# Global Shatter-Loss Identity Equation

## Purpose

This note consolidates the finite timelike emission, torsional corridor, phononic black hole analogue, and virtual Sidon fabric layers into one governing admissibility equation.

The model is an information-theoretic, phonon-only, lossy throat model. It does not claim literal spacetime wormhole engineering, DNA/matter transport, or that acoustic/Hawking equations prove Sidon injectivity by themselves.

## Core Statement

```text
shock destabilizes virtual Sidon fabric
-> transient acoustic/information horizon forms
-> phonon/spectral packets undergo greybody filtering and torsional loss
-> repair either recovers a discrete identity state or fails
-> surviving finite-window events become active indices
-> nonseparable encoding supplies the Sidon lock
```

## Global Shatter-Loss Identity Equation

For index `i` in a finite timelike emission window `T_N = [0,T_N]`, define:

```text
Psi_out(i;T_N) = Repair_{rho,mu}( Psi_in(i) * Gamma_i(omega) * exp(- integral_0^{T_N} L_total(i,t) dt) )
```

where:

```text
Gamma_i(omega) = greybody / bandgap / transmission coefficient
Repair_{rho,mu} = nonlinear coherence-and-memory repair operator
L_total = total loss density of the forming throat
```

The active index gate is:

```text
i in I_active(N) iff
  exists t in [0,T_N] such that
    W_shock(i,t) >= E_barrier(i,N)
    and |partial_x u(i,t;nu_N)| >= theta_A(N)
    and T_eff(i,t) >= T_min(N)
    and Gamma_i(omega,t) >= Gamma_min(N)
    and mode_overlap_i(t) >= eta_min(N)
    and ||Psi_out(i;T_N)|| >= Psi_min(N)
```

## Total Loss Density

Use:

```text
L_total(i,t) =
    lambda_H * k_B * T_eff(i,t) / E_scale
  + lambda_T * ||T(i,t)||^2
  + lambda_kappa * |kappa(i,t)|
  + lambda_chi * chi_mismatch(i,t)
  + lambda_mu * mu(i,t)
  + lambda_beta * max(0, beta(i,t)-beta_max)
  + lambda_M * (1 - eta_i(t))
  - lambda_G * log(max(Gamma_i(omega,t), epsilon))
```

Dimensional note:

The Hawking analogue contributes through `k_B T_eff`, an energy scale. Do not add a raw temperature directly to dimensionless geometric losses unless all terms are nondimensionalized by a common energy/action scale.

## Acoustic Horizon Term

The effective Hawking/acoustic temperature is:

```text
T_eff(i,t) = hbar/(2*pi*k_B) * kappa_eff(i,t)
```

with:

```text
kappa_eff(i,t) = |partial_x(c_s(i,t) - v_flow(i,t))| at v_flow = c_s
```

In the Burgers approximation:

```text
v_flow(i,t) ~ u(i,t)
```

and the horizon/gate occurs near:

```text
u(i,t) = c_s(i,t)
```

The shock gradient controls the acoustic horizon temperature/noise floor.

## Greybody / Bandgap Filtering

Transmission is governed by:

```text
0 <= Gamma_i(omega,t) <= 1
```

with:

```text
Gamma_i small -> bandgap blocks or localizes propagation
Gamma_i large -> mode transmits through the gate
```

The output channel law is:

```text
I_out/I_in = Gamma_i(omega) * exp(- integral_0^{T_N} L_total(i,t) dt) * R_repair(i)
```

Admissibility requires:

```text
I_out/I_in >= I_min(N)
SNR_out >= SNR_min(N)
eta_i >= eta_min(N)
RepairRate_N > DegradationRate_N
```

## Finite-Time Constraint

No actual infinite dataset or infinite emission window is allowed.

```text
T_N < infinity
E_N = { (i,t) : 1 <= i <= N, t in [0,T_N], chi_N(i,t)=1 }
I_active(N) = { i <= N : exists t, (i,t) in E_N }
```

The density target is a limit over finite receipts:

```text
limsup_{N -> infinity} |I_active(N)| / sqrt(N) = 1
```

This is not a claim that an infinite dataset was processed.

## Sidon Handoff

The material/throat equation selects recoverable active indices. The Sidon condition still belongs to the encoding layer.

Classical encoding:

```text
A_N = { Phi_N(i) : i in I_active(N) }
Phi_N(a)+Phi_N(b)=Phi_N(c)+Phi_N(d) -> {a,b}={c,d}
```

Virtual pair-state encoding:

```text
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(a,b)` must be symmetric, pair-specific, and nonseparable.

## Important Correction

Do not state that torsional loss alone is the arithmetic lock. Torsional loss is a physical/admissibility sieve. The arithmetic lock requires either:

```text
exact nonseparable encoding
or
proof that the virtual collision penalty projects to ordinary pair-sum injectivity
```

## Receipt Classification

```text
Receipt: GlobalShatterLossIdentityEquation
Status: FORMAL_EQUATION_DRAFT
Gate: U_scope
Reason: equation coherently combines finite timelike emission, acoustic horizon loss, torsional loss, repair, and active-cell gating, but needs dimensional calibration, material parameters, finite-window counting, and nonseparable Sidon encoding receipts.
```

## Required Receipts

```text
DimensionalConsistencyReceipt
AcousticMetricReceipt
EffectiveSurfaceGravityReceipt
GreybodyBandgapReceipt
TorsionLossReceipt
RepairOperatorReceipt
FiniteTimelikeEmissionReceipt
ActiveCellCountingReceipt
NonseparableEncodingReceipt
CompactDensityReceipt
```