Pick a hardware design class, set your parameters, and receive the complete
φ-derived specification — every dimension, ratio, and material determined by
the framework with zero free parameters.
Mode(r, s, k, t) — Four integers, one identity, complete bearing. Every dimension
derives from φ²=φ+1. The groove conformity f(k) = (φᵏ+φ⁻ᵏ)/2 generates Lucas numbers
at even k and √5·Fibonacci at odd k.
INPUT 1 — BORE LEVEL n
L1 WatchmakingL10 Ship propulsion
Level 5 — bore ø35.89 mm — Motors/Aero
INPUT 2 — WIDTH SERIES s
INPUT 3 — GROOVE CLASS k
INPUT 4 — MATERIAL TIER t
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RADIAL GEOMETRY
GROOVE GEOMETRY (k=3)
AXIAL & CAGE
MATERIAL SPECIFICATION
HARMONIC SELF-CHECK
ELEVEN INVARIANTS
RADIAL CROSS-SECTION
GROOVE GEOMETRY (k=3)
φ-CASCADE DIMENSIONS
BALL PACKING — 13 BALLS (F₇)
MARKET COMPARISON
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φ-LATTICE POSITION
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CONFORMITY IDENTITY
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L = π(1−ζ) · M · r⁴ — Two inputs, complete gyroscope. Specify angular momentum and
material tier. The math delivers the complete gyroscope — every bearing dimension,
rotor, housing, motor, and RPM — from φ²=φ+1 with zero free parameters.
INPUT 1 — ANGULAR MOMENTUM
10 µN·m·s10,000 N·m·s
0.500 N·m·s
INPUT 2 — MATERIAL TIER
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BEARING GEOMETRY
ROTOR & HOUSING
FIBONACCI MOTOR (13/8)
MATERIAL SPECIFICATION
EIGHT INVARIANTS
RADIAL CROSS-SECTION
ROTOR & HOUSING PROFILE
φ-CASCADE DIMENSIONS
FIBONACCI MOTOR — 13/8
φ-LATTICE POSITION
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MARKET COMPARISON
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EMERGENT SCHUMANN COINCIDENCE
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φCoherent Tri-Lobed Impeller Disc. Radial φ-cascade: D → D/φ² → D/φ⁴.
Axial √5-cascade: H = D/(√5·φ²). Lobe span 120°/φ = 74.16° · Channel 120°/φ² = 45.84°.
Bowl slope arctan(2/(√5·φ²)) = 18.86°. The F(15) instance is the Sabu Disc of Egypt,
c. 3000 BCE.
INPUT 1 — FIBONACCI LEVEL n
F(5) = 5 mmF(17) = 1597 mm
F(13) = 233 mm — Industrial circulation
INPUT 2 — MATERIAL TIER
COMPLETE SPECIFICATION — TWO φ-FAMILIES FROM ONE IDENTITY
TEN INVARIANTS — IDENTICAL AT ALL FIBONACCI LEVELS
TRI-LOBED DISC — TOP VIEW · RADIAL φ-CASCADE
Seven φ-Identities · Any Fluid · Any Scale. Select a working medium
and Fibonacci level. All seven fluid dynamics identities update live after generation.
Dimensionless ratios are identical across all media — only absolute magnitudes change.
WORKING MEDIUM
TERRESTRIAL
SPACE APPLICATIONS
FIBONACCI LEVEL F(n)
F(10)=55mm MicroF(20)=6765mm Marine
F(14) = 377 mm — Process pump
MHD MODE — φCoherent MHD System
PRESSURE RISE
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RPM
—
REYNOLDS
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MACH
—
FLOW Q
—
HEAD
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SEVEN φ-IDENTITIES — INVARIANT ACROSS ALL MEDIA
#
IDENTITY
EXPRESSION
VALUE
FLUID-FREE?
MARKET COMPARISON
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VELOCITY TRIANGLE (φCoherent Impeller, ID 1)
Q-H CHARACTERISTIC (φCoherent Impeller, ID 6)
φ-VORTEX VELOCITY PROFILE (φCoherent Impeller, ID 4)
φCoherent Hydraulic Ram Pump. Drive pipe L/D = φ³ · Efficiency η = 1/φ ·
Cascade amplification φᴺ. Water hammer in a Fibonacci-proportioned pipe delivers fluid
to arbitrary head with no external power. Every dimension from φ²=φ+1.
FIBONACCI LEVEL n
55mm garden1597mm civil
F(14) = 377 mm
SUPPLY HEAD
3 m50 m
15 m supply head
CASCADE STAGES N
N=1 · φ×N=9 · φ⁹×
3 stages · φ³ = 4.236×
MATERIAL TIER
COMPLETE SPECIFICATION
SEVEN INVARIANTS
CASCADE STAGING — N = 1 to 9
CASCADE PRESSURE BAR CHART
φCoherent Dual-Function Nozzle. Forward thrust: vane angle arctan(1/φ) = 31.717° ·
area ratio 1/(2φ) · 3-sector exit. Two material branches from φ²=φ+1: Fibonacci composite
(Al-Sc-Se) for forward thrust vanes · Lucas cermet (Cu-Ag-Os) for extreme/acoustic service.
FIBONACCI LEVEL n — INLET Dₙ = F(n) mm
F(10) = 55 mmF(15) = 610 mm
F(13) = 233 mm
MATERIAL BRANCH — VANE COMPOSITION
NOZZLE CROSS-SECTION
NOZZLE GEOMETRY
MATERIAL SPECIFICATION
VANE MICROSTRUCTURE — THREE FIBONACCI SCALES
φ-COHERENT INVARIANTS
SIX FUNCTIONS OF arctan(1/φ) — CROSS-DESIGN CONCORDANCE
FOUR ROLES OF 1/(2φ) — CROSS-DESIGN CONCORDANCE
Proof-of-concept of the φCoherent Cellular Framework. Three cell types tile
in a golden-angle spiral. Every 13 cells form a self-contained supercluster (8:5 ≈ φ).
Population grows as the Fibonacci sequence. All dimensions derive from S = λ₈/φ⁵ = 613.67 mm.