Information hidden from an observer is not the same as information destroyed by the full evolution. SMM/DTT can address the distinction by treating each completion as an isometry from retained source memory to new memory and radiation, then deriving its entropy, energy and causal behavior.
Restricted knowledge and quantum information are different
Ordinary coarse-graining forgets details that remain present in a more complete description. A quantum information-loss problem is stronger: different initial pure states may evolve into final mixed states without any remaining physical system that retains their distinctions. Hawking’s 1976 analysis formulated this issue for gravitational collapse and evaporation.
Thermal-looking radiation at an intermediate time is compatible with globally unitary dynamics if it is entangled with the remaining black hole. The decisive question concerns the entire evaporation process. If the black hole disappears, where are the correlations that distinguish the original states? A temporal theory must answer this quantum-channel question, not merely say that the inner source knows more than an outward observer.
The horizon’s causal role also matters. An explanation should recover why outward observers cannot use interior data for arbitrary signalling, while specifying how the final radiation can carry the information permitted by the full dynamics.
A source-memory formulation of SMM/DTT
The source picture developed in Chapter V of Duality of Time motivates a distinction between the process generating a spatial configuration and its completed observable record. Extending that distinction to black holes suggests retaining the formation data that an outward description no longer resolves. This article develops a new research model from that idea; it does not attribute a completed black-hole calculation to the original book.
One generative source can have a state space with many distinguishable states. Ontological unity does not mean that its physical memory has only one possible value. The source must retain enough distinctions to describe different infalling states, while its six nested inner stages update those relations and the seventh completion releases the next accessible record.
A crucial requirement is that re-creation preserve distinctions or transfer them somewhere. Resetting every internal state to the same state at the end of a cycle would erase information unless the missing distinctions had already been exported into other physical degrees of freedom.
A precise completion map
Let HB,n describe retained black-hole source memory after n emissions, and Hr,n+1 the next radiation subsystem. A coherent completion step can be modeled by
The isometry condition preserves inner products, so distinguishable input states remain distinguishable in the full output. Tracing over the retained memory can nevertheless leave the radiation mixed. Repeating the steps preserves the total quantum information. If the final memory has dimension one, the full map is an isometric encoding of the initial state into the final radiation.
This is a rigorous conditional result. It identifies a sufficient information-preservation property, but does not derive the map from gravity. A SMM/DTT construction must obtain Vn from its autonomous inner dynamics, including the clock and energy changes, rather than impose a freely chosen information-preserving circuit after describing the horizon.
A small example with a Page-shaped curve
Consider four source qubits and a fixed unitary that prepares two Bell pairs from one chosen pure input. Emit one member of each pair first, followed by their two partners. After k emissions, the entropy of the emitted subsystem is
The first two emitted qubits are individually maximally mixed and entangled with the retained memory. Later emissions transfer their partners outward, restoring a pure final radiation state. No qubit was copied, and no entropy had to be destroyed. The example uses a unitary followed by successive reassignments of subsystems, so the complete map remains reversible for arbitrary inputs; the displayed entropy values refer to the specified initial state.
This illustrates the information-transfer logic behind a rising and falling radiation entropy. Page’s analysis connects unitary evaporation with information distributed through radiation correlations. The four-qubit example does not determine a gravitational temperature, emission spectrum or horizon geometry.
The gravitational conditions on source memory
Black-hole entropy provides a physical scale for the number of accessible states. The Bekenstein–Hawking relation has dimensionless entropy SBH/kB = Ac³/(4Gℏ). A temporal source model should explain how its retained state count depends on horizon area. The relation follows the development from Bekenstein’s entropy proposal to Hawking’s thermal-radiation calculation; it is not supplied simply by counting arbitrary internal labels.
Modern gravitational calculations provide another benchmark. Penington’s entanglement-wedge calculation finds a Page-time change in the relevant quantum extremal surface for an evaporating black hole in a specified holographic setting. It demonstrates how semiclassical geometry and fine-grained entropy can be reorganized without equating local thermality with fundamental loss.
The SMM/DTT programme should derive an analogous relation among source memory, completed geometry and radiation access in its own model. An unlimited hidden memory left behind after complete evaporation would constitute a remnant sector, with its own energy, capacity and physical consequences.
A concrete next calculation
A focused model would specify finite source memory, a horizon-like restriction on outward observables, an energy-conserving emission update and a physical completion clock. It would calculate the radiation density matrix, its entropy and its correlations with a reference system initially entangled with the infalling matter. Preservation of that reference entanglement is a stronger check than following a single classical bit.
The key comparison is between two otherwise matched updates: one that exports retained distinctions coherently, and one that resets them. Their different radiation correlations make the proposed mechanism mathematically testable. Recovering an approximate Hawking spectrum, an area-dependent capacity and a causal exterior response from the same update would then turn the information-preserving construction into a substantive black-hole theory.
Books, papers and further reading
- Duality of Time — Chapter V, temporal postulate
Conceptual source for inner generation and outward completion; the explicit black-hole memory model here is a new extension.
- S. W. Hawking, Breakdown of predictability in gravitational collapse (1976)
Published in Physical Review D 14, 2460–2473; formulation of the information-loss problem.
- D. N. Page, Information in black hole radiation (1993)
Published in Physical Review Letters 71, 3743–3746; information in correlations of unitary black-hole radiation.
- J. D. Bekenstein, Black Holes and Entropy (1973)
Published in Physical Review D 7, 2333–2346; black-hole entropy and horizon area.
- S. W. Hawking, Particle creation by black holes (1975)
Published in Communications in Mathematical Physics 43, 199–220; thermal emission and the area-entropy coefficient.
- G. Penington, Entanglement wedge reconstruction and the information paradox (2020)
Published in JHEP 09 (2020) 002; a Page-time transition in a specified evaporating holographic black-hole setting.