Bertrand's postulate as a Pascal-dome balance: A geometric reading inside the prime machine

Bertrand's postulate states that for every integer n greater than 1 there is a prime p with n less than p less than or equal to 2n. The classical proof by Chebyshev (1850), simplified by Erdoes (1932), is an arithmetic balance argument on the central binomial coefficient C(2n, n). This paper gives a geometric reading of Chebyshev's balance inside the framework of the deterministic prime machine introduced in Krause (2026a) and proven complete in Krause (2026b). The reading rests on two foundations of the machine: (1) The completeness theorem: every prime is generated as an orbit minimum of the symmetry group G_k acting on the coprime residue universe U_{M_k}, with no gaps and no duplicates. The set of orbit minima at stage k equals the set of primes in the target interval (p_k, p_k^2]. (2) The full symmetry theorem: the action of G_k on U_{M_k} has three nested symmetries (inner, mirror, outer). Every orbit is a hypercube Q_{k-1}, the mirror involution a -> M_k - a is an interior element of G_k, and all orbits are pairwise isomorphic. The geometric object on which the balance is computed is the Pascal-dome with tetrahedral start. The dome is the upside-down Pascal triangle built by matchstick path-counting, with a six-matchstick tetrahedron at the apex; the central position of row 2n then has volume V_n = 6 times C(2n, n), growing exponentially as 4^n divided by square root of n. The completeness theorem makes the dome's prime content exhaustive; the full symmetry theorem makes the volume bookkeeping exact. Chebyshev's contradiction-argument is reproduced as a volume balance: the contribution of primes up to n alone is bounded by (2n)^(sqrt(2n)) times 4^(2n/3), which is sub-exponential in 4^n. The known lower bound C(2n, n) at least 4^n / (2n + 1) then forces the inequality 4^(n/3) / (2n + 1) at most (2n)^(sqrt(2n)), which fails for all n at least 468. The small cases are handled by a direct Bertrand chain (2, 3, 5, 7, 13, 23, 43, 83, 163, 317, 631). The paper is structurally honest: it does not claim a new proof of Bertrand's postulate. The arithmetic content is Chebyshev's, the bookkeeping is Erdoes'. The contribution is methodological: a geometric translation of a classical proof into the matchstick framework, showing that the two pillars of the framework (completeness, symmetry) are strong enough to carry a known result without strain. This sharpens the geometric proof-language in preparation for harder open problems such as Goldbach's conjecture. Sections of the paper---------------------1. What we set out to do2. The machine in compact form3. The completeness of the machine and why every prime in the balance is reachable4. The symmetry of the machine and why the volume balance is exact5. The Pascal-dome as a matchstick construction6. The volume balance: Chebyshev's argument as a Pascal-dome reading7. Bertrand's postulate as a Pascal-dome necessity8. Honest assessment and what the reading achieves9. Conclusion and outlook Notes for the reader----------------------This is the third paper in a methodology arc of the Geometry of Reality series. The first paper (Krause 2026a) introduces the prime machine. The second (Krause 2026b) proves its completeness. The third paper on the Devil's Game (Krause 2026c) uses the machine to read a classical logic puzzle. The present paper sharpens the proof-language on Bertrand's postulate. The next intended step is Goldbach's conjecture, where the same two-pillar structure will be tested against an open problem. Keywords---------------------Bertrand's postulate; prime machine; Pascal-dome; matchstick construction; central binomial coefficient; Chebyshev balance; geometric number theory; coprime residue universe; primorial; orbit minimum; symmetry group; geometric reading