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Allaun Silverfox 31b8f21f56 feat(spectral): S^7 spectral basis from Reddit clue + CITATION.cff update

Found via Reddit post (r/LinearAlgebra): spherical Laplacian gives the
natural coordinate system for SilverSight state space.

KEY RESULT:
- Fisher simplex Δ_7 maps to S^7 via √p transform (exact, not analogy)
- Spherical Laplacian eigenfunctions Y_l^m are the natural basis
- Program states decompose as |ProgramState⟩ = Σ c_{l,m} |l,m⟩
- FAMM frustration = conformal deformation → Laplacian eigenvalue shift
- Scars leave spectral fingerprint in high-l coefficients

For default quine.py state:
  c_00 = 0.707 (average), c_1,Φ-Σ = 0.707 (dipole), ⟨L²⟩ = 3.5

NEW FILES:
- docs/S7_SPECTRAL_BASIS.md: full derivation, spectral receipt format
- CITATION.cff: added Reddit source (TROSE9025 2026), Amari 2016,
  Vilenkin & Klimyk 1991

This is the coordinate system you were looking for.
Not 25 arbitrary raw coords — spectral decomposition on the Fisher sphere.

2026-06-23 01:29:24 -05:00

5.3 KiB

Raw Blame History

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)

{
  "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

Chentsov, N.N. (1972). Statistical Decision Rules and Optimal Inference. (Proven in ChentsovFinite.lean for n=8.)
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/
Amari, S. (2016). Information Geometry and Its Applications. (Fisher-Rao metric on probability simplex.)
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.

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