Contents (52 sections)

COVER

First Particle-Resolved Observation

EARLY UNIVERSE • PARTICLE RECORDS FROM FINITE FIELD EXCITATIONS

When can particle-resolved observations first exist?

A finite-region derivation of the first epoch at which an Actor-owned field excitation can be recorded as an individual particle-like event. No point particle is assumed; the primitive object is a normalized finite wave packet and its completed physical record.

Pass-conditional • first particle-resolved record

Candidate onset relative to the initial boundary

1.645530e-41 s

Recordable-universe clockΔt ≈ 0 at model resolution

Particle primitivefinite Actor-owned wave packet

Controlling scaleM_KK=1/R₆

Modern particle identitieslater phase-dependent milestones

Public claim

Within the finite-record protocol, particle-resolved observation can begin as soon as operationally independent records can exist. Under the frozen one-cell wave-packet branch, the bottleneck is the same record-completion bound that fixes the separability crossing: t≈1.6455×10⁻⁴¹ s after the initial boundary.

Essential distinction

At this earliest epoch “particle” means a localized, Actor-owned field excitation that produces a distinct finite record. It does not mean that the broken-phase photon, Z boson, massive W boson, electron mass eigenstate or hadron already has its present low-energy identity.

RESULT

Executive result

The framework already contains gauge and matter Actors. The new question is not whether those field sectors exist in the parent grammar, but when a finite excitation of one sector can become an individually resolvable event.

$t_{\rm particle}^{\exists}=\max\{t_{\rm sep}^{\exists},\,t_{\rm loc},\,t_{\rm record}\}$

The localization clock is t_loc=ħ/M_KK=1.047577e-41 s. The minimum completed-record clock at the admitted UV edge is t_record=πħ/(2M_KK)=1.645530e-41 s. The operational-separability calculation independently gives the same latter root.

$t_{\rm particle}^{\exists}=\frac{\pi\hbar}{2M_{\rm KK}}=1.64552989225\times10^{-41}\ {\rm s}$

Result

Under the stated finite-support packet and record instrument, no additional particle-specific delay exceeds the already-derived separability/record-completion delay. Particle-resolved records therefore begin at the recordable-universe boundary to the resolution of the present model.

CLOCK

Clock conventions

Two clock origins are useful and must not be confused.

$t_{\rm initial}=0\quad\Longrightarrow\quad t_{\rm particle}^{\exists}=1.64552989225\times10^{-41}\ {\rm s}$

$T_{\rm recordable}=t-t_{\rm sep}^{\exists}\quad\Longrightarrow\quad T_{\rm particle}^{\exists}\simeq0^{+}$

If “Big Bang” is used only as a conventional name for the initial boundary, the first particle-resolved event occurs about 1.65×10⁻⁴¹ s after it. If the clock is reset at the beginning of the recordable universe, particle-resolved observation begins essentially at that origin.

No singularity claim

The calculation does not require an exact singular spacetime point. The initial boundary is a clock convention preceding the first completed finite records.

DEFINITION

What counts as a particle in this calculation

The primitive is not a classical point particle. For an admitted field Actor with creation operators a†(k), choose a normalized finite packet f:

$a^{\dagger}[f]=\int d\mu(k)\,f(k)a^{\dagger}(k),\qquad \int d\mu(k)|f(k)|^2=1$

$|1_f\rangle=a^{\dagger}[f]|0\rangle$

A particle-resolved record is a completed record whose outcome distinguishes the packet sector from the vacuum/background sector at the frozen Granularity. This definition remains valid in curved spacetime or a thermal background whenever the relevant finite spectral packet and detector algebra are defined.

Why this is stronger than a point-particle model

Wave packets have finite support, finite energy, an Actor owner and an explicit observable algebra. They fit directly into Shape, Granularity, Dynamics and Observer without adding a singular source.

ACTOR LEDGER

Frozen Actor content before the calculation

Actor family	Parent status	Earliest particle-record meaning
SU(3) color connection	parent fundamental	color gauge excitation / gluonic packet
SU(2) weak connection	parent fundamental	weak-isospin gauge packet in symmetric phase
U(1) hypercharge connection	parent fundamental	hypercharge gauge packet in symmetric phase
quark chiral matter	parent matter Actors	quark-like chiral excitation; not a hadron
lepton chiral matter	parent matter Actors	lepton-like chiral excitation
Higgs/Wilson sector	parent/construction Actor	scalar/holonomy excitation subject to current Higgs witness scope
4D graviton	parent fundamental	finite gravitational wave packet

The calculation therefore does not manufacture particle species at t=t_particle. It asks when already-admitted field sectors acquire an individually retrievable event record.

IDENTITY

Particle identity is phase-dependent

At temperatures far above electroweak symmetry breaking, the correct gauge basis is SU(2)_L×U(1)_Y. A statement such as “the photon was observed” would therefore be premature at the first record boundary.

$W^1,W^2,W^3,B\quad\xrightarrow[\rm later]{\rm EW\ breaking}\quad W^{\pm},Z,\gamma$

Likewise, quark field excitations can exist before confinement, whereas proton, neutron and meson identities require the later QCD regime.

Public terminology

The earliest result is intentionally called particle-resolved field excitation. Modern broken-phase particle names are introduced only after the relevant phase transition is available.

SPECTRUM

Spectral admissibility below compactification

The 4D particle interpretation is taken only in the admitted low-energy branch below the compactification threshold.

$0<E_f<M_{\rm KK},\qquad M_{\rm KK}=R_6^{-1}=6.28318530718\times10^{16}\ {\rm GeV}$

At and below this threshold the heavy compact directions are not independently excited as unresolved 4D particle labels; they instead contribute controlled effective corrections and heavy-mode channels.

Central clock choice

M_KK, not M*, is the central particle-record clock because it is the spectral boundary separating the effective 4D packet grammar from explicit compactification excitations.

LOCALIZATION

Localization scale of the earliest packet

$\ell_{\rm KK}=\frac{\hbar c}{M_{\rm KK}}={ellKK:.6e}\ {\rm m}$

$t_{\rm loc}=\frac{\ell_{\rm KK}}{c}=\frac{\hbar}{M_{\rm KK}}={tauKK:.6e}\ {\rm s}$

A one-cell primitive packet cannot be resolved on a shorter spacetime support without leaving the frozen effective-particle branch. This gives the earliest localization clock.

OBSERVER

Completed particle record as an instrument

A particle observation is a physical interaction that deposits a stable, retrievable record in another finite subsystem. Represent the instrument by effects indexed by packet occupancy or another lawful particle observable:

$\mathcal M^{\rm part}=\{\mathcal M_0,\mathcal M_1,\ldots\},\qquad \sum_n\mathcal M_n\ {\rm CPTP}$

$p_n={\rm Tr}\!\left[\mathcal M_n(\rho)\right]$

The minimal binary question is whether a packet is absent or present. A more refined instrument may resolve charge, family, helicity, momentum bin or interaction channel, but those refinements can only move the first-observation time later.

DISTINGUISHABILITY

Vacuum versus one-packet distinguishability

For an ideal normalized one-packet state orthogonal to the vacuum, the state-space distinguishability is maximal:

$\langle0|1_f\rangle=0,\qquad D_{\rm tr}(|0\rangle,|1_f\rangle)=1$

The remaining problem is therefore not abstract Hilbert-space distinguishability. It is whether a finite physical record can complete and remain operationally independent at the available time and energy.

Key reduction

Particle observation reduces to the same two bottlenecks already isolated by the recordable-universe calculation: record completion and operational separability.

GRANULARITY

Granularity cell for a particle record

The public calculation uses the same action-cell normalization as the operational-separability paper. Let C be the branch-conditioned action deposited in the record channel:

$q_{\rm part}(\mathcal C)=\left\lfloor\frac{\mathcal C}{\hbar}\right\rfloor$

Two record outcomes are operationally distinct only when they occupy different frozen cells. The exact microscopic detector model may refine this map, but no refinement is allowed to create an earlier event than the quantum-speed and localization bounds.

SPEED LIMIT

Quantum speed limit for the first particle record

To write a perfectly distinguishable finite record, the record-bearing subsystem must evolve to an orthogonal outcome. For mean energy E above its reference state, the Margolus–Levitin lower bound is

$t_{\rm record}\ge\frac{\pi\hbar}{2E}$

The earliest allowed record uses the largest energy that remains inside the declared 4D particle branch, E→M_KK⁻.

$t_{\rm record,min}=\frac{\pi\hbar}{2M_{\rm KK}}={tsep:.6e}\ {\rm s}$

ROOT TEST

Why localization does not control the root

$\frac{t_{\rm record,min}}{t_{\rm loc}}=\frac{\pi}{2}=1.570796\ldots$

The packet can fit inside its minimum lawful support before a perfectly distinguishable record can finish forming. Record formation is therefore the active particle-observation constraint.

SEPARABILITY

Operational separability is the second bottleneck

A record is not an individual particle observation if its outcome cannot be operationally separated from all nearby record channels. The companion calculation gives

$t_{\rm sep}^{\exists}=\frac{\pi\hbar}{2M_{\rm KK}}={tsep:.6e}\ {\rm s}$

The particle-record and separability clocks therefore coincide in the present branch.

CENTRAL RESULT

Particle-observation root

$t_{\rm particle}^{\exists}=\max\!\left(\frac{\hbar}{M_{\rm KK}},\frac{\pi\hbar}{2M_{\rm KK}},t_{\rm sep}^{\exists}\right)$

$t_{\rm particle}^{\exists}=\frac{\pi\hbar}{2M_{\rm KK}}={tsep:.11e}\ {\rm s}$

First particle-resolved finite record

1.645530e-41 s

Meaning of the equality

The onset of particle-resolved observation is coincident with the onset of an operationally separable record universe because no additional particle-specific condition is slower than completed record formation in the one-cell admitted packet branch.

TECHNICAL

Finite-support packet normalization

A finite wave packet must be normalized in the physical spectral measure and projected through the complete constraint projector before it is called a particle record.

$|1_f\rangle=\Pi_{\rm phys}a^{\dagger}[f]|0\rangle,\qquad \langle1_f|1_f\rangle=1$

TECHNICAL

Gauge-invariant particle records

Charged or gauge-variant local fields are not themselves complete observables. The record must be attached to a lawful dressed, asymptotic, relational or finite-region observable algebra.

$O_{\rm part}\in\mathcal A_{\rm phys}(R),\qquad [O_{\rm part},C_\alpha]=0$

TECHNICAL

No point-source approximation

The source profile is the packet f and its finite causal support. Delta-function sources are neither necessary nor privileged.

$J(x)=\int d\mu(k)\,\tilde J(k)e^{-ik\cdot x}\quad\text{with finite spectral support}$

TECHNICAL

Thermal background does not erase the definition

A particle record in a hot background is a distinguishable excitation relative to the selected thermal/reference state, not necessarily a vacuum excitation.

$D_{\mathcal A_R}(\rho_{R|1},\rho_{R|0})>0$

TECHNICAL

Signal-to-background Granularity

Finite Granularity requires the particle-conditioned record to cross a cell boundary relative to the background record.

$q_R(\rho_{R|1})\ne q_R(\rho_{R|0})$

TECHNICAL

Interaction-off negative control

With the packet-record coupling set to zero, the record outcome must become independent of packet occupancy.

$H_{\rm int}=0\ \Longrightarrow\ p(r|1_f)=p(r|0)$

TECHNICAL

Product-spectator negative control

A genuinely independent spectator sector must remain unchanged when the particle record completes.

$\rho_S\mapsto\rho_S$

TECHNICAL

Reversible-ancilla control

If the putative detector record is perfectly erasable and no durable record forms, the event does not count as a completed particle observation.

$U_{\rm erase}U_{\rm record}|\Psi\rangle=|\Psi\rangle$

TECHNICAL

Finite detector response

Detector response is represented as a channel rather than an instantaneous projection.

$\Phi_{R\leftarrow P}:\rho_P\otimes\rho_R\mapsto\rho_R\prime$

TECHNICAL

Particle label versus event label

The earliest result concerns an event in an Actor sector. Species-level labels require the corresponding symmetry and phase structure to be available.

${\rm event}\subseteq{\rm Actor\ sector}\subseteq{\rm phase\ dependent\ species}$

TECHNICAL

Massless gauge packets

Massless gauge sectors do not carry a Compton localization scale, but the effective 4D branch still supplies the compactification cell and finite detector support used in this calculation.

$\ell_{\rm support}\ge\ell_{\rm KK}$

TECHNICAL

Massive packets

For a massive excitation, the packet support must also respect its Compton scale when that scale exceeds the compactification cell.

$\ell_{\rm packet}\gtrsim\max(\ell_{\rm KK},\hbar/(mc))$

TECHNICAL

Earliest energy choice is a limit

The equality uses E approaching M_KK from below. Any lower-energy packet or record necessarily appears later.

$t_{\rm record}(E)=\frac{\pi\hbar}{2E}\ge t_{\rm record}(M_{\rm KK})$

TECHNICAL

Energy-fraction sensitivity

If the available record energy is E=ηM_KK, the onset scales inversely with η.

$t_{\rm particle}(\eta)=\frac{1}{\eta}\,t_{\rm particle}(1)$

TECHNICAL

Support-size sensitivity

If a primitive packet requires λ compactification cells of diameter, geometric packing may become the active constraint when λ is sufficiently large.

$t_{\rm geom}(\lambda)\sim\lambda\frac{\hbar}{M_{\rm KK}}$

TECHNICAL

Why the result is existential

The theorem asks when at least one lawful particle packet can have an independent completed record; it does not assert that all particle channels are independently measurable at once.

$t_{\rm part}^{\exists}\le t_{\rm part}^{\forall}$

TECHNICAL

Global and topological sectors

Global constraints or topological information can remain correlated beyond the local particle-record onset and are not forced into local packet factorization.

$\mathcal A_{\rm global}\not\subseteq\mathcal A_A\vee\mathcal A_B\ \text{in general}$

TECHNICAL

Infinite KK tower residual

The local 4D particle interpretation remains conditional on the already-declared uniform control of the heavy tower. A failed tower bound can move the operational onset later.

$\sum_{n>N}\|K_n\|\le\varepsilon_{\rm tower}$

TECHNICAL

Higher-order influence residual

Non-Gaussian influence terms must remain below the residual Granularity margin inherited from the separability calculation.

$\|S_{\rm IF}^{(\ge3)}\|<0.266529\,\hbar$

TECHNICAL

Particle observation is not present-day observation

A local record at 10^-41 s need not survive to a present observer. Present recoverability is a separate history-transfer question.

$\Phi_{0\leftarrow t_{\rm part}}:\mathsf R_{t_{\rm part}}\to\mathsf R_0$

PHASE DISCIPLINE

Symmetric electroweak phase: what can be named

At the earliest particle-record boundary, the framework licenses field-sector labels inherited from the unbroken gauge grammar. It does not yet license the full broken-phase vocabulary.

Safe early label	Do not substitute prematurely
SU(2) gauge excitation	massive W or Z
hypercharge gauge excitation	photon
chiral quark excitation	proton/neutron
chiral charged-lepton excitation	fully broken-phase massive electron identity
scalar/Higgs-sector excitation	late-time Higgs vacuum excitation without phase qualification

EXTERNAL VALIDATION

Electroweak crossover as an external identity milestone

Lattice Standard Model thermodynamics places the electroweak crossover at approximately 159.5±1.5 GeV. This value is not used to derive the 10⁻⁴¹-second particle-record onset.

If one applies the conventional radiation-era clock only as a comparison, T=159.5 GeV maps to roughly

$t_{\rm EW}^{\rm std}\approx9.203e-12\ {\rm s}$

Below this regime the familiar W±, Z and photon basis and fermion mass terms become the appropriate low-energy language.

External comparison only

The temperature is a strong particle-physics result; the quoted cosmic time uses standard expansion dynamics and is not yet a native prediction of this framework.

EXTERNAL VALIDATION

QCD crossover as the hadron-identity milestone

Quark and gluon excitations can be particle-resolved long before hadrons exist as the appropriate long-lived degrees of freedom. Lattice QCD gives a crossover near 156.5±1.5 MeV.

A conventional radiation-era conversion gives an order-of-magnitude time near

$t_{\rm QCD}^{\rm std}\sim1.257e-05\ {\rm s}$

This is the natural external marker for the emergence of hadronic rather than deconfined quark/gluon particle descriptions.

HARD VALIDATION

BBN as the first hard particle-history validation

Big-Bang nucleosynthesis is the deepest empirically reliable early-universe probe based on well-understood Standard Model physics. A conventional T≈1 MeV weak-freeze-out scale maps to t of order one second:

$t_{\rm BBN}^{\rm std}\sim0.74\ {\rm s}$

Unlike the 10⁻⁴¹-second onset, BBN leaves observed abundance records. A completed native history must reproduce those records without using them to tune the onset calculation.

TIMELINE

Particle-observation ladder

Milestone	Framework statement	External physical landmark
first finite particle-like event	1.646e-41 s after initial boundary	no direct observational access; theory territory
broken electroweak identities	native history calculation still owed	T_EW≈159.5±1.5 GeV
hadronic identities	native history calculation still owed	T_QCD≈156.5±1.5 MeV
nuclear reaction history	must reproduce observed abundance records	BBN, ~first minutes
photon last scattering	must reproduce CMB transfer records	recombination ~380 kyr in standard cosmology

The first row is the new Shape/Granularity result. The later rows are validation milestones, not derivation inputs.

COMPARISON

Why the 10⁻⁴¹-second claim is not contradicted by standard cosmology

Standard high-temperature cosmology extrapolated to T≈M_KK gives a time of order a few×10⁻⁴¹ s. The framework result is 1.646e-41 s. Both occupy the same decade.

No direct experiment observes that epoch, so a factor-of-few comparison is the correct present standard: it demonstrates consistency, not empirical confirmation.

FALSIFICATION

What would falsify the particle-onset claim

A correct finite-region detector model requires more than the available Granularity residual at all t≈10⁻⁴¹ s.
The infinite KK tower or a global constraint prevents any Actor-owned packet from acquiring an independent record at the proposed root.
The packet-support minimum is larger than the sensitivity range and forces t_geom>t_record.
The record-cell action threshold is shown to be larger than the frozen one-action-cell branch used here.
A lawful source-complete Actor census shows that no matter/gauge/gravitational packet sector is physically available on the claimed 4D branch.

Asymmetric falsification

Every identified correction moves the first particle observation later. None of them provides a mechanism for a physically completed particle record earlier than the quantum-speed and separability floor.

CLAIM LEDGER

Claims established by the calculation

Claim	Status
No point particle or point mass is required	ESTABLISHED BY CONSTRUCTION
finite Actor-owned wave packets are the particle primitive	ESTABLISHED BY DEFINITION + SOURCE GRAMMAR
localization clock ħ/M_KK=1.048e-41 s	DERIVED FROM FROZEN SHAPE
completed-record lower bound πħ/(2M_KK)=1.646e-41 s	DERIVED-GIVEN QUANTUM SPEED LIMIT
first-existence separability root=1.646e-41 s	PASS-CONDITIONAL IN COMPANION PAPER
first particle-resolved record=1.646e-41 s	PASS-CONDITIONAL UNDER ONE-CELL PACKET PROTOCOL
modern broken-phase particle identities exist at same epoch	NOT CLAIMED
a present observer can recover the first particle event	OPEN HISTORY-TRANSFER QUESTION

REPRODUCIBILITY

Source hierarchy

Source class	Role
Frozen Shape / Actor census	supplies gauge, matter, graviton and Higgs-sector owners
Granularity authority	supplies finite record-cell discipline
Interdependence authority	supplies factorization firewall and conditional-influence rules
Dynamics authority	supplies instruments, channels and history evolution
Observer authority	supplies accessible particle/event record definition
Operational Separability Crossing paper	supplies t_sep^∃ and residual margins
External lattice/PDG literature	downstream validation only

BIBLIOGRAPHY

External references

M. D’Onofrio and K. Rummukainen, The Standard Model cross-over on the lattice, arXiv:1508.07161 — T_c=159.5±1.5 GeV.
HotQCD collaboration literature, The QCD crossover at zero and non-zero baryon densities from Lattice QCD, arXiv:1807.05607 — T_c(0)=156.5±1.5 MeV.
Particle Data Group, Big-Bang Nucleosynthesis, Review of Particle Physics 2024 — empirical early-universe abundance benchmark.
Particle Data Group, Cosmological Parameters — CMB-derived present-age benchmark under flat ΛCDM.

These references are comparison data. None is consumed upstream in the first-particle root.

APPENDIX

Appendix A — dimensionless form

$x=M_{\rm KK}t/\hbar,\quad x_{\rm loc}=1,\quad x_{\rm record}=\pi/2,\quad x_{\rm part}=\pi/2$

APPENDIX

Appendix B — particle packet effect algebra

$E_0+E_1\le I,\qquad p_i={\rm Tr}(E_i\rho)$

APPENDIX

Appendix C — one-packet number observable

$N_f=a^{\dagger}[f]a[f],\qquad N_f|1_f\rangle=|1_f\rangle$

APPENDIX

Appendix D — branch restart rule

$\{R_6,q_{\rm part},\Phi_{\rm record}\}\to\{R_6\prime,q_{\rm part}\prime,\Phi_{\rm record}\prime\}\Rightarrow\text{new branch}$

APPENDIX

Appendix E — present-recoverability condition

$N_{\rm today}(t)=|q_0[\Phi_{0\leftarrow t}(\mathsf H_t^{\rm particle})]|\ge2$

APPENDIX

Appendix F — final theorem statement

$t_{\rm particle}^{\exists}=1.64552989225\times10^{-41}\ {\rm s}\quad\text{under the declared finite-packet branch}$

SUMMARY

Technical summary

Question. When can an individual field excitation first leave a completed, operationally independent particle record?

$t_{\rm particle}^{\exists}=\max\{t_{\rm loc},t_{\rm record},t_{\rm sep}^{\exists}\}=\frac{\pi\hbar}{2M_{\rm KK}}$

Result

1.645530e-41 s after the initial boundary

Interpretation. The first particle-resolved event is a finite wave-packet record in an already-admitted field Actor. It begins at the same model-resolved boundary as the recordable universe. Broken-phase particle names and present-day recoverability are later, separate questions.

Claim status. PASS-CONDITIONAL under the one-cell finite-packet protocol, the operational-separability envelope, and the declared heavy/global-sector residual bounds.