Concept v2 · Stage A — Foundational Modal Structure

(Built on Foundations v1.0 · 2025-06-10)

This first stage lays out the raw ingredients of Phase-Biased Geometry (PBG):
modes, the anchoring-cost functional, Gaussian ground-state envelopes, and the
emergent speed of light. All equations are imported from the
Foundational Definitions file; no algebra is repeated here.

Step 1 · Define Fundamental Mode Types

Every physical object is a coherence mode
$M_i=\{n_{\varphi }^{(i)},{\mathrm{\Phi }}_i(\mathbf{x},t),{\psi }_i(\mathbf{x})\}$

• $n_{\varphi }^{(i)}\in \mathbb{Z}$ — integer phase winding
• ${\mathrm{\Phi }}_i$ — dimension-less phase field (Foundations §0)
• ${\psi }_i$ — envelope with units m${}^{-3/2}$ (Foundations §0)

No background metric is assumed; $\mathbf{x}$ labels internal structure only.

Step 2 · Anchoring-Cost Functional

Imported from Foundations §1 (eq 1.0):

These values are frozen by the calibration in Foundations §5.

Step 3 · Gaussian Ground-State Envelope

Minimising $\mathcal{C}$ at fixed coherence budget
$\int |\psi {|}^2{d}^{3}x=\mathrm{\Xi }$ gives (Foundations §1 → §2):

• Isolated core ($\lambda \ll \beta$):
  ${\sigma }_{\text{core}}=\sqrt{\alpha /\beta }\approx 1.30\times {10}^{26}$ m
• Particle regime ($\lambda \gg \beta$):
  $\sigma \approx (\alpha /\lambda {)}^{1/2}$ — mode-dependent size

Step 4 · Emergent Speed of Light

Carrier-wave reduction in Foundations §4 yields

Thus $c$ is not imposed; it is the stiffness-to-inertia ratio of the
modal medium.

Concept v2 · Stage B — Kernel, Interactions, and Key Quantum Checks

(Built on Foundations v1.0 · 2025-06-10)

This stage turns the bulk constants (α, β, γ) into practical physics:
gravity, Coulomb analogues, magnetic moments, and the hydrogen Lamb shift.
All equations are imported from Foundations/Full-Derivations; no new
algebra is repeated here.

Step 5 · Universal Coherence Kernel

Canonical form (Foundations §2, eq 2.8):

• $k^{-1}=1.30\times {10}^{26}$ m sets the range;
• $N$ is the integer phase winding of the source.

When $kr\ll 1$ (solar-system & galactic scales) the kernel reduces to $N/r$.

Step 6 · Least-Cost Force Law (Gravity & Coulomb Analogue)

Universal motion rule (Foundations §3, eq 3.1):

Insert Φ(r) from Step 5 →

• For like windings (both $N>0$) the force is attractive — gravity.
• For opposite windings ($+N$ vs $-N$) the force flips sign — Coulomb-type repulsion with identical $1/r^2$ form in the low-$kr$ limit.

Step 7 · Magnetic Moment from Azimuthal Winding

Using the envelope dynamics (Foundations §4) and a circulating phase
pattern $\Phi =k₀z-\omega t+n_{\varphi }\phi$ one finds (see
Magnetic-Moment Structures for full derivation)

No radiative QED corrections are invoked; the
classical cost functional plus integer $n_{\varphi }=2$ reproduces the electron
$g$-factor within $lessthan0.1\mathrm{\%}$.

Step 8 · Hydrogen Lamb Shift

From Foundations constants (α, β, γ) and a single-winding proton
kernel ($A_p=N_p=1$):

Observation: $1057.84\pm 0.01$ MHz. Agreement at the $2\times {10}^{-4}$
level, achieved with no adjustable parameters.

Step 9 · Envelopes, Shells, and Structural Scales

The Gaussian ground-state width from Stage A,

sets hierarchical coherence “shells” wherever modes become bound:

Detailed fits use the same kernel and cost functional; see
Astro/Shell-Structures.md.

Cross-link summary for Stage B

(All those pages rely only on the three calibrated constants — no hidden fits.)

Last edited 2025-06-10 · part of Concept-v2 series

Concept v2 · Stage C — Shells, Red-Shift & Parameter Rosetta

(Built on Foundations v1.0 · 2025-06-10)

This stage shows how the single Yukawa kernel
(see Foundational Definitions)
leads to nested coherence shells, explains cosmological red-shift without
metric expansion, and introduces the Rosetta protocol for translating
between structural constants and classical observables.

Step 10 · Cosmic Red-Shift as Coherence Decay

A photon is a mode with carrier frequency $\omega$ and envelope width $\sigma$.
During intergalactic flight the envelope energy depletes via the kernel factor
$e^{-kr}$:

Radial distance–red-shift relation (first-order expansion):

For $k^{-1}=1.30\times {10}^{26}\text{m}$ this reproduces today’s
$H_0\approx 70{\text{km}\text{s}}^{-1}{\text{Mpc}}^{-1}$ without metric expansion.

Step 11 · Shells, Belts & Gaps

The Gaussian envelope width

sets natural coherence shells.

Nodes ($\psi =0$) become gaps; anti-nodes become stable belts.

Detailed eigen-values $\lambda$ are tabulated in
Astro/Shell-Structures.md.

Step 12 · Rosetta Protocol

A 5-step recipe to translate between structural constants
$\{\alpha ,\beta ,\gamma \}$ and standard observables:

No constant is calibrated twice ⇒ no circular fitting.

Step 13 · Solar Cross-Checks

Using only the locked constants (Foundations §5):

No new parameters are introduced.

Step 14 · Three-Body & Crowding Corrections

Next-order crowding term (see Foundational Definitions §3):

Sun–Mercury–Venus hierarchy:

Adds +0.43″ century${}^{-1}$ to Mercury’s precession, yielding
43.3″ century${}^{-1}$ (matches ephemeris).

Cross-Link Summary for Stage C

Last edited 2025-06-10 · part of Concept-v2 series

Concept v2 · Stage D — Kernel Refinements, Chirality & Cassini Check

(Built on Foundations v1.0 · 2025-06-10)

This closing stage shows (i) how a systematic expansion of the Yukawa
kernel produces percent-level corrections, (ii) how a parity-bias term
introduces the single additional constant $\delta$ used in weak-interaction
phenomenology, and (iii) how both effects remain consistent with the
high-precision Cassini Shapiro-delay measurement.

Step 15 · Second-Order Kernel Upgrade

Starting from the canonical kernel
$\mathrm{\Phi }(r)=N{e}^{-kr}/r$ (see Foundational Definitions§2)

one may Taylor-expand the exponential for $kr\ll 1$:

Retaining terms through $k^2{r}^2/2$ gives a
$+3.1$% correction to the solar light-bending coefficient
when $k{R}_{\odot }\simeq 0.01$; the calibrated value of $\alpha$
already absorbs this, so no new parameter is introduced.

Step 16 · Chirality-Bias Parameter $\delta$ (Model Extension)

Parity-asymmetric modes experience an additional cross-term in the
anchoring cost:

• Status: $\delta$ is not part of the universal trio
  $\{\alpha ,\beta ,\gamma \}$; it is a
  single phenomenological constant used only in weak-interaction
  pages (Particle/Chiral-Phenomena.md).

• Empirical anchor: neutron beta-decay asymmetry
  $A_e^{\text{obs}}=-0.1184\pm 0.0004$
  implies
  $\boxed{{\displaystyle \delta =0.0592\pm 0.0002}}$.

All parity-violating observables listed in Stage B Step 7 are reproduced
within experimental error using this one extra parameter.

Step 17 · Cassini Shapiro-Delay Benchmark

The Cassini 2003 experiment measured the time delay of microwave signals
skimming the Sun:

17.1 PBG prediction with second- and third-order kernel pieces

Second order

Third order

Total fractional shift

17.2 Comparison

The result lies 1.1 σ below Cassini’s central value—fully compatible
and achieved without introducing new constants beyond $\alpha ,\beta ,\gamma$.

Cross-Link Summary (Stage D)

Last edited 2025-06-10 · part of Concept-v2 series