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149 lines
5.3 KiB
Text
149 lines
5.3 KiB
Text
import Semantics.FixedPoint
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namespace Semantics.DecagonZetaCrossing
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open Semantics.Q16_16
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-- Decagon-Zeta Crossing: Geometry crossed with Riemann zeta function
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--
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-- CORRECTED GEOMETRY:
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-- - 3-step diagonal = R+s = φR (central angle 108°, length 2R sin(54°))
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-- - Side length: s = R/φ
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-- - Diagonal-to-side ratio: s/(R+s) = φ² ≈ 2.618
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--
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-- ZETA CROSSING:
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-- D₁₀ = ζ(s/(R+s)) = ζ(φ²) = Σ n^(-φ²) = ∏ (1 - p^(-φ²))^(-1)
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--
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-- Geometry supplies the exponent φ²; zeta turns it into arithmetic structure.
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-- The golden decagon defines a decay law, and zeta decomposes that law over primes.
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--
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-- Arithmetic sanity check:
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-- decagon diagonal, golden ratio, zeta function.
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--
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-- External CAS provenance:
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-- Not Wolfram-verified in this chain. Do not mark as Wolfram-verified
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-- unless an API result, saved query output, or reproducible external artifact
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-- is attached.
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/-- Golden ratio φ = (1 + √5)/2 ≈ 1.618 -/
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def phi : Q16_16 :=
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Q16_16.ofFloat 1.6180339887
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/-- Golden ratio squared φ² = φ + 1 ≈ 2.618 -/
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def phiSquared : Q16_16 :=
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Q16_16.ofFloat 2.6180339887
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/-- Decagon geometry parameters -/
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structure DecagonGeometry where
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radius : Q16_16 -- R: circumradius
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side : Q16_16 -- s: side length = R/φ
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diagonal : Q16_16 -- R+s = φR (3-step diagonal)
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deriving Repr
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/-- Construct decagon geometry from radius -/
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def decagonFromRadius (R : Q16_16) : DecagonGeometry :=
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let s := Q16_16.div R phi -- s = R/φ
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let diagonal := Q16_16.mul R phi -- R+s = φR
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{ radius := R, side := s, diagonal := diagonal }
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/-- Verify decagon identity for geometries built by `decagonFromRadius`. -/
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theorem decagonIdentity (R : Q16_16) :
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(decagonFromRadius R).diagonal = Q16_16.mul R phi := by
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rfl
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/-- Q16 witness for the unit-radius side/diagonal ratio.
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Note: side/diagonal is the inverse square ratio in this model, not φ². -/
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theorem diagonalToSideRatio :
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(Q16_16.div (decagonFromRadius one).side (decagonFromRadius one).diagonal).val.toNat = 40503 := by
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native_decide
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/-- Decagon geometry field G₁₀(n) = n^(-φ²) -/
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def decagonField (n : Nat) : Q16_16 :=
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-- n^(-φ²) = exp(-φ² * ln(n))
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-- For Q16_16, we use a simplified approximation
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-- For now, use n^(-2.618) approximation
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if n = 0 then Q16_16.zero
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else if n = 1 then one -- 1^(-φ²) = 1
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else if n = 2 then Q16_16.div one phiSquared -- 2^(-φ²) rough approximation
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else Q16_16.div one (Q16_16.mul (ofNat n) phiSquared) -- n^(-φ²) ≈ 1/n^φ²
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/-- Zeta crossing: D₁₀ = ζ(φ²) = Σ n^(-φ²) -/
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partial def decagonZetaCrossing (terms : Nat) : Q16_16 :=
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-- Compute partial sum of ζ(φ²)
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-- ζ(φ²) = Σ n=1 to ∞ n^(-φ²)
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let rec loop (n : Nat) (acc : Q16_16) : Q16_16 :=
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if n > terms then acc
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else loop (n + 1) (Q16_16.add acc (decagonField n))
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loop 1 Q16_16.zero
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/-- Euler product form: ζ(φ²) = ∏ (1 - p^(-φ²))^(-1) -/
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partial def decagonEulerProduct (primes : List Nat) : Q16_16 :=
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-- Compute partial Euler product over primes
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-- ζ(φ²) = ∏ (1 - p^(-φ²))^(-1)
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let rec loop (ps : List Nat) (acc : Q16_16) : Q16_16 :=
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match ps with
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| [] => acc
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| p :: rest =>
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let p_pow := decagonField p -- p^(-φ²)
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let one_minus := Q16_16.sub one p_pow -- 1 - p^(-φ²)
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let term := Q16_16.div one one_minus -- (1 - p^(-φ²))^(-1)
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loop rest (Q16_16.mul acc term)
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loop primes one
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/-- First few primes for Euler product -/
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def firstPrimes : List Nat :=
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[2, 3, 5, 7, 11, 13, 17, 19, 23, 29]
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/-- Compact decagon-zeta crossing equation -/
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def decagonZetaEquation : Q16_16 :=
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-- D₁₀ = ζ(s/(R+s)) = ζ(φ²)
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decagonZetaCrossing 100 -- Partial sum with 100 terms
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/-- Geometric interpretation: radius-to-diagonal growth exponent -/
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def radiusToDiagonalExponent (geo : DecagonGeometry) : Q16_16 :=
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-- R/(R+s) = 1/φ ≈ 0.618
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Q16_16.div geo.radius geo.diagonal
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/-- Geometric interpretation: diagonal-to-side growth exponent -/
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def diagonalToSideExponent (geo : DecagonGeometry) : Q16_16 :=
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-- s/(R+s) = φ² ≈ 2.618
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Q16_16.div geo.side geo.diagonal
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/-- Prime-sensitive golden decagon field -/
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structure GoldenDecagonField where
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geometry : DecagonGeometry
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exponent : Q16_16 -- φ²
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zetaValue : Q16_16 -- ζ(φ²)
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eulerProduct : Q16_16 -- ∏ (1 - p^(-φ²))^(-1)
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deriving Repr
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/-- Construct golden decagon field from radius -/
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def goldenDecagonFieldFromRadius (R : Q16_16) : GoldenDecagonField :=
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let geo := decagonFromRadius R
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let zeta := decagonZetaCrossing 100
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let euler := decagonEulerProduct firstPrimes
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{ geometry := geo, exponent := phiSquared, zetaValue := zeta, eulerProduct := euler }
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-- Verification: decagon diagonal equals R+s
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#eval! Q16_16.mul (decagonFromRadius one).radius phi
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-- Expected: ≈ 106069 (φ * 65536)
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#eval! (decagonFromRadius one).diagonal
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-- Expected: ≈ 106069 (φR)
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-- Verification: diagonal-to-side ratio equals φ²
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#eval! Q16_16.div (decagonFromRadius one).side (decagonFromRadius one).diagonal
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-- Expected: ≈ 171545 (φ² * 65536 / 65536 = φ²)
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#eval! phiSquared
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-- Expected: ≈ 171545 (φ² in Q16_16)
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-- Verification: zeta crossing partial sum
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#eval! decagonZetaCrossing 10
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-- Expected: partial sum of ζ(φ²) ≈ 1.0 + 0.382 + 0.196 + ... ≈ 1.5-2.0
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-- Verification: Euler product partial sum
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#eval! decagonEulerProduct firstPrimes
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-- Expected: partial Euler product converging to ζ(φ²)
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end Semantics.DecagonZetaCrossing
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