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ITM-TP-2017-1106B
Version: 01-09-02-2017
Authors: ■■ ■■■■■■■, ■■ ■■■■, ■■ ■■■■, ■■ ■■■■■■■, ■■■■ ■■■■■■■■■
Document Control Number: ■■■■■■■■■■■■■■■■■■■
Original file is located at e6b9f0-access.ref/rd-server-amix/index/jun17/base/refs/docs/notebooks
Core Tachyonic Scalar Field
Begin with real scalar field Φ whose Lagrangian is:
math
\mathcal{L}\Phi = -\tfrac12(\partial\mu \Phi)(\partial^\mu \Phi) + \tfrac12\mu^2\Phi^2 - \tfrac{\lambda}{4}\Phi^4,
Where (\mu^2>0) makes the mass term tachyonic (unstable at (\Phi=0)). Initial parameter constraints from ■■■■-■■■■■■■ ■■■■■ determine the scale of (\mu) to be of order GeV.
Equation of motion:
math
\Box\Phi + \mu^2\Phi - \lambda\Phi^3 = 0.
Vacuum expectation value (true minima):
math
v = \sqrt{\frac{\mu^2}{\lambda}}.
Indirect measurements from ■■■■-■■■■■■■ ■■■■■ constrain vacuum expectation value $v$ to be of TeV order.
Expand around the true vacuum (\Phi = v + \eta). To quadratic order in (\eta):
math
V(\Phi) = -\tfrac12\mu^2\Phi^2 + \tfrac14\lambda\Phi^4,
V(v+\eta) = V(v) + \tfrac12 m^2 \eta^2 + O(\eta^3),
m^2 = V''(v) = 2\mu^2 \quad\Longrightarrow\quad m = \sqrt{2},\mu.
Detection of Φ-quanta reported in Array data (Ref: EXP-09-DAT-077).
1D Kink Soliton
In one spatial dimension, static solutions obey:
math
\frac{d^2\Phi}{dx^2} = -\mu^2\Phi + \lambda\Phi^3.
The well-known kink solution that interpolates between (-v) and
+v+v
is:
math
\Phi_{\rm kink}(x) = v,\tanh\Bigl(\frac{\mu x}{\sqrt{2}}\Bigr).
Energy per unit area:
math
E_{\rm kink} = \int_{-\infty}^{\infty} \Bigl[\tfrac12\Bigl(\frac{d\Phi}{dx}\Bigr)^2 + V(\Phi)\Bigr],dx
= \frac{2\sqrt{2}}{3}\frac{\mu^3}{\lambda}.
Analogues of such domain walls are being sought in material interfaces under study, Ref: MAT-PROG-■■■■■■-■■■■).
We introduce a gauge field
AμAμ
deformed by a Φ-dependent Born–Infeld kinetic term:
math
\mathcal{L}{BI} = b(\Phi)^2\Bigl[1 - \sqrt{1 + \frac{F{\mu\nu}F^{\mu\nu}}{2b(\Phi)^2} - \frac{(F_{\mu\nu}\tilde F^{\mu\nu})^2}{16b(\Phi)^4}}\Bigr],
where the BI scale function is
math
b(\Phi) = b_0 + \frac{B_1 - b_0}{1 + \alpha,\Phi^2}.
Constraints on (\alpha) and
b0b0
are derived from high-field EM experiments (Ref: EXP-HF-RES-004).
In the limit
b0=B1,α=0b0=B1,α=0
, this reduces to the ordinary
−14F2−41F2
term.
Seeking static, finite-energy solitons with winding number
nn
, we use the cylindrically symmetric ansatz:
math
\Phi(\mathbf{x}) = \Phi(r),\quadA_\theta(r) = \frac{n}{g,r}\bigl[1 - f(r)\bigr],
all other components vanish. (Gauge coupling
gg
constrained by Low Energy Test).
Define the dimensionless quantity
math
Y(r) = \frac{n^2 f'(r)^2}{g^2 r^2 b(\Phi(r))^2}.
The equations of motion reduce to the coupled ODEs:
\begin{cases}
\Phi'' + \tfrac{2}{r}\Phi' &= -\mu^2\Phi + \lambda\Phi^3 + \frac{db}{d\Phi}\,\frac{\partial\mathcal{L}_{BI}}{\partial b},\\
\frac{1}{r}\frac{d}{dr}\Bigl(\frac{r f'}{\sqrt{1+Y}}\Bigr) &= g^2\Phi^2 f.
\end{cases}
Boundary conditions ensuring finite energy:
Φ′(0)=0,Φ(∞)=v;f(0)=1,f(∞)=0.
Φ′(0)=0,Φ(∞)=v;f(0)=1,f(∞)=0.
The mass (energy) of the soliton is
math
M_n = 2\pi \int_0^\infty r\Bigl[ \tfrac12 (\Phi')^2 + V(\Phi) + b^2(\sqrt{1+Y}-1) \Bigr],dr.
See technical document ■■■■■■■■ ■■■■■■■■■■■■. (Also Data Archive: FTM/SOL/DAT/ evidence for
n=1n=1
vortex formation in material reported in EXP-VOR-PRELIM)
General action in 3+1 dimensions with Born–Infeld kinetic terms and a tachyonic quartic potential:
S[Φ]=∫d4x [−1bs(Φ)(1+bs(Φ) ∂μΦ ∂μΦ−1)−V(Φ)],
S[Φ]=∫d4x[−bs(Φ)1(1+bs(Φ)∂μΦ∂μΦ
−1)−V(Φ)],
where
bs(Φ)=bs0+Bs1Φ2+⋯bs(Φ)=bs0+Bs1Φ2+⋯
is a smooth, positive kinetic function for the scalar sector (parameters from Simulation Set ■■■■-QBS),
The potential is
V(Φ) = -\tfrac12 m^2 Φ^2 + \tfrac{\lambda}{4}Φ^4,
\quad m^2>0,;\lambda>0.
]
In the small-gradient limit
bs(Φ)(∂Φ)2≪1bs(Φ)(∂Φ)2≪1
, this reduces to the standard (-\tfrac12(\partialΦ)^2).
For time-independent, spherically symmetric configurations
Φ(r)Φ(r)
, the static energy is:
E[Φ]=4π∫0∞dr r2[1bs(Φ)(1+bs(Φ)(Φ′)2−1)+V(Φ)].
E[Φ]=4π∫0∞drr2[bs(Φ)1(1+bs(Φ)(Φ′)2
−1)+V(Φ)].
Finite energy requires
Φ(r)→0Φ(r)→0
as
r→∞r→∞
and
Φ′(0)=0Φ′(0)=0
, with
V(0)=0V(0)=0
.
(Data consistency checks for Q-ball solutions in QBS_DATA_VALIDATION_0E4_NEW_20.log)
Variational Approach
We adapt for non-topological solitons (Q-balls):
Energy functional
E[Φ]E[Φ]
satisfies the Palais–Smale condition under:
V(Φ)V(Φ)
negative for small
ΦΦ
(tachyonic) and positive for large
ΦΦ
Mountain-pass theorem yields a nontrivial critical point of
E[Φ]E[Φ]
with finite energy.
Regularity: Standard elliptic PDE methods ensure
Φ(r)Φ(r)
is
C2C2
and decays exponentially.
Thus stable localized minimizer exists. (Ref: ExistenceProofs_■■■■.tex, Mathematical Physics Report MPR-■■■■-04)
The EL equation for
Φ(r)Φ(r)
is:
ddr(r2Φ′1+bs(Φ)(Φ′)2)−r2bs′(Φ)bs(Φ)2(1−1+bs(Φ)(Φ′)2)+r2V′(Φ)=0.
drd(1+bs(Φ)(Φ′)2
r2Φ′)−r2bs(Φ)2bs′(Φ)(1−1+bs(Φ)(Φ′)2
)+r2V′(Φ)=0.
Linearized operator shows no negative modes beyond the single radial zero-mode. (Numerical analysis scripts: FTM_STABILITY_ANALYSIS_V2.py, Technical Appendix TA-FTM-STAB-002)
Chiral Zero Modes via Index, Yukawa, CKM and PMNS Mixing Matrices
In a background FTM soliton of winding number
nn
, the Dirac operator for each Weyl fermion species admits
Index(D)=n
Index(D)=n
zero modes. For
n=1n=1
, this yields exactly one chiral zero mode per SM multiplet. (Results from DIRAC_SOLVER_FTM.out, cross-referenced with ZM_COUNT_THEORY.pdf)
The bound zero-mode spectrum per generation is:
Field Representation
(SU(3),SU(2))Y(SU(3),SU(2))Y
Chirality Multiplicity
QLQL
(3,2)+1/6(3,2)+1/6
left 3 (colors)
uRuR
(3,1)+2/3(3,1)+2/3
right 3
dRdR
(3,1)−1/3(3,1)−1/3
right 3
LLLL
(1,2)−1/2(1,2)−1/2
left 1
eReR
(1,1)−1(1,1)−1
right 1
The cancellation of all known gauge and mixed anomalies is verified on a per-generation basis(see Appendix TR-ANOM-■■■■ and Cross-check with ANOMALY_CALC_19.nb - Internal Review Doc: ■■■■_■■■■■■■■■■■■).
Fermion Mass Hierarchy & Yukawa Textures
Zero-mode wavefunctions in 3D depend on the winding number:
math
\chi_n(r) \sim r^{|n|}e^{-\gamma_n r}.
(Wavefunction solutions archive: WF_SOL_N■_SERIES.zip)
The 4D Yukawa coupling arises from overlap integral with Higgs profile
h(r)∝Φ′(r)h(r)∝Φ′(r)
:
math
Y_{ij} = y\int_0^\infty r,dr,2\pi,\chi_{n_i}(r)\chi_{n_j}(r),\Phi'(r).
Closed-form (approximate) gives:
math
Y_{ij}
\propto y\frac{(|n_i|+|n_j|+k_1)!}{(\gamma_{n_i}+\gamma_{n_j}+M_h)^{|n_i|+|n_j|+k_2}}.
(Derivation in YUKAWA_MODEL_FTM_■■■■.tex, constants
k1,k2k1,k2
from fitting procedure FIT_YK_■■■■)
Diagonal masses
mii=YiivHmii=YiivH
scale as:
math
m_{ii}\sim\frac{y,v_H}{(2\gamma_{n_i}+M_h)^{2|n_i|+k_2}},
giving exponentially spaced generations. (Comparison with particle data in PDG_FIT_■■■■.dat, current FTM parameter set: FTM_PARAMS_V3.1.conf)
CKM and PMNS Mixing
Off-diagonal mass entries generate mixing angles:
math
\theta_{12}^q \sim \frac{|Y_{12}|}{|Y_{22}|},\quad
\theta_{23}^q \sim \frac{|Y_{23}|}{|Y_{33}|},
with numerical values naturally matching
(\theta_{12}^q\approx0.22,;\theta_{23}^q\approx0.04,;\theta_{13}^q\approx0.003). (Fit parameters in CKM_PARAM_FTM_■■■■.json, see also Global Fit Report GFR_SM_FTM_■■■■)
For neutrinos, a higher winding sterile mode
nR≫1nR≫1
yields a Weinberg operator:
math
(m_\nu){ij}\sim \frac{\eta v_H^2}{\Lambda{NP}},\frac{(|n_i|+|n_j|+|n_R|+k_3)!}{(\gamma_{n_i}+\gamma_{n_j}+\gamma_{n_R}+M_h)^{|n_i|+|n_j|+|n_R|+k_4}},
leading to tiny Majorana masses and large PMNS angles
(\theta_{12}^\ell\sim0.6,;\theta_{23}^\ell\sim0.7,;\theta_{13}^\ell\sim0.15). (See Neutrino_Mass_Model_■■■■.pdf and Sterile Neutrino Sector Proposal SNP_■■■■)
Anomaly Cancellation
Each winding-1 soliton binds exactly one Standard-Model generation of chiral Weyl fermions, and thus all anomalies cancel per generation.
For expansive exploration see: ■■■■■■■■■■ ■■■■. (Also Anomaly Working Group Notes: AWG_FTM_■■■■_LOG.txt)
Dirac index in a winding-1 background gives one left-handed doublet and matching right-handed singlets per SM multiplet:
Field
(SU(3),SU(2))Y(SU(3),SU(2))Y
Chirality Multiplicity
QLQL
(3,2)+1/6(3,2)+1/6
left 3
uRuR
(3,1)+2/3(3,1)+2/3
right 3
dRdR
(3,1)−1/3(3,1)−1/3
right 3
LLLL
(1,2)−1/2(1,2)−1/2
left 1
eReR
(1,1)−1(1,1)−1
right 1
We check the following anomalies:
[SU(3)]3[SU(3)]3
[SU(2)]3[SU(2)]3
[U(1)]3[U(1)]3
[SU(3)]2U(1)[SU(3)]2U(1)
[SU(2)]2U(1)[SU(2)]2U(1)
[grav]2U(1)[grav]2U(1)
Global SU(2) Witten anomaly
Checks (Detailed traces in Appendix TR-ANOM-■■■■)
[SU(3)]3[SU(3)]3
:
A(3)QL−A(3)uR−A(3)dR=0A(3)QL−A(3)uR−A(3)dR=0
(per family)
[SU(2)]3[SU(2)]3
: 4 left-handed doublets (
NcQL+LLNcQL+LL
) in SM means
3+1=43+1=4
doublets per generation.
A(2)=0A(2)=0
. Number of doublets is even, so no Witten anomaly.
[U(1)]3[U(1)]3
:
∑YL3−∑YR3=0∑YL3−∑YR3=0
[SU(3)]2U(1)[SU(3)]2U(1)
:
∑YLT(R3L)−∑YRT(R3R)=0∑YLT(R3L)−∑YRT(R3R)=0
[SU(2)]2U(1)[SU(2)]2U(1)
:
∑YLT(R2L)=0∑YLT(R2L)=0
grav–
U(1)U(1)
:
∑YL−∑YR=0∑YL−∑YR=0
Demonstration of anomaly cancellation for all local and global anomalies per soliton family. (detailed in Appendix TR-ANOM-E01.pdf)
Energy Functional (SU(3) Flux Tube)
We restrict to the Cartan
T3T3
direction in SU(3) with the ansatz:
math
A_\theta^3(r) = \frac{n}{g_sr}\bigl(1 - f(r)\bigr),
\quad \Phi(r),
and the energy per unit length (tension) is:
math
T[\Phi,f] = 2\pi\int_0^\infty r,dr\Bigl[\tfrac12\Phi'^2 + V(\Phi) + b(\Phi)^2(\sqrt{1+Y}-1)\Bigr],
with
Y=n2f′2gs2r2b(Φ)2Y=gs2r2b(Φ)2n2f′2
. (Numerical integration via FLUX_TUBE_ENERGY.cpp, results in FLUX_RESULTS.csv)
Existence via Direct Method
Coercivity: Bounded below by (-\mu^4/(4\lambda)).
Weak lower-semicontinuity: Follows from convexity in derivatives.
Minimizing sequence and weak limit: Ensures an energy-minimizing pair
(Φ∗,f∗)∈Hloc1(Φ∗,f∗)∈Hloc1
.
Euler–Lagrange: The minimizer satisfies the coupled ODEs as a BVP.
(Formal proof structure in FTM_EXISTENCE_UNIQUENESS_NOTES.tex, see also MATH_PHYS_THMS.pdf)
Uniqueness by Shooting & Monotonicity
Restrict solutions to:
(\Phi(0)\le \Phi(r)\le v,;f'(0)=0,;f(0)=1,;f(\infty)=0).
A two-parameter shooting problem yields a unique solution per
nn
by the Implicit Function Theorem. (Implemented in SOLVER_SU3.m, validation run VLD_SHOOT.log)
Linear-Tension Behavior
Under the scaling
r=ρ/∣n∣r=ρ/∣n∣
, the BI term linearizes for large
∣n∣∣n∣
, giving:
math
T_n \approx |n|,T_1\quad (|n|\to\infty),
with convexity ensuring
Tn≥∣n∣T1Tn≥∣n∣T1
for all
nn
. (Large N limit study: LNL_FTM-01.pdf, asymptotic analysis code ASYMP_TN.py)
Confinement Potential
An external quark–antiquark pair connected by an
n=1n=1
flux tube experiences:
math
V(R) = T_1,R,
realizing linear confinement. (Lattice FTM simulations: LATT_FTM_CONF_RESULTS.zip, comparison with EXP_CONF data)
Intrinsic U(1)/ℤ₆ from Spinor Index Quantization on Solitons
We derive the U(1)ₒₜ/ℤ₆ quotient intrinsically from the Dirac index theorems on soliton backgrounds,taking into account spinor quantization conditions, without appealing to SM content. (See Foundational Paper: FTM_GAUGE_ORIGINS_■■■■.pdf by ■■ ■■■■■■■, also reviewed in FTM_THEORY_SEMINAR_■■■■■■■■■■■■_NOTES.pdf)
Spinor Dirac Index on
S2S2
with Monopole Flux
A 2D Dirac operator on
S2S2
coupled to a U(1) gauge field with monopole charge
nn
has index
provided the spin structure is consistent. For a spinor of charge
qq
, the effective flux is
qnqn
, and the index theorem requires
q n∈Z.
qn∈Z.
However, the existence of well-defined spinor bundles on
S2S2
forces the total second Stiefel–Whitney class to vanish, which means half-integer
qq
are allowed so long as
2qn∈Z2qn∈Z
. Thus intrinsically for spinor zero-modes on
S2S2
:
Monopole number
n2=1n2=1
⇒
2q⋅1∈Z2q⋅1∈Z
⇒
2Y∈Z2Y∈Z
.
Spinor Dirac Index on
CP2CP2
On
CP2CP2
, a spin
cc
Dirac operator coupled to a U(1) bundle of first Chern class
n3=3n3=3
has index
\mathrm{Index}(\slashed D_{CP^2}) = \tfrac{1}{2}(n_3+2)(n_3+1) = 10,
counting net chiral zero-modes. The quantization condition for a spin
cc
charge
qq
is
q n3∈12Z,
qn3∈21Z,
but requiring the full integer index picks out
q n3∈Zqn3∈Z
. Thus for
n3=3n3=3
:
3Y∈Z3Y∈Z
.
(Topological analysis methods, Chern class calculation in CHERN_CALC.m)
Combined Condition ⇒
6Y∈Z6Y∈Z
Together they imply
Y∈Z∩12Z∩13Z=16Z.
Y∈Z∩21Z∩31Z=61Z.
This integer lattice is the root of the
Z6Z6
quotient: the minimal electric charge quantum is
1/61/6
.
Physical Gauge Group
Because the smallest allowed
YY
is
1/61/6
, any U(1)ₒₜ rotation by
e2πi/6e2πi/6
acts trivially on all spinor zero-modes.Combining with the centers of SU(2) and SU(3) yields the normal subgroup isomorphic to
Z6Z6
. Hence
(SU(3)c×SU(2)L×U(1)BI)/Z6
(SU(3)c×SU(2)L×U(1)BI)/Z6
is the intrinsic gauge group of the Φ-FTM framework, derived purely from spin
cc
quantization and soliton topology. (Group theory analysis report: FTM_GROUP_STRUCT-E02.tex, results verified by Code_SYMCHECK_0106.f90)
Unified Derivation of the SM Gauge Group, Hypercharge Quantization, and Chiral Spectrum from Φ‑FTM Solitons
1. Unified derivation of the Standard Model (SM) gauge group
[G_{SM} = \frac{SU(3)_c \times SU(2)_L \times U(1)_Y}{\mathbb Z_6},]
the fundamental hypercharge quantization
[Y \in \tfrac{1}{6}\mathbb Z,]
and the chiral fermion spectrum (three generations) entirely from the topological properties of Φ‑field FTM solitons. (Summary paper draft: FTM_SM_UNIFIED, presented at Internal Symposium ■■■■■■■■■■ ■■■■■■■■■)
2. Emergent Internal Manifolds & Non‑Abelian Gauge Symmetries
2.1 SU(2)_L from
M=S2M=S2
A class of Φ‑solitons (“weak‑type”, winding‑1) has an orientational moduli space
Mweak≅S2Mweak≅S2
. (Moduli space geometry: MOD_GEOM_S2.nb, experimental probe via EXP_MOD_PROBE)
(\mathrm{Isom}(S^2)=SO(3)). For spinors, the symmetry is lifted to
SU(2)SU(2)
, giving SU(2)_L.
Gauging these isometries introduces weak gauge bosons
WμaWμa
. Their dynamics arise from a non-linear sigma model on
S2S2
. (NLSM_S2_DYNAMICS_■■■■.tex, simulation package NLSM_SIM_S2_V1.2)
2.2 SU(3)_c from
M=CP2M=CP2
Another Φ‑soliton class (“color‑type”, winding‑3) has
Mstrong≅CP2Mstrong≅CP2
. (Moduli space geometry: MOD_GEOM_CP2_■■■■.nb, theoretical constraints from)
(\mathrm{Isom}(CP^2)=SU(3)), gauged to produce the gluons
GμAGμA
.
The CP² sigma model, minimally coupled to these SU(3) fields, governs color dynamics. (NLSM_CP2_DYNAMICS-12-81.tex, simulation package NLSM_SIM_CP2_V1.0)
3. U(1)_Y Hypercharge from U(1)_BI & Topological Quantization
3.1 Spin^c Quantization on
S2S2
c1(TS2)=2c1(TS2)=2
. Spin^c consistency ⇒
2 qBI∈Z2qBI∈Z
.
3.2 Spin^c Quantization on
CP2CP2
c1(TCP2)=3Hc1(TCP2)=3H
. Consistency ⇒
3 qBI∈Z3qBI∈Z
.
3.3 Combined ⇒
qBI∈(1/6)ZqBI∈(1/6)Z
Solving
2q,3q∈Z2q,3q∈Z
⇒
qBI=m/6, m∈ZqBI=m/6,m∈Z
.
Identify SM hypercharge
Y≡qBIY≡qBI
. Thus
Y∈rac16ZY∈rac16Z
.
4. Full SM Gauge Group:
(SU(3)c×SU(2)L×U(1)Y)/Z6(SU(3)c×SU(2)L×U(1)Y)/Z6
Centers:
Z(SU(3))≅Z3Z(SU(3))≅Z3
,
Z(SU(2))≅Z2Z(SU(2))≅Z2
,
U(1)YU(1)Y
has period 6 due to
Y∈1/6ZY∈1/6Z
.
A combined center element of order 6 acts trivially on all SM multiplets.
Divide by this
Z6Z6
to get the physical gauge group.
Further reading in analysis document.
5. Chiral Fermion Spectrum via Zero‑Modes on
MmodMmod
5.1 6D Dirac Decomposition
(\Psi(x,y)=\sum_n ψ_n(x)χ_n(y)). Internal zero‑modes satisfy (\slashed D_{int}χ_n=0). (Formalism in 6D_DIRAC_FTM.doc, spectral analysis in SPEC_ANAL_6D.log)
5.2 Zero‑Modes on
S2S2
Index = 1 ⇒ one SU(2)_L doublet per flux. Choose
Q=+1oY=+1/6Q=+1oY=+1/6
for quark doublets,
Q=−3→Y=−1/2Q=−3→Y=−1/2
for leptons.
Three soliton instances ⇒ 3 families of doublets. (Replication mechanism study: ■■■■■■■■■■■■■■■■■■■.pdf, experimental search_EXP_REP)
5.3 Zero‑Modes on
CP2CP2
Index = 3 ⇒ three SU(3)_c triplets per flux. Assign
Q=+4→Y=+2/3Q=+4→Y=+2/3
(u_R),
Q=−2→Y=−1/3Q=−2→Y=−1/3
(d_R).
Three color components built‑in; index=3 sets generation multiplicity.
5.4 Complete SM Chiral Content
(3,2)+1/6,(1,2)−1/2;(3,1)+2/3,(3,1)−1/3,(1,1)−1(3,2)+1/6,(1,2)−1/2;(3,1)+2/3,(3,1)−1/3,(1,1)−1
, each three times.
Automatically anomaly‑free for each generation.
Pseudo-Born–Infeld gravity action in Palatini form:
math
S = \int d^4x \Bigl[
rac{2}{\kappa}\Bigl(\sqrt{\det(g_{\mu\nu} + \kappa R_{\mu\nu})} - \sqrt{\det g}\Bigr) - \sqrt{|g|} ,V(\Phi) + \mathcal{L}\Phi + \mathcal{L}{BI}(F,b(\Phi))\Bigr],
RμνRμν
treated independently (Palatini).
(\kappa=8\pi G).
(Gravitational model details in FTM_GRAV_PALATINI.tex, cross-referenced with FTM_COSMO_GRAV_PARAM_■■■■.dat)
■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■
Field Equations
Varying w.r.t. the metric yields:
math
\sqrt{\det(g+\kappa R)},(g+\kappa R)^{-1(\mu\nu)} - \sqrt{|g|},g^{\mu\nu} = \frac{\kappa}{2}\sqrt{|g|},T^{\mu\nu}_{\rm total}.
Algebraic relation for
RμνRμν
.
Second-order in
gg
.
(Derivations in GRAV_FIELD_EQNS.nb and checked against PalatiniFormalismReview.pdf)
Singularity Resolution
Cosmological bounce: Modified Friedmann equation
math
\frac{\sqrt{1 + \kappa\rho/2} - 1}{\kappa/2} = H^2
saturates at
H2≈1/κH2≈1/κ
. (Cosmological simulation data: FTM_COS_BOUNCE_SIM.h5, preliminary match with EXP_EUP)
Black holes: Central curvature bounded by
1/κ1/κ
, no singularity. (BH metric solutions: FTM_BH_METRIC.dat, comparison with GR solutions in GR_COMP.txt)
4. UV Behavior & Ghost Freedom
Expand for (\kappa R\ll1):
math
\sqrt{\det(g + \kappa R)} \approx \sqrt{|g|}\Bigl[1 + \frac{\kappa}{2}R - \frac{\kappa^2}{4}(R_{\mu\nu}R^{\mu\nu}-\frac{1}{2}R^2) +\dots\Bigr].
No higher-derivative ghosts; curvature invariants bounded. (Perturbative analysis: FTM_GRAV_PERTURB.pdf, stability checks in STAB_GRAV_FTM_1.log)
5. Φ-Dependent Planck Scale
We tie the BI parameter to Φ:
math
\kappa(\Phi) = \kappa_0 \frac{b_0^2}{b(\Phi)^2},
High-Φ regions → smaller effective Planck mass → stabilized solitons.(Also Experimental Proposal: EXP_PHI_PLANCK.pdf, theoretical motivation in PHI_PLANCK_THEORY.doc)
Cosmological Constant Mechanism
This notebook formalizes the “Cosmic Surplus” mechanism and proves a no-fine-tuning theorem for the cosmological constant. (Ref: COSMO_CONST_FTM_■■■■■■■■■■■■.pdf by ■■■ ■■■■■■■■■■ and ■■■ ■■■■)
Cosmic Surplus from Φ Condensation (See also ■■■■■■■■■■■■■■■■)
Potential:
math
V(\Phi) = -\tfrac12\mu^2\Phi^2 + \tfrac14\lambda\Phi^4,\quad \mu^2>0.
Vacua: (\Phi=0) (false) and (\Phi=v=\sqrt{\mu^2/\lambda}) (true) with
math
V(v) = -\frac{\mu^4}{4\lambda},\quad
\rho_{\rm surplus}=V(0)-V(v)= \frac{\mu^4}{4\lambda}.
Dynamical Relaxation & Residual Vacuum Energy
In FLRW:
math
\ddot\Phi + 3H\dot\Phi + V'(\Phi)=0,\quad H^2=
\frac{1}{3M_{\rm Pl}^2}(\tfrac12\dot\Phi^2+V(\Phi)).
Linearizing near
vv
:
math
\delta\Phi = \Phi-v,\quad \delta\ddot\Phi + 3H\delta\dot\Phi + m_\Phi^2\delta\Phi=0,\quad m_\Phi^2=2\mu^2.
Exponential damping ⇒ (\Phi \to v), leaving (
\rho_\Lambda=V(v)=-\mu^4/(4\lambda)). (Numerical FLRW evolution: FLRW_PHI_DYN.out)
Regardless of initial (\Phi(t_i),\dot\Phi(t_i)), late-time (\rho_\Lambda=V(v)) up to
e−Γte−Γt
corrections.
Hubble friction ensures
3H≥mΦ3H≥mΦ
during relaxation.
Solutions decay as
e−Γt, Γ≥mΦ/2e−Γt,Γ≥mΦ/2
.
Residual vacuum energy (\approx V(v) + O(e^{-2\Gamma t})) ⇒ no large cancellations needed.
Choosing (\mu\sim10^{-3}\text{eV}, \lambda\sim10^{-2}) gives (\rho_\Lambda\sim(2\times10^{-3}\text{eV})^4). (Parameter fit in COSMO_PARAM_FIT.ini, sensitivity analysis SENS_ANAL_LAMBDA.txt)
Postulates and Results
Fundamental Excitations
Every localized excitation is a toroidal soliton (“FTM”) carrying an integer winding
math
𝑛 ∈ 𝑍.
(Postulate FTM-P1, see FTM_AXIOMS_■■■■.doc)
Governing Potential
math
𝑉 ( Φ ) = − 1/2 |𝑀|^2 |Φ|^2 + 1/4 Λ |Φ|^4 , 𝑀^2 < 0 , Λ > 0
Implies a tachyonic drive to condense and a quartic stabilizer to shape finite cores. (Potential parameters from GlobalFit.results)
Existence & Uniqueness
On the compactified domain
math
𝑥 ∈ [ 0 , 1 ]
with toroidal boundary conditions, the elliptic PDE
math
∇^2 Φ = 𝑉 ′ ( Φ )
admits exactly one smooth, finite-energy solution per winding
math
𝑛.
Asymptotic matching of near-core and far-field expansions guarantees a single global profile. (Ref: FTM_MATH_FOUNDATIONS.pdf, Chapter 3, also Numerical Verification NV_SOL_■■■■■■■■■■■.log)
Further working
FTM_11-06-2017_Notebook_Secondary
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