Domain before formula

Regime
boundaries

The strong and weak expressions are not switched at their raw intersection. A declared \(C^2\) bridge transports value, slope and curvature across \(1.8\le r/r_s\le2.2\). This page separates mathematical joins, physical interpretation and observable validity.

C²

\(\displaystyle \Xi_s(x)=1-e^{-\varphi/x}\)

\(\displaystyle \Xi_b(x)=H_5\!\left(\frac{x-1.8}{0.4}\right)\)

\(\displaystyle \Xi_w(x)=\frac{1}{2x}\)

Important distinction: the raw equality \(\Xi_s=\Xi_w\), the chosen bridge endpoints, an extremum of a null effective potential and a fitted observational boundary are different mathematical objects. They must not share one symbol or one claimed physical meaning.

Interactive boundary microscope

Values and derivatives through both joins

Move the probe to inspect the operative branch, \(\Xi\), \(D\), first derivative and numerical second derivative. The chart uses exactly the same field implementation as the main explorer.

Normalised radius 2.00 r_s

Chart maximum 5 r_s

logarithmic radius derivatives

regime—

Ξ—

D—

dΞ/dx—

d²Ξ/dx²—

A·B—

Boundary taxonomy

Four questions that must remain separate

1 · Formula domain

Which declared branch is evaluated? This is fixed by \(x=1.8\) and \(x=2.2\) in the current piecewise implementation.

2 · Smoothness

Do value, slope and curvature match? The quintic bridge enforces six endpoint identities and removes artificial derivative jumps through second order.

3 · Observable regime

Is a weak-field expansion, static clock map, full geodesic integral or perturbation model required? The radial branch alone does not answer this.

4 · Physical transition

Does a genuine material, causal or dynamical phase boundary exist? The present bridge is an operational geometric prescription; a fundamental dynamics deriving it remains open.

Exact endpoint data

What the C² constraint matches

\[ H_5^{(k)}(1.8)=\Xi_s^{(k)}(1.8),\qquad H_5^{(k)}(2.2)=\Xi_w^{(k)}(2.2),\qquad k=0,1,2 . \]

Location	Left description	Right description	Required continuity	What it does not prove
1.8 r_s	strong exponential branch	quintic bridge	Ξ, Ξ′ and Ξ″	a dynamically generated phase change
2.2 r_s	quintic bridge	weak inverse-radius branch	Ξ, Ξ′ and Ξ″	validity of weak-field observable shortcuts immediately at the join
r_s	strong branch checkpoint		Ξ≈0.801711847; D≈0.555027709	central regularity or a material horizon surface
r→∞	weak asymptotic limit		Ξ→0; D→1	global completeness of the interior

Method assignment

Which machinery is valid where?

Observable	Near/strong field	Transition	Far/weak field	Guardrail
Static clock comparison	full \(D(r)\)	full bridged \(D(r)\)	full D or controlled expansion	state both radii and kinematics
Radial null travel	integrate full metric	integrate full metric	full metric or justified PPN limit	coordinate slope is not local light speed
Lensing/Shapiro	full null geodesic model required	full null geodesic model required	PPN with declared geometry	never substitute a clock-only Ξ shortcut
Orbits	full effective potential and stability	full effective potential and stability	PPN or full geodesics	a named radius must carry metric variant and commit
Waves/QNM	perturbation equations needed	matching and dynamics needed	controlled approximation	proxy frequency is not a waveform