5.8 KiB
Finite Timelike Emission Set Audit
Purpose
This note records the temporal correction to the lossy throat / phonon selector model:
Because the signal is emitted through a physical, timelike process, the temporal coordinates cannot be treated as infinite free coordinates.
They are finite, bounded, and constrained by causal support, energy budget, lifetime, bandwidth, and observation window.
This strengthens the model by preventing an unphysical escape route where a construction hides arbitrarily many active events in unlimited temporal degrees of freedom.
Core Statement
finite timelike emission set
-> finite temporal support
-> finite number of recoverable phonon packets
-> finite active-cell window per scale N
-> no infinite temporal cheating
-> density proof must count recoverable events inside the finite causal window
Corrected Doctrine
The throat is already modeled as a lossy torsional corridor, not a stable bridge. The new constraint is that the corridor also has finite timelike support.
For a physical emission event, define:
T_emit(N) = [t_0(N), t_1(N)]
with finite duration:
Delta T_N = t_1(N) - t_0(N) < infinity
A cell can be active only if it lies inside both the spatial/material gate and the finite temporal emission window:
chi_N(i,t) = 1 iff
t in T_emit(N)
and BurgersGradientGate_N(i,t)
and BandGapPhononGate_N(i,t)
and PhononModeOverlap_i(t) >= eta_min(N)
and Survivability_i(t) >= I_min(N)
Timelike Emission Set
Define the finite emission set:
E_N = { (i,t) : 1 <= i <= N, t in T_emit(N), chi_N(i,t)=1 }
The projected active index set is:
I_active(N) = { i <= N : exists t in T_emit(N), (i,t) in E_N }
This is the set that can be passed into the non-separable algebraic encoding layer.
Physical Constraints
The temporal coordinate is finite because of:
causal propagation speed <= c or material sound speed v_s
finite shock-front lifetime
finite energy in the emission packet
finite phonon coherence time
finite relaxation time before dissipation
finite observation and decoding window
finite bandwidth / sampling resolution
finite material fatigue and damage budget
Lossy Throat Update
The previous information law remains:
I_out = I_in * exp(-L_throat) * R_repair
but the loss integral is now taken only over finite timelike support:
L_throat(N) = integral_{gamma restricted to T_emit(N)} [
lambda_T ||T(p,t)||^2
+ lambda_kappa |kappa(p,t)|
+ lambda_chi chi_mismatch(p,t)
+ lambda_mu memory_strain(p,t)
+ lambda_beta boundary_stress(p,t)
] dp dt
Admissibility requires:
I_out / I_in >= I_min(N)
rho_out >= rho_min(N)
mu_out <= mu_max(N)
beta_out <= beta_max(N)
RepairRate_N > DegradationRate_N
within the finite time interval.
Consequence for Active-Cell Counting
The density target is still:
|I_active(N)| ~ sqrt(N)
but this must now be proven under finite temporal support:
|{ i <= N : exists t in T_emit(N), chi_N(i,t)=1 }| / sqrt(N) -> 1 in limsup
This separates two receipts:
TemporalFinitenessReceipt: Delta T_N < infinity
ActiveCellCountingReceipt: finite-window active count scales like sqrt(N)
The first is physically mandatory. The second is a nontrivial scaling theorem.
No Infinite Temporal Cheating Rule
The construction may not claim density by allowing unlimited emission time for fixed N.
Forbidden:
For fixed N, let T_emit -> infinity until enough cells activate.
Allowed:
Choose a scale-dependent but finite T_emit(N), viscosity nu_N, threshold theta_N, and shock schedule, then prove the active count inside that finite window.
Handoff to Algebraic Layer
The temporal layer only selects recoverable events:
finite lossy throat + phonon gate + timelike emission window -> I_active(N)
The Sidon property still requires:
A_N = { Phi(i) : i in I_active(N) }
Phi(a)+Phi(b)=Phi(c)+Phi(d) -> {a,b}={c,d}
If the model uses a virtual pair state, the condition is:
S_N(a,b) = Phi_N(a)+Phi_N(b)+Lambda_N(a,b)
S_N(a,b)=S_N(c,d) -> {a,b}={c,d}
where Lambda_N must be non-separable and collision-penalizing.
Lean-Oriented Skeleton
structure FiniteTimelikeEmissionReceipt where
T_emit : Nat -> Set Nat
finite_time : forall N, (T_emit N).Finite
structure RecoverableEventReceipt where
active : Nat -> Nat -> Prop
survivable : Nat -> Nat -> Prop
active_implies_survivable : forall i t, active i t -> survivable i t
structure FiniteActiveSetReceipt where
I_active : Nat -> Set Nat
finite_active : forall N, (I_active N).Finite
projected_from_time : forall N i,
i in I_active N -> exists t, t in T_emit N ∧ active i t
structure ActiveCellCountingReceipt where
sqrt_density_limsup_one : Prop
Audit Classification
Receipt: FiniteTimelikeEmissionSet
Status: FORMAL_CONSTRAINT_DRAFT
Gate: U_scope
Reason: finite timelike support is physically mandatory, but scale functions T_emit(N), nu_N, theta_N, eta_min(N), and the active-cell counting theorem still need explicit definitions and proof.
Required Receipts
TemporalFinitenessReceipt
CausalSupportReceipt
FiniteEnergyReceipt
FiniteBandwidthReceipt
PhononCoherenceTimeReceipt
FiniteWindowActiveCountingReceipt
ScaleTuningReceipt
NonseparableEncodingReceipt
Boundary
This note does not prove the Sidon theorem. It prevents overclaiming by forcing every active cell to arise from a finite recoverable timelike emission event.
Correct doctrine:
The throat is lossy, torsional, and finite in time.
Only recoverable phonon events inside the finite timelike emission window may be counted.
The resulting active indices must still be passed through a non-separable algebraic lock to obtain Sidon pair-sum injectivity.