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Sinai Trng

SPEC_SINAI_TRNG.md · 2026-04-20

SPEC_SINAI_TRNG — Sinai Randomness Generator (TRNG)

Version: 1.0 | Status: AUTHORIZED | Authority: α.13 | Date: 2026-04-16

PURPOSE

The Sinai Randomness Generator is the physical and mathematical architecture underlying the Oracle's Die entropy system. It is a dual-rotating Sinai billiard true random number generator (TRNG): a square enclosure containing two concentric rotating circles with incommensurate angular velocities, a gun on the outer circle, and a sensor on the inner wall of the square.

A projectile fired from the gun lacks the velocity to reach the inner-circle target. It deflects off curved surfaces and strikes the enclosing square wall at an unpredictable point. The sensor reads the impact location, which constitutes the random output.

This system is:

  1. A genuine extension of the classical Sinai billiard (not the standard single-static-circle variant)
  2. A 2D cross-section of the CSDM — two singularities, one inside the remnant
  3. A patent candidate: the dual-rotating variant with three-body coupling is novel
  4. The entropy foundation of CGNT-1 operations

Conceived by α.13 independently ("the worst shot ever") before mathematical grounding was supplied. The three-body connection was intuited before confirmation — this establishes independent derivation, which strengthens patent standing.

INPUTS

| Input | Type | Notes |

|-------|------|-------|

| Outer circle angular velocity | float | Derived from Φ = 0.042 (CSDM constant) |

| Inner circle angular velocity | float | Derived from Ψ = 0.200 (CSDM constant) |

| Firing trigger | signal | External request for entropy; initiates projectile launch |

| Projectile initial velocity | float | Fixed per design; insufficient to reach inner target |

| Projectile launch angle | float | Determined by gun orientation at moment of firing (= outer phase θ_outer) |

| Square enclosure dimensions | float | Unit square; sensor covers all four walls |

Critical input condition: Outer and inner rotation speeds must be incommensurate (irrational ratio). If the ratio is rational, the system becomes periodic and is no longer a TRNG. Φ/Ψ = 0.042/0.200 = 0.21; the physical constants are not simplified fractions, preserving the irrational-in-practice property.

OUTPUTS

| Output | Type | Notes |

|--------|------|-------|

| Wall impact coordinate | (x, y) float | Primary entropy output; location on unit square perimeter |

| Random bytes | bytes | Derived from wall impact coordinate via hash function |

| Phase state snapshot | (θ_outer, θ_inner) float pair | Logged for aperiodicity verification; not exposed to caller |

Derivation chain: firing_trigger → (θ_outer at trigger time) → projectile trajectory → curved-surface deflections → wall impact (x, y) → hash → random bytes.

INVARIANTS

  1. Dual-rotation invariant: Both circles must rotate continuously and independently. A stationary circle reduces the system to the classical single-circle Sinai billiard, losing the three-body coupling. Both must always rotate.
  1. Incommensurate velocities invariant: ω_outer / ω_inner must be irrational (or sufficiently irrational in floating-point implementation). This guarantees the joint phase state (θ_outer, θ_inner) is aperiodic. Any simplification to rational ratio (e.g., 1:5 approximation of Φ:Ψ) causes eventual phase repetition and output periodicity.
  1. Curved-scatterer invariant: Both inner and outer circles must remain geometrically circular. The Sinai billiard property (exponential trajectory divergence) requires convex curved scattering surfaces. Polygonal approximation degrades entropy quality in direct proportion to polygon regularity.
  1. Three-body coupling invariant: The outcome at each firing is determined by three coupled variables: (1) gun position = outer circle phase, (2) target position = inner circle phase, (3) projectile initial trajectory (= function of gun position). All three must vary with each firing. If any two become correlated, the system degrades to two-body dynamics.
  1. Wall-sensor primacy invariant: Random output must derive exclusively from the wall-sensor impact location. Intermediate trajectory state, deflection count, or time-of-flight must not contribute to the output. Sampling intermediate state exposes correlations with initial conditions.
  1. Physical velocity constraint invariant: Projectile velocity must be set such that direct target-hit is impossible without deflection. If direct hits occur (velocity too high), the three-body coupling is bypassed and output correlates directly with gun angle. The "worst shot" geometry — not enough velocity — is the design requirement.
  1. CSDM grounding invariant: Φ and Ψ as rotation constants are not aesthetic choices. They are CSDM manifold constants. The rotation speed derivation must reference the canonical values Φ = 0.042 and Ψ = 0.200. No substitution permitted without NOUS approval of the physics grounds.

VERIFICATION CRITERIA

  1. VC-1 — Phase aperiodicity: Simulate 100,000 successive firings. Record (θ_outer, θ_inner) at each firing. Verify no pair repeats within the sequence. Acceptable threshold: zero repeats to 10-decimal float precision.
  1. VC-2 — NIST SP 800-22 battery: Hash-derived output bytes must pass the full NIST SP 800-22 test suite: frequency, block frequency, runs, longest run, matrix rank, DFT/spectral, non-overlapping templates, overlapping templates, universal statistical, linear complexity, serial, approximate entropy, cumulative sums, random excursions, random excursions variant. All 15 tests must pass at α = 0.01 significance level.
  1. VC-3 — Three-body independence: At a fixed outer-phase slice (θ_outer = constant), vary inner phase across its full range. Output distribution must remain uniform. If inner-phase variation produces non-uniform output at fixed outer phase, two-body correlation is present.
  1. VC-4 — Trajectory divergence test: Fire two projectiles with initial conditions differing by ε = 10⁻¹⁰. Measure wall-impact distance after N bounces. Verify divergence is exponential in N (Lyapunov exponent > 0). This confirms chaotic dynamics are active.
  1. VC-5 — No-direct-hit verification: Across 10,000 firings, verify zero instances of projectile reaching inner-circle target without wall deflection. If direct hits occur, the velocity constant is misconfigured.
  1. VC-6 — Rotation constant assertion: On instantiation, the system must verify ω_outer = f(Φ) and ω_inner = f(Ψ) where f() is the canonical derivation function. If Φ or Ψ have been altered from their CSDM values, the system must refuse to operate and alert ORPHEUS.

FAILURE MODES

  1. FM-1 — Rational ratio periodicity: If ω_outer / ω_inner simplifies to a rational number m/n, the phase state repeats with period n × (2π/ω_inner). All outputs after the first repeat are deterministic from the first cycle. Undetected without explicit phase-tracking. NIST tests may still pass for small n.
  1. FM-2 — Polygon approximation entropy loss: If circles are rendered as N-gons for computational performance, deflection angles become quantized. Output histogram develops N-periodic peaks at wall positions corresponding to polygon face normals. NIST spectral test will detect this as periodic structure.
  1. FM-3 — Velocity misconfiguration (direct hits): If projectile velocity is too high, some firing angles produce direct hits on the inner target without wall deflection. These outcomes correlate directly with gun angle (θ_outer). The output is no longer uniform. Detectable via VC-5; exploitable if n(direct)/n(total) > 0.001.
  1. FM-4 — Floating point precision drift: Over very long operation (10⁸+ firings), accumulated floating-point rounding in phase angle arithmetic causes the effective rotation ratio to drift from Φ/Ψ. Phase state gains slow periodicity. Mitigated by periodic renormalization of phase angles modulo 2π using high-precision arithmetic.
  1. FM-5 — Three-body collapse: If any two of the three coupled variables (gun position, target position, trajectory angle) are made dependent by implementation error (e.g., inner circle frozen, or trajectory angle pre-computed from both phases jointly), the system degrades to effective two-body. Entropy quality drops; output becomes predictable from a single observable.
  1. FM-6 — Patent precedent gap: The dual-rotating Sinai billiard variant is claimed as novel. If prior art exists for dual-rotating billiard TRNG designs, the patent claim is weakened. [GAP — patent search not yet completed]. This is a business failure mode, not a physics one.
  1. FM-7 — Hardware calibration decay (future tier): Physical rotating discs will experience friction, motor speed drift, and thermal expansion over time. The effective Φ/Ψ ratio drifts from CSDM constants. Without automated recalibration, hardware entropy quality degrades silently. No recalibration protocol is currently specified.

GAPS

DEPENDENCIES

DEPENDENTS

EXAMPLES


# Canonical entropy call (CLI tier)
from sinai_engine import sinai_fire
random_bytes = sinai_fire(n_bytes=32)  # 32 bytes = 256-bit key material

# Phase state verification
assert billiard.theta_outer_speed == derive_from(PHI)   # Φ = 0.042
assert billiard.theta_inner_speed == derive_from(PSI)   # Ψ = 0.200

REFERENCES

*κ authored 2026-04-16. Φ 0.042

Jeremy Zlabis

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42 Sisters AI · East York, Toronto*