A Unified Algorithm for Codimension-2 Spacetime Reductions

Dimensional Collapse Theory: A Unified Algorithm for Codimension-2 Spacetime Transitions Release Tag: v1.0-dct_algorithm Overview This record archives the first public release of the manuscript “Dimensional Collapse Theory A Unified Algorithm for Codimension-2 Spacetime Reductions”.The paper consolidates the full DCT algorithm that maps a Lorentzian $D$-dimensional spacetime with a horizon into a $D-2$-dimensional child spacetime via: - Ledger core formation at a universal curvature scale,- Null-Pair Removal (NPR) as the microscopic snap mechanism,- Infinity Bits $(W,X,Y,Z)$ as the discrete record written on the ledger, and- A global master evolution equation that tracks how entropy, curvature, and dimension evolve across snaps. The manuscript unifies DCT components (ledger placement, NPR gate, Infinity bits, SU(5)/HaPPY-inspired ledger structure, radion phenomenology, etc.) into a single coherent algorithmic picture of how spacetime “steps down” $D \to D-2$ while remaining consistent with horizon thermodynamics and an emergent Newton constant. Main contributions of this paper This release focuses on the algorithmic backbone of DCT rather than on any single physical application. The key contributions are: 1. Global D→D−2 evolution algorithm - Formulates a step-by-step algorithm for dimensional collapse: $$ \text{(horizon + entropy flux)}\;\Rightarrow\;\text{NPR snap}\;\Rightarrow\;\text{ledger write}\;\Rightarrow\;\text{D→D−2 transition}. $$ - Packages this into a master evolution equation that tracks the state of the parent spacetime, the child spacetime, and the ledger across snaps. 2. Ledger core and horizon structure - Treats the ledger surface at radius $r_{\mathcal{L}} = \sqrt{\mathcal{T}}R_S$ as the true “black hole interior”, distinct from the event horizon $R_S$. - Uses the curvature invariant $$ \mathcal{I}(r) = r^4\,K(r) $$ to fix the ledger location by the universal value $\mathcal{I}(r_{\mathcal{L}}) = 48\ln 2$, with $\mathcal{T} = 1/(4\ln 2)$. - Emphasizes that the region between event horizon and ledger core is just vacuum GR; there is no thermal “cavity” with its own independent entropy. 3. Null-Pair Removal (NPR) gate and Infinity Bits - Describes the NPR mechanism as an elementary operation that deletes a null pair in the interior and writes a discrete Infinity nibble $(W,X,Y,Z)$ to the ledger. - Clarifies the roles: - $W$: irreversible, thermodynamic “write/branch” payload bit; - $X,Y,Z$: informational / geometric / gauge metadata bits. - Shows that each NPR event carries a fixed entropy quantum $$ \Delta S_{\text{NPR}} = 4\ln 2,k_B, $$ matching a single ledger cell of area $\mathcal{A}_{\text{cell}} = 4\ln 2\,L_{\mathcal{T}}^{D-2}$. 4. HaPPY / SU(5) inspired ledger structure - Interprets the ledger as effectively a quantum gauge field rather than just a static tensor network picture. - Uses the HaPPY [5,1,3] intuition and SU(5)-like structure to motivate how a single ledger “unit” can couple to multiple Standard Model–like degrees of freedom in the emergent $D-2$ spacetime. - Crucially, in this setup the SU(5)-like degrees of freedom live on the ledger and the dangerous leptoquark $X,Y$ channels are sequestered from low–energy 4D bulk physics, so the construction naturally avoids the catastrophic proton–decay rates that rule out conventional 4D SU(5) GUTs. - Discusses how repeated NPR events + ledger updates give rise to an effective lower-dimensional field theory. 5. Corrected picture of Hawking radiation and BH growth - Reinterprets Hawking-like radiation as waste heat of snap events, not as autonomous evaporation of a finite “cavity.” - Emphasizes that Hawking radiation in this framework is evidence of ongoing growth and entropy processing, not of complete black hole evaporation. - Discusses implications for primordial black holes (PBHs), null results in gamma-ray burst searches for PBH evaporation, and possible femtolensing signatures. 6. Cosmological and higher-D extensions - Outlines how the same $D\to D−2$ algorithm applies in cosmological settings (e.g. dimensional collapse at early times) and in higher-dimensional parents. - Adopts the canonical DCT stance that all physically realized spacetimes are Lorentzian $(-,+,+,\ldots,+)$ prior to NPR, and that NPR removes the timelike and radial directions to leave a Euclidean child ledger (with a new time direction later selected). Structure of the manuscript The paper is organized as follows (section numbering may vary slightly depending on the final compiled version): 1. Introduction and motivation - Conceptual background of Dimensional Collapse Theory. - Why an explicit **algorithmic formulation** is needed: to track $D\to D−2$ transitions, NPR events, and ledger updates in a single unified story. 2. Ledger core and geometric setup - Definition of the ledger core radius $r_{\mathcal{L}}$. - Curvature invariants and the universal value $\mathcal{I}(r_{\mathcal L})=48\ln 2$. - Distinction between event horizon and ledger core. 3. Null-Pair Removal and Infinity Bits - Definition and properties of the NPR operation. - Role of $(W,X,Y,Z)$, entropy quantum per snap, and ledger cell area. - Relation between NPR events and dimensional collapse steps. 4. Ledger as a quantum gauge field (HaPPY/SU(5) layer) - Discussion of tensor-network intuition versus true ledger dynamics. - SU(5)-like structure as a toy model for how a ledger unit couples to Standard-Model-like data. 5. Master evolution equation - Construction of the unified evolution equation governing: - parent spacetime geometry, - ledger state, - child spacetime geometry. - Interpretation of the equation as a “DCT Schrödinger + bookkeeping” law for $D\to D−2$ transitions. 6. Phenomenology and consistency checks - Black hole thermodynamics, Hawking waste heat, PBHs, and dark matter. - Consistency with null results on PBH evaporation bursts. - Qualitative discussion of gravitational-wave echoes / near-horizon structure as possible probes of a finite-radius ledger. 7. Higher-dimensional and cosmological $D\to D−2$ chains - Conceptual extension of the algorithm to higher-dimensional parents. - Outline of how repeated NPR and ledger updates could drive an early-universe dimensional cascade. 8. Summary and outlook - Recap of the DCT algorithm as a complete pipeline: $$ \text{GR horizon + flux} \to \text{NPR snap} \to \text{ledger write} \to \text{D→D−2 child} $$ - How this paper connects to other DCT works (interior surface, NPR gate, neutrino mass, etc.). - Open problems and next steps (e.g. more detailed SU(5)/Standard Model mapping, numerical toy models, observational tests). Relation to other DCT releases This algorithm paper fits into the wider DCT series as: - A unifying backbone that stitches together: - the earlier ledger placement / interior surface paper (Transdimensional constant $\mathcal{T}$ and $r_{\mathcal{L}})$; - the NPR mechanism & Infinity Bits work; - follow-up phenomenology (e.g. radion-induced neutrino mass, cosmology, PBHs). It is intended to be the reference document for how all the DCT components operate together as one algorithm.