Following the Dirac program for constrained systems, we distinguish a kinematic state space ℋ kin —carrying all degrees of freedom before constraints are imposed—from the physical state space of the DT-CPS. Let ℋ DT-CPS denote the latter. Unlike conventional quantum Hilbert spaces, ℋ DT-CPS is not equipped with a unitary time-evolution operator. Instead, it is defined as the physical subspace of ℋ kin annihilated by all dense-time constraints:

where the constraints C ^ a encode:

saturation of the time-density field ( t ^ ( x ) = t ^ sat ),absence of an ordinary finite-frequency radiative propagating sector,conservation of total deformation energy.

Throughout this paper ℋphys ≡ ℋDT-CPS ≡ ℋDT; the abbreviated forms are used wherever no ambiguity arises. For the constraint system to be consistent, the constraints must close under commutation. We state the algebraic closure condition:

where f a b       c and g a     b are structure functions. Writing this condition establishes that the interior theory is a genuine Dirac constraint system. The structure functions can be computed directly from the explicit constraint operators (54) and the canonical equal-time commutation relations of the timeon field.

The canonical equal-time commutation relations inherited from the parent Lagrangian (38) are

with all other equal-time commutators vanishing. The two dense-time constraint operators are (Equation (54)):

a)Inter-constraint commutator [ C ^ 1 , C ^ 1 ] . Since t ^ ( x ) is a configuration-space operator,

b) Inter-constraint commutator [ C ^ 2 i , C ^ 2 j ] . Since θ ^ ( x ) is also a configuration-space operator,

c)Cross commutator [ C ^ 1 , C ^ 2 i ] . Since t ^ and θ ^ are independent field degrees of freedom,

d) Hamiltonian-constraint commutators. On the dense-time constraint surface,

the static Hamiltonian density (48) reduces to V ( t ^ sat 2 ) in the bulk interior (gradient terms vanish by (49)). The integrated Hamiltonian ℋ ^ DT depends on t ^ 2 through the potential V ( t ^ 2 ) and on θ ^ through t ^ 2 ( ∇ θ ^ ) 2 . Both are configuration-space functionals; therefore

e) Result: Abelian closure. All structure functions vanish:

The DT-CPS constraint algebra is Abelian: the constraints are first-class and mutually commuting, and they commute with the Hamiltonian strongly (not merely on the constraint surface). This is the strongest possible closure result—it means the constraint system is internally consistent without requiring any second-class constraints or Dirac brackets, and the physical Hilbert space ℋ phys (Equation (8)) is well-defined without any additional regularization. The Abelian structure is a direct consequence of the saturation constraint ( t ^ = t ^ sat ) being a function only of t ^ , and the phase-quench constraint ( ∇ θ ^ = 0 ) being a function only of θ ^ —the two fields are canonically independent.

Inner products are defined as usual,

but expectation values correspond to static observables rather than time-dependent operators.

We take the interior to be described by a complex timeon field,

but in the dense phase the following physical constraints apply:

1) Saturation: the amplitude is driven to a plateau value,

2) No phase-transport: extended phase gradients cannot be supported, so the interior cannot sustain coherent spatial phase structure:

3) No time-parametrized dynamics: time derivatives are not physically meaningful inside the DT-CPS. Hence, the interior action contains no ∂ t terms.

Because the dense-time phase cannot support time-parametrized propagation, the appropriate interior functional is a three-dimensional Euclidean action defined on spatial slices of the interior volume V int :

where h is the determinant of the induced spatial metric (in the spherical ansatz of Sec. 12 this factor is absorbed into the radial measure r 2 d r ).

The minimal dense-time Lagrangian density consistent with: 1) finite energy, 2) saturation, 3) absence of propagating modes, is

where Z t and Z θ are the stiffness moduli (units of Force) inherited from the BPS theory, and V DT ( t ^ ) is a dense-phase potential that enforces saturation at t ^ sat . Its exact algebraic form from first principles remains an open function of the theory (see Section 15.12); a model-agnostic expansion about the saturation point is:

so that deviations from saturation carry an energetic penalty. The coefficients λ 2 , λ 4 are not free parameters to be fit but stand-ins for a derivation from the microscopic BPS sector; until that derivation is completed, the precise energetic penalty for deviations from saturation and the dynamical timescale τ C S of the Chrono-Shear transition (Equation (114)) cannot be computed from first principles.

To impose dense-time physics as hard constraints rather than soft penalties, we introduce Lagrange multiplier fields μ ( x ) and ν ( x ) :

Here:

μ enforces saturation pointwise, ν enforces phase-quench (no phase texture), S Σ is the boundary action at the Chrono-Quantum Mirror Σ.

a) Saturation constraint:

b) Phase-quench constraint:

Under the dense-time constraints (29) and (30), this reduces to a static balance condition determining μ :

Under ∇ θ = 0 , (33) becomes a compatibility condition for ν .

To establish the CQM as a proper matching surface rather than only a narrative interface, we impose Israel junction conditions at Σ [14][15]. Let h a b denote the induced metric on Σ and K a b its extrinsic curvature. The two junction conditions are:

where [ ⋅ ] Σ denotes the jump across Σ and S a b ( Σ ) is the effective surface stress-energy of the CQM shell. The first condition requires the induced metric to be continuous across the boundary. The second relates the jump in extrinsic curvature to the shell stress-energy, which encodes the interface tension σ ( t ^ ) (one of the three closing functions of Section 15.12). Even with S a b ( Σ ) left model-dependent, writing these conditions establishes that the boundary is a proper matching surface in the sense of general relativity, and that its energetics are fully determined once σ ( t ^ ) is specified.

Start from a standard Lorentz-invariant parent Lagrangian for the timeon field,

so that (in t ^ , θ variables)

The canonical momenta are then

In the dense-time phase, the timeon field has reached the dense phase discrete state critical limit: t ^ = t ^ sat . At saturation the field carries little remaining phase-gradient budget to support propagating excitations—the available degrees of freedom are bounded deformations of the lattice, redshifted waves. The physical statement is therefore that time-parametrized transport and time-parametrized wave dynamics in conventional radiative bands have little physical support. Operationally, the allowed interior configurations approach the static limits

so the canonical momenta vanish on the dense-time constraint surface:

The vanishing of the canonical momenta does not imply the complete disappearance of wave dynamics. Rather, the extreme temporal dilation of the dense-time phase shifts all propagating modes to ultra-low frequencies—far below conventional electromagnetic or radiative bands, but not to zero.

The timeon phase field θ carries propagating excitations with bare dispersion relation ω 0 ( k ) in the exterior vacuum phase. In the dense-time interior these modes are dressed by the local time-density field through a temporal dilation factor Z t :

where Z t is the stiffness modulus of the dense-phase Lagrangian (Section 15.12). As gravitational compression deepens into the dense-time regime, Z t ≫ 1 , so that

Modes are not eliminated; they are stretched to extremely long periods.

The dense-time state is further organized into discrete stable phase levels t ^ n , analogous to quantized orbital states, which correspond to local minima of the effective dense-phase potential V eff ( t ^ ) . Under continued gravitational compression the system does not collapse continuously but settles into successive stable configurations:

where δ t ^ is a small oscillatory perturbation about the stable minimum. These perturbations are slow temporal oscillations of the dense-time field itself. Because the underlying temporal scale is strongly dilated at each level t ^ n , such oscillations manifest as ultra-low-frequency modes—potentially in the kilohertz, hertz, or sub-hertz regime—rather than as conventional high-frequency radiation.

The dense-time interior may therefore be understood as a redshift-saturated slow-wave resonant cavity: oscillatory dynamics persist, but at periods so long that the interior produces no signal in any conventional radiative band. Whether these ultra-low-frequency modes couple to observable channels outside the CQM depends on the low-frequency behavior of the reflectivity G ( ω ) at ω ≪ ω 0 —a quantity that is not yet computed and constitutes an open prediction of the framework. A complete calculation of G ( ω ) requires specifying the CQM constitutive law σ ( t ^ ) and solving the wave equation across the finite-thickness shell; the result would determine whether ULF interior modes produce any imprint on exterior gravitational-wave strain at frequencies accessible to current or planned detectors such as LISA or the Einstein Telescope. What the framework does establish is that the absence of high-frequency radiative signatures from black hole interiors is not evidence for the absence of all dynamics, but for the extreme redshifting of those dynamics below detectable thresholds.

The Hamiltonian density obtained from (39) is

Restricting to the dense-time constraint surface (43) gives the static Hamiltonian density:

Inside the DT-CPS, the dense-phase constraints are:

To enforce these as hard constraints, define the constrained Hamiltonian functional

Here H Σ is the boundary energy associated with the Chrono-Quantum Mirror Σ. A minimal penalty form (kept general) is:

In a star or planet, pressure does work on wave-supporting microscopic degrees of freedom: collisional and radiative channels exist, enabling thermalization and temperature rise. In the DT-CPS, those channels do not exist.

On the dense-time constraint surface, (49) implies

so the gradient contributions in (48) vanish, leaving

Thus, the interior energy is stored primarily as latent chrono-pressure (potential-sector energy), not as thermal occupation of propagating modes.

Because unitary time evolution is not the organizing principle in the t ^ 2 phase, the quantum description is naturally phrased as a sector projection. Let ℋ Φ be the timeon Hilbert space from the BPS theory. Define the dense-time constraint operator(s) schematically as

and define the dense-time physical subspace as

Throughout this paper ℋ DT ≡ ℋ DT-CPS ; the abbreviated form is used wherever no ambiguity arises.

Assume an effective spherical symmetry about the compact object. Let r denote the areal radius and define three regions:

1) Interior (DT-CPS): 0 ≤ r < r Σ − , where the dense-time constraints hold:

2) CQM shell: r Σ − ≤ r ≤ r Σ + , a finite-thickness layer of width Δ Σ ≡ r Σ + − r Σ − in which the timeon field transitions between dense and exterior phases.

3) Exterior: r > r Σ + , where the timeon field relaxes to the low-density phase:

Write the timeon in polar form,

and adopt a purely radial ansatz in the shell:

To tie the dense-time matter sector directly to the spherical geometry, define the interior mass function m ( r ) by

so that the standard Schwarzschild-like redshift factor

plays its usual role in the interior line element. Even though the full metric is not solved in closed form here, this relation establishes exactly how the dense-time matter sector couples into the spherical geometry and recovers r Σ = 2 G M / c 2 at the CQM boundary when m ( r Σ ) = M .

The relevant functional to minimize in the shell is the static energy functional:

where V Σ is the effective shell potential.

Varying (62) yields the shell equations:

a) Amplitude equation

b) Phase equation

where J is a conserved radial phase-current constant across the shell.

The DT-CPS interior approaches zero phase current. Therefore,

With d θ / d r = 0 , (63) simplifies to

a) (A) Center regularity(interior)

b) (B) Inner shell matching at r = r Σ −

c) (C) Outer shell matching at r = r Σ +

d) (D) Far-fieldasymptotics

We impose the operational anchoring:

We treat the DT-CPS interior as a coarse-grained continuum with four-velocity u μ ( u μ u μ = − 1 ) and rest-frame energy density ρ DT . To leading order it is a perfect fluid:

To capture departures from equilibrium during rapid merger/relaxation, we add standard first-order dissipative terms following the formalism of Landau and Lifshitz [16]:

is the shear tensor.

When the interior saturates,

so the free-energy cost of transverse displacement vanishes to leading order. We therefore model an effective shear modulus G eff collapsing,

We introduce an EOS parameterization:

with corresponding adiabatic sound speed

The value of w DT is uniquely determined by the DT-CPS constraint structure; the derivation is given in Section 14.4.

Define an effective surface tension for the CQM shell:

A robust timescale in relativistic compact-object dynamics is t g ≡ G M / c 3 . In the dense-time framework, the interior does not dynamically “flow” to a new equilibrium because internal time evolution is suppressed ( ∂ t t ^ ≈ 0 ). Instead, the relaxation is driven by the exterior boundary dynamics and the shedding of angular momentum via gravitational radiation.

The interior updates quasi-statically, instantly satisfying the new boundary constraints imposed by the relaxing Chrono-Quantum Mirror. The observable ringdown timescale is therefore determined by the CQM response stiffness, not internal viscosity:

where κ characterizes the geometric response of the shell.

We distinguish radiative temperature from effective bookkeeping temperature:

T rad (radiative/Planck temperature) versus

If photon modes are nearly unsupported and redshifted near zero, a Planck bath cannot form [17]:

In this sense, the onset of the dense-time phase does not correspond to heating or cooling in the conventional thermodynamic meaning. Rather, when a collapsing object crosses into the dense-time phase, its interior loses access to heat as a physical concept: wave-mediated equilibration channels nearly cease to exist, radiative temperature becomes undefined, and gravitational work is stored as latent chrono-pressure instead of thermal excitation. Nevertheless, we introduce an effective temperature by a local Gibbs relation:

In the simplest closure, d n eff = 0 in the bulk:

Let s μ = s DT u μ . The local second law is

A standard viscous form gives

Local energy conservation is

We split the bulk energy density into thermalizable and latent parts:

Define the latent chrono-pressure:

Here P χ (capital) denotes the total pressure (force per area) and p χ (lower case) the energy-density form; the two are related by P χ = p χ in the bulk uniform limit.

The “no heating” condition is:

The DT-CPS interior obeys p DT = w DT ρ DT (Equation (77)), and all accreted energy is stored as latent chrono-pressure (Equation (90)):

With r Σ = 2 G M / c 2 (Equation (104)):

giving the chrono-pressure

P χ ∝ M − 2 : the chrono-pressure falls as mass grows because interior volume grows as M 3 while stored energy grows as M . The product P χ ⋅ V int is, however,

which requires

the Zeldovich stiff-fluid limit [18][19]. The equation-of-state parameter w DT ≡ p DT / ρ DT is dimensionless in every unit system; here ρ DT denotes the dense-time energy density (so that p DT and ρ DT carry the same units), and the identification above is unit-independent.

To see this as a compact derivation rather than a persuasive argument: in the dense phase, all energy is stored as latent compression energy with little or no thermalizable remainder, so

hence

The causal consistency remark follows immediately:

identifying w DT = 1 as the limiting causal equation of state—not an arbitrary phenomenological choice. This value is selected not merely because it is convenient, but because it is the unique barotropic value consistent with: 1) maximally stored latent compression energy, 2) absence of an ordinary radiative heat bath, and 3) causal saturation.

This EOS is not assumed; it is the unique value consistent with both the DT-CPS structure and the deconfinement condition, and it corresponds to the maximally compact saturated condensate expected on independent grounds. Equation (95) then holds identically for all M : the DT-CPS is always at the energy threshold of its own deconfinement. The CQM prevents the CSE from firing until the geometric condition (105) is met.

Introduce an effective free-energy density ℱ DT ( t ^ ) :

Identify

Allow an interface first-law form:

where T Σ is not a radiative Hawking temperature but the conjugate to boundary entropy in the CQM effective theory.

1) Radiative darkness is intrinsic: T rad is not a meaningful bulk state variable.

2) Merger energy partitions gravitationally: free energy prefers GW/ringdown over EM.

3) Compression stores latent energy: accretion increases the integrated latent energy E χ and the global lattice strain load, even though the local chrono-pressure density scales as P χ ∝ M − 2 under the spherical volume law of Equation (94).

The Chrono-Shear Event is the dynamical failure of a metastable dense-time configuration once gravitational compression exceeds the stability scale of the Timeon Lattice Potential (TLP) landscape. Within this landscape the DT-CPS—including the full Baryon Partner State sector—occupies a metastable basin whose depth sets the binding energy of the configuration. As mass accumulates through hierarchical coalescence, the stored energy grows until the compression energy becomes comparable to the basin depth:

at which point the metastable configuration can no longer remain confined. The geometric expression of this condition is Equation (105): the CQM radius reaches the domain horizon, the confining exterior ceases to exist, and Δ E TLP is released as E CSE = M crit c 2 .

Once the threshold is crossed the time-density field evolves rapidly away from the metastable minimum t ^ 2 toward the shear phase t ^ 3 . The transition proceeds on a dynamical timescale set by the curvature of the potential at the saddle:

which is rapid relative to the cosmological epoch but whose precise value depends on the open function V eff ( t ^ ) (Section 15.12).

The released energy E CSE = M crit c 2 drives rapid expansion of the previously confined configuration. To observers embedded within the expanding daughter domain, this transition appears as the sudden onset of expansion of an extremely dense state—a global Chrono-Shear instability that restores propagating vacuum conditions and initiates a new causal expansion phase. It is not energy created from nothing; it is energy released from the metastable TLP basin and reorganized into propagating vacuum-wave modes. The CSE therefore provides a physical mechanism for a new causal onset that requires no initial singularity, no inflation, and no fine-tuning of initial conditions beyond the geometry of the preceding domain.

The following subsections develop the formal quantum mechanics underlying this picture.

We take the timeon field Φ to remain the fundamental order parameter,

with t ^ the local time-density amplitude. The phases relevant here are:

Atomic phase t ^ a : localized BPS texture exists.Dense phase t ^ 2 : DT-CPS saturation, ∇ t ^ ≈ 0 , wave-mediated transport suppressed.Shear phase t ^ 3 : a transient reconfiguration regime in which global causal foliation fails, but local ordering remains meaningful in patches. During this phase the standard global Hilbert space ℋ DT loses its support (the constraint operators C ^ 1 and C ^ 2 cannot be simultaneously satisfied), yet local conservation laws are preserved in causally connected subregions. A rigorous mathematical treatment of this regime likely requires the machinery of algebraic quantum field theory or non-commutative geometry to formalize how patchwise Hilbert sectors are defined and how unitarity is maintained locally while global time-ordering fails; this formal framework constitutes a major open direction for the theory and is left for future work.

The CSE is the transition t ^ 2 → t ^ 3 .

To describe the CSE as a decay of a metastable state, we introduce an effective field theory for the t ^ amplitude in the dense regime. The minimal Lorentz-invariant (or near-invariant) bulk action is

where Z t , Z θ are effective stiffness parameters in the dense regime and V eff ( t ^ ) is an effective potential with (at least) a metastable minimum at t ^ = t ^ 2 and a deeper or runaway channel associated with the shear phase. We emphasize that in the dense phase the phase degree θ is nearly unsupportive of long-range coherent transport; nevertheless, its local gradients still contribute to energy.

A minimal metastable potential structure is:

with parameters chosen so that t ^ = t ^ 2 is a local minimum separated by a barrier from a decay channel toward a shear configuration.

The CSE is not purely a bulk instability; it is precipitated by failure of the confining interface Σ. We encode the interface energetics with an effective surface action

where γ a b is the induced metric on Σ, σ ( t ^ ) is an effective surface tension, and κ Σ is a surface stiffness. The net effect is that the DT-CPS interior behaves as a high-pressure medium confined by a finite-strength membrane. The CSE occurs when the effective membrane fails:

Here P yield is the maximum confining stress supportable by the CQM given the ambient lattice strain environment.

To compute the rate of a metastable decay, we utilize the Euclidean bounce formalism developed by Coleman and Callan [20][21]. We pass to Euclidean time τ = i t and define the Euclidean action S E :

The semiclassical decay rate per four-volume takes the standard bounce form

where t ^ bounce is the Euclidean saddle (bounce) that connects the metastable dense minimum to the shear channel.

A standard vacuum decay in scalar field theory is mediated by an O ( 4 ) -symmetric bounce, t ^ = t ^ ( ρ ) , ρ 2 = τ 2 + r 2 , satisfying

with boundary conditions

However, the Chrono-Shear Event differs from ordinary bubble nucleation in one crucial way: the dense phase does not support stable intermediate gradients over macroscopic scales. As argued earlier, the dense regime suppresses wave-mediated propagation and long-lived phase interfaces. Consequently, the dominant saddle is not generically a slowly expanding bubble wall but a near-simultaneous global failure mode. We capture this by allowing the effective stiffness Z t to diminish as t ^ → t ^ 2 , so that gradient energy is strongly penalized or becomes ill-defined:

which drives the bounce toward a “thin” (or effectively nonlocal) transition. In this limit, the decay resembles a collective tunneling of the interior into t ^ 3 rather than a propagating front.

We use acausal in a technical sense: during the CSE there is no single global time slicing that orders the entire transition. Nevertheless, local evolution remains well-defined within finite patches. We encode this by treating the transition as unitary in a patchwise Hilbert decomposition:

(125)

where

is a covering of the transition region by patches over which an approximate local Hamiltonian H p exists. Global foliation fails because the patch couplings are non-adiabatic and the boundary conditions at Σ change discontinuously. This provides a controlled meaning to “acausal”: no global time-ordering, but no violation of local conservation laws.

When the confining geometry fails, E CSE = M crit c 2 (Equation (112)) deconfines instantaneously. This energy partitions across two channels by causal domain.

In the parent domain, the observable channel is gravitational radiation:

where ϵ GW is the fraction coupling to exterior gravitational modes. The remainder ( 1 − ϵ G W ) M crit c 2 crosses the t ^ 3 causal boundary as pure deconfined timeon energy seeding the daughter domain [11][12]. The full E CSE = M crit c 2 is conserved across both channels; ϵ GW ≪ 1 is not in contradiction with total energy release but reflects the partition between domains. In the parent domain the event registers as an impulse-like gravitational transient at Σ. In the daughter domain the same event is the causal-onset expansion—the new causal expansion phase of the next domain, its low entropy guaranteed by the maximal order of the precursor DT-CPS ground state.

The shear phase t ^ 3 following the CSE is a pure vacuum-wave configuration of the timeon lattice—structurally equivalent to the pre-onset state t ^ 1 of the present domain [11]. Chrono-Emergence, which operates continuously on a timescale vastly exceeding any causal-domain lifetime, now acts on a lattice reset to that vacuum configuration and seeded with E CSE = M crit c 2 of pure field energy. The new causal expansion—initiated by the global Chrono-Shear instability—proceeds from this reset state. Matter does not form instantaneously at the CSE; it accumulates through the same slow Chrono-Emergence that populated the parent domain, now operating on a fresh lattice.

The causal reset is closed and exact:

where the CSE arrow is instantaneous and the causal expansion arrow spans the entire lifetime of a domain. Chrono-Emergence runs beneath both arrows on its own independent, far longer timescale—the continuous creative background against which causal domains are born, age, and reset. The low entropy of each new causal onset follows from the maximal order of the saturated DT-CPS ground state that immediately preceded it.

The CSE energy partition, and therefore the initial conditions of the daughter domain, depends on the geometry of the final merger. Two properties of that merger are determined by the long prior history of hierarchical coalescence; one is intrinsically variable.

a) Spin cancellation in hierarchical mergers. Each merger adds an angular momentum contribution whose orientation is set by the orbital geometry of that encounter. Over N mergers with uncorrelated orientations, the net spin follows a three-dimensional random walk:

where J 0 is a characteristic single-merger angular momentum scale. With M = N M 0 , the dimensionless spin parameter is

Spin is suppressed as N − 3 / 2 : the two black holes entering the final merger are very nearly Schwarzschild. This is a consequence of the merger history, not a special assumption.

b) The grace parameterG. Near-zero spin does not fix the merger geometry completely. Mass ratio q = M 2 / M 1 ≤ 1 and impact parameter b are set by the stochastic encounter of the last two surviving objects in the domain. We define a dimensionless grace parameter

where 4 q / ( 1 + q ) 2 is the symmetric mass ratio normalized to unity at q = 1 [22], g ( b / r Σ ) is a geometric efficiency with g ( 0 ) = 1 falling for oblique encounters, and h ( a ) ≤ 1 captures residual-spin decoherence. Equal-mass, head-on, non-spinning gives G = 1 ; unequal, grazing, spinning gives G ≪ 1 .

c)Yield and daughter-domain entropy. The GW energy fraction radiated into the parent domain is

where ϵ 0 is the maximum timeon-field radiative coupling. The complementary fraction ( 1 − ϵ G W ) seeds the daughter domain. A merger with G ≈ 1 releases E CSE coherently: the daughter domain receives a homogeneous energy density at causal onset, a low-entropy initial state, and a near-uniform timeon vacuum from which an isotropic expansion proceeds. A merger with G ≪ 1 imprints spatial gradients on that energy: the daughter domain begins with higher entropy and a less uniform causal onset.

The isotropy of a daughter domain is therefore encoded in a single number—the grace of the terminal merger of its parent. Our own universe provides a calibration: the cosmic microwave background has δ T / T ~ 10 − 5 , implying extraordinarily uniform initial conditions. Within this framework that uniformity is a retrodiction: