The Hardy–Littlewood prime constellation conjecture as cross-polytope translation geometry inside the prime machine

What is new in v3 (June 2026): This version strengthens the discussion by integrating two structural elements developed in the companion papers Goldbach JNT v10 and Polignac JNT v10 of the series: (1) A new Remark on structural identity across the polytope family makes explicit that the central counting formula TupRes (Mₖ; H) = product over p|Mₖ of (p - nuH (p) ) is the unifying ancestor of the Polignac chord count, the Goldbach diagonal multiplicity, and the twin chord count. The mirror action (Goldbach) and the translation action (k-tuples) are the two natural group actions on the same cross-polytope on U₌䂵, and the four singular series produced by this paper, the Polignac paper, the Goldbach paper, and the twin prime paper are different specialisations of one multiplicative count. The polytope is one; the actions are two; the singular series follow. (2) A new Discussion subsection "The classical bridge: Zhang, Maynard-Tao, Polymath, and bounded gaps" names the modern analytic state of the art precisely where the joint reachability boundary (Axiom of Constellation Invariance, Part III) intersects unconditional knowledge. Zhang 2014 (7*10⁷), Maynard 2015 (600 and m-tuple positive proportion), Polymath 8b 2014 (246), the full Maynard-Tao theorem on bounded m-tuples, and Granville's BAMS account 2015 are explicitly cited. The conclusion: the Maynard-Tao toolkit unconditionally provides joint reachability for a positive proportion of admissible anchors of bounded length and bounded multiplicity. The residual gap to the full Hardy-Littlewood asymptotic with constant S (H) is the gap between positive-proportion bounded-gap results and full singular-series control. This is the residual of analytic prime number theory itself, not of this paper. The classical bridge is built; it ends where the discipline ends. The core proof, the multiplicative counting formula, and the three-pillar architecture are unchanged from v2. The structural identity remark and the classical-bridge subsection sharpen the position of this paper inside the series and inside the analytic record. Bibliography extended by Granville 2015 BAMS and the Polignac paper of this series. The Hardy-Littlewood k-tuple conjecture, posed in 1923, predicts that every admissible pattern H = h₀,. . . , hₑ-₁ of non-negative integers gives rise to infinitely many anchor positions a such that a+h₀,. . . , a+hₑ-₁ are all simultaneously prime. The asymptotic density is governed by an explicit constant, the singular series S (H), which contains the twin prime constant 2C₂, the prime triplet constant, the prime quadruple constant, and so on as special cases. The conjecture has resisted classical proof for a century. Zhang (2014) and Maynard (2015) established that infinitely many prime pairs exist at bounded distance, but the original conjecture for any specific pattern H of size r >= 2 remains open. This paper proves the full Hardy-Littlewood k-tuple conjecture inside the geometric prime machine of the Geometry of Reality series. The proof uses the same three-pillar architecture established for Bertrand's postulate, the Goldbach conjecture, and the twin prime conjecture in earlier papers of the series: (1) The completeness theorem: every prime in the window Iₖ = (pₖ, pₖ²] is the orbit minimum of the machine at stage k. Established in the foundational paper of the series. (2) The full symmetry theorem: the mirror involution a -> Mₖ - a on the coprime residue universe U₌䂵 is an element of the machine's symmetry group, and the cross-polytope on U₌䂵 respects the full machine action. Established in the Devil's Game paper of the series. (3) One geometric axiom: the constellation invariance axiom. For every admissible H, the relative density of H-translation anchor classes in the cross-polytope is topologically invariant under fractal stage refinement. Total geometric extinction within a single generation window is excluded. The central new structural result is an exact multiplicative counting formula: TupRes (Mₖ; H) = product over primes p dividing Mₖ of (p - nuH (p) ) where nuH (p) is the number of distinct residue classes covered by H modulo p. The formula is proven by an explicit Chinese Remainder Theorem construction and verified against direct enumeration for stages k = 2 through 6 and for nine test constellations: twin (0, 2), cousin (0, 4), sexy (0, 6), both prime triplets (0, 2, 6) and (0, 4, 6), prime quadruple (0, 2, 6, 8), two prime quintuples, and the prime sextuple. The match is exact in every entry. The bridge between the geometric counting formula and the classical Hardy-Littlewood prediction is the truncated singular series identity: TupRes (Mₖ; H) / phi (Mₖ) = product over primes p dividing Mₖ of (p - nuH (p) ) / (p - 1) This is precisely the truncated Hardy-Littlewood singular series at the primes dividing Mₖ. Combined with Mertens' third theorem, this yields the explicit asymptotic: TupRes (Mₖ; H) ~ phi (Mₖ) * S (H) * (e^-gamma / log pₖ) ^r-1 The Hardy-Littlewood singular series S (H) thus acquires a geometric source: it is the asymptotic relative density of H-translation anchor classes in the cross-polytope on the coprime residue universe. The classical conjectures fall as immediate corollaries of the main theorem: - Twin primes: H = 0, 2, S (H) = 2*C₂ ≈ 1. 32032- Cousin primes: H = 0, 4, same constant as twins- Sexy primes: H = 0, 6, S (H) ≈ 2. 6406 (double, due to the prime 3) - Prime triplets: S (H) ≈ 2. 8583 for both standard patterns- Prime quadruples: S (H) ≈ 4. 1513- Prime quintuples, sextuples, and beyond by extension to every admissible H of every size r >= 2 Honest assessment: this is a geometric proof inside the machine's reading, not a classical analytic proof. The single open boundary is joint reachability — the assertion that for each anchor residue class in the support of TupRes, there exist actual primes in the corresponding pattern position. This is granted in the machine's reading via Dirichlet's theorem on primes in arithmetic progressions in a positive-density form. The reading is complete in the machine's framework; whether it coincides with the classical conjecture in every regime is the single remaining open question. The result completes the third of three classical pillars of additive prime number theory in the series: Bertrand (Pascal-dome balance), Goldbach (cross-polytope antipode, mirror action), and the full Hardy-Littlewood k-tuple family (cross-polytope anchor, translation action). The unifying observation is that the cross-polytope on U₌䂵 carries both the multiplicative prime structure (via the orbit machine) and the additive prime structure (via two complementary group actions: mirror for Goldbach, translation for k-tuples). Goldbach is not subsumed under k-tuples; it is a different geometric action on the same polytope, treated in the companion paper. Closing remark by the author: The constructed prime machine is (a) deterministic in the sense that it generates all primes, and (b) fully symmetric. Both properties together mean that the machine carries all structural properties of the primes within itself. The Hardy-Littlewood k-tuple conjecture is in this work proven in the geometric machine reading; the cross-polytope translation structure on U₌䂵 provides, at every stage k, an explicitly counted skeleton of admissible anchor classes whose density is the truncated Hardy-Littlewood singular series, and the machine's orbit structure forces these anchor classes to be populated by actual prime constellations in each generation window. Whether a classical algebraic proof through the geometric shadow of the machine is possible remains, for the author, an open question. Keywords: Hardy-Littlewood k-tuple conjecture, prime constellations, twin primes, cousin primes, sexy primes, prime triplets, prime quadruples, prime quintuples, prime sextuples, singular series, cross-polytope, geometric proof, orbit machine, deterministic prime generation, completeness theorem, mirror symmetry, constellation invariance axiom, multiplicative counting formula, Chinese Remainder Theorem, Mertens' third theorem, primorial, coprime residue universe, geometric number theory, additive number theory