Add q-desic wormhole throat equation update

6-Documentation/docs/avmr/q_desic_wormhole_throat_equations.md

@@ -0,0 +1,388 @@
+# 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.
+```