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

The general symbolic calibration factors are

\Tcal=\rho F(q),
\qquad
\Lcal=\rho^2F(q),
\qquad
\Ccal=\rho.

The path and point-local factors are

\kpath=\frac1\rho,
\qquad
\kop=\frac{1}{\rho F(q)},
\qquad
1-\kop^2=1-\frac{1}{\rho^2F(q)^2}.

Under phase-tension invariance,

\mu_{\Tph}=F(q),
\qquad
 c_{\Hcal}^{(\Tph)}=\rho c_\oplus,
\qquad
 \hbar_{\Hcal}^{(\Tph)}=\rho^3F(q)^2\hbar_\oplus,
\qquad
 G_{\Hcal}^{(\Tph)}=\rho^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 formalism, electroweak non-identity firewall, strong-sector tension boundary, and formally specified 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.