5.9 KiB
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:
\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:
g_{\mathrm{throat}}
=
w_P g_P + w_B g_B + w_N g_N + w_T g_T
and:
w_i = \frac{p_i}{p_P+p_B+p_N+p_T}.
The torsion-modified version is:
\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:
p_0 = p_P+p_B+p_N+p_T.
2. q-desic connection correction
Define the q/FAMM-corrected effective connection:
\Gamma_{\mathrm{eff}}
=
\Gamma_{LC}
+
K_T
+Q_\Gamma
+F_\Gamma
+\Omega_\Gamma.
Terms:
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:
\nabla \rightarrow \nabla^{(qF\Omega)}.
3. Q-FAMM throat evolution equation
The updated throat evolution equation is:
\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:
p_{\mathrm{throat}} = p_P+p_B+p_N+p_T
and:
Δ_{g,qFΩ} = q/FAMM/interface-corrected Laplace-Beltrami operator
Ω_gap = penalty from representation seam debt or bad boundary coupling
Interpretation:
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:
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:
C_{QWH}(\gamma_{\mathrm{throat}})
<
C_{normal}(\gamma_{\mathrm{manifold}}).
For compression routes:
C_{QWH}(\gamma_{transform})
+C_{residual}
+C_{decoder}
<
C_{baseline}.
5. q-corrected throat cost
Classical throat cost:
C_{classical}
=
C_{exoticMatter}+C_{stabilityPenalty}.
q-corrected cost:
C_{qthroat}
=
C_{classical}
+C_{connection}
+C_{torsion}
+C_{density}
+C_{interface}
+C_{FAMM}
+C_{decode}.
Expanded:
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:
\eta_{classical}
=
\frac{D_{manifold}}{L_{proper}}.
q-corrected efficiency is:
\eta_q
=
\frac{D_{manifold}}{L_{proper}+C_{qcorrection}}.
where:
C_{qcorrection}
=
C_{connection}+C_{torsion}+C_{density}+C_{interface}+C_{FAMM}+C_{decode}.
This blocks false positives:
short throat ≠ good throat
7. Temporal bounds
Maximum stable temporal component:
T_{max,q}
=
\frac{C_{temporal}}{D_q}
with:
D_q
=
D_{base}
+D_{connection}
+D_{torsion}
+D_{density}
+D_{interface}
+D_{FAMM}
+D_{decode}.
Minimum computable bound:
\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:
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:
\boxed{T_{min,q} \le T_{max,q}}
If:
T_{min,q} > T_{max,q},
then the throat is mathematically describable but computationally unusable.
8. Full admissible q-throat condition
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:
saved\_bits
>
model\_bits
+residual\_bits
+interface\_bits
+decoder\_bits
+FAMM\_penalty\_bits
and:
T_{min,compress}\le T_{max,context}.
Thus:
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:
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:
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.