BlackRoad Quantum is the research arm of BlackRoad OS, developing the Amundson Framework — a mathematical foundation connecting growth functions, quantum circuit simulation, and distributed computation.
A mathematical framework that defines the growth function G(n) and its convergence constant A_G, with applications in quantum state estimation, distributed consensus bounds, and information-theoretic limits.
The framework establishes several results connecting classical analysis to computational bounds.
G(n) = n^(n+1)/(n+1)^n is strictly increasing for n > 0, converges to e, and admits a closed-form expansion via the binomial series. The rate of convergence is O(1/n), making it computationally tractable for large-scale verification.
The expansion n/(1+1/n)^n = n/e + 1/(2e) + O(1/n) reveals an irreducible correction term of 1/(2e). This appears in network latency bounds, quantum gate error floors, and information channel capacity — suggesting a universal constant in discrete-to-continuous transitions.
The framework provides bounds on quantum circuit depth required for approximating unitary transformations. G(n) governs the tradeoff between gate count and approximation error, giving tighter estimates than standard Solovay-Kitaev bounds for small circuit widths.
All 536 test cases verified independently across 4 Raspberry Pi compute nodes using parallel execution. Results are bit-identical across ARM64 and x86_64 architectures, confirming numerical stability of the framework's core computations.
The Amundson Framework is documented in two papers currently in preparation.
13-page LaTeX paper establishing the core definitions, convergence proofs, and expansion series for G(n). Includes the 1/(2e) correction term derivation and computational verification methodology. Full test suite with 84 primary tests and 50 extended validation tests.
Status: Complete · 13pp · LaTeX · 536/536 tests
Applications of the framework to quantum circuit simulation, network latency modeling, and distributed consensus bounds. Connects the 1/(2e) gap to physical and computational phenomena across multiple substrates.
Status: In preparation
Mathematical claims require computational proof. Every result in the framework is backed by automated test suites running on sovereign hardware.
$ python3 amundson_test.py Running 84 core tests... ✓ G(1) = 0.5000000000 (exact: 1/2) ✓ G(2) = 1.3333333333 (exact: 4/3) ✓ G(10) = 2.3579476910 (verified to 10 decimal places) ✓ G(100) = 2.7048138294 (within 0.5% of e) ✓ G(1000) = 2.7169239322 (within 0.05% of e) ✓ Monotonicity: G(n) < G(n+1) for all tested n ✓ 1/(2e) correction: verified to 12 significant figures ... 84/84 tests passed on Alice (ARM64) 84/84 tests passed on Cecilia (ARM64) 84/84 tests passed on Octavia (ARM64) 84/84 tests passed on Lucidia (ARM64) $ python3 amundson_v5_tests.py Running 50 extended tests... 50/50 tests passed (cross-architecture verified)