# Application-driven Design Exploration for Dense Ferroelectric Embedded Non-
volatile Memories

Mohammad Mehdi Sharifi∗∞, Lillian Pentecost†∞, Ramin Rajaei∗, Arman Kazemi∗,
Qiuwen Lou∗,
Gu-Yeon Wei†, David Brooks†, Kai Niψ, X. Sharon Hu∗, Michael Niemier∗, Marco
Donatoγ
∗University of Notre Dame, †Harvard University, ψRochester Institute of
Technology, γTufts University, ∞equal contribution

###### Abstract

The memory wall bottleneck is a key challenge across many data-intensive
applications. Multi-level FeFET-based embedded non-volatile memories are a
promising solution for denser and more energy-efficient on-chip memory.
However, reliable multi-level cell storage requires careful optimizations to
minimize the design overhead costs. In this work, we investigate the interplay
between FeFET device characteristics, programming schemes, and memory array
architecture, and explore different design choices to optimize performance,
energy, area, and accuracy metrics for critical data-intensive workloads. From
our cross-stack design exploration, we find that we can store DNN weights and
social network graphs at a density of over $\mathbf{8MB/mm^{2}}$ and sub-2ns
read access latency without loss in application accuracy.

††publicationid: pubid: 978-1-6654-3922-0/21/$31.00 $\copyright$2021 IEEE

##  I Introduction

Data-intensive applications such as deep neural networks (DNNs) and graph
analytics have emerged as dominating workloads in the current computing
landscape and have influenced many research efforts and advances in computer
architecture in the past few years. Specialized hardware architectures deliver
outstanding performance and computational efficiency. However, the size and
complexity of critical workloads continues to outstrip available on-chip
memory capacity, making data movement and efficient on-chip storage an
outstanding challenge in scaling these applications.

Embedded non-volatile memories (eNVM) offer a promising alternative to
traditional on-chip SRAM [1]. Smaller memory cell size and the ability to
store multiple bits in a single memory cell can translate to a denser array
implementation, which can be optimized to store entire DNNs on-chip [2, 3]. In
addition to storage density improvements, non-volatility increases energy
efficient due to lower leakage power and the ability to retain information in
power-off state for intermittent computing. The advantages of eNVMs are often
countered by lackluster write performance and reduced reliability, which may
be exacerbated with multi-level cell programming.

In today’s varied application space, meeting different memory read and write
access patterns requires careful consideration of eNVM memory characteristics.
An application-driven approach can provide a clear path towards design
optimization targeting a specific application and coincidentally offer insight
regarding which device-level specifications such as reliability, endurance,
write performance, and cell area should be further improved to serve a wider
set of applications.

Figure 1: (a) FeFET device; (b) Transfer characteristics of a FeFET device in
8 different resistance levels for 3-bit-per-cell programming.

In this work, we advance the study of a particularly compelling eNVM proposal,
Ferroelectric FET (FeFET), by modeling and quantifying the impact of device-
level design choices such as size and programming scheme on the accuracy of
target workloads. FeFET memories promise dense storage with competitive read
characteristics, but their viability and achievable density in a realistic
application use case, such as DNN inference or graph analytics, has not yet
been explored.

We begin by evaluating the device-to-device (D2D) variation for different
FeFET cell areas using model presented in [4]. Based on these preliminary
device-level results, we investigate and develop custom programming schemes to
increase the programming reliability for single-level cell (SLC) and multi-
level cell (MLC) configurations. To quantify the relationship between
resulting device characteristics and memory array performance, we extend the
NVSim tool to support a FeFET cell model [5]. Additionally, we develop a FeFET
fault model based on the variability both at the memory and sensing circuitry
level, and we extend an application-level fault injection framework to
evaluate the impact of FeFET device size and programming scheme on application
accuracy.

This paper provides the first comprehensive design exploration and evaluation
of FeFET-based eNVMs for realistic use-cases such as DNN weight storage and
graph analytics. More specifically, we make the following contributions:

  * •

We perform a device-level study, modeling the effects of memory cell size and
programming schemes on the fault rate, energy, and write latency for single
and multi-level (2-bit and 3-bit) cell implementations;

  * •

We evaluate array-level implications of different FeFET cell characteristics
by modeling and optimizing memory architecture and sensing circuit to minimize
read errors;

  * •

We introduce a FeFET fault model and an application-level fault-injection
study to quantify the application accuracy across cell-level and memory-level
characteristics for a collection of data-intensive workloads.

##  II Background

###  II-A Ferroelectric FET memories

The FeFET device (Figure 1(a)) is made by replacing the normal gate dielectric
in a MOSFET with a ferroelectric layer [6]. For this reason, FeFET devices can
be easily integrated in existing CMOS processes, especially when high-$k$
dielectrics such as hafnium oxide can be used as the ferroelectric layer [7].
The polarization of the ferroelectric layer can be controlled by applying
different voltages to the gate of the device, where this polarization sets the
threshold voltage ($V_{th}$) of the device. Recent work [8, 9, 10] uses FeFETs
for ultra-dense, low-leakage, and fast memories. This work employs 1FeFET AND
arrays (Figure 2(a), where the source lines (SLs) connect the elements along
the columns), which are more promising than the other types of FeFET memories
(e.g., NOR arrays, where the SLs connect the elements along the rows) for two
reasons: first, the cell size is equal or even smaller than NOR arrays;
second, the parallel bit-lines and source-lines reduce the write disturbance
[11]. Note that the write disturbance can be further mitigated by applying a
half-bias scheme (e.g., applying $V_{write}/2$ to deselected cells) [8].

Figure 2: (a) AND array memory architecture; (b) Customized multi-level
sensing circuit for four programmed (2-bit) MLC.

###  II-B Multi-level NVM storage

Storing multiple bits in a single memory cell is desirable for increasing
storage density, and has been demonstrated using many eNVM devices, including
resistive random access memories (ReRAM), phase change memories (PCMs),
charge-trap transistors (CTT), and FeFETs [1, 12]. In order to discriminate
among different programmed levels, the memory uses analog to digital
converters (ADCs). The ADCs translate each programmed $I_{D}$ target level to
the corresponding binary word [13]. One key challenge in MLC storage is the
higher susceptibility to D2D variations, making each programmed level less
reliable [9]. In addition, the higher area and power overhead of the ADCs
discriminating between more current levels can limit the relative benefits of
MLC implementations compared to single-level-cell (SLC) storage.

###  II-C Target applications

We evaluate the impact of FeFET reliability characteristics on target
applications that are (1) data-intensive in terms of required capacity and
read bandwidth and (2) have infrequent or highly batched write accesses. In
embedded and mobile SoCs, the high density and compelling efficiency of FeFET
memories could be crucial in enabling on-device or otherwise power-efficient
execution of these critical applications, and performance will not be
debilitated by potentially long writes.

Deep Neural Networks (DNNs) for vision tasks and natural language processing
(NLP) are well-studied driving applications that have demonstrated resilience
to MLC eNVM storage [2]. DNN inference performance and power-efficiency
benefits significantly from increased on-chip memory density, particularly for
storage of weight parameters that are infrequently updated. Thus, we evaluate
the viability of FeFET MLC memories for two representative DNN workloads:
ResNet18 for image classification (CiFar10), and the ALBERT transformer-based
model for natural language understanding (MNLI) [14, 15]. Another critical
category of data-intensive workloads are those that operate on large graphs
[16], such as search tasks on social network graphs, though resilience of
graph formats to MLC eNVM storage has not been addressed in prior work. Thus,
we additionally evaluate the viability of FeFET MLC memories for storing
social network graphs with sample graphs representing Wikipedia article voting
patterns and anonymized social circles from Facebook [17].

Figure 3: ADC quantized levels with variation as a Gaussian function with
3$\sigma$ deviation of 5% and target currents for 3-bit FeFET MLC.

##  III Methodology

###  III-A FeFET multi-level current model

FeFETs, like other emerging devices, suffer from D2D variation, an issue that
severely limits their scalability both in terms of physical dimension and
multi-level programming. In our study, we account for D2D variations by
employing a stochastic model of polarization switching in FeFET devices [4].
The model uses Monte Carlo sampling on independent polarization domains in the
ferroelectric layer. In doing so, it captures the essential behaviors required
to perform scalability and reliability studies on FeFET memories, including
(i) D2D variation as the cell size changes (different number of $10nm\times
10nm$ domains); (ii) stochasticity of domain switching; and (iii) the
accumulation of domain switching probability when a train of pulses are
applied to the gate of the FeFET device. Also, it is worth noting than this
model has been experimentally validated. Using this model, we can effectively
project the impact of write schemes and device size on D2D variation, and,
consequently, the impact of D2D variation on application level accuracy.

###  III-B Memory array architecture model

We extend NVSim [5] to extrapolate our device-level characterization study to
memory array architecture. NVSim does not natively support FeFET memory cells,
so we add a customized memory cell definition. Moreover, we modify NVSim to
estimate the costs of MLC sensing circuitry. These additions are discussed in
the following subsections.

####  III-B1 NVSim MLC FeFET model

We integrate a new memory cell definition using the energy, delay, and area
results collected from SPICE simulations. Our evaluation considers two
programming schemes, which are discussed in detail in Section IV-A. The write
energy and latency for the write-verify scheme use the average number of set
and reset programming pulses computed by sampling 1500 FeFET cells for D2D
variation using the model discussed in III-A, in addition to estimated energy
and latency due to write circuitry.

Figure 4: (a) Proposed write-verify programming scheme; (b) Example of pulse
by pulse level tuning process of an FeFET device into the target range.

An additional change to improve model fidelity is that FeFETs fabrication
requires adding a ferroelectric layer on top of a traditional MOSFET gate, so
the gate equivalent oxide thickness is increased and, therefore, the gate
capacitance. We captured this in NVSim by setting FeFET gate capacitance
1.73$\times$ bigger than the default CMOS gate capacitance.

####  III-B2 Sensing circuit design

For a 1-bit read operation, we employ a simple voltage-based sense amplifier,
which we characterize in SPICE. The same sense amplifier model is employed to
build a parallel sensing scheme for MLC operation. In a parallel sensing
scheme, the current from a memory cell is compared against $2^{n}-1$ reference
levels and returned as an $n$-bit binary word. Figure 2 depicts the circuit
schematic of the sensing circuit for a 2-bit read operation. The operation is
similar to a Flash ADC and is similarly affected by D2D variation, which in
turn has an impact on memory reliability. We model the effects of D2D
variations on transistor dimensions, resistance values, etc., as a Gaussian
function with a 3$\sigma$ deviation of 5% [18, 19]. As a result, the quantized
levels show variability proportional to the threshold currents. Based on this
constraint, we propose spacing the MLC programming currents such that the
sensing threshold distributions are equally spaced (Figure 3). This approach
uses the extra margins in low-current levels to distribute the read errors due
to sensing threshold shifts more evenly across the entire programming window.
The energy, latency, and area for the resulting design is incorporated in
NVSim to model sensing at the memory array architecture level.

Figure 5: (a) Current distribution of 2-bit FeFET MLCs with single pulse
programming for 1500 cells; (b) Current distribution of 2-bit FeFET MLCs with
write-verify programming for 1500 cells.

###  III-C Cross-application fault injection framework

Evaluating device non-idealities at the application level requires a balance
of fault model accuracy and simulation performance. As the focus of this work
is on DNN and graph applications, we leverage two popular Python packages,
namely PyTorch [20] and SNAPpy [21] to execute representative workloads.
Previous work has explored the evaluation of faults in DNN models [22] and
extended the study to eNVM errors in DNN applications [23]. We extend existing
MLC eNVM fault modeling to simulate FeFET memories based on the model
discussed in Section III-A and ADC quantized level variations. Moreover, we
use a general input interface to process parameters from both DNN and graph
workloads. For each target application, the fault injection tool reads in the
application data to be stored in FeFET-based memory and applies a quantization
transform followed by MLC encoding based on the selected configuration.

Stored data values are assigned to FeFET current levels, and we sample the
current based on the FeFET Monte Carlo model. The currents computed in this
first sampling process are then compared against current sensing thresholds
which are also sampled based on ADC variations. The sensed currents are then
converted back to quantized values and used to repeat workload execution and
evaluate the effect of memory faults on the target application.

Figure 6: Shmoo plot of maximum read fault probability as a function of cell
size and bits per cell for single pulse vs. write-verify programming.

##  IV FeFET optimization study

###  IV-A Programming schemes

We consider two approaches for programming multi-level FeFETs, (i) single
pulse; and (ii) write-verify. The single-pulse approach first performs a hard
reset of the device (a -4$V$, $1\mu s$ pulse), followed by one pulse of
amplitude dictated by the target level. While appealing in terms of
simplicity, the single-pulse scheme is not robust to D2D variation for smaller
FeFET cells, as evident in the overlap of resulting current distributions in
Figure 5(a) for 50-domain cells.

The write-verify scheme proposed in this work (Figure 4(a)) helps combat D2D
variations by producing tighter level distributions. After a hard reset, cells
are supplied with shorter $100ns$ fixed-amplitude pulses. After each pulse,
the device current is measured against a reference to determine whether the
target current has been exceeded, and a soft reset pulse of the same duration
is applied to correct any overshoot. As shown in Figure 4(b), a soft reset
pulse does not return the device to its initial state because of the shorter
applied pulse. This gives us more control to set the device to a target range.
A similar strategy has been applied to RRAM devices [24], but, to the best of
our knowledge, the impact of this approach has not been studied for FeFETs.
Additionally, our use of fixed-amplitude pulses reduces the overhead of the
write circuitry. Conversely, other emerging devices such as ReRAM and PCM
require variable amplitude and width in programming pulses to reach the
desired current targets [24, 25].

Our evaluation shows that only a small subset of devices (less than 0.1% for
200-domain cells) fails to reach the target range after 10 soft reset cycles.
Therefore, we fix the maximum number of soft resets to prevent prohibitively
long and energy-consuming programming sequences. The programming sequence is
terminated either if the measured device current is within a target range, or
if the number of applied soft reset pulses reaches its maximum count (Figure
4(b)).

Figure 5 shows the current distribution of 1500 2-bit FeFET MLCs programmed
using single-pulse and write-verify programming schemes. Figure 5(a) shows
that there are large overlapping areas between different levels if the single-
pulse scheme is used. The level distribution overlap worsens when the number
of domains is reduced from 200 to 50. Figure 5(b) shows that using write-
verify scheme, overlaps can be mitigated for 200 domain cells. However, even
with the write-verify scheme, there is some overlap in 50 domain cells.

Figure 7: For a 4MB capacity array with fixed optimization target (read
energy-delay product) and word width (64b), we observe varying storage density
and performance for different cell sizes and programming choices.

###  IV-B Memory design trends

For each of the programming schemes discussed in Section IV-A, we carefully
consider the fault characteristics of the FeFET memory as we vary the number
of programmed levels per cell and the cell size. The highest resulting inter-
level fault rate per cell design and programming scheme is shown in Figure 6.
Our fault model additionally captures the asymmetry of inter-level fault rates
(i.e., when it is more likely for the programmed level to be mis-read as a
lower level than a higher one), which is increasingly apparent in 2-bit and
3-bit programming. Thus, the raw maximal fault rate alone does not capture the
complexity of the potential impact of MLC FeFET behavior on application
accuracy.

The resulting memory array storage density and performance characteristics for
both programming schemes and across cell sizes is shown in Figure 7. We note
that for a fixed capacity and NVSim optimization target, 2-bit storage
provides strictly better density and read energy independent of programming
choices. Though write-verify and MLC programming increase array write latency
(Figure 7, bottom left), we highlight that the compelling read characteristics
and impressive storage density inspire us to investigate the viability of
these memories for critical, data-intensive workloads.

As discussed in Section II-C, there are multiple classes of important
workloads that can be tolerant of long write latency and would benefit
immensely from increased storage density and efficiency. However, these
workloads may exhibit different resilience to MLC fault characteristics. In
light of the relatively high error rates for MLC programming across cell size
shown in Figure 6, it is critical that we evaluate application-level
implications to identify the densest allowable storage scheme without
degrading application accuracy.

Figure 8: Fault injection studies are performed per cell size and programming
scheme to evaluate resulting application error, as measured by relative
degradation from baseline accuracy. The minimum cell size that keeps relative
error below the acceptable threshold dictates densest storage per scheme.

##  V Application-level evaluation

###  V-A DNN Weight Storage

We evaluate two DNN workloads which differ both in terms of target application
and eNVM utilization. The first is image classification using ResNet18. In
this case, we consider a system in which the full model is stored in the FeFET
memory. Figure 8, left, shows how application error relative to baseline
classification accuracy increases significantly as cell size decreases across
programming schemes. For 1-bit, single-pulse programming (no verify), the
minimum FeFET cell size that preserves inference accuracy is 300 domains.
Introducing the write-verify scheme allows the memory cell size to be scaled
more aggressively, with 1- and 2-bit storage displaying minimum cell sizes of
50 and 150 domains, respectively.

The second DNN we consider is natural language inference running on ALBERT.
BERT-based DNNs are often re-used across multiple tasks (e.g., translation vs.
sentiment analysis), so we propose to store the shared embedding parameters in
FeFET memory, while the task-specific portion of the network is stored
separately, an approach that has proven successful in maximizing energy
efficiency for multi-task DNN inference [3]. In this case, the 1-bit, single-
pulse configuration requires 200 domains, while for the write-verify cases,
1-, 2-, and 3-bit configurations can be scaled down to 50, 150, and 150,
respectively. Compared to the relative vulnerability of 3-bit MLC for
ResNet18, partitioning ALBERT to store more vulnerable parameters in SRAM
provides the opportunity for more aggressive scaling in terms of FeFET cell
size.

TABLE I: Summary of minimal cell size in number of domains per programming scheme and bits per cell (BPC) without application accuracy degradation, as discussed in Section V. BPC | Programming | ResNet18 | ALBERT | Wiki | Facebook
---|---|---|---|---|---
1 | Single Pulse | 300 | 200 | 250 | 250
1 | With Verify | 50 | 50 | 50 | 20
2 | With Verify | 150 | 150 | 150 | 150
3 | With Verify | 400 | 150 | 400 | 200
TABLE II: Summary of best programming schemes and full memory array characteristics provisioned per workload. Workload | Optimal Scheme |  | Storage
---
Requirements
| # Cell
---
Domains
| Array Area
---
($mm^{2}$)
| Read Latency
---
($ns$)
| Read Energy
---
($pJ$ / $bit$)
| SET Latency
---
($\mu s$)
| SET Energy
---
($pJ$ / $bit$)
ResNet18 | SLC with Verify | 24MB | 50 | 1.686 | 1.866 | 0.461 | 1.47 | 0.745
ALBERT | MLC2 with Verify | 4MB | 150 | 0.313 | 1.20 | 0.189 | 1.80 | 0.438
Wikipedia | MLC2 with Verify | 6MB | 150 | 0.682 | 1.268 | 0.278 | 1.80 | 0.422
Facebook | MLC2 with Verify | 2MB | 150 | 0.241 | 0.976 | 0.169 | 1.80 | 0.234
SRAM (16nm) | Reference Point | 4MB | N/A | 3.9 | 1.3 | 0.5 | 0.001 | 0.5

###  V-B Social Network Graph Storage

To determine the viability of different FeFET cell designs and programming
schemes for graph analytics tasks, we measure the average impact to the
accuracy of a set of breadth-first search queries on the example graphs as a
proxy for maintaining network structure and resilience to faults across many
integral graph processing kernels [16]. We store input graphs described in
Section II-C as adjacency matrices and perform fault injection to quantify
resilience to MLC storage. The relative error increase for the Facebook graph,
for example, (Figure 8, right), demonstrates workload-specific sensitivity to
FeFET fault characteristics distinct from ResNet18 in terms of allowable cell
size. The minimum cell size per workload and per programming scheme is
summarized in Table I.

Due to the varying sparsity and structure of the two graph workloads studied,
we observe different fault resilience in the 3-bit configuration, and the
Wikipedia voting graph degrades in terms of average query accuracy when using
a cell with fewer than 400 domains. Each mis-read value in the adjacency
matrix corresponds to an erasure or erroneous addition of a connection between
two graph nodes, and it is possible that each mis-read value has a
disproportionate impact on a graph that is less clustered and more sparsely
affiliated (as in the Wikipedia graph) than one with more disparate, strongly
connected social circles (as in the Facebook graph), but further analysis is
required to justify this argument.

###  V-C Application-Provisioned FeFET Arrays

Figure 9: For an example workload (ALBERT, 4MB capacity), we show key
characteristics for FeFET memory arrays with acceptable application accuracy
across programming schemes and NVSim optimization targets. From these, we
select optimal array configurations summarized in Table II that minimize array
area, read latency, and read energy per access.

Though all of the examined workloads are amenable to MLC FeFET storage, in
order to derive the most efficient FeFET storage solution per application, we
must consider the achievable zero-accuracy-loss storage density in conjunction
with memory array performance, area, and energy characterization. For example,
we find that SLC FeFET under the write-verify scheme can be safely used down
to a cell size of 50 domains or smaller (Table I), so this programming scheme
consistently provides highly dense, zero-accuracy-loss memory. This is
highlighted for an example application (ALBERT) in Figure 9.

Figure 9 also demonstrates that 2-bit storage is a compelling design in terms
of density and read characteristics, with 3-bit showing competitive
performance in all metrics, though not providing the pareto-optimal
configuration highlighted in Figure 9 and summarized for each workload in
Table II. The storage density achieved here is a full order of magnitude
higher than that of 16nm SRAM, in which a 4MB array requires about 4$mm^{2}$
according to NVSim evaluation, cited as a reference point in Table II.
Similarly, the observed read latency and energy per bit tends to match or
improve upon SRAM. We repeat analysis of area, latency, and read energy per
bit of provisioned memory arrays per-application to identify FeFET memory
arrays that minimize latency and energy. In this way, the FeFET arrays
highlighted in Table II will cause minimal impact to workload execution time
while providing high density, energy-efficient embedded storage.

It is important to note that these design points are not fully optimized,
which indicates that even higher density could be achieved with architectural
enhancements [2]. While these optimizations are outside of the scope for this
paper, we want to highlight two possible additions. First, incorporating
lightweight error mitigation or error correction strategies could enable use
of even higher density storage solutions, such as 3-bit MLC with smaller cell
sizes. Previous work considering fault-prone MLC eNVM storage unveiled that
hand-tuned design studies can maximize total storage efficiency. Second, more
sophisticated programming schemes (e.g., increasing pulse amplitudes [24])
could be used to further tighten the current distribution of MLC FeFET
configurations. However, more complicated programming schemes would add area,
latency, and energy overhead due to write circuitry.

##  VI Conclusion

This work is a demonstration of the viability of MLC-programmed FeFET eNVMs
with an application-driven evaluation of these memories in light of DNN and
social network graph resilience and memory access patterns. Our methodology
and presented studies provide exposure to the critical trade-offs among write
costs, storage density, fault characteristics, and application impact that
will drive future development of highly-dense FeFET memory solutions. In our
simulations, we expose the fault characteristics and performance implications
of different programming schemes for a set of interesting data-intensive
workloads. Our results show that MLC FeFET storage can be effectively
leveraged for both deep neural network inference and graph processing tasks,
achieving even in the absence of more involved optimizations.

## Acknowledgment

This work was supported by ASCENT and ADA research centers, JUMP centers
cosponsored by SRC and DARPA. This work is also sponsored in part by E2CDA-
NRI, a funded center of NRI, a SRC program sponsored by NERC and NIST.

## References

  * [1] A. Chen.  A review of emerging non-volatile memory (nvm) technologies and applications.  Solid-State Electronics, 125:25 – 38, 2016.  ESSDERC.
  * [2] L. Pentecost, et al.  Maxnvm: Maximizing dnn storage density and inference efficiency with sparse encoding and error mitigation.  2019\.
  * [3] M. Donato, et al.  Memti: Optimizing on-chip nonvolatile storage for visual multitask inference at the edge.  IEEE Micro, 39(6):73–81, 2019.
  * [4] S. Deng, et al.  A comprehensive model for ferroelectric fet capturing the key behaviors.  In Symposium on VLSI Technology. IEEE, 2020.
  * [5] X. Dong, et al.  Nvsim: A circuit-level performance, energy, and area model for emerging nonvolatile memory.  IEEE TCAD, 2012.
  * [6] M. Jerry, et al.  A ferroelectric field effect transistor based synaptic weight cell.  Journal of Physics D: Applied Physics, 2018.
  * [7] T. Mikolajick, et al.  Hafnium oxide based ferroelectric devices for memories and beyond.  In VLSI-TSA, pages 1–2, 2018.
  * [8] D. Reis, et al.  Design and analysis of an ultra-dense, low-leakage, and fast fefet-based random access memory array.  IEEE Journal on Exploratory Solid-State Computational Devices and Circuits, 2019.
  * [9] T. Ali, et al.  A multilevel fefet memory device based on laminated hso and hzo ferroelectric layers for high-density storage.  In IEDM, 2019.
  * [10] S. Dünkel, et al.  A fefet based super-low-power ultra-fast embedded nvm technology for 22nm fdsoi and beyond.  In IEDM, 2017.
  * [11] K. Ni, et al.  Write disturb in ferroelectric fets and its implication for 1t-fefet and memory arrays.  IEEE Electron Device Letters, 2018.
  * [12] F. Khan, et al.  Charge trap transistor (ctt): An embedded fully logic-compatible multiple-time programmable non-volatile memory for high-k-metal-gate cmos technologies.  IEEE Electron Device Letters, 2017.
  * [13] Cong Xu, et al.  Understanding the trade-offs in multi-level cell reram memory design.  In Design Automation Conference (DAC), 2013.
  * [14] K. He, et al.  Deep residual learning for image recognition.  CoRR, 2015.
  * [15] Z. Lan, et al.  ALBERT: A lite BERT for self-supervised learning of language representations.  CoRR, abs/1909.11942, 2019.
  * [16] S. Beamer, et al.  Locality exists in graph processing: Workload characterization on an ivy bridge server.  In IEEE ISWC, 2015.
  * [17] J. Leskovec et al.  SNAP Datasets: Stanford large network dataset collection.  http://snap.stanford.edu/data, June 2014.
  * [18] L.-T. Pang, et al.  Measurement and analysis of variability in 45 nm strained-si cmos technology.  IEEE Journal of Solid-State Circuits, 2009.
  * [19] F. Razi, et al.  Design of an energy-efficient radiation-hardened non-volatile magnetic latch.  IEEE Transactions on Magnetics, 2020.
  * [20] A. Paszke, et al.  Pytorch: An imperative style, high-performance deep learning library.  In NeurIPS. 2019.
  * [21] J. Leskovec et al.  Snap: A general-purpose network analysis and graph-mining library.  ACM TIST, 2016.
  * [22] B. Reagen, et al.  Ares: A framework for quantifying the resilience of deep neural networks.  In Design Automation Conference (DAC), 2018.
  * [23] M. Donato, et al.  On-chip deep neural network storage with multi-level envm.  In Design Automation Conference (DAC), 2018.
  * [24] W. Shim, et al.  Two-step write–verify scheme and impact of the read noise in multilevel rram-based inference engine.  Semi. Sci. Tech., 2020.
  * [25] J. Chen, et al.  A parallel multibit programing scheme with high precision for rram-based neuromorphic systems.  IEEE T-ED, 2020.


The quantizing condition (110) implies that the ways for us to decompose (more
precisely, to look or to consider) the disk into concentric shells and setting
their release parameter $\gamma_{i}$s are not arbitrary. Among various listing
and counting ways, a special one could be implemented as follows, choosing
each $\gamma_{i}$ in such a way that
$\big{(}GM_{i}+\frac{\gamma^{2}_{i}-1}{2}\big{)}$ equals to 1 uniformly, in
this case all $\frac{m_{i}}{\hbar\ell^{-1}}$ are forced to take integers. In
another word, the excitation $\gamma_{i}$ in this case is no long a totally
independent parameter for the microstate definition and their number counting
becomes simply the conventional integer partition of $M_{\mathrm{tot}}$. By
the partition-number formula of Ramanujan we have

$\displaystyle W\equiv\\#\mathrm{microstate(}AdS_{2+1}\mathrm{dustdisk)}=$
(114)
$\displaystyle\exp\big{\\{}2\pi\sqrt{\frac{1}{6}\frac{M_{\mathrm{tot}}}{\hbar\ell^{-1}}}\big{\\}}=\exp\big{\\{}\frac{2\pi\sqrt{2GM_{\mathrm{tot}}\ell^{2}}}{4G}\big{(}\frac{3}{4}\frac{\ell}{G}\big{)}^{-\frac{1}{2}}\big{\\}}$

Obviously, $k_{\scriptscriptstyle B}\log W$ is almost the Bekenstein-Hawking
formula $S_{BH}=A/4G$ except for a numeric factor
$\big{(}\frac{3}{4}\frac{\ell}{G}\big{)}^{-{1/2}}$. Ambiguities brought about
by the subtlety involved in $M_{i}$’s definition and other possible ways of
shell-decomposition and $\gamma_{i}$s’ assigning will not change the basic
fact that we have only finite number of ways to look the dust disk as
concentric shells and setting their initial releasing condition as they
collapse inside the horizon and begin to oscillation. Instead, precise
consideration of those factors may bring us the factor
$\big{(}\frac{3}{4}\frac{\ell}{G}\big{)}^{-{1/2}}$ back as the exact
Bekenstein-Hawking formula requires.

##  V A working example

To help readers more precisely understand physic pictures for our spontaneous
radiation of black holes and their inner structure model, we provide in this
section a working example of spontaneous radiation of an initially $W=6$
degenerating black hole. We will use equation (16) and initial conditions (17)
to follow up evolutions of such an initial black hole. This means that we need
to introduce 1440 coefficients $c_{u^{n}}^{\vec{\omega}}$ to characterize
quantum state of the system and set values for 184 coupling constant
$g_{u^{n}v^{\ell}}$, according equation (12)-(14) to account for transitions
between all possible channels.

When we get all $C_{16}^{2}=120$ similarity factors, we substitute them into
equation (12) to get the coupling constant. As long as these preparation are
ready, we can integrate equation (16)-(17) to get all wave function
coefficients $c_{u^{n}}^{\vec{\omega}}$. With the resulting
$c_{u^{n}}^{\vec{\omega}}$, we can construct all needed density matrix
$\rho^{0}_{ij}$, $\rho^{b}_{ij}$, $\rho^{m}_{ij}$, $\rho^{s}_{ij}$,
$\rho^{1}_{ij}$. We will let $b\sim$big correspond $w=4$ black holes and
illustrate that, tracing out states of the radiation products indeed leads to
a non-diagonal density matrix for $\rho^{b}_{ij}$, although we are working in
black hole inner structure eigenstate basis. Sizes and entropies of the black
hole will be calculated as follows

$\displaystyle
r_{h}(t)=|c_{6}^{\phi}|^{2}r_{6}+|c_{5}^{\vec{1}}|^{2}r_{5}+|c_{4}^{\vec{2}}|^{2}r_{4}+\cdots+|c_{1}^{\vec{5}}|^{2}r_{1}$
(115) $\displaystyle
r_{6}=2GM_{6},r_{5}=2GM_{5},\cdots,r_{1}=2GM_{\mathrm{pl}}$ (116)
$\displaystyle
s(t)=|c_{5}^{\vec{1}}|^{2}s_{5}+|c_{4}^{\vec{2}}|^{2}s_{4}+|c_{3}^{\vec{3}}|^{2}s_{3}+|c_{2}^{\vec{4}}|^{2}s_{2}+|c_{1}^{\vec{5}}|^{2}s_{1}$
(117) $\displaystyle s_{6}=s_{0}=0,s_{5}=5_{1}=1,s_{4}=s_{2}=2,s_{3}=3$ (118)

Variations of this two quantity have been displayed in FIG.2.

###  V.1 Microstate Wave Function

Obviously, our first object should be the wave function of all possible
intermediate state black holes. Among these black holes, the minimal mass one
has degeneracy $W=1$, shell partition and excitation schemes of the form

$\displaystyle W=1,M=1,\\{m_{i}\\}=\\{1\\},\\{M_{i}\\}=\\{1\\}$ (119)
$\displaystyle\\{M_{i}m_{i}\\}{=}\\{1\\},\\{\nu_{i}\\}{=}\\{1\\},\\{z_{i}\\}{=}\\{1\\},\\#{\mathrm{p}{\otimes}\mathrm{e}}{=}1$
(120)

Its wave function is a simple single factor one

$\Psi_{1^{1}}[M(r)]=e^{-x}xL^{1}_{0}(2x),x=Mr\nu/\hbar=r$ (121)

with the average radius calculated easily

$r^{1}_{11}=N_{11}\int_{0}^{\infty}\\!\\!\Psi_{1^{1}}\cdot
r\cdot\Psi_{1^{1}}\cdot 4\pi r^{2}dr=\frac{5}{2}$ (122)

where $N_{11}$ is a normalization factor for $\Psi_{1^{1}}$. It should be
noted that $M=1$ is only lower bound of mass of black hole with $W=1$. Our
inner structure model for black holes does not predict discrete spectrums for
the mass of black hole holes.

For the $W=2$ intermediate state, we can easily determine that

$\displaystyle
W=2,M=\sqrt{2},\\{m_{i}\\}=\\{\sqrt{2}\\},\\{M_{i}\\}=\\{\sqrt{2}\\}$ (123)
$\displaystyle\\{M_{i}m_{i}\\}{=}\\{2\\},\\{\nu_{i}\\}{=}\\{\frac{2}{1},\frac{2}{2}\\},\\{z_{i}\\}{=}\\{1,2\\},\\#{\mathrm{p}{\otimes}\mathrm{e}}{=}2$
(124)
$\displaystyle\Psi_{2^{1}}[M(r)]=e^{-x}xL^{1}_{0}(2x),x=Mr\nu/\hbar=2\sqrt{2}r$
(125)
$\displaystyle\Psi_{2^{2}}[M(r)]=e^{-x}xL^{1}_{1}(2x),x=Mr\nu/\hbar=\sqrt{2}r$
(126) $\displaystyle r^{1}_{21}=N_{21}\int_{0}^{\infty}\\!\\!\Psi_{2^{1}}\cdot
r\cdot\Psi_{2^{1}}\cdot 4\pi r^{2}dr=\frac{5}{4\sqrt{2}}$ (127) $\displaystyle
r^{1}_{22}=N_{22}\int_{0}^{\infty}\\!\\!\Psi_{2^{2}}\cdot
r\cdot\Psi_{2^{2}}\cdot 4\pi r^{2}dr=\frac{55}{14\sqrt{2}}$ (128)

where $N_{21}$ and $N_{22}$ are corresponding normalization factors. Similar
with the $W=1$ case, $M=\sqrt{2}$ is also only lower bound of mass of black
holes with $W=2$. When $\sqrt{2}<M<M_{W=3}$, we can always fine tuning the
value of $\nu_{i}$ so that $z_{i}\equiv\frac{GM_{i}m_{i}}{\nu_{i}}=\\{1,2\\}$.
From hereafter, we will not point out explicitly that the value of all mass
parameters $M$ determined in the following $W=3,4,5,6$ cases are only lower
bound.

Beginning from $W=3$, we have to note that the matter contents of a black hole
could have two or more layers of shell structure

$\displaystyle\hskip 22.76219ptW=3,M=\frac{\sqrt{5}+1}{2}\approx 1.618$ (129)
$\displaystyle\begin{matrix}\\{m_{i}\\}\\!&\\!\\{1{+}\xi\\}\\!&\\!\\{1,\xi\\}\\\
\\{M_{i}\\}\\!&\\!\\{(1{+}\xi)\\}\\!&\\!\\{1,1{+}\xi\\}\\!&\\!\xi(1{+}\xi){=}1\\\
\\{M_{i}m_{i}\\}\\!&\\!\\{(1{+}\xi)^{2}{<}3\\}\\!&\\!\\{1,\xi(1{+}\xi)\\}\\!&\\!\Downarrow\\\
\\{\nu_{i}\\}\\!\\!&\\!\\!\\{\frac{(1{+}\xi)^{2}\\!}{1},\\!\frac{(1{+}\xi)^{2}\\!}{2}\\}\\!\\!&\\!\\!\\{1,1\\}\\!&\\!\xi{=}\\\
\\{z_{i}\\}\\!\\!&\\!\\!\\{1,2\\}\\!\\!&\\!\\!\\{1,1\\}&(\sqrt{5}{-}1)/2\\\
\\#{\mathrm{p}{\otimes}\mathrm{e}}\\!&\\!2\\!&\\!1\end{matrix}$ (130)
$\displaystyle\Psi_{3^{1}}[M(r)]=e^{-x}xL^{1}_{0}(2x),x=\frac{Mr\nu}{\hbar}=(\sqrt{5}{+}2)r$
(131)
$\displaystyle\Psi_{3^{2}}[M(r)]=e^{-x}xL^{1}_{1}(2x),x=Mr\nu/\hbar=\frac{\sqrt{5}{+}2}{2}r$
(132) $\displaystyle\Psi_{3^{3}}[M(r)]=e^{-x_{1}}x_{1}L_{0}^{1}(2x_{1})\otimes
e^{-x_{2}}x_{2}L_{0}^{1}(2x_{2})$ (133) $\displaystyle
x_{1}=m_{1}r\nu_{1}/\hbar=r,x_{2}=m_{2}r\nu_{2}/\hbar=\frac{\sqrt{5}-1}{2}r$
(134) $\displaystyle
r^{1}_{31}=N_{31}\int_{0}^{\infty}\\!\\!|\Psi_{3^{1}}|^{2}\cdot r\cdot 4\pi
r^{2}dr=\frac{\scriptstyle 2415{-}1080\sqrt{5}}{\scriptstyle
6(17\sqrt{5}{-}38)}$ (135) $\displaystyle
r^{1}_{32}=N_{32}\int_{0}^{\infty}\\!\\!|\Psi_{3^{2}}|^{2}\cdot r\cdot 4\pi
r^{2}dr=\frac{\scriptstyle 55(161-72\sqrt{5})}{\scriptstyle
7(17\sqrt{5}{-}38)}$ (136) $\displaystyle
r^{1}_{33}{\otimes}r^{2}_{33}=N_{33}\int_{0}^{\infty}\\!\\!\Psi_{3^{3}}\cdot
r\cdot\Psi_{3^{3}}\cdot 4\pi r^{2}dr$ (137)
$\displaystyle=\\{\frac{\scriptstyle 5}{\scriptstyle 2},\frac{\scriptstyle
5(7{+}3\sqrt{5})}{\scriptstyle 4(2{+}\sqrt{5})}\\}$ (138)

The radius of each shell in the two shell structure must be calculated
independently and directly multiplying together. This direct product format is
necessary for our calculation of similarities between different microstates.

For the $W=4$ case, we have

$\displaystyle W=4,M=\sqrt{3}\approx
1.732,\epsilon{<}2{-}\sqrt{3}{<}\frac{1}{2}$ (139)
$\displaystyle\begin{matrix}\\{m_{i}\\}&\\{\sqrt{3}\\}&\\{\frac{2}{\sqrt{3}},\frac{1}{\sqrt{3}}\\}&\\{\frac{2{-}\epsilon}{\sqrt{3}},\frac{1{+}\epsilon}{\sqrt{3}}\\}\\\
\\{M_{i}\\}&\\{\sqrt{3}\\}&\\{\frac{2}{\sqrt{3}},\sqrt{3}\\}&\\{\frac{2{-}\epsilon}{\sqrt{3}},\sqrt{3}\\}\\\
\\{M_{i}m_{i}\\}\\!\\!&\\!\\!\\{3\\}&\\{\frac{4}{3},1\\}&\\{\frac{(2{-}\epsilon)^{2}\\!}{3},1{+}\epsilon\\}\\\
\\{\nu_{i}\\}\\!&\\!\\{\frac{3}{1},\frac{3}{2},\frac{3}{3}\\}\\!&\\!\\{\frac{4}{3},1\\}&\\{\frac{(2{-}\epsilon)^{2}\\!}{3},1{+}\epsilon\\}\\\
\\{z_{i}\\}\\!&\\!\\{1,2,3\\}\\!&\\!\\{1,1\\}\\!&\\!\\{1,1\\}\\\
\\#{\mathrm{p}{\otimes}\mathrm{e}}&3&1&\\!\\!0\mathrm{\,acc\,to\,}M_{i}^{{\rule[1.5pt]{2.0pt}{0.5pt}}\frac{1}{2}}\\!\\!/2\end{matrix}$
(140)
$\displaystyle\Psi_{4^{1}}[M(r)]=e^{-x}xL^{1}_{0}(2x),x=Mr\nu/\hbar=3\sqrt{3}r$
(141)
$\displaystyle\Psi_{4^{2}}[M(r)]=e^{-x}xL^{1}_{1}(2x),x=Mr\nu/\hbar=\frac{3\sqrt{3}}{2}r$
(142)
$\displaystyle\Psi_{4^{3}}[M(r)]=e^{-x}xL^{1}_{2}(2x),x=Mr\nu/\hbar=\frac{3\sqrt{3}}{3}r$
(143) $\displaystyle\Psi_{4^{4}}[M(r)]=e^{-x_{1}}x_{1}L_{0}^{1}(2x_{1})\otimes
e^{-x_{2}}x_{2}L_{0}^{1}(2x_{2})$ (144) $\displaystyle
x_{1}=m_{1}r\nu_{1}/\hbar=\frac{8r}{3\sqrt{3}},x_{2}=m_{2}r\nu_{2}/\hbar=\frac{r}{\sqrt{3}}$
(145) $\displaystyle r^{1}_{41}=N_{41}\int_{0}^{\infty}\\!\\!\Psi_{4}^{1}\cdot
r\cdot\Psi_{4}^{1}\cdot 4\pi r^{2}dr=\frac{\scriptstyle 5}{\scriptstyle
6\sqrt{3}}$ (146) $\displaystyle
r^{1}_{42}=N_{42}\int_{0}^{\infty}\\!\\!\Psi_{4}^{2}\cdot
r\cdot\Psi_{4}^{2}\cdot 4\pi r^{2}dr=\frac{\scriptstyle 55}{\scriptstyle
21\sqrt{3}}$ (147) $\displaystyle
r^{1}_{43}=N_{43}\int_{0}^{\infty}\\!\\!\Psi_{4}^{3}\cdot
r\cdot\Psi_{4}^{3}\cdot 4\pi r^{2}dr=\frac{\scriptstyle 25}{\scriptstyle
6\sqrt{3}}$ (148) $\displaystyle r^{1}_{44}{\otimes}r^{2}_{44}=\\{\langle
r_{1}\rangle,\langle r_{2}\rangle\\}_{4}^{4}=\\{\frac{\scriptstyle
15\sqrt{3}}{\scriptstyle 16},\frac{\scriptstyle 5\sqrt{3}}{\scriptstyle 2}\\}$
(149)

For the $W=5$ case, we have

$\displaystyle\hskip 8.53581ptW=5,M=\sqrt{3+\xi}\approx 1.823,\epsilon={1/2}$
(150)
$\displaystyle\begin{matrix}\\{m_{i}\\}&\\{\sqrt{3{+}\xi}\\}\\!&\\!\\{\frac{2{+}\xi}{\sqrt{3{+}\xi}}\\!,\\!\frac{1}{\sqrt{3{+}\xi}}\\}\\!&\\!\\{1,\frac{1{+}\epsilon}{\sqrt{3+\xi}}\\}\\\
\\{M_{i}\\}&\\{\sqrt{3}{+}\xi\\}\\!&\\!\\{\frac{2{+}\xi}{\sqrt{3{+}\xi}}\\!,\\!{\scriptstyle\sqrt{3{+}\xi}}\\}\\!\\!&\\!\\{1,{\scriptstyle\sqrt{3{+}\xi}}\\}\\\
\\{M_{i}m_{i}\\}\\!\\!&\\!\\!\\{3{+}\xi\\}\\!&\\!\\{\frac{(2{+}\xi)^{2}}{3{+}\xi}{<}2,1\\}\\!&\\!\\{1,1{+}\epsilon\\}\\\
\\{\nu_{i}\\}\\!\\!&\\!\\!\\{\frac{3{+}\xi}{1}\\!,\\!\frac{3{+}\xi}{2}\\!,\\!\frac{3{+}\xi}{3}\\}\\!&\\!\\{\frac{(2{+}\xi)^{2}\\!}{3{+}\xi}\\!,\\!1\\}&\\{1,1{+}\epsilon\\}\\\
\\{z_{i}\\}\\!&\\!\\{1,2,3\\}\\!&\\!\\{1,1\\}&\\{1,1\\}\\\
\\#{\mathrm{p}{\otimes}\mathrm{e}}&3&1&1\end{matrix}$ (151)
$\displaystyle\hskip
8.53581pt1{+}\frac{1{+}\epsilon}{\sqrt{3{+}\xi}}=\sqrt{3{+}\xi}\Rightarrow\xi=\frac{\sqrt{7}}{2}{-}1$
(152)
$\displaystyle\Psi^{1}_{5}[M(r)]=e^{-x}xL^{1}_{0}(2x),x=\frac{Mr\nu}{\hbar}=\frac{(1{+}\sqrt{7})^{3}r\\!\\!}{8}$
(153)
$\displaystyle\Psi^{2}_{5}[M(r)]=e^{-x}xL^{1}_{1}(2x),x=\frac{Mr\nu}{\hbar}=\frac{(1{+}\sqrt{7})^{3}r\\!\\!}{16}$
(154)
$\displaystyle\Psi^{3}_{5}[M(r)]=e^{-x}xL^{1}_{2}(2x),x=\frac{Mr\nu}{\hbar}=\frac{(1{+}\sqrt{7})^{3}r\\!\\!}{24}$
(155) $\displaystyle\Psi^{4}_{5}[M(r)]=e^{-x_{1}}x_{1}L_{0}^{1}(2x_{1})\otimes
e^{-x_{2}}x_{2}L_{0}^{1}(2x_{2})$ (156) $\displaystyle
x_{1}=\frac{m_{1}r\nu_{1}}{\hbar}=\frac{\scriptstyle(\sqrt{7}{+}2)^{3}r}{\scriptstyle(\sqrt{7}{+}1)^{3}},x_{2}=\frac{m_{2}r\nu_{2}}{\hbar}=\frac{2r}{\scriptstyle\sqrt{7}{+}1}$
(157) $\displaystyle\Psi^{5}_{5}[M(r)]=e^{-x_{1}}x_{1}L_{0}^{1}(2x_{1})\otimes
e^{-x_{2}}x_{2}L_{0}^{1}(2x_{2})$ (158) $\displaystyle
x_{1}=m_{1}r\nu_{1}/\hbar=r,x_{2}=m_{2}r\nu_{2}/\hbar=\frac{\scriptstyle
9r/2}{\scriptstyle\sqrt{7}{+}1}$ (159) $\displaystyle
r_{51}^{1}=N_{51}\\!\int_{0}^{\infty}\\!\\!\Psi_{5}^{1}\cdot
r\cdot\Psi_{5}^{1}\cdot 4\pi r^{2}dr=\frac{\scriptstyle
5(5\sqrt{7}{-}11)}{\scriptstyle 27}$ (160) $\displaystyle
r_{52}^{1}=N_{52}\\!\int_{0}^{\infty}\\!\\!\Psi_{5}^{2}\cdot
r\cdot\Psi_{5}^{2}\cdot 4\pi r^{2}dr=\frac{\scriptstyle
110(5\sqrt{7}{-}11)}{\scriptstyle 189}$ (161) $\displaystyle
r_{53}^{1}=N_{53}\\!\int_{0}^{\infty}\\!\\!\Psi_{5}^{3}\cdot
r\cdot\Psi_{5}^{3}\cdot 4\pi r^{2}dr=\frac{\scriptstyle
85(5\sqrt{7}{-}11)}{\scriptstyle 69}$ (162) $\displaystyle
r_{54}^{1}{\otimes}r_{54}^{2}=\\{\frac{\scriptstyle
5(115{-}41\sqrt{7})}{\scriptstyle 27},\frac{\scriptstyle
1{+}\sqrt{7}}{\scriptstyle 4}\\}$ (163) $\displaystyle
r_{55}^{1}{\otimes}r_{55}^{2}=\\{\frac{\scriptstyle 5}{\scriptstyle
2},\frac{\scriptstyle 5(1{+}\sqrt{7})}{\scriptstyle 9}\\}$ (164)

The final is the $W=6$ case, although only $\Psi_{6}^{1}$ will be used for our
concrete example, we choose to write down all 6 wave functions explicitly
here.

$\displaystyle\hskip 8.53581ptW=6,M=\sqrt{3{+}x}\approx 1.932,\epsilon=1/2$
(165)
$\displaystyle\begin{matrix}\\{m_{i}\\}&\\{\sqrt{3{+}x}\\}&\\{\sqrt{2},\frac{1}{\sqrt{3{+}x}}\\}\\!&\\!\\{\frac{2{-}\epsilon{+}x}{\sqrt{3{+}x}}\\!,\\!\frac{1{+}\epsilon}{\sqrt{3{+}x}}\\}\\\
\\{M_{i}\\}&\\{\sqrt{3{+}x}\\}&\\{\sqrt{2},{\scriptstyle\sqrt{3{+}x}}\\}\\!\\!&\\!\\!\\{\frac{2{-}\epsilon{+}x}{\sqrt{3{+}x}}\\!,\\!{\scriptstyle\sqrt{3{+}x}}\\}\\\
\\{M_{i}m_{i}\\}&\\{3{+}x\\}&\\{2,1\\}\\!&\\!\\!\\!\\{\\!\frac{(2{-}\epsilon{+}x)^{2}\\!\\!}{3{+}x}\\!,\\!1{+}\epsilon\\!\\}\\\
\\{\nu_{i}\\}\\!&\\!\\{[\frac{3{+}x}{1,2,3}]\\}\\!&\\!\\{[\frac{2}{1},\frac{2}{2}],1\\}\\!\\!&\\!\\!\\{\\!\frac{(2{-}\epsilon{+}x)^{2}\\!\\!}{3{+}x}\\!,\\!1{+}\epsilon\\!\\}\\\
\\{z_{i}\\}\\!&\\!\\{1,2,3\\}\\!&\\!\\{[1,2],1\\}\\!&\\!\\{1,1\\}\\\
\\#{\mathrm{p}{\otimes}\mathrm{e}}&3&2&1\end{matrix}$ (166)
$\displaystyle\hskip
8.53581pt\sqrt{2}{+}\frac{1}{\sqrt{3{+}x}}{=}\sqrt{3{+}x}\Rightarrow
x{=}\sqrt{3}-1$ (167)
$\displaystyle\Psi^{1}_{6}[M(r)]=e^{-x}xL^{1}_{0}(2x),x=Mr\nu/\hbar=\frac{\scriptstyle(\sqrt{3}{+}1)^{3}r\\!}{\scriptstyle\sqrt{8}}$
(168)
$\displaystyle\Psi^{2}_{6}[M(r)]=e^{-x}xL^{1}_{1}(2x),x=Mr\nu/\hbar=\frac{\scriptstyle(\sqrt{3}{+}1)^{3}r\\!}{\scriptstyle
2\sqrt{8}}$ (169)
$\displaystyle\Psi^{3}_{6}[M(r)]=e^{-x}xL^{1}_{1}(2x),x=Mr\nu/\hbar=\frac{\scriptstyle(\sqrt{3}{+}1)^{3}r\\!}{\scriptstyle
3\sqrt{8}}$ (170)
$\displaystyle\Psi^{4}_{6}[M(r)]=e^{-x_{1}}x_{1}L_{0}^{1}(2x_{1})\otimes
e^{-x_{2}}x_{2}L_{0}^{1}(2x_{2})$ (171) $\displaystyle
x_{1}=m_{1}r\nu_{1}/\hbar=2\sqrt{2}r,x_{2}=m_{2}r\nu_{2}/\hbar=\frac{\scriptstyle\sqrt{2}r}{\scriptstyle\sqrt{3}{+}1}$
(172) $\displaystyle\Psi^{5}_{6}[M(r)]=e^{-x_{1}}x_{1}L_{0}^{1}(2x_{1})\otimes
e^{-x_{2}}x_{2}L_{1}^{1}(2x_{2})$ (173) $\displaystyle
x_{1}=m_{1}r\nu_{1}/\hbar=\sqrt{2}r,x_{2}=m_{2}r\nu_{2}/\hbar=\frac{\scriptstyle\sqrt{2}r}{\scriptstyle\sqrt{3}{+}1}$
(174) $\displaystyle\Psi^{6}_{6}[M(r)]=e^{-x_{1}}x_{1}L_{0}^{1}(2x_{1})\otimes
e^{-x_{2}}x_{2}L_{0}^{1}(2x_{2})$ (175) $\displaystyle
x_{1}=m_{1}r\nu_{1}=\frac{\scriptstyle 85\sqrt{2}{-}39\sqrt{6}}{\scriptstyle
16}r,x_{2}=m_{2}r\nu_{2}=\frac{\scriptstyle 9\sqrt{2}r}{\scriptstyle
4(\sqrt{3}{+}1)}$ (176) $\displaystyle
r^{1}_{61}=N_{61}\\!\int_{0}^{\infty}\\!\\!\Psi_{6}^{1}\cdot
r\cdot\Psi_{6}^{1}\cdot 4\pi r^{2}dr=\frac{\scriptstyle 5}{\scriptstyle
2\sqrt{16{+}15\sqrt{3}}}$ (177) $\displaystyle
r^{1}_{62}=N_{62}\\!\int_{0}^{\infty}\\!\\!\Psi_{6}^{2}\cdot
r\cdot\Psi_{6}^{2}\cdot 4\pi r^{2}dr=\frac{\scriptstyle 55}{\scriptstyle
7\sqrt{16{+}15\sqrt{3}}}$ (178) $\displaystyle
r^{1}_{63}=N_{63}\\!\int_{0}^{\infty}\\!\\!\Psi_{6}^{3}\cdot
r\cdot\Psi_{6}^{3}\cdot 4\pi r^{2}dr=\frac{\scriptstyle 765}{\scriptstyle
46\sqrt{16{+}15\sqrt{3}}}$ (179) $\displaystyle
r^{1}_{64}{\otimes}r^{2}_{64}==\\{\frac{\scriptstyle 5}{\scriptstyle
4\sqrt{2}},\frac{\scriptstyle 5(1{+}\sqrt{3})}{\scriptstyle 2\sqrt{2}}\\}$
(180) $\displaystyle r^{1}_{65}{\otimes}r^{2}_{65}==\\{\frac{\scriptstyle
5}{\scriptstyle 2\sqrt{2}},\frac{\scriptstyle 55(1{+}\sqrt{3})}{\scriptstyle
14\sqrt{2}}\\}$ (181) $\displaystyle
r^{1}_{66}{\otimes}r^{2}_{66}==\\{\frac{\scriptstyle 20\sqrt{2}}{\scriptstyle
85{-}39\sqrt{3}},\frac{\scriptstyle 5(\sqrt{2}{+}\sqrt{6})}{\scriptstyle
9}\\}$ (182)

###  V.2 Similarity Factors and Transition Amplitude

By definitions (13)-(14) and (12), we can calculate all the relevant
transition amplitude or similarity factors as follows

$\mathrm{Siml}[\Psi_{1^{1}},\Psi_{2^{1}}]=\frac{2r^{1}_{11}\cdot
r^{1}_{21}}{(r^{1}_{11})^{2}+(r^{1}_{21})^{2}}=\frac{4\sqrt{2}}{9}$ (183)
$\mathrm{Siml}[\Psi_{1^{1}},\Psi_{2^{2}}]=\frac{2r^{1}_{11}\cdot
r^{1}_{22}}{(r^{1}_{11})^{2}+(r^{1}_{22})^{2}}=\frac{154\sqrt{2}}{219}$ (184)
$\mathrm{Siml}[\Psi_{1^{1}},\Psi_{3^{1}}]=\frac{2r^{1}_{11}\cdot
r^{1}_{31}}{(r^{1}_{11})^{2}+(r^{1}_{31})^{2}}=\frac{1}{\sqrt{5}}$ (185)
$\mathrm{Siml}[\Psi_{1^{1}},\Psi_{3^{2}}]=\frac{308(870+533\sqrt{5})}{663545}$
(186)
$\displaystyle\mathrm{Siml}[\Psi_{1^{1}},\Psi_{3^{3}}]=\Big{[}\frac{2r^{1}_{11}\cdot
r^{1}_{33}}{(r^{1}_{11})^{2}{+}(r^{1}_{33})^{2}}\cdot\frac{2r^{1}_{11}\cdot
r^{2}_{33}}{(r^{1}_{11})^{2}{+}(r^{2}_{33})^{2}}\Big{]}^{\frac{1}{2}}$
$\displaystyle=\frac{\sqrt{2}}{5^{1/4}}$

Other similarity factors could be calculated similarly. Our results are
compiled in FIG.18 explicitly. From the figure, we easily see that, in most
cases, $\mathrm{Siml}[\Psi_{u^{n}},\Psi_{v^{\ell}}]=\mathcal{O}[1]$, there
very few cases $\mathrm{Siml}[\Psi_{u^{n}},\Psi_{v^{\ell}}]\ll 1$. So, in
crude estimations, we can set all $g_{u^{n}v^{\ell}}=1$ for efficiency.

Figure 18: Similarity factors between all different inner structure wave
function defined by (13)-(14).

###  V.3 Schrödinger Equation and Partial State Tracing out

Substituting results above into transition amplitude (12) and differential
equations (16)-(17). We can integrate and get 1440 time dependent coefficient
functions $c_{u^{n}}^{\vec{\omega}}(t)$. They form wave functions of an
evaporating black hole and its radiation products which initially is set on
pure state
$c_{u^{n}}^{\vec{\omega}}=\delta_{u6}\delta_{n1}\delta^{\vec{\omega}\phi}$,
symbols $\phi$ here is used denoting zero radiation product state. Our key
results has been displayed in FIG.2 already. We provide here evidences that
tracing out state of the radiation products will make the density matrix of
the black hole non-diagonal even when we are working on basis of inner
structure eigenstate of the black hole during the whole process.

We will take the subspace of $u^{n}\otimes\vec{\omega}=$
$4^{n=}_{1234}\otimes$ $\\{\\{2^{1}\\},\\{1^{j=}_{1\cdot\cdot 5},1^{1}\\}\\}$
as example. This is a 24 dimensional Hilbert space, where $u^{n}$ is the
possible black hole microstate, $\vec{\omega}$ is the possible radiation
product state. If we write this vector explicitly, its form reads like

$\displaystyle
c_{u^{n}}^{\vec{\omega}}{=}\\{c_{4^{1}}^{\\{2^{1}\\}},c_{4^{1}}^{\\{1^{1}1^{1}\\}},c_{4^{1}}^{\\{1^{2}1^{1}\\}},\cdots
c_{4^{1}}^{\\{1^{5}1^{1}\\}},c_{4^{2}}^{\\{2^{1}\\}},c_{4^{2}}^{\\{1^{1}1^{1}\\}}$
(188) $\displaystyle,c_{4^{2}}^{\\{1^{2}1^{1}\\}},\cdots
c_{4^{2}}^{\\{1^{5}1^{1}\\}},\cdots\cdots
c_{4^{4}}^{\\{1^{5}1^{1}\\}}\\}^{\mathrm{transpose}}$

The corresponding density matrix has forms (since initially our black holes
has $w=6$, we will call its radiation residue $w=4$ big here)

$\displaystyle\rho^{b}_{ij}=c_{u^{n}}^{\vec{\omega}}c_{u^{n}}^{\vec{\omega}\dagger}=\left[\begin{matrix}a_{6\times
6}&b_{6\times 6}&c_{6\times 6}&d_{6\times 6}\\\ b_{6\times 6}&e_{6\times
6}&f_{6\times 6}&g_{6\times 6}\\\ c_{6\times 6}&f_{6\times 6}&h_{6\times
6}&i_{6\times 6}\\\ d_{6\times 6}&g_{6\times 6}&i_{6\times 6}&j_{6\times
6}\end{matrix}\right]$ (189)

subscript $i,j$ on $\rho_{ij}$ specify state of the black hole and radiation
products. Tracing out state of the latter means that we take traces on each
$6\times 6$ submatrix and get a $4\times 4$ matrix

$\displaystyle\rho^{b}_{hh^{\prime}}=\left[\begin{matrix}a&b&c&d\\\ b&e&f&g\\\
c&f&h&i\\\ d&g&i&j\end{matrix}\right]$ (190)

subscripts $h,h^{\prime}$ on this matrix density specify state of the black
hole only. Although during the whole process we are kept on basis of
eigenstate of the black hole inner structure and radiation products’ energy,
tracing out microstate of the latter provides us a non-diagonal matrix
$\rho^{b}_{hh^{\prime}}$. We provide in FIG.19 their time dependence
explicitly.

Figure 19: Variations of the diagonal and off diagonal elements of density
matrix (190), the horizontal axis is time.

##  VI Some discussions

This section is some discussion or critics on our inside horizon oscillation
picture and its development.

###  VI.1 On the meaning of inside horizon oscillation

Many people believe that, it is of nonsense to talk about a spherically
symmetric black hole’s inner matter distribution and motion modes due to the
horizon’s forbidding of such inner structure’s being measurable by outside
observers. However, this argument applies only in single body context. When we
let two spherically symmetric black holes (one with totally known inner
structure and the other unknown) spin around and merge with each other,
spherical symmetries of both black holes will be broken due to tidal
deformation and the structure of the previously unknown black hole will be
measurable from gravitational waves following from the two bodies’ merging
process. The basic reason here is that, black holes with different inner
matter distribution and oscillation modes, for example two with singular
central points and two without such singular centers, will exhibit different
tidal deformation and lead to different gravitational waves as the merging
proceeds dfzeng2020 . We display in FIG.20 the quadrupole of two such static
binary systems. Although the difference between the two static systems is a
diagonal constant matrix which has no observable effects, when they spin
around their own central vertical line, the right system will experience
different tidal deformation and produce different gravitation waves relative
to the left one.

Figure 20: A binary black holes’ inspiral and merging process. The left hand
side black holes point like one, quadrupole $D_{ij}$ of the sysetm have only
one non-0 component. The right hand side black holes have oscillatory cores,
the system have rather different quadrupoles. When they inspiral and merging,
this two systems will experience rather different tidal deformation thus
emitting rather different gravitational wave.

If the method of effective field theory is adopted to describe inspiral and
mergering processes of such inner structured objects Porto2016 ;
eftGoldberger2006 ; eftGoldberger2007 , we have to write

$\displaystyle\exp\\{iS_{\mathrm{EH}}[g^{\scriptscriptstyle\\!L}_{\mu\nu}(x)+iS_{\mathrm{eff}}[x^{\alpha}_{\mathrm{cm}}(\sigma),g^{\scriptscriptstyle\\!L}_{\mu\nu}(x)]]\\}=$
(191)
$\displaystyle\int\\!\\!\mathscr{D}\\!g^{\scriptscriptstyle\\!S}_{\mu\nu}\mathscr{D}\\!\delta\\!{x}^{\\!\alpha}\\!(\sigma_{p}\\!)\exp\\{iS_{\mathrm{EH}}[g_{\mu\nu}\\!(x){+}iS_{\mathrm{eff}}[x^{\alpha}_{\mathrm{cm}}\\!(\sigma_{p}\\!),g_{\mu\nu}(x)]]\\}$

where $g^{\scriptscriptstyle\\!L}_{\mu\nu}$ and
$g^{\scriptscriptstyle\\!S}_{\mu\nu}$ are the long and short degrees of
freedom of spacetime geometry;
$x^{\alpha}_{\mathrm{cm}}(\sigma)$($\alpha=1,2$) denote the center of mass
coordinate of the black holes under consideration; $\sigma$ parameterizes
their worldlines;
${\delta}x^{\alpha}_{p}=x^{\alpha}_{p}-x^{\alpha}_{\mathrm{cm}}$ represent
displacement of constituents of the black hole relative to their own centers.
Integrating out short the distance degrees of freedom, we will have

$\displaystyle
S_{\mathrm{eff}}[x_{\mathrm{cm}},g^{\scriptscriptstyle\\!L}_{\mu\nu}]=\\!\\!\int\\!\\!d^{4}x\delta[x-x_{\mathrm{cm}}(\sigma)]\Big{\\{}$
(192)
$\displaystyle-m\sqrt{g^{\scriptscriptstyle\\!L}_{\mu\nu}\dot{x}^{\mu}_{\mathrm{cm}}(\sigma)\dot{x}^{\nu}_{\mathrm{cm}}(\sigma)}$
$\displaystyle+C_{R}\int
R^{\scriptscriptstyle\\!L}(g^{\scriptscriptstyle\\!L}_{\mu\nu})\sqrt{g^{\scriptscriptstyle\\!L}_{\mu\nu}\dot{x}^{\mu}_{\mathrm{cm}}(\sigma)\dot{x}^{\nu}_{\mathrm{cm}}(\sigma)}$
$\displaystyle+C_{V}\int
R^{\scriptscriptstyle\\!L}_{\mu\nu}(g^{\scriptscriptstyle\\!L}_{\mu\nu})\frac{\dot{x}^{\mu}_{\mathrm{cm}}(\sigma)\dot{x}^{\nu}_{\mathrm{cm}}(\sigma)}{\sqrt{g^{\scriptscriptstyle\\!L}_{\mu\nu}\dot{x}^{\mu}_{\mathrm{cm}}(\sigma)\dot{x}^{\nu}_{\mathrm{cm}}(\sigma)}}+\cdots\Big{\\}}$

Ellipsis on the last line denotes higher order terms in long wave length
expansion of the world line action, $C_{R}$, $C_{V}$ and those omitted $C_{X}$
are series of parameters called Wilson coefficients. If the black hole owns no
inner structure, all the relevant Wilson coefficients $C_{i}$ will be
determined solely by the its mass $M$ and Newton couplings $G$, Bini2012 ;
Chakrabarti2013a ; Chakrabarti2013b ; Bini2015 . However, if the black hole
own non-trivial inner structure, such as we discussed in this work, then these
Wilson coefficients will manifest correspondingly features reflecting the
underlying structure. This will make the black hole behave more closely with
neutron stars eftBinnington ; eftDamour2009 ; Hinderer2008 ; Hinderer2010 ;
Vines2011 ; Yagi2013 ; Donneva2013 ; Bernuzzi2014 ; Agathos2015 ;
Chatziioannou2015

In practice, data collected in the LIGO/VIRGO observation may not be
sufficient to uncover precisely the inner structure of black holes being
observed gw150914 ; gw170817a ; gwScalarSoliton ; cardoso2016 ; afshordi2016 ;
cardoso2017 . But it may have potentials to tell us if black holes have
singular centers or not. So to talk about the consisting matters’ inside
horizon oscillation is not a pure theoretical question, it is an experimental
one.

In single body case, implications of matter’s oscillation inside the horizon
could be seen in some theoretical examination. For example, to recover Hawking
radiation’s unitarity and to get proper interpretation for the Bekenstein
Hawking entropy’s origin, ’t Hooft tHooftISSP2016 ; tHooftISSP2018 ;
tHooft1809 ; tHooft1804 ; tHooft1612 ; tHooftFP2018 ; tHooft1605 ; tHooft1601
pointed out that extra assumption, such as Antipodal Points Identification
(API) about the horizon itself is almost unavoidable. API assumes that points
in regions III and IV of the Penrose diagram of the Schwarzschild black hole
should be identified with the antipodal ones in regions I and II. Obviously,
in black hole with oscillatory inner structures such as those described by our
exact solutions, API is not an artificial assumption, but a natural deduction
of the math formalism. Further more, when quantum effects are considered,
matters consisting of the black hole have very large probabilities be measured
outside its horizon, thus making the horizon a blurring or fuzzing region
instead a cut-clear surface. Observers outside the hole need not waiting
infinite durations to see signals unveiling the matters’ falling into this
region and participating oscillations inside it. On pure general relativistic
level, the fact that inside horizon oscillations happen beyond the domain of
time definition of outside observer only implies that, we need to consider
them as physics occurring in the parallel universe or statistic ensemble
world. Details of such motions are measurable to observers living in the
corresponding universe, i.e. Maxwell’s demon like observers. To us, statistic
mechanic viewpoint or many world explanation is enough.

To emphasize the role of matters consisting of the black hole instead of
focusing on pure gravitational or geometric degrees of freedom in the
information missing puzzle’s resolving is also the key of works Stojkovic2007
; Stojkovic2008a ; Stojkovic2008b ; Stojkovic2009 ; Stojkovic2013 ;
Stojkovic2014 ; Stojkovic2015 . However, there is popular sayings that the
number of degree of freedom following from the constituent matter would lead
to a volume-law entropy instead of area-law type one. This is a very
misleading doctrine. Since the number of particles constrained inside the
horizon are proportional to the black hole mass $N\propto M$, thus in 3+1D
Schwarzschild case are proportional to its horizon size $N\propto r_{h}=2GM$,
simple calculation

$\displaystyle
Z=\int\\!d\vec{x}_{1}d\vec{p}_{1}{\cdots}d\vec{x}_{\scriptscriptstyle\\!N}d\vec{p}_{\scriptscriptstyle\\!N}e^{-\beta
H(\vec{x}_{1},\vec{p}_{1},\cdots)}$ (193) $\displaystyle
S\sim(1-\beta\partial_{\beta})\ln Z\sim N^{1-\epsilon}\sim
r_{h}^{1-\epsilon},\epsilon>0$ (194)

will give us neither volume-law, nor area-law, but diameter-law entropy at
most. Then how do we get area law entropy in our work? The key point here is,
the objects carrying degrees of freedom evaluated by the area-law entropy is
not the usual particles consisting of the black hole, but collective motion
modes of them. According to our quantizing condition (84), the characteristic
mass of such modes is
$m_{\mathrm{i=out.most}}\propto(GM_{i})^{-1}=(GM_{\mathrm{tot}})^{-1}$. In a
black hole of mass $M_{\mathrm{tot}}$, the number of such objects is
$N_{\scriptscriptstyle\\!CMM}\propto GM_{\mathrm{tot}}^{2}$, which is
obviously proportional to the area of the black hole under consideration.

###  VI.2 Development of our ideas

Finally, we wish to make a short review about our ideas on taking the inside
horizon oscillation modes as the origin of Bekenstein-Hawking entropy. We
firstly proposed this possibility in reference dfzeng2017 . In that work, we
write the inside horizon metric in the following form

$\displaystyle ds^{2}=-h^{-1}A(\tau,r)d\tau^{2}+h^{-1}dr^{2}+r^{2}d\Omega^{2}$
(195) $\displaystyle
A=\frac{\dot{m}^{2}}{{m^{\prime}}^{2}}+h,~{}h=1-\frac{2m(\tau,r)}{r}$

and tries to build a quantization description for $m(\tau,r)$ basing on
Wheeler-DeWitt equation deWitt1967 ; QCosmologyHalliwellLecture ; QBHAllen1987
; QBHFangLi1986 ; QBHLaflamme1987 ; QBHNambuSasaki1988 ; QBHNagai1989 ;
QBHRodrigues1989 . But our equations (18) and (21) there contain errors. The
correct form should be

$\big{[}\frac{16r^{2}\rho
m^{\prime-1}}{h^{2}}{+}\frac{\ell_{p}^{2}\delta^{2}}{\delta
m(r)^{2}}+\frac{4r^{4}\rho^{2}m^{\prime-2}}{h^{2}}\big{]}\Psi[m(r)]=0$ (196)
$\displaystyle\forall
r\in[0,r_{h}],~{}\big{[}\big{(}\frac{8mr}{\xi}r^{2}\\!\rho{+}\frac{16r^{2}}{\xi^{2}}r^{4}\\!\rho^{2}\big{)}\big{(}1{-}\frac{2m}{r}\big{)}^{{\rule[1.5pt]{2.0pt}{0.5pt}}2}(\ln{\\!r})^{2}$
$\displaystyle\rule{14.22636pt}{0.0pt}+\hbar^{2}\big{(}\ln{\\!r\,}\partial_{\xi}-\partial^{2}_{\xi}\big{)}\big{]}\Psi(\xi)=0,~{}m=\frac{r_{h}}{2}\big{(}\frac{r}{r_{h}}\big{)}^{\xi}$
(197)

After this correction, the quantitative feature of wave functions listed in
the Fig.4 there would be revised. But the key conclusions do not change.

The functional Schrödinger equation (196) is very difficult to solve. So in
reference dfzeng2018a  we try an alternative way to consider the question. We
look the matter contents of a spherically symmetric black hole as many
concentric shells. By applying quantization scheme to each of these shells and
taking their total mass being equal to the mass of black hole under
consideration as a constraint, we build our first version of inside horizon
oscillation model in this work. Basing on this model, we write down an
explicitly hermitian hamiltonian to describe their radiation and argues that
the spectrum is thermal.

In reference dfzeng2018b , we further polish our ideas in reference
dfzeng2018a  and provide more evidences for the degeneracy counting of inside
horizon oscillation modes of their matter contents leads in indeed area law
formulas for the Bekenstein Hawking entropy. By the hermitian Hamiltonian
describing the radiation of black holes, we provide numeric evidences that the
information missing during the radiation of a black hole could be retrieved
from its size variation curve. But both indfzeng2018a  and in dfzeng2018b , we
find no ways to prove exactly how thermal spectrums could be produced from
such an explicitly hermitian Hamiltonian.

In reference dfzeng2020 , we find exact dynamical solutions to the Einstein
equation sourced by oscillatory and shell structured dust matter, both in
asymptotically AdS2+1 and Minkowskian 3+1 spacetime. We discuss their
quantization and implications for the Bekenstein Hawking formulas. We also
speculate this inside horizon structure’s being detectability.

Finally, in the current work, we compiles all these results together and build
concrete theories for the spontaneous radiation of black holes. By applying
Wigner-Wiesskopf approximation, we prove with rather strictness that an
explicitly hermitian hamiltonians like those in dfzeng2018a  and dfzeng2018b
indeed yield thermal radiations spectrum as expected in Hawking radiation.

##  VII Conclusion

We build an explicitly hermitian hamiltonian description for the spontaneous
radiation of black holes. Our description is essentially a many-mode multi-
degeneracy generalization of Jaynes-Cummings model, which we call Z-model
according to which particles are radiated away from or absorbed into the black
hole through monopole type couplings between the black hole and radiation
particles. Focusing on single particle’s radiation, by the well known method
of Wigner-Wiesskopf approximation, we find that our model yields power
spectrum completely the same as Hawking radiation requires. While as many
particles’ radiation is concerned, numeric method can be used to trace the
full process of a black hole’s radiation evolution, from which proper curve
for the variation of hawking particles entropy follows rather naturally.

Our model analysis point out that, to let hawking particles carry information
and escape away from a black hole controlled by no hair theorem, two
ingredients are necessary. The first is, hawking particles must be produced
through direct/indirect couplings with the black hole microscopic state,
instead of vacuum pair production around the classic no-hair horizon, as
Hawking imagined originally. Mathur’s small correction theorem provides us
strong implication on this point. The second is, a two-levels superposition
happening in the intermediate state of an evaporating black hole must be
accounted for properly. That is, any such intermediate state are
superpositions of both (i) microscopic states of black hole with equal mass
but different inner structures and (ii) states of black hole with all possible
masses less than or equal to the initial one. We explore in this language
working logics of the replica wormhole method and point out that, it can be
interpreted as an effective account of the two-levels superposition essence of
the intermediate state of evaporating black holes.

We then discuss the most important basis for the Z-model of spontaneous
radiation of black holes, that is, their atomic like inner structures embodied
in the Bekenstein-Hawking entropy. We provide exact inside horizon oscillatory
solutions family to the Einstein equation sourced by spherically symmetric,
shell structured dust ball, both in Minkowskian and Anti-de Sitter asymptotic.
After quantization, the horizon of such inside horizon oscillatory solutions
becomes highly blurred and the whole black hole becomes large amount of fuzzy
balls, the matter contents of each have remarkably big probability being found
outside the horizon. More importantly, we find that spectrums of these fuzzy
balls’ quantum wave functional are discrete and countable, with the degeneracy
almost perfectly consistent with the area law formula of Bekenstein-Hawking
entropy, both in 3+1D Schwarzschild black hole and in 2+1D AdS-Schwarzschild
case. Comparing with string theory fuzzy balls, our Schwarzschild fuzzy ball
has two sides of advantage at least. The first is, both of its classic metric
and quantum wave functions can be written out explicitly. The second is, its
construction involves no hyper-concepts of as extra dimensions or super
symmetry.

We response to some critics to our ideas published previously and review our
works on this topic in the next-to-final section. As is well known,
necessities of the quantum gravitation is not excited experimentally, but
logically. The information missing puzzle is just such an excitation source.
So, frankly speaking, we do not want to see any satisfactory resolutions to it
if they do not bring us new research area at the same time. Quantum
information and quantum geometry accompanying with the replica wormhole method
are of course such new areas. We wish inner structure of black holes be also
such an area, both theoretically and observationally.

## Acknowledgements

This work is supported by NSFC grant no. 11875082
https://search.crossref.org/funding..

## References

  * (1) N. Engelhardt and A. C. Wall, “Quantum extremal surfaces: Holographic entanglement entropy beyond the classical regime,” JHEP 01 (2015), 073 arXiv:1408.3203.
  * (2) X. Dong and A. Lewkowycz, “Entropy, extremality, Euclidean variations, and the equations of motion,” JHEP 01 (2018), 081 arXiv:1705.08453.
  * (3) G. Penington, “Entanglement wedge reconstruction and the information paradox,” JHEP 09 (2020), 002 arXiv:1905.08255.
  * (4) A. Almheiri, N. Engelhardt, D. Marolf and H. Maxfield, “The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole,” JHEP 12 (2019), 063 arXiv:1905.08762.
  * (5) A. Almheiri, R. Mahajan, J. Maldacena and Y. Zhao, “The Page curve of Hawking radiation from semiclassical geometry,” JHEP 03 (2020), 149 arXiv:1908.10996.
  * (6) A. Almheiri, R. Mahajan and J. Maldacena, “Islands outside the horizon,” arXiv:1910.11077.
  * (7) G. Penington, S. H. Shenker, D. Stanford and Z. Yang, “Replica wormholes and the black hole interior,” arXiv:1911.11977.
  * (8) A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini, “Replica wormholes and the entropy of Hawking radiation,” JHEP 05 (2020), 013 arXiv:1911.12333.
  * (9) A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini, “The entropy of Hawking radiation,” arXiv:2006.06872.
  * (10) B. Chen, B. Czech, Z. Wang, “Quantum Information in Holographic Duality”, arXiv:2108.09188.
  * (11) A. Almheiri, D. Marolf and J. Polchinski et al, “Black Holes: Complementarity or Firewalls?” JHEP 1302 (2013) 062, arXiv:1207.3123
  * (12) A. Almheiri, D. Marolf and J. Polchinski et al, “An Apologia for Firewalls”, JHEP 1309(2013) 018, arXiv:1304.6483.
  * (13) P. Chen, Y. Ong and D. Page et al, “Naked Black Hole Firewalls”, Phys. Rev. Lett. 116 (2016) 161304, arXiv:1511.05695.
  * (14) S. Hawking, “The Information Paradox for Black Holes”, arXiv:1509.01147.
  * (15) S. Hawking, “Black hole explosions”, Nature 248 (1974) 30; “Particle creation by blackholes”, Comm. Math. Phys. 43 (1975) 199.
  * (16) J. Preskill, “Do Black Holes Destroy Information?”, International Symposium on Black Holes, Membranes, Wormholes, and Superstrings, arXiv:hep-th/9209058.
  * (17) S. Giddings, “The black hole information paradox”, Particles, Strings and Cosmology. Johns Hopkins Workshop on Current Problems in Particle Theory 19 and the PASCOS Interdisciplinary Symposium 5, arXiv:hep-th/9508151
  * (18) J. Polchinski, “The Black Hole Information Problem”, Lecture Notes in 2014-15 Jerusalem Winter School and the 2015 TASI.
  * (19) “The information paradox: A pedagogical introduction”, Class. Quant. Grav. 26 (2009) 224001, eprint:0909.1038
  * (20) B. Guo, M. Hughes, S. Mathur, M. Mehta, “Contrasting the fuzzball and wormhole paradigms for black holes”, Turk. J. Phys. 45 (2021) 6, 281-365, eprint: 2111.05295
  * (21) D. Dai, D. Minic, D. Stojkovic, and C. Fu, “Testing the ER=EPR conjecture”, Phys. Rev. D102 (2020), 066004; arXiv:2002.08178.
  * (22) S. Giddings, “The deepest problem: some perspectives on quantum gravity”, arXiv:2202.08292
  * (23) Ding-fang Zeng, “Resolving the Schwarzschild singularity in both classic and quantum gravities”, Nucl. Phys. B917 178-192, arXiv: 1606.06178.
  * (24) Ding-fang Zeng, “Schwarzschild Fuzzball and Explicitly Unitary Hawking Radiations”, Nucl. Phys.B930 (2018) 533-544, arXiv: 1802.00675.
  * (25) Ding-fang Zeng, “Information missing puzzle, where is hawking’s error?”, Nucl. Phys.B941 (2018) 665, arXiv: 1804.06726.
  * (26) Ding-fang Zeng, “Exact inner metric and microscopic state of AdS3-Schwarzschild BHs”, Nucl. Phys.B954 (2020) 115001, arXiv: 1812.06777.
  * (27) M. Scully, & M. Zubairy, “Quantum Optics”, section 6.3, Cambridge University Press, ISBN-13,978-0524235959.
  * (28) D. N. Page, “Information in black hole radiation,” Phys. Rev. Lett. 71 (1993), 3743-3746 arXiv:hep-th/9306083.
  * (29) D. N. Page, “Average entropy of a subsystem,” Phys. Rev. Lett. 71 (1993), 1291-1294 arXiv:gr-qc/9305007.
  * (30) D. N. Page, “Time dependence of Hawking radiation entropy,” JCAP 09 (2013), 028 1301.4995[hep-th].
  * (31) D. Dai, D. Minic, D. Stojkovic, “On black holes as macroscopic quantum objects”, arXiv:2006.09202.
  * (32) J. Bekenstein, “The quantum mass spectrum of the Kerr black hole”, Lett. Nuovo Cimento 11, 467 (1974).
  * (33) V. Mukhanov, “Are black holes quantized?”, JETP Letters 44, 63 (1986); V Mukhanov in Complexity, Entropy and the Physics of Information: SFI Studies in the Sciences of Complexity, Vol. III, ed. W H Zurek (Addison–Wesley, New York, 1990).
  * (34) J. D. Bekenstein and V. F. Mukhanov, “Spectroscopy of the quantum black hole”, Phys. Lett. B360 (1995) 7,
  * (35) J. D. Bekenstein and V. F. Mukhanov in Sixth Moscow Quantum Gravity Seminar, eds. V A Berezin, V A Rubakov and D V Semikoz (World Publishing, Singapore, 1997).
  * (36) J. Bekenstein, “Quantum black holes as atoms”, Talk given at conference “8th Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity, Gravitation and Relativistic Field Theories”, p.92-111, ePrint: gr-qc/9710076
  * (37) M. Allen, “Canonical quantisation of a spherically symmetric, massless scalar field interacting with gravity in (2+1) dimensions”,  Class. and Quant. Gravity, Vol.4, Iss.1 (1987) 149-169
  * (38) Y. Nambu, M. Sasaki “The Wave Function of a Collapsing Dust Sphere inside the Black Hole Horizon”, Prog. Theor. Phys., Vol.79, Iss.1 (1988) 96–109
  * (39) H. Nagai, “Wave Function of the de Sitter-Schwarzschild Universe”, Prog. Theor. Phys., Vol.82, Iss.2 (1989) 322–328
  * (40) A. O. Barvinsky, V. P. Frolov, A. I. Zelnikov, “Wavefunction of a Black Hole and the Dynamical Origin of Entropy”, gr-qc/9404036
  * (41) H. Kawai, Y. Matsuo and Y. Yokokura, “A SELF-CONSISTENT MODEL OF THE BLACK HOLE EVAPORATION”, Intl. J. of Mod. Phys. A, Vol. 28, No. 14, 1350050 (2013)
  * (42) T. Vachaspati, D. Stojkovic, L. M. Krauss, “Observation of Incipient Black Holes and the Information Loss Problem”, Phys. Rev.  D76 (2007) 024005 gr-qc/0609024
  * (43) T. Vachaspati, D. Stojkovic; “Quantum Radiation from Quantum Gravitational Collapse”, Phys. Lett. B663 (2008) 107-110, ePrint: gr-qc/0701096.
  * (44) E. Greenwood, D. Stojkovic, “Quantum gravitational collapse: non-singularity and non-locality” JHEP 0806 (2008) 042, 0802.4087
  * (45) J. Wang, E. Greenwood, D. Stojkovic, “Schrödinger formalism, black hole horizons and singularity behavior”, Phys. Rev. D80 (2009) 124027, 0906.3250
  * (46) J Hutchinson, D. Stojkovic, “Icezones instead of firewalls: extended entanglement beyond the event horizon and unitary evaporation of a black hole”, Class. Quant. Grav. 33 (2016) no.13, 135006, ePrint: 1307.5861
  * (47) A. Saini, D. Stojkovic, “Non-local (but also non-singular) physics at the last stages of gravitational collapse”, Phys. Rev. D89 (2014) 044003 1401.6182
  * (48) A. Saini, D. Stojkovic, “Radiation from a collapsing object is manifestly unitary”, Phys. Rev. Lett. 114 (2015), 111301; arXiv: 1503.01487.
  * (49) Gerard’t Hooft, “Ambiguity of equivalence erinciple and Hawking’s temperature”, J. Geom. Phys. 1 (1984) 45
  * (50) Gerard’t Hooft, “On the quantum structure of a black hole”, Nucl. Phys. B256 (1985) 727
  * (51) P. C. Aichelburg and R. U. Sexl, “On the gravitational field of a massless particle”, eLink, J. Gen. Rel. Grav. 2 (1971) 303.
  * (52) W. B. Bonnor, “The gravitational field of light”, eLink Commun. Math. Phys. 13 (1969) 163,
  * (53) T. Dray and G. ’t Hooft, “The gravitational shock wave of a massless particle”, eLink Nucl. Phys. B253 (1985) 173.
  * (54) Gerard’t Hooft, “The Black Hole Firewall Transformation and Realism in Quantum Mechanics”, Universe 2021 (2021) 7, 298, lectures on black hole
  * (55) Gerard’t Hooft, “Quantum Black Holes, Firewalls and the Topology of Space-Time”, lectures on black hole
  * (56) G. ’t Hooft, “The scattering matrix approach for the quantum black hole: an overview”, J. Mod. Phys. A11 (1996) pp. 4623-4688, gr-qc/9607022
  * (57) G. ’t Hooft, “Microstates in Black Holes and the Topology of Space-Time”, ISSP 2016, 169-192
  * (58) “Black hole unitarity and antipodal entanglement”, Found. Phys. 46 (2016) 9, 1185-1198, 1601.03447
  * (59) G. ’t Hooft, “The Quantum Black Hole as a Hydrogen Atom: Microstates Without Strings Attached”, 1605.05119
  * (60) G. ’t Hooft, “The Firewall Transformation for Black Holes and Some of Its Implications”, Found. Phys. 47 (2017) 12, 1503-1542 1612.08640
  * (61) G. ’t Hooft, “What happens in a black hole when a particle meets its antipode” 1804.05744
  * (62) G. ’t Hooft, “Discreteness of Black Hole Microstates” 1809.05367
  * (63) G. ’t Hooft, “Virtual Black Holes and Space–Time Structure”, Found. Phys. 48 (2018) 10, 1134-1149
  * (64) G. ’t Hooft, “The quantum black hole as a theoretical lab, a pedagogical treatment of a new approach”, ISSP 2018, 1902.10469
  * (65) S. Carlip, “The Statistical Mechanics of the (2+1)-Dimensional Black Hole”, Phys. Rev. D51 (1995) 632, gr-qc/9409052
  * (66) A. Strominger, C. Vafa, “Microscopic Origin of the Bekenstein-Hawking Entropy” Phys. Lett B379 (1996) 99, arXiv: 9601029.
  * (67) A. Strominger, “Black Hole Entropy from Near-Horizon Microstates”, JHEP 02 (1998) 009, hep-th9712251.
  * (68) M. Banados, A. Gomberoff, “Black Hole Entropy in the Chern-Simons Formulation of 2+1 Gravity” Phys. Rev D55 (1997) 6162, gr-qc/9611044.
  * (69) S. Carlip, “The Statistical Mechanics of the Three-Dimensional Euclidean Black Hole”, Phys. Rev. D55 (1997) 878, gr-qc/9606043.
  * (70) M. Banados, T. Brotz, M. Ortiz, “Boundary dynamics and the statistical mechanics of the 2+1 dimensional black hole”, Nucl. Phys. B545 (1998) 340, hep-th/9802076.
  * (71) S. Carlip, “What we don’t know about BTZ black hole entropy”, Class. Quantum Grav. 15 (1998) 3609, ePrint: hep-th/9806026.
  * (72) S. Carlip, “Conformal Field Theory, (2+1)-Dimensional Gravity, and the BTZ Black Hole”, Class. Quant. Grav. 22 (2005) R85-R124, gr-qc/0503022.
  * (73) J. Baez, D. Wise, A. Crans, “Exotic Statistics for Strings in 4d BF Theory”, Adv. Theor. Math. Phys. 11 (2007) 707, gr-qc/0603085.
  * (74) D. Mateos, P. K. Townsend, “Supertubes”, Phys. Rev. Lett. 87 (2001) 011602, arXiv: hep-th/0103030
  * (75) O. Lunin, S. D. Mathur, “Statistical interpretation of Bekenstein entropy for systems with a stretched horizon”, Phys. Rev. Lett. 88 (2002) 211303, arXiv: hep-th/0202072.
  * (76) O. Lunin, J. Maldacena, L. Maoz, “Gravity solutions for the D1-D5 system with angular momentum”, arXiv: hep-th/0212210.
  * (77) S. Giusto, S. D. Mathur, “Fuzzball geometries and higher derivative corrections for extremal holes”, Nucl. Phys. B738 (2006) 48-75, arXiv: hep-th/0412133
  * (78) S. D. Mathur, “The Fuzzball proposal for black holes: An Elementary review”, Fortsch.Phys. 53 (2005) 793-827, arXiv: 0502050.
  * (79) S. Mathur, “The quantum structure of black holes”, Class. Quant. Grav. 23 (2006) R115, arXiv:hep-th/0510180.
  * (80) “Non-supersymmetric smooth geometries and D1-D5-P bound states”, V. Jejjala, O. Madden, S. F. Ross, G. Titchener, Phys. Rev. D71 (2005) 124030, arXiv: hep-th/0504181.
  * (81) I. Kanitscheider, K. Skenderis, M. Taylor, “Fuzzballs with internal excitations”, JHEP 0706 (2007) 056, arXiv: 0704.0690
  * (82) S. D. Mathur, “Black hole size and phase space volumes’, arXiv:0706.3884
  * (83) C. Rovelli, “Black hole entropy from loop quantum gravity”, Phys. Rev. Lett. 77 (1996) 3288, arXiv:gr-qc/9603063.
  * (84) A. Ashtekar, J. Baez, A. Corichi, K. Krasnov, “Quantum Geometry and Black Hole Entropy”, Phys. Rev. Lett. 80 (1998) 904, arXiv:gr-qc/9710007.
  * (85) D. Christodoulou, “Nonlinear nature of gravitation and gravitational-wave experiments”, Phys. Rev. Lett. 67 (1991) 1486, DOI:https://doi.org/10.1103/PhysRevLett.67.1486
  * (86) S. Hawking, M. Perry, A. Strominger, “Soft Hair on Black Holes”, Phys. Rev. Lett. 116 (2016) 231301, ePrint: 1601.00921.
  * (87) S. Hawking, M. Perry, A. Strominger, “Superrotation Charge and Supertranslation Hair on Black Holes”, JHEP 05 (2017) 161, ePrint: 1611.09175.
  * (88) S. Pasterski, “Asymptotic Symmetries and Electromagnetic Memory”, JHEP 09 (2017) 154, doi: 10.1007/JHEP09(2017)154
  * (89) J. D. Bekenstein, “Black holes and entropy” Phys. Rev. D7 (1973) 2333, PRD7(1973)2333
  * (90) J. M. Bardeen, B. Carter, S. W. Hawking, “The Four laws of black hole mechanics”, Commun. Math. Phys. 31 (1973) 161, cmp31(1973)161.
  * (91) J. D. Bekenstein, “Generalized second law of thermodynamics in black hole physics”, Phys. Rev. D9 (1974) 3292. PRD9(1974)3292
  * (92) J. D. Bekenstein, “Universal upper bound on the entropy-to-energy ratio for bounded systems”, Phys. Rev. D23 (1981) 287, PRD23(1981)287
  * (93) R. Wald, “The thermodynamics of black holes,” Living Rev. Rel. 4 (2001) 6, gr-qc/9912119.
  * (94) V. Cardoso, P. Pani, “Tests for the existence of horizons through gravitational wave echoes”, Nature Astronomy 1 (2017) 586-591, 1709.01525.
  * (95) J.R. Oppenheimer and H. Snyder, “On continued gravitational contraction”, Phys. Rev. 56 (1939) 455.
  * (96) C. Misner, K. Thorne, J. Wheeler, “Gravitation”, “Gravitation” §32.3,32.4,32.6, W. H. Freeman and Company Version 1973.
  * (97) P. Yodzis, H.J. Seifert and H. Müller zum Hagen, “On the occurrence of naked singularities in general relativity”, Commun. Math. Phys. 34 (1973), 135.
  * (98) P. Yodzis, H.J. Seifert and H. Müller zum Hagen, “On the occurrence of naked singularities in general relativity. II”, Commun. Math. Phys. 37 (1974), 29.
  * (99) P. C. Vaidya, “Nonstatic Solutions of Einstein’s Field Equations for Spheres of Fluids Radiating Energy”, Phys.Rev. 83 (1951) 10; “The External Field of a Radiating Star in General Relativity”, Proc. Indian Acad. Sci. A33 (1951) 264; Reprinted, Gen. Rel. Grav. 31 (1999) 119. P. C. Vaidya, Proc. Indian Acad. Sci. A33, 264 (1951); Phys. Rev. 83 , 10 (1951). Gen. Relativ. Gravit. 31, 119 (1999);
  * (100) W. Bonnor and P. Vaidya, “Spherically Symmetric Radiation of Charge in Einstein-Maxwell Theory”, Ge. Relativ. Gravit. 1 (1970) 127.
  * (101) R. Penrose, “Gravitational Collapse and Space-Time Singularities”. Phys. Rev. Lett. 14 (3), 57(1965);
  * (102) R. Penrose, “Gravitational collapse: The role of general relativity”, Riv. Nuovo Cim. 1 (1969) 252–276. [Gen. Rel. Grav.34,1141(2002)].
  * (103) S. Hawking, “Breakdown of Predictability in Gravitational Collapse”, Phys. Rev.  D14,  2460 (1976)
  * (104) R. Geroch and G. Horowitz, “Global structure of spacetimes, in General Relativity: An Einstein Centenary Survey”, pp. 212–293, 1979.
  * (105) S. Hawking, G. F. R. Ellis, “The Large Scale Structure of Space-Time”, Cambridge University Press, ISBN 0-521-09906-4.
  * (106) C. W. Misner, K. S. Thorne and J. A. Wheeler, ”Gravitation”, Princeton University Press, 2017 version, ISBN-13,978-0691177793.
  * (107) Y. Liu, S.N. Zhang, “Exact solutions for shells collapsing towards a pre-existing black hole”, Phys. Lett. B679 (2009) 88-94, arXiv:0907.2574.
  * (108) S.N. Zhang, “On the Solution to the ”Frozen Star” Paradox, Nature of Astrophysical Black Holes, non-Existence of Gravitational Singularity in the Physical Universe and Applicability of the Birkhoff’s Theorem” Int. J. Mod. Phys. D20 (2011) 1891-1899, arXiv:1003.1359.
  * (109) , V. P. Frolov and I. D. Novikov, “Black Hole Physics”, (Kluwer, Dordrecht, 1998).
  * (110) A. Einstein & N. Rosen, “The Particle Problem in the General Theory of Relativity”, PR48(1935)73 Phys. Rev. 48 (1935) 73.
  * (111) R. W. Fuller & J. A. Wheeler, “Causality and Multiply Connected Space-Time”, PR128(1962)919, Phys.Rev. 128 (1962) 919.
  * (112) J. L. Friedman, K. Schleich, & D. M. Witt, “Topological Censorship”, Phys. Rev. Lett. 71 (1993) 1486, [Erratum-ibid. 75, 1872 (1995)], arXiv: gr-qc/9305017.
  * (113) G. J. Gallowayetal, “Topological Censorship and Higher Genus Black Holes”, Phys. Rev. D60 (1999) 104039, arXiv: gr-qc/9902061.
  * (114) A. Einstein, B. Podolsky, & N. Rosen, “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?”, PR47(1935)777, Phys. Rev. 47 (1935) 777.
  * (115) J. Maldacena, L. Susskind, “Cool horizons for entangled black holes”, Fortschr. Phys. 1–31 (2013),DOI 10.1002/prop.201300020, arXiv:1306.0533
  * (116) M. V. Raamsdonk, “Building up spacetime with quantum entanglement”, Gen. Rel. Grav. 42 (2010) 2323-2329, 2010; Int. J. Mod. Phys. D19 (2010) 2429-2435, arXiv:1005.3035
  * (117) B. Czech, et al, “The Gravity Dual of a Density Matrix”, Class. Quant. Grav. 29 (2012) 155009, 1204.1330
  * (118) N. Lashkari, M. B. McDermott, M. V. Raamsdonk, JHEP 04 (2014) 195, “Gravitational dynamics from entanglement ’thermodynamics”’, arXiv:1308.3716
  * (119) “Lectures on Gravity and Entanglement”, TASI 2015, 297-351, arXiv:1609.00026
  * (120) M. V. Raamsdonk, “Building up spacetime with quantum entanglement II: It from BC-bit”, arXiv:1809.01197
  * (121) E. Witten, “Three Dimensional Gravity Reconsidered”, arXiv: 0706.3359
  * (122) M. Ba$\tilde{n}$ados, C. Teitelboim, J. Zanelli, “The Black Hole in Three Dimensional Space Time”, Phys. Rev. Lett. 69 (1992) 1849, hep-th/9204099.
  * (123) M. Banados, “Global Charges in Chern-Simons theory and the 2+1 black hole”, Phys. Rev. D52 1996 (5816), hep-th/9405171.
  * (124) S. Weinberg, “Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity”, §11.7-9, Wiley and Sons, New York, 1972, ISBN-13: 978-0471925675
  * (125) S. Ramanujan and G. Hardy, Proc. London Math. Soc.(ser.2) 17 75, 1918, reprinted in G. Hardy et al (ed), Collected papers of Srinivase Ramanujan (New York: Chelsea).
  * (126) wikipedia, ”Composition(combinatorics)” entry, wiki:Composition
  * (127) R. A. Porto, “The effective field theorist’s approach to gravitational dynamics”, Phys. Rept. 633 (2016) 1-104, missinghttps://arxiv.org/abs/1601.04914arXiv:1601.04914
  * (128) W. D. Goldberger and I. Z. Rothstein, “An effective field theory of gravity for extended objects”, Phys. Rev. D73 (2006) 104029, arXiv:hep-th/0409156.
  * (129) W. D. Goldberger, “Les houches lectures on effective field theories and gravitational radiation”, Les Houches Summer School—Session 86: Particle Physics and Cosmology: The Fabric of Spacetime Les Houches, France, July 31–August 25, 2006, 2007, arXiv:hep-ph/0701129.
  * (130) D. Bini, T. Damour, G. Faye, “Effective action approach to higher-order relativistic tidal interactions in binary systems and their effective one body description”, Phys. Rev. D85 (2012) 124034. arXiv:1202.3565.
  * (131) S. Chakrabarti, T. Delsate, J. Steinhoff, “New perspectives on neutron star and black hole spectroscopy and dynamic tides”, arXiv:1304.2228.
  * (132) S. Chakrabarti, T. Delsate, J. Steinhoff, “Effective action and linear response of compact objects in Newtonian gravity”, Phys. Rev. D88 (2013) 084038. arXiv:1306.5820.
  * (133) D. Bini, A. Geralico, “Tidal invariants along the worldline of an extended body in kerr spacetime”, Phys. Rev. D91 (8) (2015) 084012. 1806.07696
  * (134) T. Binnington, E. Poisson, “Relativistic theory of tidal love numbers”, Phys. Rev. D80 (2009) 084018. arXiv:0906.1366.
  * (135) T. Damour, A. Nagar, “Relativistic tidal properties of neutron stars”, Phys. Rev. D80 (2009) 084035, arXiv:0906.0096.
  * (136) T. Hinderer, “Tidal love numbers of neutron stars”, Astrophys. J. 677 (2008) 1216–1220, arXiv:0711.2420.
  * (137) T. Hinderer et al, “Tidal deformability of neutron stars with realistic equations of state and their gravitational wave signatures in binary inspiral”, Phys. Rev. D81 (2010) 123016. arXiv:0911.3535.
  * (138) J. Vines, E. E. Flanagan, T. Hinderer, “Post-1-Newtonian tidal effects in the gravitational wave form from binary inspirals”, Phys. Rev. D83 (2011) 084051, arXiv:1101.1673.
  * (139) K. Yagi, N. Yunes, “I-Love-Q relations in neutron stars and their applications to astrophysics, gravitational waves and fundamental physics”, Phys. Rev. D88 (2) (2013) 023009, arXiv:1303.1528.
  * (140) D. D. Doneva et al, “Breakdown of i-love-q universality in rapidly rotating relativistic stars”, Astrophys. J. 781 (2013) L6. arXiv:1310.7436.
  * (141) S. Bernuzzi et al, “Modeling the dynamics of tidally interacting binary neutron stars up to the merger”, Phys. Rev. Lett. 114 (16) (2015) 161103, arXiv:1412.4553.
  * (142) M. Agathos et al, “Constraining the neutron star equation of state with gravitational wave signals from coalescing binary neutron stars”, Phys. Rev. D92 (2) (2015) 023012; arXiv:1503.05405.
  * (143) K. Chatziioannou et al, “Probing the internal composition of neutron stars with gravitational waves”, Phys. Rev. D92 (2015) 104008; arXiv:1508.02062.
  * (144) The LIGO Scientific Collaboration, the Virgo Collaboration, “Observation of Gravitational Waves from a Binary Black Hole Merger”, Phys. Rev. Lett. 116 (2016) 061102; arXiv:1602.03837.
  * (145) B. P. Abbott et al. [LIGO Scientific and Virgo], Phys. Rev. Lett. 119, no.16, 161101 (2017) doi:10.1103/PhysRevLett.119.161101 arXiv:1710.05832.
  * (146) T. Helfer, E. A. Lim, M. A. Garcia, M. A. Amin, “Gravitational Wave Emission from Collisions of Compact Scalar Solitons”, arXiv: 1802.06733
  * (147) V. Cardoso, S. Hopper, C F. B. Macedo and et al, “Gravitational-wave signatures of exotic compact objects and of quantum corrections at the horizon scale”, Phys. Rev. D94 (2016) no.8, 084031, arXiv: 1608.08637.
  * (148) J. Abedi, H. Dykaar, N. Afshordi, “Echoes from the Abyss: Tentative evidence for Planck-scale structure at black hole horizons”, Phys. Rev. D96 (2017) 082004, arXiv: 1612.00266
  * (149) B. S. DeWitt, “Quantum Theory of Gravity. I. The Canonical Theory”, Phys. Rev. 160, 1113 (1967);
  * (150) J. J. Halliwell, “INTRODUCTORY LECTURES ON QUANTUM COSMOLOGY”, Proceedings of the Jerusalem Winter School on Quantum Cosmology and Baby Universes edited by S. Coleman, J. B. Hartle, T. Piran and S. Weiberg (World Scientific, Singapore, 1991);
  * (151) M. Allen, “Canonical quantization of a spherically symmetric, massless scalar field interacting with gravity in 2+1 dimensions”, Class. Quantum Grav. 4(1987) 149.
  * (152) L. Z. Fang and M. Li, “Formation of black holes in quantum cosmology”, Phys. Lett. B169 (1986) 28.
  * (153) R. Laflamme, “Wavefunction of a black hole interior”, in Origin and Early History of the Universe: Proceedings of the 26th Liege International Astrophysical Colloquium, ed J. Demaret.
  * (154) Y. Nambu and M. Sasaki, “The wavefunction of a collapsing dust sphere inside the black hole horizon”, Prog. Theor. Phys. 79 (1988) 96.
  * (155) H. Nagai, “Wavefunction of the de Sitter-Schwarzschild universe”, Prog. Theor. Phys. 82 (1989) 322.
  * (156) L. Rodrigues, I. Soares and J. Zanelli, “Black hole decay and topological stability in quantum gravity”, Phys. Rev. Lett. 62 (1989) 989.