Conditional $\mathcal{P}_{\text{GET}} = \mathcal{NP}_{\text{GET}} = \text{TIME}_{\text{GET}}(O(1))$
Interactive Formal Machine Tuple $\mathfrak{M}_{\text{GET}}$ & Complexity Class Animator
Click any component of the machine tuple $\mathfrak{M}_{\text{GET}}$ to inspect its governing operator, or switch between the three complexity regimes to watch how state evolution and witness verification behave in real time.
Interactive Theorem 3A: Spatial Synthesis in the Hyperbolic $\{4,3,5\}$ Honeycomb
In hyperbolic space $\mathbb{H}^3$, sequential traversal $E(N) = T_1 \otimes \dots \otimes T_N \cdot E(0)$ requires $N$ serial multiplications across exponentially diverging cells. Select any of the **4 Proof Stages** below to isolate and highlight its exact geometric action on the hyperbolic disk and inspect the synchronized complexity graph in **Theorem 3B**.
Click Any Proof Stage Below to Spotlight It in Both Visual Graphs:
Shared procedural parameters $P$ map step index $N$ directly to 3 orthogonal basis coordinates in $\Gamma = \langle \gamma_1, \gamma_2, \gamma_3 \rangle$:
Pre-computed basis states along the 3 axes are fetched in parallel from direct-access tables $L_1, L_2, L_3$:
Multiply the three 32-bit quaternion cores and apply the 16-bit DSMEM volumetric hyperbolic depth scaling:
The 16-bit Halo/Ghost regions verify neighbor connectivity locally via bitwise AND/XOR without secondary network fetches:
Interactive Theorem 3B: Complexity Scaling Graph ($\log_{10}\text{Ops}$ vs. $\log_{10}N$) & Stage Dissection
This dual-panel chart plots total operation count on a logarithmic scale ($\log_{10}\text{Ops}$) against horizon depth ($\log_{10}N$) on the left, and dissects the **constant $\le 60$-operation budget** across Stages 1β4 on the right. Click any stage button to highlight its exact contribution on the graph and read the mathematical explanation of what is shown.
Five Integrability Axioms ($\mathcal{C}_{\text{GET}}$ Collapse) vs. Five Proven Obstructions (Classical Lower Bounds)
Click any of the **5 Integrability Axioms (`C1`β`C5`)** or **5 Unconditional Obstructions (`O1`β`O5`)** to launch its dedicated real-time mathematical animation and inspect its governing proof.
The 64-Bit Einstein Tile & Table-Free Garner CRT Reconstruction
Interactive 64-Bit Einstein Tile Bit-Map Inspector
Select a Form ID (`Bits 40β43`) or click any bit-field segment to inspect its exact role in the $O(1)$ pipeline.
Lemma 9 Interactive Calculator: Table-Free $O(1)$ Garner CRT
[PROVEN]Every prime triplet $(p, p+2, p+6)$ for $p > 5$ lies in $(11,13,17)$ or $(17,19,23) \pmod{30}$. The maximal 10-bit triplet is $(881, 883, 887)$, bijectively indexing $M = 690,017,701 \approx 2^{29.36}$ cubic states with closed-form polynomial inverses:
Step 6 Torsional Twistor Collapse, $\{4,3,5\}$ Boosts & Null-Cone Linearization
Interactive 3D Klein Ball Model: $\{4,3,5\}$ Cell $\mathcal{K} = [-a,a]^3$, $\varphi$-Trefoil & 6 Ghosts
Drag on canvas to rotate 3D view β’ Use the Collapse Slider to pull back the 3 positive ghosts via $B_{+k}^{-1}$.
1. The Rational Ghost Map $G = \Phi \circ \mathcal{C}_{\text{rat}}$ (Prop. 6.2)
Composing the rational Cayley map $\mathcal{C}_{\text{rat}}(p) = \frac{(1-\|p\|^2, 2p)}{1+\|p\|^2} \in S^3 \cap \mathbb{Q}^4$ with the Lorentz lift $\Phi(w,m) = (1/w, m/w)$ gives an exact, root-free PoincarΓ©-ball isometry onto $\mathbb{H}^3$:
2. $\{4,3,5\}$ Honeycomb & Golden Face Boosts $B_{\pm k}$ (Prop. 6.4)
In the $\{4,3,5\}$ honeycomb, 5 cubes meet at each edge with dihedral angle $\theta = 72^\circ$, fixing the Klein half-edge $a$ and face-to-face boost rapidity $\vartheta = 2r_{\text{in}}$:
3. Conformal Null-Cone Sphere Vector & $O(1)$ Ghost Recovery (Β§7.5)
Lifting $x \in \mathbb{R}^3$ to the null cone $X(x) = x + \frac{1}{2}\|x\|^2 e_\infty + e_0 \in \mathbb{R}^{4,1}$ linearizes Euclidean quadrance ($-2X(x)\cdot X(y) = \|x-y\|^2$). All 10 vertices of $\Delta_1 \cup \Delta_2 \cup \Delta_3$ and Stella Octangula $\Delta_1 \cup (-\Delta_1) = \{\pm 1\}^3$ lie on $\|v\|^2 = 3$, satisfying the linear sphere coincidence:
The Torsional Markov Memory-Offload Pump ($720 \to 72 \to 10$) & Its Systemic Role
Interactive 3-Gear Multi-Resolution Pipeline: Downward Offload ($720\to 72\to 10$) & Upward SLERP Lift ($10\to 72\to 720$)
Scrub through any outer frame $n \in [0, 719]$ to inspect its exact tooth-and-frame coordinates across the 12-point outer gear, 10-point phase-shearing inverse middle gear ($7.2\text{ frames/tooth}$), and 8-point inner equilibrium gear ($1.25\text{ frames/tooth}$), or switch pipeline modes to see how downward block telescoping, upward principal-branch SLERP interpolation, and Lie-group holonomy renormalization operate.
Three-Scale Cyclic Gear Hierarchy & Active Frame Locus
Flow: Downward P_{12β10} β P_{10β8}The Five Architectural Roles of the Torsional Markov Pump in $\mathfrak{M}_{\text{GET}}$ & Subexponential `#k-SAT`
Click any of the 5 Systemic Roles below to launch its dedicated real-time mathematical animation and inspect how the pump resolves bit-growth, non-commutative drift, thermodynamic circulation, floating-point quarantine, and polynomial-space `#k-SAT` recursion.
The Four-Structure Sandwich ($A_j I_j A_j^{-1} E_j = 1$) & Interactive $64/70$ Peeling Decoder
1. Sandwich Closure & Angle Conservation
Flanking Input $I_j$ with Balancers $A$ and $\bar{A}=A^{-1}$ fixes the Emergent structure $E_j$ while preserving the exact rotation angle $\theta_{I_j}$:
2. Reverse-Ordered Emergent Closure
When a planar 4-engine input group closes ($I_1 I_2 I_3 I_4 = 1$), adjacent $A^{-1}A$ factors cancel so $E$ closes in reverse order:
3. Exact Combinatorial Peeling Spectrum
Across the 8 engines $\mathcal{P}_j = \{I_j, E_j\}$ ($j=1..4$), $O(1)$ peeling recovers $100\%$ of $\le 3$ losses and $64/70$ of $4$ losses:
Interactive 8-Engine Planar Sandwich Peeling Simulator
Click any engine tile below to toggle its state between INTACT and ERASED, or test a preset 4-erasure topology class.
Each structure $S \in \{I, A, \bar{A}, E\}$ logs 12 perimeter engine streams (Slots `0β11`) and 4 interior orthogonal projection streams $(w,x,y,z)$ of its net holonomy $Q_{\text{net}}^{(S)}$ (Slots `12β15`).
Hyperbolic Tree $S_5$ Compression, Ramanujan $X(5,q)$ Mixing & Stator Shells
1. Norm-5 Hyperbolic Tree $S_5$ vs. Cubic Shadow $\mathbb{Z}^3$
Multiplier-free shift-and-add $q \mapsto q + 2(uq)$ over $S_5 = \{1\pm 2i, 1\pm 2j, 1\pm 2k\}$.
2. Diffusive Consensus: Ramanujan $X(5,q)$ vs. 3-Torus
Spectral gap $\gamma_{X(5,q)} \ge 6 - 2\sqrt{5} \approx 1.528$ ($O(\log n)$) vs. $\gamma_{\text{torus}} \approx 4\pi^2/n^{2/3}$.
3. Global Spatial Nesting (Lemma 10): Disjoint Toroidal Shells & $7.536\,C_0$ Separation Gap
Major radii $R_n = (5n^2+15n+15)C_0$, minor radii $r_n = (2n+4)C_0$, windings $W_n = 16-2n$ ($r_n + W_n = 20, W_\Sigma = 60$), anchored to $C_0 = 53.748\text{ cm}$.
Refutation of the Exponential Time Hypothesis: $2^{O_k(n\log\log n/\sqrt{\log n})}$ Exact `#k-SAT`, Lean 4 Formalization & $\mathfrak{M}_{\text{GET}}$ Equivalence
How the Deterministic Subexponential `#k-SAT` Algorithm Refutes ETH, SETH, Gap-ETH, NSETH & `#ETH`
For any fixed $k \ge 2$ and any $k$-CNF $F(x_1,\dots,x_n) = \bigwedge_{e=1}^m [(x_{i_{e,1}}\oplus b_{e,1})\vee\dots\vee(x_{i_{e,k}}\oplus b_{e,k})]$, the algorithm computes the exact satisfying count $\#F = |\{x\in\{0,1\}^n : F(x)=1\}|$ in polynomial space $(n+m)^{O_k(1)}$ and subexponential time $(n+m)^{O_k(1)} 2^{O_k(n\log\log n/\sqrt{\log n})}$ (and $2^{O_{k,D}(n/\sqrt{\log n})}$ for bounded density $m \le Dn$). Click any of the 6 Algorithmic Stages below to animate its exact algebraic mechanism.
Interactive Swap-Matrix $M_0 = V_0 f_{\mathcal{I}}(W_0)$ Eigenbasis Concentrator & Remark 4.4 Explicit 4-State Toy Lab
Why can low-degree replacement forms $\vec{P}_{f,v}(T)$ decouple all $r$ variable groups simultaneously across $Q$ evaluation points using only $2\tau n \ll n$ measurements? Explore the binomial eigenvalue concentration of $W_0 = \frac{1}{2}\sum_{s=1}^Q (I_D - \Pi_s)$ on the left, and test the exact 4-coordinate $(Q=R=2, r=2)$ algebraic identity from Remark 4.4 on the right.
How $\langle \vec{P}_u(T), \zeta \rangle$ Isolates Group 1 at $T=1,2$ Using One Measurement $c(\zeta)$
Take $Q=R=2, r=2$, and $\mathcal{I}=\{0,1\} \implies f_{\mathcal{I}}(z) = 1 - 2z$ and $M_0 = \Pi_1 + \Pi_2 - \Pi_1\Pi_2$. Starting from label $u = (00, 11)$, the swaps $\Pi_1, \Pi_2$ generate 4 coordinates:
Interactive Lean 4 Codebase Inspector & Formal Proof Decomposition of `exact_count_and_sat`
Inspect the exact Lean 4 definitions (`SAT`, `Encoding`, `PrimitiveMachine`, `LocatedCfg`, `runtimeScale`, and `theorem exact_count_and_sat`) and see how the top-level `sorry` decomposes into 6 machine-checkable Mathlib lemmas corresponding to Sections 3β6 and Appendices AβB.
Six Formal Equivalences Between the Subexponential `#k-SAT` System and the Geometric Engine of Time
Why do both frameworks share the exact constant $r = 64$, commuting involution swap algebras, Cayley expander parity graphs, and 2-adic valuation hierarchies? Click any of the 6 Structural Bridges below to inspect the algebraic equivalence and see how the Geometric Engine's $O(1)$ traversal emerges as the exact zero-entropy integrable limit ($\tau \to 0, R/Q \to 0, n/r \to O(1)$) of an $(r,\tau,D)$-design.
Experimental Benchmarking & Algorithmic Reductions for General $\mathcal{NP}$ Problems: 3-Coloring & Unbounded-Clause CNF-SAT
8A. Interactive $\mathcal{NP}$ Benchmark Simulator: 3-Coloring & Unrestricted-Width CNF-SAT
While Theorem 1.1 proves an asymptotic $2^{o(n)}$ runtime for fixed $k$, practical benchmarking on **3-Coloring** and **Unrestricted CNF-SAT** (where clause width $w_e$ can reach $n$) faces two obstacles: (1) variable/degree blowup when reducing long clauses or vertex colors, and (2) the large $(n+m)^{O(1)}$ prefactor of conditional-expectation schedule search and big-integer Bareiss elimination. Select a problem class, adjust instance scale $n$, and toggle the **5 Geometric-Algebraic Accelerators** below to inspect their impact on the empirical crossover point $n^*$ and effective base $b_{\text{eff}}(n) = 2^{\text{Ops}/n}$.
From $3v$-Variable 3-CNF to $2v$-Wire $C_3$ Chiral Tensor Contraction
In Corollary 1.3 of `paper.pdf`, proper 3-coloring on $G=(V,E)$ is encoded using $3v$ Boolean variables $x_{u,c}$ ($c\in\{1,2,3\}$) and $4v+3e$ clauses of width $\le 3$. While this proves $2^{o(v)}$ time, tripling the variable count ($n = 3v$) cubically inflates the subexponential base for moderate $v$. Our benchmark reduction compresses this in three steps:
At the canonical 3-Coloring phase transition ($c = 2e/v \approx 4.69$, so $e \approx 2.345 v$ has bounded average degree), the bipartite tensor network of **Theorem D.1** has $s = O(v)$ wires and maximum degree $O(1)$, bypassing both the Appendix A sparsification tree and the Section 6 Beigel-Tarui parity lift to run in **$v^{O(1)} 2^{O(v/\sqrt{\log v})}$** time!
Bypassing Tseitin Blowup & $2^{w_e}$ Monomial Expansion via Rank-1 Group Factoring
When clause width $w_e = |C_e|$ is unbounded (up to $n$), naive Tseitin splitting adds $O(m n)$ auxiliary variables, and Section 6's $\mathbb{F}_2$ expansion $v_e(x) = \prod_{j=1}^{w_e}(x_{i_j}\oplus b_j)$ explodes into $2^{w_e}$ monomials. We eliminate both bottlenecks via three exact reductions:
Because the clause violation indicator $v_e(x) = \prod_{g=1}^r u_{e,g}(x_g)$ is already a **rank-1 product of local group functions**, any clause of width $w_e = n$ collapses inside an $(r,\tau,D)$-design to a clause of width **at most $r$** over the group indicators $u_{e,g}$ with **zero cross-group linear measurements $C_{t,g}$**!
8C. Preregistered Experimental Benchmark Protocols (Protocols D, E & F)
Designed under the Stage 1 Registered Report methodology of Hoard (2026d Β§8): fixed instance generators, baseline solvers, primary scaling metrics, and falsifiable decision rules.