Four levels of explanation

SSZ theory
from A to Z

Intuition, physics, mathematics, implementation and validation are kept together so that no formula floats free of its assumptions.

ds²

Canonical lock: P0-2026.08 •Geometry: static spherical effective metric •Evidence: derived, implemented and tested are distinct •Reviewed: 3 August 2026

Reviewer first: no complete action, field equations, regular global interior, rotating solution, perturbation theory or independently confirmed strong-field discriminator currently exists. Current stationary candidates are bridge-sensitive diagnostics.

Start here

SSZ in plain language

Before reading equations, keep the central idea and its limits in view.

What is being changed?

SSZ uses a declared static spherical geometry in which a dimensionless field Ξ describes how the local clock and radial scale are related to a central mass. The primary input is the normalised radius x = r/r_s. From Ξ the implementation derives D, then the metric coefficients and observables.

In simple terms: far from the source, the profile becomes weak and approaches flat spacetime; near the strong-field region, the profile changes the relation between coordinate time, proper time and radial distance.

What does SSZ claim today?

The public programme is a concrete, falsifiable effective geometry with calculable limits and reproducible software. It is not presented here as a completed fundamental theory or as an empirical replacement for general relativity.

Definitions come before calculations.
Each result is tied to a branch, coordinate scope and evidence class.
Internal tests establish implementation consistency, not nature's choice of model.
Finite coefficients at the horizon do not establish a regular centre or a complete global spacetime.

Reading rule: distinguish a definition, a mathematical consequence, a numerical result, a data-compatible comparison and an open hypothesis. The same word “prediction” must not be used for all five.

Foundations

What “segmented” means

Segmentation is a model description of how local radial and temporal scaling is encoded by a dimensionless field Ξ. It is not, by itself, proof that spacetime is literally made of discrete blocks.

Intuitive

Imagine comparing identical local clocks and rulers at different gravitational radii. SSZ summarises the comparison with one local scaling field. More Ξ means a smaller D and therefore fewer local proper-time seconds per coordinate-time second.

Physical

A static observer at radius r is assigned dτ=D(r)dt. Radial proper length is scaled inversely in the diagonal ansatz. Local measurements still use local proper time and proper distance.

Mathematical

\[D(r)=\frac{1}{1+\Xi(r)},\qquad s(r)=1+\Xi(r)=\frac{1}{D(r)}\]

Ξ, D and s are dimensionless. The field depends on the dimensionless ratio r/r_s once a central mass fixes r_s.

Implementation

Code must declare the observable class, select the applicable branch or PPN method, retain units, and test limits before producing a number.

observable → class → method → scope → calculate

Model declaration

Assumptions that precede every formula

The canonical static metric is an effective ansatz. Its results are conditional on the symmetry, coordinate and branch choices below.

Staticity

Metric components do not depend on coordinate time. Collapse, inspiral and ringdown require additional dynamics.

Spherical symmetry

The public canonical metric has no spin or multipolar deformation. A Kerr-like completion is not implied.

Areal radius

The sphere at fixed \(r\) has area \(4\pi r^2\). Centre divergences cannot be dismissed as a harmless relabelling.

Metric compatibility

Calculations use the Levi-Civita connection: torsion-free and compatible with the declared metric.

Piecewise Xi lock

Strong, derivative-matched bridge and weak branches have explicit inclusive domains. Historic global or smoothstep variants are not canonical.

Effective status

No complete action currently derives this geometry, source coupling, perturbations and interior as one theory.

Traceability: see SSZ-CLAIM-METRIC-DIAGONAL-001 and SSZ-CLAIM-SCOPE-001.

Language before calculation

Coordinates, units and epistemic status

Every SSZ statement depends on what is being measured, which branch is used and whether the result is a definition, a mathematical consequence, a numerical test or an empirical claim.

Coordinates

t: static chart time
r: areal radius
θ, φ: spherical angles

The symmetry sphere at fixed r has the invariantly defined area 4πr². Calling r an areal radius is therefore a geometric statement, not merely a coordinate label.

Natural scale

\(r_s=\dfrac{2GM}{c^2},\qquad x=\dfrac{r}{r_s}\)

The public implementation evaluates a universal dimensionless profile in \(x\), then restores physical units through \(r_s\).

Units

Ξ, D, A and B are dimensionless. Radius carries length, coordinate time carries time, and the curvature invariants scale as R ∼ L⁻² and K ∼ L⁻⁴.

\(G\)Newtonian gravitational constant

\(c\)locally measured invariant light speed

\(\varphi\)golden ratio \((1+\sqrt5)/2\), dimensionless model constant

\(x\)normalised areal radius \(r/r_s\)

Reading rule: “implemented” means that reproducible code exists. “Tested” means that an assertion or comparison was executed. Neither word, by itself, means that nature empirically prefers SSZ.

Differential geometry

From one radial function to curvature

The metric is not merely a clock formula. Once \(D(r)\) is fixed, the Levi-Civita connection, geodesics and curvature tensors follow from ordinary differential geometry within the declared static spherical ansatz.

Connection

\( \Gamma^{\rho}_{\ \mu\nu}=\frac{1}{2}g^{\rho\sigma}(\partial_\mu g_{\sigma\nu}+\partial_\nu g_{\sigma\mu}-\partial_\sigma g_{\mu\nu}) \)

The connection contains first radial derivatives of \(D\). This is why a value-continuous branch join is insufficient: derivative jumps would create non-smooth geometry or distributional structure.

Curvature

\( R^{\rho}_{\ \sigma\mu\nu}=\partial_\mu\Gamma^\rho_{\nu\sigma}-\partial_\nu\Gamma^\rho_{\mu\sigma}+\Gamma^\rho_{\mu\lambda}\Gamma^\lambda_{\nu\sigma}-\Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\sigma} \)

Ricci and Kretschmann scalars depend on first and second derivatives. The C² bridge makes these finite through the transition, but says nothing by itself about the limit at the areal centre.

Effective source diagnostic

\( G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu},\qquad T^{\rm eff}_{\mu\nu}=G_{\mu\nu}/(8\pi G) \)

This can diagnose what effective stress tensor would support the geometry. It is not a derivation of matter dynamics and must not be mistaken for a fundamental SSZ field equation.

Regularity ladder

finite metric coefficients;
non-degenerate signature;
finite connection in a suitable chart;
finite curvature invariants;
extendible geodesics and causal curves.

The current continuation fails the invariant-curvature rung at \(r\to0\).

Exact transition construction

The derivative-matched quintic Hermite bridge

The transition is the unique degree-five polynomial that transports the value, slope and curvature of both neighbouring branches across \(x_0=1.8\) and \(x_1=2.2\). It is an explicit model prescription, not a fitted curve and not a hidden interpolation toward Schwarzschild.

\(t=\dfrac{x-x_0}{h},\qquad h=x_1-x_0=0.4,\qquad 0\le t\le1\)

\[ \begin{aligned} H_5(t)={}&h_{00}(t)y_0+h\,h_{10}(t)y'_0+h^2h_{20}(t)y''_0\\ &+h_{01}(t)y_1+h\,h_{11}(t)y'_1+h^2h_{21}(t)y''_1 . \end{aligned} \]

Left basis

\(\begin{aligned}h_{00}&=1-10t^3+15t^4-6t^5\\h_{10}&=t-6t^3+8t^4-3t^5\\h_{20}&=\tfrac12(t^2-3t^3+3t^4-t^5)\end{aligned}\)

Right basis

\(\begin{aligned}h_{01}&=10t^3-15t^4+6t^5\\h_{11}&=-4t^3+7t^4-3t^5\\h_{21}&=\tfrac12(t^3-2t^4+t^5)\end{aligned}\)

Evaluate the physical branches

\(y_0=\Xi_{\rm strong}(x_0)\), \(y_1=\Xi_{\rm weak}(x_1)\), together with their first and second derivatives with respect to \(x\).

Apply the interval scale

The factors \(h\) and \(h^2\) convert endpoint derivatives in \(x\) into derivatives in the normalised coordinate \(t\).

Verify all six identities

\[H_5^{(k)}(0)=\Xi_{\mathrm{strong}}^{(k)}(x_0),\qquad H_5^{(k)}(1)=\Xi_{\mathrm{weak}}^{(k)}(x_1),\qquad k=0,1,2\]

These are six scalar endpoint conditions: value, first derivative and second derivative at each boundary.

Interactive audit: the metric explorer and continuity microscope evaluate this construction directly. The Schwarzschild trace is shown only as a reference curve.

Time, frames and light

Proper time is not coordinate time

Proper time τ

The time recorded by a clock travelling on a timelike worldline. It is local and invariant along that worldline.

Coordinate time t

A coordinate label used to compare separated events within a chosen chart. It is not directly what every observer measures.

Light

A photon follows a null path with dτ=0. Every local freely falling observer measures light at c; coordinate speeds may differ because coordinates and metric coefficients differ.

Local statement: c is locally constant. A radial coordinate expression such as dr/dt is not automatically a locally measured speed. Confusing the two produces false superluminal or “slowed photon” claims.

Operational observables

How metric quantities become measurements

A coordinate expression becomes physical only after an observer, apparatus and comparison procedure are specified.

Static redshift between two radii

\(1+z=\dfrac{\nu_{\rm emit}}{\nu_{\rm obs}}=\dfrac{D(r_{\rm obs})}{D(r_{\rm emit})}\)

This assumes static emitter and observer in the same stationary geometry. Doppler motion, cosmological expansion and propagation through matter require additional terms.

Radial null travel time

\(\Delta t=\dfrac1c\int_{r_1}^{r_2}\dfrac{dr}{D^2(r)}\)

The integral describes the selected static coordinates. A local observer still measures light at \(c\).

Proper radial distance

\(\ell(r_1,r_2)=\int_{r_1}^{r_2}\dfrac{dr}{D(r)}\)

This is the length on a constant-\(t\) slice of the diagonal ansatz; it is not the proper time of an infalling observer.

Impact-parameter geometry

\(b^2=\dfrac{r_{\rm turn}^2}{A(r_{\rm turn})},\qquad A=D^2\)

Photon-ring and lensing claims require the full null geodesic problem, not only the value of \(\Xi\) at a selected radius.

Do not mix observable classes: static redshift, radial coordinate slope, lensing deflection, orbital stability and wave ringdown are related through the geometry, but they are not interchangeable formulas.

Geometry

The canonical diagonal metric ansatz

\[ds^2=-D^2(r)c^2dt^2+\frac{dr^2}{D^2(r)}+r^2\left(d\theta^2+\sin^2\theta\,d\phi^2\right)\]

ds²: invariant spacetime interval
t, r, θ, ϕ: static spherical coordinates; r is the areal radius
D(r): dimensionless static time factor, derived from Ξ
c: local invariant speed of light

Components

g_tt	−D²c²	static clock scaling
g_rr	D⁻²	radial proper-length scaling
g_θθ	r²	areal angular radius
g_ϕϕ	r²sin²θ	azimuthal angular term

Areal radius

The area of a symmetry sphere is 4πr². This makes r=0 a geometrically meaningful areal centre, not merely a coordinate label that can be dismissed when curvature invariants diverge.

The determinant and inverse metric being finite at one radius do not guarantee global regularity.

Scope boundary: this metric is a mathematically concrete effective geometry. A fundamental action and complete field equations that dynamically generate it are still missing.

SSZ-METRIC_COMPLETE implementation chain

Metric, inverse, determinant and coordinate forms

The repository computes every component from the primary field. This provides several exact algebraic checks, but coordinate-form equivalence and component finiteness must remain distinct from curvature regularity and global completeness.

Diagonal tensor

\[ g_{\mu\nu}=\operatorname{diag}\!\left(-D^2c^2,\;s^2,\;r^2,\;r^2\sin^2\theta\right),\qquad s=1+\Xi=D^{-1}. \]

For \(r>0\) and \(0<\theta<\pi\), the eigenvalue signs are \((-+++)\). Zeros of spherical angular factors on the axis are coordinate degeneracies of spherical coordinates.

Exact inverse

\[ g^{\mu\nu}=\operatorname{diag}\!\left(-\frac{1}{D^2c^2},\;\frac1{s^2},\;\frac1{r^2},\;\frac1{r^2\sin^2\theta}\right). \]

The repository test multiplies \(g_{\mu\alpha}g^{\alpha\nu}\) and compares with \(\delta_\mu^{\ \nu}\) at \(10^{-10}\) relative/absolute tolerance.

Determinant and volume element

\[ \det g=-c^2D^2s^2r^4\sin^2\theta=-c^2r^4\sin^2\theta,\qquad \sqrt{-g}=cr^2|\sin\theta|. \]

The cancellation follows from \(Ds=1\). It is an algebraic coupling check; it neither removes the areal centre nor forces curvature invariants to remain finite.

Local orthonormal coframe

\[ \vartheta^{\hat0}=Dc\,dt,\quad \vartheta^{\hat1}=s\,dr,\quad \vartheta^{\hat2}=r\,d\theta,\quad \vartheta^{\hat3}=r\sin\theta\,d\phi. \]

Then \(ds^2=-(\vartheta^{\hat0})^2+\sum_i(\vartheta^{\hat i})^2\). This is the clean route from coordinate components to local measurements.

Repository flow form

\[ ds^2=-c^2(1-\beta^2)dt^2+2\beta c\,dt\,dr+dr^2+r^2d\Omega^2, \qquad \gamma=1+\Xi,\quad \beta=\sqrt{1-\gamma^{-2}}. \]

Radial null directions

In the diagonal chart, \(dr/(c\,dt)=\pm D^2\). In the flow chart the quadratic null condition gives \(dr/(c\,dt)=-\beta\pm1\). These are chart slopes, not a change of locally measured \(c\).

What is tested

The pure-metric suite checks construction from \(\Xi\), inverse multiplication, determinant, signature, branch routing and dynamic tensor dependence. Each is an implementation property of the declared model.

What remains open

A complete proof that the diagonal and flow forms cover the same global manifold, plus an independently derived interior, action, field equations and boundary conditions, is not supplied by these component identities.

Inspect interactively: use the rebuilt metric, inverse and local-frame explorer to switch coordinate form, inverse components, radial scale and polar angle.

From calculation to evidence

Prediction, uncertainty and model comparison

Numerical agreement is meaningful only when inputs, uncertainties, nuisance parameters and a comparison model are explicit.

Claimscope and status

Equationbranch and assumptions

Codeversion and units

Testtolerance and data

Evidencelikelihood and alternatives

Residual and uncertainty

\(\rho_i=\dfrac{y_i-f(x_i;\theta)}{\sigma_i}\)

A residual without \(\sigma_i\) cannot state statistical tension. Correlated data require a covariance matrix rather than independent denominators.

Model comparison

\(\chi^2=(\mathbf y-\mathbf f)^{\mathsf T}C^{-1}(\mathbf y-\mathbf f)\)

A smaller \(\chi^2\) is not enough if models have different flexibility, priors or effective parameter counts. The comparison protocol must be fixed before interpreting a preference.

Current limit: several public applications demonstrate computable mappings or compatibility checks. A unified uncertainty-aware likelihood pipeline across all strong-field observables remains open.

Kinematic closure

Escape velocity and the complementary fall scale

The SSZ corpus pairs the usual escape scale with a reciprocal quantity called v_fall. The word “velocity” here names a kinematic scale; it must not be confused with a locally measured material 3-velocity.

Escape scale

\[v_{\rm esc}(r)=c\sqrt{\frac{r_s}{r}}\]

In the weak field this is the Newtonian escape-speed expression √(2GM/r).

Dual fall scale

\[v_{\rm fall}(r)=c\sqrt{\frac{r}{r_s}}=\frac{c^2}{v_{\rm esc}(r)}\]

Values above c express the reciprocal scale and do not represent local faster-than-light motion.

Exact closure

\[v_{\rm esc}(r)\,v_{\rm fall}(r)=c^2\]

The identity is repeatedly tested numerically and follows algebraically from the paired definitions.

Terminology boundary: some documents also call c√(1−D²) an infall velocity. That is a different metric-derived proxy. The portal labels it separately so the two definitions cannot be silently mixed.

Open the interactive escape–fall visualization

Canonical correction map

What P0 changed—and what it did not

The internal long-form synthesis is used only to identify topics that deserve explanation. Where its pre-P0 conclusions conflict with the current canonical geometry, the current derivation and reproducible tests win.

Singularity scope in plain language: the public test corpus repeatedly reproduces finite \(\Xi(r_s)\), finite \(D(r_s)\) and finite static metric coefficients at the Schwarzschild radius. That is a real tested result of the declared formula and removes a divergence of this static clock factor at that radius. The separate \(R\) and \(K\) divergence follows only when the present diagonal exterior formula is extrapolated all the way to the areal centre. It shows that this particular extrapolation is not yet a regular global interior; it does not prove that Segmented Spacetime cannot supply a different inner solution, matching surface or boundary geometry. Whether such a completion resolves the remaining centre question is open research.

Question	Current canonical statement	Superseded overstatement	Reason
Horizon clock factor	Finite: \(D(r_s)\approx0.555027709\)	Vanishing or infinite static dilation	Direct evaluation of the canonical strong branch
Inner limit of Ξ	\(\Xi(r)\to1\) as \(r\to0^+\)	\(\Xi\to0\) at the centre	\(1-\exp(-\varphi r_s/r)\to1\)
Metric coefficient	\(A=D^2\to1/4\)	Metric finiteness implies regularity	Finite components do not control invariants
Central curvature	\(R\sim3/(2r^2)\), \(K\sim9/(4r^4)\)	Complete singularity freedom	Invariant curvature diverges at the areal centre
Theory status	Concrete falsifiable effective geometry	Completed fundamental gravity theory	No complete covariant action, dynamics or global interior yet

Publication boundary: the private reference book is neither copied nor linked from this portal. Its historical claims cannot override P0, the current source hierarchy or reproducible mathematics.

Piecewise field

Strong, transition and weak regimes

r/r_s < 1.8

Strong/inner branch

\[\Xi_{\mathrm{strong}}(r)=1-\exp\!\left(-\varphi\frac{r_s}{r}\right)\]

Ξ decreases outward. At r_s it is about 0.8017; as the formula is formally continued inward it approaches 1.

1.8 ≤ r/r_s ≤ 2.2

Blend zone

\[\Xi(x)=H_5\!\left(\frac{x-1.8}{0.4}\right)\]

A quintic Hermite polynomial matches value, first derivative and second derivative of both branches. This is an operational interpolation, not yet a derived phase transition.

r/r_s > 2.2

Weak branch

\[\Xi_{\mathrm{weak}}(r)=\frac{r_s}{2r}=\frac{1}{2x}\]

It tends to zero at infinity, so D tends to one and the metric tends to Minkowski form.

Why C² rather than a simple smoothstep?

A cubic 3t²−2t³ forces zero endpoint slopes and only guarantees C¹ matching when the neighbouring functions are not themselves flat. The current requirement transports the actual endpoint slope and curvature of both physical branches. That removes artificial derivative jumps through second order.

Implementation conflict found: the current segmented-calculation-suite/segcalc/methods/xi.py uses the quintic smoothstep 6t⁵−15t⁴+10t³ to weight the two branch functions. Calling the weight C² does not make the resulting piecewise field C² when the neighbouring branch slopes are non-zero. The portal therefore follows the higher-priority P0 endpoint-derivative match and records this repository drift explicitly.

Construction and continuity

From mass scale to the C² field

The current portal treats the branch definitions and the transition prescription as declared model structure. It does not present the bridge as a solution of an undiscovered field equation.

1 · Fix the radial scale

\[r_s=\frac{2GM}{c^2},\qquad x=\frac{r}{r_s},\qquad r=xr_s\]

For a positive central mass M, r_s supplies the length scale and x is dimensionless. The universal curve is written in x; physical metres return through r=xr_s.

2 · Derive the scalings

\[\Xi(x)\longmapsto D(x)=\frac{1}{1+\Xi(x)}\longmapsto A(x)=D^2(x),\quad B(x)=D^{-2}(x)\]

Ξ is primary in the canonical implementation. D, the temporal coefficient A, and radial coefficient B are derived quantities—not separately fitted functions.

3 · Match six endpoint conditions

With t = (x − 1.8)/0.4, the quintic matches Ξ, dΞ/dx and d²Ξ/dx² at both boundaries. Six conditions determine six coefficients.

\[H_5^{(k)}(0)=\Xi_{\mathrm{strong}}^{(k)}(1.8),\qquad H_5^{(k)}(1)=\Xi_{\mathrm{weak}}^{(k)}(2.2),\qquad k=0,1,2\]

4 · Test identities and limits

\[D(x)s(x)=1,\qquad \lim_{x\to\infty}\Xi(x)=0,\qquad \lim_{x\to\infty}D(x)=1\]

Continuity, monotonicity within the declared branches, horizon values and asymptotic flatness are software-testable. Their success establishes implementation consistency, not empirical truth.

Horizon checkpoint: Ξ(r_s)=1−e^−φ≈0.801711847 and D(r_s)≈0.555027709. The older value Ξ(r_s)=1 is not canonical.

Weak field

Newtonian and PPN limits

The weak-field branch gives Ξ≈GM/(rc²). Different observable classes must still use different complete formulas.

Static clocks

\[D(r)=1-\frac{GM}{rc^2}+\mathcal O\!\left(\frac{r_s^2}{r^2}\right)\]

At first order, this yields the familiar gravitational clock shift. Clock observables can use the direct D mapping.

Null paths

\[\alpha=(1+\gamma)\frac{r_s}{b},\qquad \gamma=1\]

Lensing and Shapiro delay depend on both temporal and spatial metric structure. They must use the PPN completion, not an Ξ-only shortcut.

Mercury perihelion

\[\Delta\omega=\frac{6\pi GM}{a(1-e^2)c^2}\]

The declared β=γ=1 PPN machinery reproduces the reference weak-field formula.

What that means

Passing GPS, Pound–Rebka, Cassini, lensing, and perihelion reference tests demonstrates consistency of the implemented weak-field limit. It does not test the centre, strong-field stability, or a new fundamental dynamics.

Motion in the effective geometry

Worldlines, null paths and conserved quantities

Static clock

For dr=dθ=dϕ=0, the metric directly gives dτ=D(r)dt. Comparing two static radii requires a stated emitter and observer.

\[\frac{d\tau}{dt}=D(r)\]

Radial light coordinate slope

Setting ds²=0 for a radial null path gives a coordinate relation. It is not a locally measured light speed.

\[\frac{dr}{dt}=\pm cD^2(r)\]

Spherical symmetries

Stationarity and rotational symmetry permit conserved energy- and angular-momentum-like quantities for geodesic calculations. Circular photon and matter orbits must be derived from the full effective potential.

\[E=-g_{tt}u^t,\qquad L=g_{\phi\phi}u^\phi\]

Strong-field caution: a radius quoted by an individual repository is not automatically universal across every historic SSZ metric variant. The branch, metric convention, commit and numerical method must accompany it.

Sagnac and rotation

Direction comparison, not a conveyor-belt analogy

Classical structure

On a rotating platform, counter-propagating signals accumulate different travel times. The effect is extracted by comparing directions; common, direction-symmetric contributions cancel in the subtraction.

\[\Delta t_{\mathrm{Sagnac}}\simeq\frac{4\mathcal A\Omega}{c^2}\]

This leading expression assumes the standard rotating-loop geometry.

SSZ research question

Potential-dependent segment or phase contributions must be classified as one-way or two-way, local or global, and symmetric or antisymmetric under path reversal. Synchronisation conventions must be stated.

A rim-speed or conveyor-belt picture cannot replace the spacetime path integral.

One-way versus two-way measurement

One-way timing requires a synchronisation convention between separated clocks. Round-trip timing can avoid that convention but measures a different observable. A factor of two from path duplication is separate from a factor such as (1+γ) in a PPN formula.

Fields and propagation

Electromagnetism: what follows, what is proposed, what is missing

Several SSZ papers use frequency, radial scaling and rotating-wave language. These layers must be separated from standard electromagnetism on the effective metric.

Standard geometric layer

Local Maxwell propagation

Given a metric, Maxwell equations can be written covariantly as \(\nabla_\mu F^{\mu\nu}=\mu_0J^\nu\) and \(\nabla_{[\alpha}F_{\beta\gamma]}=0\). Geometric optics then follows null characteristics while every local inertial frame measures \(c\).

SSZ proposal

Radial scaling gauge

Amplitude, phase or radial field scalings proposed by the paper programme require an explicit map between coordinate components, orthonormal measurements and conserved flux. A scaling rule must preserve units and boundary conditions.

Interpretation

Rotating-space picture

A visual or topological interpretation of Maxwell waves as rotating space is not equivalent to a covariant action. Its empirical content begins only when it yields a distinct, gauge-consistent observable.

\[\displaystyle \nabla_\mu F^{\mu\nu}=\mu_0J^\nu,\qquad \nabla_{[\alpha}F_{\beta\gamma]}=0,\qquad k_\mu k^\mu=0\ \text{(geometric optics)}\]

No in-flight retuning shortcut: gravitational redshift is an endpoint comparison of observer frequencies in a stationary geometry. It need not be described as a photon continuously losing locally measured energy along its path.

P0 correction

SSZ Interior and Global Structure

Finite time dilation at r=r_s removes the Schwarzschild-style vanishing of the static time factor there. It does not settle what happens at the areal centre.

Derived limit

Metric coefficient

\[A(r)=D^2(r)\longrightarrow\frac14\qquad(r\to0^+)\]

The temporal coefficient remains finite in the formal diagonal continuation.

Curvature divergence

Areal centre

\[R(r)\sim\frac{3}{2r^2},\qquad K(r)\sim\frac{9}{4r^4}\]

The Ricci scalar R and Kretschmann scalar K diverge. Because the sphere area is 4πr², this is not removed merely by relabelling the coordinate.

Not established

Central regularity, geodesic completeness, causal completeness and a globally regular continuation are not proved.

Needed for completion

A new interior solution or rigorously defined boundary geometry, matching conditions, regularity analysis and suitable matter/energy content.

Global questions

Maximal extension, trapping structure, junction behaviour, rotation and dynamical collapse remain separate research problems.

Continue to the dedicated Interior and Global Structure audit, including invariants, affine completeness, null expansions, trapped surfaces, junction conditions and concrete completion tasks.

Foundational status

Effective metric versus fundamental dynamics

What exists

A defined scalar profile, a concrete diagonal geometry, numerical implementations, observable pipelines and falsifiable comparisons. These are sufficient to study an effective spacetime model.

What is missing

No complete covariant action or field equations currently derive the metric, source coupling, conservation laws, perturbations, rotating solution and interior from a single dynamical principle.

Consequence: energy conditions, wave propagation and stability cannot be inferred solely from the static profile Ξ(r). Each requires the declared effective stress tensor or underlying dynamics and its domain of validity.

Observable assignment

One field does not mean one shortcut

Class	Examples	Method	Primary caution
Static timelike	clock rate, static redshift	direct D/Ξ mapping	include emitter/observer and kinematics
Null path	lensing, Shapiro, VLBI	full metric or PPN completion	do not use Ξ-only factor
Timelike orbit	perihelion, ISCO, epicycles	geodesic/effective-potential machinery	do not infer from D at one radius
Wave dynamics	QNM, inspiral, ringdown	perturbation equations from declared dynamics	proxy frequency is not a full waveform

Theory → literature → tests

How the public paper record maps onto the model

Foundational proposals

Segmentation, structural constants, emergent axes and frequency language motivate the programme. Several of these are hypotheses rather than consequences of the canonical metric.

Metric and strong field

The metric, dual-velocity, boundary and singularity papers contain the claims most affected by the P0 correction. Historical claims are retained as history, not repeated as current results.

Observable applications

Redshift, curvature detection, radio-wave and nebular papers propose mappings to observations. Each requires an explicit likelihood, uncertainties and comparison model before it can count as empirical evidence.

Audit the sequence: use the paper and claim catalogue for all 25 numbered public SSZ manuscripts, verified item links, topics and P0 status.

Theory architecture

From the locked static metric to the wider research programme

The public corpus contains several layers with different maturity. They should cooperate without being treated as one already-derived fundamental theory.

Layer	Current object	What is reproducible	What remains open
Static geometry	piecewise \(\Xi\), derived \(D,A,B\), diagonal metric	limits, C² bridge, components, geodesic diagnostics	fundamental derivation and global interior
Weak-field compatibility	β=γ=1 PPN assignment	clock, perihelion, lensing and Shapiro reference calculations	independent novel weak-field discriminator
Strong-field phenomenology	finite horizon clock factor and effective orbit calculations	metric-specific numerical candidates	rotation, perturbations, transfer and full forward imaging
Astrophysical pipelines	mass projection, nebulae, energy and starmap analyses	captured scripts, residuals, plots and test snapshots	selection control, uncertainty hierarchy and external replication
Foundational extensions	structural constants, emergent axes, coherence and wave interpretations	specific algebraic constructions and code experiments	derivation from one covariant action and decisive experiments

Canonical reading order: definitions → metric → limits → observable-specific forward model → code → test → data comparison → interpretation. Reversing that order turns numerical agreement into circular support.

Foundations expanded

How the pieces fit together

This is the explanatory bridge between the symbols and the later test pages.

1 · Choose the scope

Declare the central mass, the static spherical symmetry, the areal radius and the coordinate time. These assumptions define what can be calculated and what cannot.

2 · Build the profile

Evaluate the canonical Ξ branch for the declared radius. Use the C² bridge only between 1.8 and 2.2 r_s; use the weak branch outside. Do not mix historic crossover rules with the current lock.

3 · Derive, do not refit

Compute D = 1/(1+Ξ), A = D² and B = D⁻². They are derived coefficients. A separate observable needs its own forward model; a clock formula is not automatically a lensing or orbit formula.

4 · Interpret locally first

At the horizon, the static coefficient is finite in the declared geometry. Near the areal centre, the current continuation has divergent curvature asymptotics. Neither local statement settles the global causal structure.

5 · Compare with a reference

Weak-field agreement with GR is compatibility. A discriminating claim needs a complete observable, uncertainty model, nuisance treatment and an independent reproduction.

6 · Record provenance

Every public result should point from definition to formula, implementation, test, data and limitation. The Evidence and Reproducibility tabs expose that chain.

Suggested reading order: this overview → Coordinates and units → Metric → Branches → Weak field → Strong field → Interior and Global Structure → Tests → Evidence Ledger.

Scientific decision points

How the model can fail

Internal falsification

The declared metric fails if its branch matching violates C² continuity, its code does not reproduce its formulas, a claimed limit is mathematically inconsistent, or a supposedly universal result depends on an undeclared historic variant.

Observational falsification

A precision observation excludes a specified SSZ version when the predicted observable—with uncertainty and nuisance parameters fixed in advance—lies outside the measured confidence region while a valid comparison model remains compatible.

Discriminating targets

Strong-field redshift, shadow morphology, photon-ring structure, orbital frequencies, lensing time delays and ringdown are candidates only where SSZ provides a complete forward model distinct from GR.

Non-discriminating success

Recovering β=γ=1 in the weak-field PPN limit is necessary compatibility evidence. It is not a novel SSZ prediction and cannot by itself select SSZ over GR.

Guardrails

Common misreadings

“Finite at r_s means no singularity anywhere.”

False. Horizon regularity and central regularity are different mathematical questions. Curvature diverges at the areal centre in the current diagonal continuation.

“A green test proves nature uses SSZ.”

False. A test can prove a code path satisfies an assertion or reproduces a dataset transformation. Empirical preference requires uncertainty-aware comparison against alternatives.

“Segmentation means a literal lattice.”

Not established. Ξ is an effective field in the present formalism; the microphysical origin remains open.