Appendix sections

A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13 A14 A15 A16 A17 A18 A19

Appendix

Quantitative Appendix

Finite-Boundary Redshift-Distance Framework

Plain-language companion: For a plain-language orientation before the calculations, see What the book is really asking and When rulers stop agreeing.

This appendix collects the equations, variables, fitted forms, diagnostic quantities, and replication steps used in the finite-boundary redshift-distance analysis. The purpose is reproducibility: a reader should be able to follow the calculation sequence from redshift data through finite-response coordinates, BAO/CMB diagnostics, projection targets, and register-scaling constants.

A1. Purpose, inputs, and notation

Calculation purpose: define the minimum data inputs, numerical settings, and fitting conventions needed to reproduce the diagnostic calculations.

All distance-dependent calculations require one consistent distance coordinate. If the plotted coordinate is \(d/1000\,\mathrm{Mpc}\), then \(d_b\) uses the same scaled unit and \(g_0\) uses the reciprocal unit. Dimensionless combinations such as \(g_0d_b\), \(A\), \(X\), and \(\chi\) are unchanged when the unit conversion is handled consistently.

Table A1-1. Minimum data inputs.

Probe	Required quantities	Used for
Type Ia supernovae	\(z\), distance modulus or distance coordinate \(d\), uncertainties where available	A2-A8 response fits, residuals, stability diagnostics
Supernova sky sectors	\(z\), \(d\), right ascension, declination	A9 directional partitioning
BAO standard-ruler data	redshift, observable type \(D_M/r_d\), \(D_H/r_d\), or \(D_V/r_d\), value, uncertainty	A10, A14-A17 ruler diagnostics
Cosmic chronometers	redshift, \(H(z)\), uncertainty, method group	A11 derivative-shape comparison
Time-dilation constraints	published exponent \(b\), uncertainty, redshift range, object class	A12 arrival-time constraint
Compressed CMB priors	\(R\), \(\ell_A\), \(\omega_bh^2\), covariance matrix, \(z_*\)	A13-A17 CMB and projection diagnostics
Register-scaling tests	\(D=AG\), \(X=A/A_t\), \(\eta=g_{\mathrm{bar}}/g^\dagger\)	A19 scaling constants and response-engine weighting

Table A1-2. Numerical settings.

Setting	Representative value or rule	Controls
Train/test split	0.80 training fraction	held-out RMSE and overfit gap
Stability stress	minimum rows per window 80; multiple upper-redshift windows; training fractions 0.7, 0.8, 0.9	spread in \(d_b\) and finite-boundary win rate
BAO range	maximum BAO redshift about 2.3; Ly\(\alpha\) points included where stated	BAO shape comparison
Chronometer filter	maximum redshift about 1.97; locked \(d_b\approx2.04\) where stated	derivative-shape comparison
CMB prior	\(z_*\approx1089.92\); covariance-weighted compressed-prior statistic	CMB distance-prior diagnostic
Shared-ruler test	one common calibration scale \(S\) for BAO and CMB terms	shared-ruler failure statistic
Separate-scale test	independent \(S_{\mathrm{BAO}}\) and \(S_{\mathrm{CMB}}\)	CMB-to-BAO scale ratio
Projection target	preserve BAO range while reaching the CMB-side compressed scale	sharp transition and preferred projection law

Fitting convention

For one-parameter scale fits of the form \(Y_i=S F_i\), the diagnostic scale is selected by weighted least squares:

\[\chi^2(S)=\sum_i\left(\frac{Y_i-SF_i}{\sigma_i}\right)^2 \tag{A1.1}\]

When the prediction is linear in \(S\), the equivalent analytic solution is:

\[S=\frac{\sum_i Y_iF_i/\sigma_i^2}{\sum_i F_i^2/\sigma_i^2}. \tag{A1.2}\]

For mixed BAO/CMB tests where compressed CMB priors introduce covariance terms, the same principle is used, but the total objective may be evaluated by one-dimensional numerical minimization over the shared or separate scale.

Table A1-3. Master glossary.

Symbol	Meaning	Status
\(z\)	observed redshift	dimensionless direct observable
\(d\)	supplied distance coordinate used in supernova fitting	Mpc or scaled Mpc
\(A=\ln(1+z)\)	accumulated path response	dimensionless
\(G(d)\)	local response rate	inverse distance
\(g_0\)	near-origin response-rate scale	inverse distance
\(d_b\)	finite-boundary distance scale	same units as \(d\)
\(p=g_0d_b\)	dimensionless accumulated-response exponent	dimensionless
\(x(z)\)	inverted finite-boundary distance coordinate	same units as \(d_b\)
\(x\prime(z)\)	derivative of model coordinate with respect to redshift	distance per redshift
\(S\)	nuisance calibration scale	maps model coordinate into BAO/CMB ratio units
\(S_{\mathrm{BAO}},S_{\mathrm{CMB}}\)	separate effective calibration scales	diagnostic scales
\(T(z),P_\star(z)\)	projection factors	dimensionless
\(D=AG\)	raw morphology response coordinate	dimensionless times inverse distance, normalized before engine use
\(D_n=D/D_0\)	normalized morphology response	dimensionless
\(X=A/A_t\)	normalized projection coordinate	dimensionless
\(\eta=g_{\mathrm{bar}}/g^\dagger\)	weak-field compliance coordinate	dimensionless
\(u=2GM/(Rc^2)\)	compactness coordinate	dimensionless
\(K_D,K_X,K_\eta,K_u\)	register-scaling constants and compactness working exponent	dimensionless

Table A1-4. Calculation map.

Section	Calculation	Primary output
A2	redshift-to-response transform	\(A(d)=\ln(1+z)\)
A3	empirical redshift-distance forms	fit parameters and residuals
A4	candidate accumulated-response laws	candidate \(A(d)\) and \(G(d)\) forms
A5	finite-boundary local-response law	\(G(d)\), \(A(d)\), \(z(d)\)
A6	inverted model coordinate	\(x(z)\), \(x\prime(z)\)
A7	fit statistics	RMSE, \(\chi^2\), AIC, BIC
A8	stability diagnostics	overfit gap and \(d_b\) spread
A9	directional partitioning	sector-specific \(d_b\) values
A10	BAO standard-ruler shape	BAO ratio predictions and \(\chi^2_{\mathrm{BAO}}\)
A11	chronometer derivative shape	calibrated derivative comparison
A12	time-dilation exponent	constraint on \(b\)
A13	compressed CMB priors	\(\chi^2_{\mathrm{CMB}}\)
A14	joint BAO+CMB shared ruler	shared-scale statistic
A15	separate BAO/CMB scales	\(S_{\mathrm{CMB}}/S_{\mathrm{BAO}}\)
A16	redshift-dependent projection	candidate \(T(z)\) behavior
A17	sharp transition and preferred projection	BAO preservation plus CMB/JWST target behavior
A18	replication chain	end-to-end calculation sequence
A19	unified registers and scaling constants	\(C(X,\eta,u,D_n)\) and \(K\) values

A2. Accumulated redshift response

Calculation purpose: transform observed redshift into accumulated path response and define the local response derivative.

Inputs are observed redshift \(z\) and the distance coordinate \(d\). The transformation below is independent of the later choice of local response law.

\[A(d)=\ln(1+z) \tag{A2.1}\]

\[1+z=e^{A(d)} \tag{A2.2}\]

\[z(d)=e^{A(d)}-1 \tag{A2.3}\]

The path-response interpretation treats \(A(d)\) as the integral of a local response rate along the signal path:

\[A(d)=\int_0^d G(s)\,ds \tag{A2.4}\]

\[G(d)=\frac{dA}{dd}. \tag{A2.5}\]

If \(d\) is measured in Mpc, then \(G(d)\) has units of Mpc\(^{-1}\). The product \(G(s)ds\) is dimensionless.

A3. Empirical redshift-distance forms

Calculation purpose: fit compact empirical redshift-distance curves before interpreting the result as accumulated response.

Inputs are supernova redshift and distance coordinate. Outputs are fitted curve parameters, predicted redshifts, residuals, and ranking statistics. These forms are empirical fits, not dynamical derivations.

\[z_{\mathrm{lin}}(d)=a d \tag{A3.1}\]

\[z_{\exp}(d)=e^{a d}-1 \tag{A3.2}\]

\[z_{\mathrm{pow}}(d)=(1+a d)^n-1 \tag{A3.3}\]

\[z_{\mathrm{quad}}(d)=a d+b d^2 \tag{A3.4}\]

A flat \(\Lambda\)CDM luminosity-distance reference is written schematically as:

\[d_L(z;H_0,\Omega_m)=\frac{c(1+z)}{H_0}\int_0^z \frac{du}{\sqrt{\Omega_m(1+u)^3+1-\Omega_m}}. \tag{A3.5}\]

Here \(a\), \(b\), and \(n\) are fitted empirical parameters; \(u\) is a dummy integration variable.

Table A3-1. Representative direct-fit performance.

Model form	Representative RMSE
Power-law form	0.0213
Linear Hubble form	0.0488
Quadratic proxy	0.0709
Exponential form	0.1303

Minimal recipe: read \(z,d\); fit Equations (A3.1)-(A3.4); compute predicted redshifts and residuals using A7; rank by RMSE, AIC, BIC, and residual structure.

A4. Candidate accumulated-response laws

Calculation purpose: compare candidate forms for \(A(d)\) and their implied local response rates \(G(d)\).

Constant accumulation

\[A(d)=kd \tag{A4.1}\]

\[G(d)=k \tag{A4.2}\]

Quadratic deformation

\[A(d)=kd+\beta d^2 \tag{A4.3}\]

\[G(d)=k+2\beta d \tag{A4.4}\]

Logarithmic power-law shadow

\[A(d)=n\ln(1+a d) \tag{A4.5}\]

\[G(d)=\frac{na}{1+a d} \tag{A4.6}\]

Equations (A4.5)-(A4.6) bridge the empirical power-law fit and the finite-boundary interpretation: accumulated response grows logarithmically while the local rate declines with distance.

Cubic deformation

\[A(d)=kd+\beta d^2+\gamma d^3 \tag{A4.7}\]

\[G(d)=k+2\beta d+3\gamma d^2. \tag{A4.8}\]

Inputs are \(d\) and \(A(d)\) from A2. Fit each candidate response law, differentiate to obtain \(G(d)\), and compare residual structure.

A5. Finite-boundary local-response law

Calculation purpose: define the finite-boundary local-response law, integrate it, and recover \(z(d)\).

\[G(d)=\frac{g_0}{1+d/d_b} \tag{A5.1}\]

\[G(d)=\frac{g_0d_b}{d_b+d} \tag{A5.2}\]

\(g_0\) is the near-origin response-rate scale and \(d_b\) is the finite-boundary distance scale. If \(d\) and \(d_b\) are in Mpc, then \(g_0\) is in Mpc\(^{-1}\).

\[A(d)=\int_0^d \frac{g_0}{1+s/d_b}\,ds \tag{A5.3}\]

\[A(d)=g_0d_b\int_1^{1+d/d_b}\frac{du}{u} \tag{A5.4}\]

\[A(d)=g_0d_b\ln\!\left(1+\frac{d}{d_b}\right) \tag{A5.5}\]

\[z(d)=\exp\!\left[g_0d_b\ln\!\left(1+\frac{d}{d_b}\right)\right]-1 \tag{A5.6}\]

\[z(d)=\left(1+\frac{d}{d_b}\right)^{g_0d_b}-1. \tag{A5.7}\]

The dimensionless product \(g_0d_b\) controls the curvature of the redshift-distance relation.

A6. Inverted distance-redshift coordinate

Calculation purpose: invert the finite-boundary redshift relation to obtain \(x(z)\) and \(x\prime(z)\).

\[1+z=\left(1+\frac{d}{d_b}\right)^{g_0d_b} \tag{A6.1}\]

\[(1+z)^{1/(g_0d_b)}=1+\frac{d}{d_b} \tag{A6.2}\]

\[x(z)=d_b\left[(1+z)^{1/(g_0d_b)}-1\right] \tag{A6.3}\]

\[x'(z)=\frac{dx}{dz}=\frac{1}{g_0}(1+z)^{1/(g_0d_b)-1}. \tag{A6.4}\]

\(x(z)\) is a model coordinate, not automatically a physical BAO or CMB distance. BAO and CMB comparisons require a calibration scale such as \(S\).

A7. Fit residuals and model-ranking quantities

Calculation purpose: compute residuals, goodness-of-fit quantities, and information criteria.

\[r_i=y_i-\widehat{y}_i \tag{A7.1}\]

\[\mathrm{RMSE}=\sqrt{\frac{1}{n_{\mathrm{data}}}\sum_{i=1}^{n_{\mathrm{data}}}r_i^2} \tag{A7.2}\]

\[\chi^2=\sum_{i=1}^{n_{\mathrm{data}}}\left(\frac{y_i-\widehat{y}_i}{\sigma_i}\right)^2 \tag{A7.3}\]

\[\chi^2_\nu=\frac{\chi^2}{n_{\mathrm{data}}-k_{\mathrm{par}}} \tag{A7.4}\]

\[\mathrm{RSS}=\sum_{i=1}^{n_{\mathrm{data}}}r_i^2 \tag{A7.5}\]

\[\mathrm{AIC}=2k_{\mathrm{par}}+n_{\mathrm{data}}\ln\!\left(\frac{\mathrm{RSS}}{n_{\mathrm{data}}}\right) \tag{A7.6}\]

\[\mathrm{BIC}=k_{\mathrm{par}}\ln(n_{\mathrm{data}})+n_{\mathrm{data}}\ln\!\left(\frac{\mathrm{RSS}}{n_{\mathrm{data}}}\right). \tag{A7.7}\]

Inputs are observed values, predicted values, optional uncertainties, and parameter counts. Outputs are residuals, RMSE, chi-square statistics, AIC, and BIC.

A8. Cross-validation and stability diagnostics

Calculation purpose: test prediction error and boundary-scale stability under split or redshift-window changes.

\[\mathrm{RMSE}_{\mathrm{train}}=\sqrt{\frac{1}{n_{\mathrm{train}}}\sum_{i\in\mathrm{train}}r_i^2} \tag{A8.1}\]

\[\mathrm{RMSE}_{\mathrm{test}}=\sqrt{\frac{1}{n_{\mathrm{test}}}\sum_{i\in\mathrm{test}}r_i^2} \tag{A8.2}\]

\[\Delta_{\mathrm{overfit}}=\mathrm{RMSE}_{\mathrm{test}}-\mathrm{RMSE}_{\mathrm{train}} \tag{A8.3}\]

\[\{d_{b,1},d_{b,2},\ldots,d_{b,N}\} \tag{A8.4}\]

\[\widetilde{d}_b=\mathrm{median}(d_{b,j}) \tag{A8.5}\]

\[\mathrm{CV}_{d_b}=\frac{\sigma(d_{b,j})}{\mu(d_{b,j})}. \tag{A8.6}\]

Representative stability values include median \(d_b\approx1.57\), coefficient of variation \(\approx0.26\), and finite-boundary win rate \(\approx97.6\%\).

A9. Directional partitioning

Calculation purpose: fit the same finite-boundary accumulated-response law independently by sky sector.

\[A_j(d)=g_{0,j}d_{b,j}\ln\!\left(1+\frac{d}{d_{b,j}}\right) \tag{A9.1}\]

\[\mu_{d_b}=\frac{1}{N}\sum_{j=1}^N d_{b,j} \tag{A9.2}\]

\[\mathrm{CV}_{d_b}=\frac{\sqrt{\frac{1}{N-1}\sum_{j=1}^N(d_{b,j}-\mu_{d_b})^2}}{\mu_{d_b}}. \tag{A9.3}\]

Representative right-ascension-slice values are mean \(d_b\approx2.25\), coefficient of variation \(\approx0.077\), and max/min \(\approx1.23\).

A10. BAO standard-ruler diagnostics

Calculation purpose: compare BAO standard-ruler ratios against finite-boundary shape predictions.

The BAO comparison uses dimensionless standard-ruler ratios:

\[\frac{D_M(z)}{r_d} \tag{A10.1}\]

\[\frac{D_H(z)}{r_d} \tag{A10.2}\]

\[\frac{D_V(z)}{r_d}. \tag{A10.3}\]

Using a nuisance calibration scale \(S\), the diagnostic BAO predictions are:

\[\widehat{D_M/r_d}=Sx(z) \tag{A10.4}\]

\[\widehat{D_H/r_d}=Sx'(z) \tag{A10.5}\]

\[\widehat{D_V/r_d}=S\left[zx^2(z)x'(z)\right]^{1/3} \tag{A10.6}\]

\[\chi^2_{\mathrm{BAO}}=\sum_i\left(\frac{B_i-\widehat{B}_i}{\sigma_{B,i}}\right)^2 \tag{A10.7}\]

For a BAO point \(i\), let \(F_i\) be the corresponding model-shape term: \(x(z_i)\), \(x\prime(z_i)\), or \([z_i x^2(z_i)x\prime(z_i)]^{1/3}\).

\[\chi^2_{\mathrm{BAO}}(S)=\sum_i\left(\frac{B_i-SF_i}{\sigma_{B,i}}\right)^2 \tag{A10.8}\]

\[S_{\mathrm{BAO}}=\frac{\sum_i B_iF_i/\sigma_{B,i}^2}{\sum_i F_i^2/\sigma_{B,i}^2}. \tag{A10.9}\]

Representative simplified-shape values are \(\chi^2=11.8514\) for flat \(\Lambda\)CDM and \(\chi^2=28.1117\) for the finite-boundary case.

A11. Chronometer-style derivative comparison

Calculation purpose: compare chronometer \(H(z)\) trends with the inverse finite-boundary derivative shape.

\[H(z)=-\frac{1}{1+z}\frac{dz}{dt} \tag{A11.1}\]

\[H_{\mathrm{shape}}(z)\propto \frac{1}{x'(z)} \tag{A11.2}\]

\[H_{\mathrm{shape}}(z)\propto g_0(1+z)^{1-1/(g_0d_b)}. \tag{A11.3}\]

A calibration constant is required to express the diagnostic in km s\(^{-1}\) Mpc\(^{-1}\). This section compares shape, not a full expansion-history likelihood.

A12. Supernova time-dilation exponent

Calculation purpose: compare published supernova time-dilation exponents with the \(b=1\) and \(b=0\) cases.

\[\Delta t_{\mathrm{obs}}=\Delta t_{\mathrm{emit}}(1+z)^b \tag{A12.1}\]

\[b=1 \quad \text{standard arrival-time stretching} \tag{A12.2}\]

\[b=0 \quad \text{no arrival-time stretching} \tag{A12.3}\]

\[b\approx1.0062. \tag{A12.4}\]

The time-dilation result is an external constraint: a path-response redshift model must account for arrival-time stretching as well as wavelength shift.

A13. CMB compressed distance-prior quantities

Calculation purpose: evaluate compressed CMB distance-prior consistency using \(x(z_*)\) and the prior covariance.

\[\mathbf{p}=\left(R,\ell_A,\omega_bh^2\right)^T \tag{A13.1}\]

\[\ell_A=\pi\frac{D_M(z_*)}{r_s(z_*)} \tag{A13.2}\]

\[x(z_*)=d_b\left[(1+z_*)^{1/(g_0d_b)}-1\right] \tag{A13.3}\]

\[D_{M,\mathrm{model}}(z_*)=S_{\mathrm{CMB}}x(z_*) \tag{A13.4}\]

\[\ell_{A,\mathrm{model}}=\pi\frac{S_{\mathrm{CMB}}x(z_*)}{r_s(z_*)} \tag{A13.5}\]

\[R_{\mathrm{model}}=\frac{\sqrt{\Omega_m}H_0}{c}S_{\mathrm{CMB}}x(z_*) \tag{A13.6}\]

\[\chi^2_{\mathrm{CMB}}=(\mathbf{p}_{\mathrm{model}}-\mathbf{p}_{\mathrm{obs}})^TC^{-1}(\mathbf{p}_{\mathrm{model}}-\mathbf{p}_{\mathrm{obs}}). \tag{A13.7}\]

\(z_*\) is held near 1089.92 in these diagnostics. This is a compressed-prior comparison, not a full anisotropy-spectrum fit.

A14. Joint BAO+CMB shared-ruler test

Calculation purpose: test whether one calibration scale can satisfy both BAO ratios and compressed CMB-prior distances.

\[S_{\mathrm{BAO}}=S_{\mathrm{CMB}}=S \tag{A14.1}\]

\[D_M/r_d\approx Sx(z) \tag{A14.2}\]

\[\ell_A\approx\pi Sx(z_*) \tag{A14.3}\]

\[\chi^2_{\mathrm{total}}=\chi^2_{\mathrm{BAO}}+w_{\mathrm{CMB}}\chi^2_{\mathrm{CMB}} \tag{A14.4}\]

\[\chi^2_{\mathrm{total}}(S)=\chi^2_{\mathrm{BAO}}(S)+w_{\mathrm{CMB}}\chi^2_{\mathrm{CMB}}(S). \tag{A14.5}\]

Representative shared-ruler values are total \(\chi^2=11.8573\) and reduced \(\chi^2=0.9121\) for flat \(\Lambda\)CDM. For the locked finite-boundary case, BAO \(\chi^2=7730.6214\), CMB \(\chi^2=2.1744\), and total \(\chi^2=7732.7958\). The failure is dominated by the BAO term under one shared scale.

A15. Separate BAO and CMB effective scales

Calculation purpose: fit BAO-side and CMB-side effective scales separately and compute their ratio.

\[S_{\mathrm{ratio}}=\frac{S_{\mathrm{CMB}}}{S_{\mathrm{BAO}}} \tag{A15.1}\]

\[S_{\mathrm{ratio}}=0.390906. \tag{A15.2}\]

The separate-scale diagnostic repeats the scale minimization twice: once for BAO-only terms and once for CMB-prior terms. The ratio is the recovery target for projection functions.

A16. Redshift-dependent projection factor

Calculation purpose: test redshift-dependent projection factors that preserve BAO ratios while reaching the CMB-side scale target.

\[S_{\mathrm{eff}}(z)=S_0T(z) \tag{A16.1}\]

\[T(z)\approx1\quad\text{for BAO redshifts} \tag{A16.2}\]

\[T(z_*)\approx0.390906 \tag{A16.3}\]

\[T_1(z)=(1+z)^{-q} \tag{A16.4}\]

\[T_2(z)=\frac{1}{1+q\,x(z)/d_b} \tag{A16.5}\]

\[T_3(z)=\left(1+\frac{x(z)}{d_b}\right)^{-q}. \tag{A16.6}\]

Broad, gentle transformations tend to disturb the BAO redshift range before reaching the CMB-side target. The useful target keeps \(T(z_i)\approx1\) across BAO while driving \(T(z_*)\) toward the separated-scale value.

A17. Sharp transition and preferred projection law

Calculation purpose: define projection laws that preserve the BAO range while reaching high-redshift CMB/JWST targets.

BAO-preserving high-redshift recovery target

\[T(z)=1-\frac{\Delta z^m}{z^m+z_t^m} \tag{A17.1}\]

\[\lim_{z\ll z_t}T(z)\approx1 \tag{A17.2}\]

\[\lim_{z\gg z_t}T(z)\approx1-\Delta \tag{A17.3}\]

\[1-\Delta\approx0.390906 \tag{A17.4}\]

\[\Delta\approx1-0.390906=0.609094. \tag{A17.5}\]

A schematic fitting objective combines BAO preservation, CMB-prior behavior, and a penalty for disturbing the BAO range:

\[\Phi=\chi^2_{\mathrm{BAO}}(T)+w_{\mathrm{CMB}}\chi^2_{\mathrm{CMB}}(T)+\lambda_{\mathrm{BAO}}\sum_{z_i\in\mathrm{BAO}}[T(z_i)-1]^2. \tag{A17.6}\]

Preferred JWST-centered projection law

Chapter 4 also uses a preferred projection law centered in the high-redshift galaxy regime. This construction preserves BAO behavior, shifts the JWST mass-size register, and approaches the compressed CMB-side band:

\[P_\star(z)=1-\frac{\Delta z^n}{z^n+z_s^n} \tag{A17.7}\]

\[P_\star(z)=1-\frac{0.627714\,z^8}{z^8+8^8}. \tag{A17.8}\]

\[M_{\star,\mathrm{FD}}\approx P_\star^2M_{\star,\mathrm{standard}},\qquad R_{e,\mathrm{FD}}\approx\frac{R_{e,\mathrm{standard}}}{P_\star}. \tag{A17.9}\]

Table A17-1. Preferred projection-law anchor behavior.

Regime	\(z\)	\(P_\star\)	Mass ratio \(P_\star^2\)	Radius ratio \(1/P_\star\)
BAO high end	2.3	approximately 1.000	approximately 1.000	approximately 1.000
JWST shoulder	8	0.686143	0.4708	1.4574
CMB side	1089.92	0.372286	0.1386	2.6861

The transition laws are recovery targets and diagnostic constructions. They specify the scale and redshift behavior that a physical mechanism would need to produce.

A18. Compact replication chain

Calculation purpose: list the minimum end-to-end dependency chain for reproducing the appendix calculations.

The finite-boundary calculation can be summarized as:

\[G(d)=\frac{g_0}{1+d/d_b}\]

\[A(d)=\int_0^dG(s)\,ds\]

\[A(d)=g_0d_b\ln\!\left(1+\frac{d}{d_b}\right)\]

\[1+z=e^{A(d)}\]

\[z(d)=\left(1+\frac{d}{d_b}\right)^{g_0d_b}-1\]

\[x(z)=d_b\left[(1+z)^{1/(g_0d_b)}-1\right]\]

\[\frac{D_M}{r_d}=Sx(z),\qquad \frac{D_H}{r_d}=Sx'(z),\qquad \frac{D_V}{r_d}=S\left[zx^2(z)x'(z)\right]^{1/3}\]

\[S_{\mathrm{eff}}(z)=S_0T(z)\]

\[T(z)=1-\frac{\Delta z^m}{z^m+z_t^m}\]

\[T(z)\approx1\ \text{across BAO},\qquad T(z_*)\approx0.390906\ \text{near the CMB-prior regime}.\]

A complete implementation should reproduce the empirical supernova fit, accumulated-response transform, finite-boundary mechanism fit, stability tests, BAO shape comparison, chronometer and time-dilation checks, CMB-prior comparison, shared-ruler failure, separate-scale ratio, projection target, and register-scaling constants.

A19. Unified registers, scaling constants, and limits

Calculation purpose: collect the dimensionless response registers, the register-scaling constants, and the main interpretation limits.

The chapter model organizes projection depth, morphology, weak-field gravity, and compactness as separate but linked registers:

\[D=A\,G,\qquad D_n=\frac{D}{D_0} \tag{A19.1}\]

\[X=\frac{A}{A_t},\qquad A_t=\ln(1+z_t) \tag{A19.2}\]

\[\eta=\frac{g_{\mathrm{bar}}}{g^\dagger} \tag{A19.3}\]

\[u=\frac{2GM}{Rc^2} \tag{A19.4}\]

\[C=C(X,\eta,u,D_n). \tag{A19.5}\]

The register-scaling analysis gives the following recovered dimensionless constants:

\[K_D\approx0.504869,\qquad K_X\approx0.980376,\qquad K_\eta\approx0.666348 \tag{A19.6}\]

\[K_D\sim\frac{1}{2},\qquad K_X\sim1,\qquad K_\eta\sim\frac{2}{3}. \tag{A19.7}\]

An initial compactness-sector test was attempted using black-hole mass and lag-radius information, but it did not isolate a stable empirical \(K_u\). Two compactness interpretations remain: \(K_u\sim1/3\) for radialized collapse scaling, or \(K_u\sim0\) if compactness behaves mainly as a saturation threshold. The full cosmology adopts \(K_u=1/3\) as the working compactness exponent.

\[K_u=\frac{1}{3}\quad\text{(working compactness value)} \tag{A19.8}\]

Table A19-1. Register-scaling constants.

Register	Quantity	Recovered \(K\)	Simple form	Interpretation
Morphology/path response	\(D=A\,G\)	0.504869	\(1/2\)	transport-like weakening response
Projection coordinate	\(X=A/A_t\)	0.980376	\(1\)	coherent projection ordering
Weak-field compliance	\(\eta=g_{\mathrm{bar}}/g^\dagger\)	0.666348	\(2/3\)	dimensional coupling in the low-acceleration register
Compactness	\(u=2GM/(Rc^2)\)	not empirically isolated	working \(1/3\); alternate \(0\)	radial collapse scaling or saturation threshold

A diagnostic scaled-response form can be written as:

\[C_{\mathrm{scaled}}=C\!\left(X^{K_X},\eta^{K_\eta},u^{K_u},D_n^{K_D}\right),\qquad K_u=\frac{1}{3}. \tag{A19.9}\]

Equation (A19.9) is a response-weighting diagnostic. It does not replace the final physical field equation; it records how the measured registers and the working compactness exponent enter the finite-response engine after their characteristic scaling has been identified or predicted.

Limits of interpretation

The supernova distance coordinate \(d\) is used as supplied. Changes in distance calibration alter fitted scale values.

BAO \(\chi^2\) values in this appendix are shape-compatibility diagnostics unless full covariance matrices are included.

The compressed CMB-prior comparison uses priors rather than a full CMB anisotropy-spectrum fit.

The nuisance scale \(S\) absorbs ruler calibration and separates redshift-dependent shape from absolute sound-horizon derivation.

The projection functions \(T(z)\) and \(P_\star(z)\) identify required recovery behavior. They do not by themselves derive the underlying physical mechanism.

The compactness exponent \(K_u=1/3\) is a working value for the full cosmology, not a recovered empirical constant. A stronger black-hole compactness test must distinguish this radial-collapse scaling interpretation from the \(K_u\sim0\) saturation-threshold interpretation.

The chronometer comparison is a derivative-shape comparison unless an explicit calibration is supplied.

The time-dilation result constrains interpretation: a path-response redshift model must account for arrival-time stretching as well as wavelength change.

Substituting a different public compilation, calibration, covariance treatment, or split protocol is a new numerical test and will not reproduce the representative values exactly.