Research-Stack/docs/S7_SPECTRAL_BASIS.md

# S⁷ Spectral Basis — Fisher Sphere → Spherical Harmonics

## Discovery Source

Reddit post: "From Spherical Gradients to Dirac Kets: The Hidden Linear
Algebra of the Laplacian" — r/LinearAlgebra, user TROSE9025.
URL: https://www.reddit.com/r/LinearAlgebra/comments/1ucglpd/

The post shows how the spherical Laplacian decomposes into angular momentum
operators, giving the discrete eigenbasis |l,m⟩. This is the coordinate
system for SilverSight's state space.

## The Map: Δ₇ → S⁷

The Fisher-Rao metric on the 7-simplex maps EXACTLY to the round metric on
the 7-sphere via the square-root transform:

```
  Δ₇ (probability simplex)     --√p-->     S⁷ (unit sphere in ℝ⁸)
  p_i ≥ 0, Σp_i = 1                       x_i = √p_i, Σx_i² = 1

  g^Fisher_ij = δ_ij/p_i + 1/p_8    →    g^round_μν = δ_μν
```

This is not an analogy. The Fisher metric IS the round metric in √p
coordinates. Chentsov's theorem (proven in ChentsovFinite.lean) guarantees
this metric is unique — there is no other choice.

## The Laplacian on S⁷

The Laplace-Beltrami operator on S⁷ has eigenfunctions Y_l^m (spherical
harmonics) with eigenvalues l(l+6) for l = 0, 1, 2, ...

| Mode | Eigenvalue | Physical Meaning | SilverSight Interpretation |
|------|-----------|------------------|---------------------------|
| l=0 | 0 | Constant (average) | Background state, no structure |
| l=1 | 7 | Dipole | Φ vs Σ imbalance, stack bias |
| l=2 | 16 | Quadrupole | Basin structure, FAMM curvature |
| l=3 | 27 | Octupole | Fine structure, scar details |
| l≥4 | l(l+6) | Higher multipoles | Memory patterns, complexity |

## Program State as Spectral Decomposition

Instead of raw 25-dim coordinates, expand in the |l,m⟩ basis:

```
|ProgramState⟩ = Σ_{l=0}^∞ Σ_m c_{l,m} |l,m⟩

c_{l,m} = ⟨l,m|ProgramState⟩ = ∫_{S⁷} Y_l^m*(x) · state(x) dΩ
```

For the default quine.py state (stack=['Φ', 'Σ'], 2 FAMM cells, 1 scar):

```
|stack⟩ = (1/√2)(|Φ⟩ + |Σ⟩)
        = (1/√2)|l=0,m=0⟩ + (1/√2)|l=1,m=Φ-Σ⟩ + 0|l≥2⟩

Spectrum: c_00 = 1/√2 ≈ 0.707  (average)
          c_1,Φ-Σ = 1/√2 ≈ 0.707  (dipole)
          all other c_{l,m} = 0

Laplacian expectation: ⟨L²⟩ = Σ l(l+6)|c_{l,m}|² = 7 × 0.5 = 3.5
```

## Why This Basis Is Natural

| Property | Raw Coords (25-dim) | Spectral (|l,m⟩) |
|----------|-------------------|------------------|
| Orthonormal | No | Yes (Y_l^m are orthonormal on S⁷) |
| Physical meaning | None | l = curvature scale |
| Geodesic path | Curved in Δ₇ | Rotation in |l,m⟩ space |
| State comparison | Euclidean distance | Spectral overlap |
| Scars | Point masses | High-l coefficients |
| FAMM frustration | Metric distortion | Laplacian eigenvalue shift |

## FAMM Frustration = Laplacian Eigenvalue Shift

When FAMM cells compete for delay lines, the metric stretches. This is
a **conformal deformation** of S⁷:

```
g' = e^{2σ(x)} · g_round

where σ(x) = Σ_{scars} pressure_k · G(x, x_k)

G(x, x_k) = Green's function on S⁷ (log-distance kernel)
```

The Laplacian eigenvalues shift:

```
λ'_l = λ_l + ⟨Y_l|σ|Y_l⟩ + O(σ²)
```

High-l modes shift MORE (they probe finer structure). So the scar field
creates a **spectral fingerprint**: the eigenvalue shifts encode the
manifold's wound pattern.

## Receipt Format (Spectral)

```json
{
  "receiptID": "sha256(...)",
  "spectralDecomposition": {
    "basis": "spherical_harmonics_S7",
    "manifold": "Fisher_sphere_sqrt_p",
    "lmax": 3,
    "coefficients": {
      "c_00": 0.707,
      "c_1_mPhiSigma": 0.707,
      "c_2m": [0.0, 0.0, ...],
      "c_3m": [0.0, 0.0, ...]
    },
    "laplacianSpectrum": {
      "eigenvalues": [0, 7, 16, 27],
      "expectation": 3.5,
      "frustrationShift": 0.0
    },
    "scarImprint": {
      "count": 1,
      "totalPressure": 0.1,
      "spectralFingerprint": "low-l dominant"
    }
  }
}
```

## Connection to Prior Work

| This Result | Prior SilverSight Work |
|------------|----------------------|
| S⁷ from Δ₇ via √p | ChentsovFinite.lean (metric uniqueness) |
| Spherical harmonics Y_l^m | UniversalMathEncoding.lean (8-state basis) |
| Laplacian eigenvalues | FAMM_BAKER_ANALOGUE.md (curvature bounds) |
| Spectral decomposition | STATE_SPACE_EMBEDDING.md (25-dim coords) |
| Frustration = eigenvalue shift | FAMM.lean (delay-line competition) |
| Scar spectral fingerprint | COEVOLUTION_MODEL.md (scar accumulation) |

## References

1. Chentsov, N.N. (1972). Statistical Decision Rules and Optimal Inference.
   (Proven in ChentsovFinite.lean for n=8.)

2. TROSE9025 (2026). "From Spherical Gradients to Dirac Kets: The Hidden
   Linear Algebra of the Laplacian." r/LinearAlgebra, Reddit.
   https://www.reddit.com/r/LinearAlgebra/comments/1ucglpd/

3. Amari, S. (2016). Information Geometry and Its Applications.
   (Fisher-Rao metric on probability simplex.)

4. Vilenkin, N.J. & Klimyk, A.U. (1991). Representation of Lie Groups
   and Special Functions. (Spherical harmonics on Sⁿ.)

## One-Line Summary

The Fisher information simplex IS the 7-sphere in √p coordinates. The
spherical Laplacian gives the natural |l,m⟩ spectral basis. Program
states are superpositions of spherical harmonics. FAMM frustration is a
conformal deformation that shifts the Laplacian spectrum. Scars leave a
spectral fingerprint.

This is the coordinate system you were looking for.