Key points are not available for this paper at this time.
Version 4. 7: Part VI: Trans-Generational Continuity of the Singularity and the Singularity Sphere Recent TOE5 work began as an attempt to understand the relationship between ΛIR and a possible black-hole counterpart ΛBH. Surprisingly, the investigation led to the Kretschmann scalar (K), which appears to connect black-hole and de Sitter limits through a common geometric structure. What started as a study of vacuum energy and RG flow gradually evolved into a question about the physical role of singularities. This led to the introduction of the Singularity Sphere and, ultimately, to the hypothesis that the singularity and the Singularity Sphere may represent the same topological point viewed from opposite directions on a closed S³ geometry. The result is a possible framework in which trans-generational continuity is not merely a dynamical process, but a consequence of topology itself. Version 4. 6: Zero II: Fundamental Energy (FE): A Minimal Axiomatic Framework— Background Geometry from Primitive Energy Structure This manuscript presents FE as the pre-geometric background sector of TOE5. The framework is intentionally minimal, based on three primitives: state space, monotonic flow, and projection operators. Cosmological observables are interpreted as emergent projections rather than fundamental inputs. The current version incorporates conceptual refinements developed through the broader TOE5 program and clarifies the role of FE as an infrared background sector. Several open problems remain active and are explicitly identified. Version 4. 5: TOE5 - Λ Scaling and Emergent Cosmological Reconstruction FM–FE Exchange Structure and Infrared Fixed-Point Interpretation TOE5 is a phenomenological cosmology framework based on an FM–FE (Fundamental Matter–Fundamental Energy) exchange structure. The framework reinterprets the cosmological constant Λ not as a direct ultraviolet vacuum-energy sum, but as an emergent infrared reconstruction quantity associated with coarse-grained spacetime geometry and fixed-point behavior. Paper I focuses on Λ scaling, FM stress-energy structure, gravitational boundary conditions, infrared fixed points, and ΛCDM recovery in the IR limit using Planck 2018 cosmological parameters. Version 4. 3: Cosmological Constant We reformulate the cosmological constant Λ as an infrared fixed point of a horizon–entropy renormalization flow defined on a coarse-grained FM/FE cosmological state space. In this framework, Λ is not treated as a vacuum energy parameter but as an emergent eigenvalue of a self-consistency condition between spacetime geometry and entropy partitioning. We define a nonlinear mapping Λ → 𝒢 (Λ) induced by horizon-scale entropy constraints and show that, under boundedness and entropy saturation assumptions, this map admits at least one fixed point. The observed small positive Λ corresponds to a stable infrared attractor of this flow. We summarize this result as a Fixed-Point Closure Theorem for Λ in FM/FE spacetime, replacing vacuum-energy fine-tuning with a geometric–entropic consistency condition. Abstract Part Zero I Self-gravitating systems exhibit persistent structure despite expectations ofhomogenization under entropy maximization. We propose that such systems are morenaturally described by an entropy extremization condition rather than global entropymaximization. The resulting configurations admit non-uniform density and velocitydispersion profiles. By linking the variational formulation to observable quantities, weobtain a scaling relation between velocity dispersion and density. This framework providesa consistent structural interpretation of effective sources compatible with gravitationalfield equations. Part Zero II We propose Fundamental Energy (FE) as the background sector of the TOE5 framework. Un-like the FM sector, which is associated with entropy-extremizing gravitational structure, FE isdefined as a primitive energy-like state space equipped with a monotonic flow and a projec-tion rule. FE is not identified with matter, radiation, or spacetime itself. Instead, it provides the back-ground substrate from which effective geometric observables emerge. The framework is in-tentionally minimal and consists of three primitives: 𝐹𝐸 = (ℋ𝐹𝐸, 𝐹, Π) · ℋ 𝐹𝐸: primitive state space· 𝐹: monotonic flow· Π: projection operatorsAll observable quantities arise as effective projections of the underlying FE dynamics. Part I We call it dark matter because we cannot see it, and dark energy becausewe do not understand it. Both names reflect our limitations rather than thenature of the things themselves. This paper proposes that what isconventionally termed dark matter is more precisely Fundamental Matter (FM) — the basic form of matter, interacting solely through gravity — andthat what is conventionally termed dark energy is more preciselyFundamental Energy (FE) — the basic expansive property of space, symmetric to gravity. We propose that what is conventionally termed dark matter is moreprecisely Fundamental Matter (FM) — the basic form of matter, interactingsolely through gravity — and that dark energy is more preciselyFundamental Energy (FE) — the basic expansive property of space, symmetric to gravity. Both are entropic invariants: FM occupies themaximum entropy state available to gravity-only matter; FE has zeroentropy as a uniform property of space. This entropic inertness is thestructural reason why FM and FE persist across cosmological transitionswhile ordinary matter does not. The cosmological constant problemdissolves once vacuum energy (matter sector, extractable via Hawkingradiation) and FE (spatial property, not extractable) are recognised ascategorically distinct — reducing the 10¹²² discrepancy to a boundarycondition, not a fine-tuning problem. Baryonic asymmetry emerges as astructural consequence of the inherited inhomogeneous FM distribution. The fine structure constant α is derived from the FM/FE energy inventory asa fixed-point relation, recovering the observed value to seven significantfigures. The framework makes falsifiable predictions testable with JWST, Euclid, DESI, and LISA. Part II Part I established that Fundamental Matter (FM) and Fundamental Energy (FE) persist across cosmological generations while ordinary matter (OM) cycles through black holes via Hawking radiation. Part II extends thisframework to the terminal state of the universe. We propose that the finaluniverse — in which all OM has been converted to FM and only a singleclass of black holes remains — satisfies two distinct trigger conditions thattogether initiate a new Big Bang: a geometric condition, in which externalFE volume exceeds internal volume, and a thermodynamic condition, inwhich black hole entropy converges on horizon entropy. These twoconditions constitute a chorus trigger: they need not arrive simultaneously, but both must be met. We further propose that our universe is the firstcosmological generation (n = 0), and derive that OM depletion reachescompletion within approximately 4 generational cycles under f = 0. 844, derived from the observed OM fraction. The framework requires no inflatonfield, no external fine-tuning, and no appeal to structures beyond FM, FE, and classical thermodynamics. Fine-tuning is not imposed from outside —each generation, the universe converges upon its own constants throughthe self-referential structure of FM, FE, and α. Part III The terminal universe satisfies two conditions — geometric andthermodynamic — that together constitute the Chorus Trigger. Whatfollows is the mechanism by which the universe closes, contracts, and opens again. We propose that Fundamental Energy (FE), identified as the Hawking quantum vacuum at the cosmologicalhorizon, undergoes pair production at the de Sitter boundary; atthe cosmological horizon, pair production manifests vacuumproperties (matter and antimatter), converting to FM state —which then enters the trans-horizon layer, a physicallyinaccessible but physically real space admitting only gravity-only matter. Following the Chorus Trigger, this trans-horizon FMundergoes Horizon Equivalent Exchange (HEX): FM traverses thecosmological horizon inward — material state preserved, noentropic exchange — returning to Domain 1. As trans-horizon FMreturns, the accessible FE volume shrinks. The de Sitter horizoncontracts. The approximately 18 causally isolated terminal blackholes — one per de Sitter bubble, N = (Rₒbs / RdS) ³ ≈ 18 —converge toward a black-hole-dominated asymptotic terminalconfiguration. The Big Bang is not an explosion into externalspace. It is an inward event — an entropic seeding phasetransition into pre-existing, pre-defined space between theterminal black-hole-dominated state and the contracting cosmichorizon. The universe closes upon itself, and opens again. Thisis the universe's memory. Part IV Two Languages, One UniverseWhat is conserved, and how it is carried — these are not thesame question, but they share the same answer. The thermody-namic argument traces what persists across the cosmologicaltransition. The information-theoretic argument traces themechanism by which it is delivered. The two formalisms are in-dependent in their language and mutually compatible in theirconclusions. Together they describe a universe that does notforget. > **What is the minimal information structure from which geom-etry, entropy, and cosmological vacuum emerge? **> We propose it is FM — Fundamental Matter. The argument un-folds below. One proposition anchors everything that follows: > **There is one ledger. ** The black hole event horizon andthe cosmological horizon do not maintain separate accounts —they are two faces of the same boundary, governed by the samerelation: S = A / 4Gℏ. FM moves within this unified ledger asthe sole carrier that traverses both boundaries without en-tropic mediation. Once this is established, the rest is not achoice. It follows. Part V TOE5 is a phenomenological emergent-cosmology framework in whi
Jaehwa Hong (Fri,) studied this question.