Add Möbius-Apollonius chord partition gate documentation

6-Documentation/famm/MOBIUS_APOLLONIUS_CHORD_PARTITION_GATE.md

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+# Möbius-Apollonius Chord Partition Gate
+
+## Purpose
+
+Add the uploaded visualization topic — Möbius transforms, Circles of Apollonius, spirals, angles, and trigonometric chord partitions — as a conformal-geometry witness gate in the FAMM/BraidStorm/Universal Shortcut Center stack.
+
+The useful project shape is:
+
+```text
+complex-plane configuration
+→ Möbius transform
+→ Apollonius circle / chord-ratio witness
+→ angle-preserving conformal transport
+→ spiral / coaxal-family projection
+→ FAMM residual or receipt
+```
+
+This gate is especially useful because it gives the project a precise way to move circle/line/spiral structures through a lawful complex transform while preserving the invariants that matter: cross-ratio, angle, generalized-circle structure, and distance-ratio loci.
+
+## Core Möbius transform
+
+A Möbius transformation has the form:
+
+```math
+T(z)=\frac{az+b}{cz+d},\qquad ad-bc\ne0
+```
+
+It maps generalized circles to generalized circles:
+
+```text
+circle/line → circle/line
+```
+
+and is conformal wherever its derivative is nonzero:
+
+```math
+T'(z)=\frac{ad-bc}{(cz+d)^2}
+```
+
+so local angles are preserved away from the pole.
+
+## Apollonius circle witness
+
+A Circle of Apollonius is the locus:
+
+```math
+\mathcal A(a,b;k)
+=
+\left\{z\in\mathbb C:\frac{|z-a|}{|z-b|}=k\right\}
+```
+
+For `k != 1`, this is a circle. For `k = 1`, it degenerates to a line/perpendicular bisector.
+
+Project meaning:
+
+```text
+fixed source pair (a,b)
++ constant ratio k
+→ exact distance-ratio witness curve
+```
+
+Under a Möbius transformation, the Apollonius family is transported into another generalized-circle family, while cross-ratio and angle structure provide the guard conditions.
+
+## Cross-ratio invariant
+
+The primary exact witness is the cross ratio:
+
+```math
+[z_1,z_2;z_3,z_4]
+=
+\frac{(z_1-z_3)(z_2-z_4)}{(z_1-z_4)(z_2-z_3)}
+```
+
+Möbius transformations preserve it:
+
+```math
+[T(z_1),T(z_2);T(z_3),T(z_4)]
+=
+[z_1,z_2;z_3,z_4]
+```
+
+This makes the cross-ratio a clean Judge receipt:
+
+```math
+R_{\mathrm{cr}}
+=
+\left|
+[T(z_1),T(z_2);T(z_3),T(z_4)]-[z_1,z_2;z_3,z_4]
+\right|
+```
+
+Pass condition:
+
+```math
+R_{\mathrm{cr}}\le \Theta_{\mathrm{tol}}
+```
+
+## Chord / unit-circle partition witness
+
+On the unit circle, a chord between two points with angular separation `theta` has length:
+
+```math
+L(\theta)=2\sin\left(\frac{\theta}{2}\right)
+```
+
+A trigonometric partition of a chord can therefore be treated as an angle-to-length witness:
+
+```math
+\theta\mapsto 2\sin(\theta/2)
+```
+
+Project meaning:
+
+```text
+angle partition
+→ chord length
+→ circle-boundary witness
+→ conformal transport through Möbius map
+```
+
+## Spiral connection
+
+Möbius transforms can turn simple circle/line pencils into visually spiral-like families under parameterized motion or composition. Project use:
+
+```text
+circle pencil / chord partition
+→ conformal transport
+→ spiral-like projected trajectory
+→ chirality / angle / ratio receipt
+```
+
+The Warden must distinguish actual logarithmic spiral structure from a parameterized family of transformed circles that merely appears spiral-like in projection.
+
+## Universal Shortcut Center packet
+
+```math
+\Gamma_{\mathrm{MobiusApollonius}}
+=
+(
+X_{\mathbb C},
+\pi_T,
+W_{\mathrm{circle/ratio}},
+R_{\mathrm{cr}},
+I_{\mathrm{angle,ratio}},
+G_{ad-bc\ne0},
+K,
+\epsilon
+)
+```
+
+| Packet term | Meaning |
+|---|---|
+| `X_C` | original complex-plane configuration |
+| `pi_T` | Möbius projection `T(z)` |
+| `W_circle/ratio` | transformed generalized circle / Apollonius witness |
+| `R_cr` | cross-ratio or angle-preservation receipt |
+| `I_angle,ratio` | preserved conformal/ratio invariant |
+| `G_ad-bc_nonzero` | guard that the transform is valid |
+| `K` | cost of tracking full geometry versus witness family |
+| `epsilon` | residual from numerical/visual/projection error |
+
+## FAMM object
+
+```math
+\mathfrak C_{\mathrm{MobiusApollonius}}
+=
+A_{16}(u_{\mathrm{mobius}})
+\otimes
+[
+\Sigma_z
++
+\Sigma_T
++
+\Sigma_{\mathrm{circle}}
++
+\Sigma_{\mathrm{Apollonius}}
++
+\Sigma_{\mathrm{crossRatio}}
++
+\Sigma_{\mathrm{angle}}
++
+\Sigma_{\mathrm{chord}}
++
+\Sigma_{\chi}
++
+\Sigma_{\epsilon}
++
+\Sigma_{\mathrm{receipt}}
+]
+```
+
+## BraidStorm use
+
+Each strand can carry a conformal geometry state:
+
+```math
+s_i=(z_i,T_i,\mathcal A_i,\theta_i,L_i,\chi_i,\rho_i)
+```
+
+A crossing may now be tested by whether its conformal invariants survive:
+
+```text
+strand crossing
+→ Möbius transport
+→ cross-ratio receipt
+→ Apollonius ratio receipt
+→ chord-angle receipt
+→ FAMM scar if invariant drifts
+```
+
+## Anti-FAMM / Warden checks
+
+The Warden should check:
+
+```text
+invalid Möbius determinant ad-bc = 0
+pole crossing / infinity handling
+confusing visual spiral with proven spiral
+failure to preserve cross-ratio
+angle preservation claimed at a singular point
+circle/line degeneration not recorded
+unit-circle chord formula used off the unit circle without normalization
+```
+
+Anti-FAMM shadow test:
+
+```text
+a perturbation invisible in the rendered curve
+but visible in cross-ratio or chord-ratio residual
+is a conformal shadow.
+```
+
+## Stack placement
+
+```text
+MOBIUS_APOLLONIUS_CHORD_PARTITION_GATE
+→ Universal Shortcut Center Manifold
+→ BraidStorm conformal-strand receipt
+→ Golden Braid Centering / chirality check
+→ FAMM Scar Ledger
+→ Anti-FAMM conformal-shadow attack
+→ NUVMAP Delta-DAG geometry receipt
+```
+
+## Warden boundary
+
+This gate imports the conformal-geometry structure, not the visual animation as proof.
+
+Allowed claim:
+
+```text
+Möbius-Apollonius geometry gives the project a lawful conformal transport gate: circles/lines and ratio loci move through a Möbius map while cross-ratio and angle receipts check invariant preservation.
+```
+
+Disallowed claim:
+
+```text
+A GeoGebra visualization alone proves a new Riemann, spiral, or partition theorem.
+```
+
+## Project sentence
+
+The Möbius-Apollonius gate turns complex-plane geometry into a receipt-bearing conformal transport layer: Möbius maps move circles, lines, Apollonius ratio loci, and chord partitions while cross-ratio, angle, and chord-length witnesses tell FAMM whether the projected geometry stayed lawful or became a scarred conformal shadow.