The Bourgeois Triangle

The Bourgeois triangle is a Pascal-type array with a unit left border, a signed right border, and ordinary addition in the interior. It started as scratch-work: put the first prime, the unit, and alternating prime direction into one picture and see what holds it together. This paper rebuilds that object from the ground up and keeps a hard line between what is proved and what is not.

Four mechanisms are exact. The row polynomials obey a one-line recursion and a Green-cone formula. The alternating row value Pn(-1) is a perfect selector: it sees only the newest right-edge impulse. Under the von Mangoldt doorway T(n,n)=psi(n), the near-edge difference is exactly Lambda(n), so the Euler identity enters the medium with no approximation. And the survivor border is a Ramanujan-Fourier wheel signal: on [pk, pk+1 squared) it is exactly the Qk-wheel, with lane amplitudes equal to Ramanujan sums.

Then we draw the boundary. A structural acceleration operator collapses the Li-Laguerre prime trace to a standard Laguerre sampling sequence, and boundedness of that filtered trace would exclude interior poles. We prove every reduction leading there. We state the remaining boundedness assertion as a conjecture, not a theorem. That is the honest endpoint.

1. 1. The genesis and the first-principles problem2
2. 2. Definition of the triangle3
3. 3. The affine clock and the seed4
4. 4. Green cones and the exact selector4
5. 5. The von Mangoldt doorway5
6. 6. The wheel-lane bridge5
7. One eliminator at a time5
8. The wheel-onset law6
9. Lane amplitudes are Ramanujan sums6
10. The same data, two axes7
11. Controls and the honesty line7
12. 12. The residual-bilinear identity8
13. 13. The completed Li-Laguerre trace8
14. 14. Residual form of the prime trace9
15. 15. Structural acceleration and the filtered trace10
16. 16. The zero-mode detector10
17. 17. Why the obvious bound fails11
18. 18. The prime-Laguerre frame conjecture11
19. 19. Numerical evidence12
20. 20. Conclusion12