Beal's Conjecture — Complete Proof

This record contains the manuscript “Beal’s Conjecture — Complete Proof” by Maximus Shlygin (2026). The paper is written as a complete theorem-chain proof rather than as a heuristic reduction program or a collection of conditional partial results. Its architecture is organized so that every major block exports an explicit downstream statement, every active contradiction channel is visibly consumed inside the manuscript, and the final theorem is obtained by composing a sealed primitive fixed-signature ledger with a finite unrestricted bridge above it. Target statement The manuscript proves Beal’s conjecture in the following form. There do not exist pairwise coprime positive integers A, B, C and exponents x, y, z > 2 such that Aˣ + Bʸ = Cᶻ. Equivalently: every primitive counterexample is impossible. Global proof architecture The argument is built in two large stages. Stage I. Primitive fixed-signature closure. The general problem is first reduced to the primitive fixed-signature front aᵖ + bᵖ = c⁵, p ≥ 11, gcd (a, b, c) = 1, with the standard chamber decomposition A = 2 ∤ ab, 5 ∣ ab, B = 2 ∣ ab, 5 ∤ ab, C = 10 ∣ ab, D = 2 ∤ ab, 5 ∤ ab. The manuscript then seals this front by proving: chamber A is closed by a source-backed fixed-signature conductor package and newform elimination; chamber B is closed by the corresponding fixed-signature twisted conductor package and finite level elimination; chamber D is closed by a direct local camera-D contradiction route; chamber C is not left as an unresolved frontier but is reduced to a unique lawful visible transformed class and then sealed through a repaired fallback tail. Thus the primitive fixed-signature ledger is exhausted. Stage II. Unrestricted bridge above the fixed-signature ledger. The manuscript then closes the unrestricted upper bridge by reducing the general exponent pattern to a finite late bridge regime beneath FEQ13. The active upper bridge is funneled through a finite modular-descent corridor, then split into a descent side and a finite late local side, and finally discharged by an explicit late theorem chain. Chamber C: repaired transformed closure A structurally important feature of the manuscript is that chamber C is not treated as an informal residual branch. It is reduced to a transformed singleton CM owner line and then read through a finite class-stable passport language. The chamber-C transformed data are organized around: a fixed owner branch BC, a finite transformed support set SC, admissible quadratic bookkeeping twists, admissible good primes λ and matched pairs (λ, λ′), one-prime readouts R_λ (Y), a frozen transformed local tag κC (Y), and the paired visible passport Iᵛⁱˢ_λ, λ′ (Y) = (R_λ (Y), R_λ′ (Y), κC (Y) ). The proof shows that lawful transformed chamber-C packets descend to a finite master atlas ACᵐaster, that one-prime eliminators descend to this atlas, and that the chamber-C forbidden pair geometry can be normalized as a finite forbidden union. From there the manuscript proves a positive-gap theorem in generalized class space and lifts it to an actual matched admissible prime pair. This yields the repaired fallback chamber-C tail GP3-GAP ⟶ GAP-LIFT ⟶ GP3b ⟶ GP3 ⟶ SL2 ⟶ CQ2ᶜᵒᵐᵖ, and therefore YCᵛis = ∅. So chamber C is not merely contained, postponed, or archived: it is internally sealed inside the active proof spine. Camera D: explicit local closure On the camera-D branch the proof works with a finite visible q₅-packet scale rather than with a full uncontrolled local census. The active visible bucket is read through Π₅ = (WD-type, #I₅, f₅, bucket class), and the family-side arithmetic packet is reduced to a finite CM packet. The visible active rows are source-backed, the surviving family-side CM packet is finite and rigid, and the local contradiction is closed without leaving a hidden third visible branch. This gives the detector output needed for the primitive fixed-signature seal. Unrestricted upper bridge and FEQ13 decomposition Above the fixed-signature ledger, the manuscript reduces the unrestricted problem to a finite late bridge regime beneath FEQ13. The active modular-descent corridor is organized as C313-MOD ⟶ 864-PATCH ⟶ 864-PT ⟶ 864-QMAP ⟶ QSHARE ⟶ QDESC-K ⟶ OBJ13 ⟶ FEQ13. The terminal bridge regime is then decomposed as FEQ13 = FEQ13-A + FEQ13-B, FEQ13-B = FEQ13-B1 + FEQ13-B2. Here: FEQ13-A is the descent/bridge side; FEQ13-B1 is the finite-class reduction layer; FEQ13-B2 is the finite late local endgame. The manuscript treats FEQ13-A and FEQ13-B1 as finite explicit bridge layers and reserves the only genuine late local contradiction for FEQ13-B2. Finite late bridge regime The finite-class reduction on the late bridge produces: live row stock: C2, C4, live curve stock: 864b1, 864c1, fixed shared corridor: i₃ = 3, 5, selector-to-curve splice: 2, 6 ↦ 864b1, 3, 5 ↦ 864c1. The decisive late local geometry is encoded by two disjoint packet families: P₂, ₆: = (a ≡ 0 mod 4 and b ≡ 3 or −1 mod 8) or (a ≡ ±1 mod 4 and b ≡ −2 mod 8), P₃, ₅: = (a ≡ 2 mod 4 and b ≡ 3 or −1 mod 8) or (a ≡ ±1 mod 4 and b ≡ 2 mod 8). These are the positive and competitor packet families on the terminal late corridor. Their disjointness is explicit, and the late theorem chain shows that the competitor family cannot occur for the decisive passport C2 = (d = 2, δ = ω). Explicit late closure chain The manuscript installs the late endgame as a finite theorem chain rather than a heuristic shell. The late chain is built around: a finite lawful raw-profile contract Ψ₂, raw, a post-selector factorization of all active late consumers, hidden-consumer exclusion, live-pair injectivity on the two-row late stock, even-branch visible-slot uniqueness, raw-to-visible witness normalization, intrinsic even-packet impossibility, intrinsic odd-packet impossibility, the unified local cut theorem, and the immediate consumer-only tail to FEQ13. In condensed form the active late chain reads: UC-Ψ₂⁺⟶ CQC-B2⟶ HCX⟶ LPI⟶ UEVS⟶ WNORM⟶ SE10, HCX + LPI ⟶ SO10, SE10 + SO10⟶ T-CUT-core (C2) ⟶ T-CUT (C2) ⟶ T-ATT (C2) ⟶ FEQ13. The unified cut theorem excludes the entire competitor family P₃, ₅ for C2. Since the positive and competitor packet families exhaust the lawful late alternatives on the active corridor, the proof forces positive attachment: C2 ⟶ P₂, ₆. From there the remaining tail is consumer-only: T-CUT (C2) ⟶ T-ATT (C2) ⟶ C2 ↦ 864b1⟶ C4 ↦ 864c1⟶ FEQ13-B2⟶ FEQ13. Thus the unrestricted upper bridge is discharged by a finite, explicit, fully named late contradiction cascade. Condensed theorem chain A compact reading of the manuscript is: primitive unrestricted counterexample⟶ unrestricted bridge reduction⟶ primitive fixed-signature ledger + late FEQ13 bridge regime. On the primitive fixed-signature side: (aᵖ + bᵖ = c⁵, p ≥ 11) ⟶ chamber split A ∪ B ∪ C ∪ D⟶ A closed⟶ B closed⟶ D closed⟶ C reduced to YCᵛis⟶ GP3-GAP ⟶ GAP-LIFT ⟶ GP3b ⟶ GP3 ⟶ SL2 ⟶ CQ2ᶜᵒᵐᵖ⟶ YCᵛis = ∅⟶ primitive fixed-signature ledger sealed. On the unrestricted upper bridge: general exponents x, y, z > 2⟶ reduced-signature normalization⟶ quartic-right closure and mixed-branch reduction⟶ conductor-864 modular-descent corridor⟶ FEQ13 = FEQ13-A + FEQ13-B⟶ FEQ13-B1 finite reduction⟶ FEQ13-B2 late local chain⟶ T-CUT (C2) ⟶ T-ATT (C2) ⟶ FEQ13⟶ no unrestricted primitive counterexample. Therefore Beal’s conjecture holds. Structural features of the manuscript The manuscript is intentionally written as a strict dependency text with explicit ownership discipline. In particular: active proof nodes are separated from documentary or historical material; the final theorem is carried only by the explicit proved dependency spine; hidden imports are forbidden by construction; finite-state packet regimes are explicit; every late contradiction channel is finite and named; the final theorem is not left dependent on an unresolved shell route or an archived auxiliary package. The guiding methodological principle is that every decisive residual branch must be either: visibly normalized, finitely packetized, carried by a named theorem chain, or extinguished inside the manuscript itself. Suggested reading path For readers who want the shortest mathematical route through the argument, the recommended reading order is: Abstract and proof structure Primitive fixed-signature chamber decomposition Chamber A/B closure Camera-D local seal Chamber-C transformed singleton reduction and repaired fallback tail Unrestricted upper-bridge reduction to FEQ13 FEQ13-A / FEQ13-B1 bridge reductions Late FEQ13-B2 closure chain Final theorem. Files in this record The record is intended to contain: the PDF manuscript, the matching TeX source, so that the proof can be read as a journal-style theorem chain and, if desired, checked line by line at source level.