# Q-Desic Wormhole Throat Equations

Status: EQUATION_UPDATE
Claim state: FORMAL_SCAFFOLD / ANALOGY_BOUNDED

This note updates the wormhole throat equation system with q-desic routing, interface-gap pricing, FAMM route scars, decode/recovery constraints, and computable temporal bounds.

This is not a claim that the stack implements physical quantum gravity. The q-desic source pattern is used as an operator-corrected route-selection analogy: visible geometry is not enough; the route must be corrected by hidden connection, torsion, density, interface, scar, and recovery terms.

---

## 1. Existing throat center equation

The existing contested-center throat equation is:

```math
\frac{\partial H}{\partial t}
=
\Delta_{g_{\mathrm{throat}}}H
+
2\left\langle
\nabla\log(p_P+p_B+p_N+p_T),\nabla H
\right\rangle
```

with:

```math
g_{\mathrm{throat}}
=
w_P g_P + w_B g_B + w_N g_N + w_T g_T
```

and:

```math
w_i = \frac{p_i}{p_P+p_B+p_N+p_T}.
```

The torsion-modified version is:

```math
\frac{\partial H}{\partial t}
=
\Delta_g H
+
2\left\langle
\nabla\log p_0
+\nabla\log\det(e)
+\widetilde{T},
\nabla H
\right\rangle
```

where:

```math
p_0 = p_P+p_B+p_N+p_T.
```

---

## 2. q-desic connection correction

Define the q/FAMM-corrected effective connection:

```math
\Gamma_{\mathrm{eff}}
=
\Gamma_{LC}
+
K_T
+Q_\Gamma
+F_\Gamma
+\Omega_\Gamma.
```

Terms:

```text
Γ_LC = Levi-Civita connection of the current throat metric
K_T  = contorsion / torsion correction
Q_Γ  = q-desic operator-level connection correction
F_Γ  = FAMM route-scar correction
Ω_Γ  = interface-gap / seam correction
```

Use the corrected derivative:

```math
\nabla \rightarrow \nabla^{(qF\Omega)}.
```

---

## 3. Q-FAMM throat evolution equation

The updated throat evolution equation is:

```math
\boxed{
\frac{\partial H}{\partial t}
=
\Delta_{g,qF\Omega}H
+
2\left\langle
\nabla^{(qF\Omega)}\log p_{\mathrm{throat}},
\nabla^{(qF\Omega)}H
\right\rangle
-
\Omega_{\mathrm{gap}}H
}
```

where:

```math
p_{\mathrm{throat}} = p_P+p_B+p_N+p_T
```

and:

```text
Δ_{g,qFΩ}  = q/FAMM/interface-corrected Laplace-Beltrami operator
Ω_gap      = penalty from representation seam debt or bad boundary coupling
```

Interpretation:

```text
A throat is not admissible merely because it is short.
It is admissible only if the q-corrected route remains stable, recoverable,
and cheaper than the normal manifold path.
```

---

## 4. Q-desic throat action

For a route γ through or around a throat:

```math
C_{QWH}(\gamma)
=
\int_\gamma
\left[
  ds_g
  +\lambda_Q\lVert Q_\Gamma\rVert
  +\lambda_T\lVert T\rVert
  +\lambda_p\lVert\nabla\log p_{\mathrm{throat}}\rVert
  +\lambda_F L_{FAMM}
  +\lambda_\Omega\Omega_{gap}
  +\lambda_R R_{decode}
\right]ds.
```

A q-desic throat is route-admissible iff:

```math
C_{QWH}(\gamma_{\mathrm{throat}})
<
C_{normal}(\gamma_{\mathrm{manifold}}).
```

For compression routes:

```math
C_{QWH}(\gamma_{transform})
+C_{residual}
+C_{decoder}
<
C_{baseline}.
```

---

## 5. q-corrected throat cost

Classical throat cost:

```math
C_{classical}
=
C_{exoticMatter}+C_{stabilityPenalty}.
```

q-corrected cost:

```math
C_{qthroat}
=
C_{classical}
+C_{connection}
+C_{torsion}
+C_{density}
+C_{interface}
+C_{FAMM}
+C_{decode}.
```

Expanded:

```math
C_{qthroat}
=
C_{exoticMatter}
+C_{stabilityPenalty}
+C_{Q_\Gamma}
+C_T
+C_{\nabla\log p}
+C_{\Omega}
+C_{FAMM}
+C_R.
```

---

## 6. q-efficiency

Current visible efficiency is:

```math
\eta_{classical}
=
\frac{D_{manifold}}{L_{proper}}.
```

q-corrected efficiency is:

```math
\eta_q
=
\frac{D_{manifold}}{L_{proper}+C_{qcorrection}}.
```

where:

```math
C_{qcorrection}
=
C_{connection}+C_{torsion}+C_{density}+C_{interface}+C_{FAMM}+C_{decode}.
```

This blocks false positives:

```text
short throat ≠ good throat
```

---

## 7. Temporal bounds

Maximum stable temporal component:

```math
T_{max,q}
=
\frac{C_{temporal}}{D_q}
```

with:

```math
D_q
=
D_{base}
+D_{connection}
+D_{torsion}
+D_{density}
+D_{interface}
+D_{FAMM}
+D_{decode}.
```

Minimum computable bound:

```math
\boxed{
T_{min,q}
=
\max\left(
\Delta t_{tick},
\left\lceil\frac{P}{F}\right\rceil,
\left\lceil\frac{W_d}{R_d}\right\rceil,
\left\lceil\frac{G_i}{R_i}\right\rceil,
\left\lceil\frac{L_{FAMM}}{R_f}\right\rceil,
\epsilon_{Q16}
\right)
}
```

Definitions:

```text
P       = payload size
F       = flux capacity
W_d     = decode work
R_d     = decode rate
G_i     = interface gap cost
R_i     = interface crossing rate
L_FAMM  = route-scar / frustration load
R_f     = repair or stabilization rate
ε_Q16   = smallest Q16.16 representable quantum
```

Temporal admissibility condition:

```math
\boxed{T_{min,q} \le T_{max,q}}
```

If:

```math
T_{min,q} > T_{max,q},
```

then the throat is mathematically describable but computationally unusable.

---

## 8. Full admissible q-throat condition

```math
AdmissibleQThroat(\gamma,P)
\iff
Traversable(\gamma,P)
\land
C_{QWH}(\gamma) < C_{normal}
\land
T_{min,q}\le T_{max,q}
\land
DecodeConnected(\gamma)
\land
NoOverflow_{Q16}(\gamma).
```

---

## 9. Compression interpretation

For compression, a transform throat is admissible iff:

```math
saved\_bits
>
model\_bits
+residual\_bits
+interface\_bits
+decoder\_bits
+FAMM\_penalty\_bits
```

and:

```math
T_{min,compress}\le T_{max,context}.
```

Thus:

```text
A compression wormhole is useful only when its q-corrected transform shortcut
beats the ordinary route after model, residual, interface, decoder, and FAMM
scar costs are all counted.
```

---

## 10. Warden boundary

Allowed:

```text
Use q-desic as an operator-corrected route-selection analogy.
Use the equations to price hidden route, interface, torsion, and decode costs.
Use the temporal bound as a computable admissibility gate.
```

Blocked:

```text
Do not claim physical wormhole construction.
Do not claim quantum gravity validation of OTOM/FAMM/AVMR.
Do not promote any transform throat unless q-corrected cost, temporal bounds,
and exact recovery are computed or formally gated.
```