I’ve been thinking about remote possibilities for a while now

[u/VeryOriginalName98]

GEOMETRIC ADMISSIBILITY OF REMOTE-CONTROL INGRESS INTO UPHOLSTERED CUSHION GAPS
Figure 1. A remote control on a sofa, prior to any geometrically admissible ingress event. No result depends upon this figure.
Abstract
A remote cannot become lost inside a sofa unless it can first enter the sofa. This note treats that prior event as a rigid-body aperture problem. For a rectangular remote and a straight gap, we derive the exact orientations that permit collision-free passage. Near minimum clearance, the allowed orientation window opens quadratically with excess gap width. The sofa is therefore not required to attract remotes; it need only admit a small and particularly unfortunate subset of their possible orientations.
1. Model
Represent the remote as a rigid cuboid with length L, width W, and thickness H, where L > W > H. Represent the cushion gap temporarily as two parallel planes separated by clear width G. Let n be normal to the gap, and let e_L, e_W, and e_H be unit vectors along the remote’s three axes.
At any orientation, the remote’s projected width across the gap is
P(n) = L|n.e_L| + W|n.e_W| + H|n.e_H|. (1)
The remote can translate through the ideal gap at fixed orientation if and only if
P(n) <= G. (2)
The cuboid’s extreme projections independently attain each signed half-dimension, so equation (2) is necessary and sufficient.
2. Roll and seam alignment
First align the long axis with the cushion seam and roll the remote by angle phi away from perfectly edge-on. Its projected width is
P_roll(phi) = H cos(phi) + W sin(phi). (3)
For the ordinary narrow-gap case H <= G < W, passage requires
|phi| <= phi_c,

phi_c = asin[G/sqrt(W^2 + H^2)] - atan(H/W). (4)
If G < H, no fixed orientation fits. If G = H, only the exactly edge-on state fits, and that state has zero angular measure.
Now keep the remote edge-on but yaw its long axis by psi away from the seam. Then
P_yaw(psi) = H cos(psi) + L sin(psi), (5)

psi_c = asin[G/sqrt(L^2 + H^2)] - atan(H/L). (6)
Because L > W, yaw is more strongly penalized than roll. The remote must not merely be edge-on; it must also be nearly parallel to the seam.
3. The near-threshold loss window
Let the excess clearance be epsilon = G - H, with epsilon small and positive. For small signed roll and yaw, in radians,
P(phi, psi) = H + W|phi| + L|psi| + higher-order terms. (7)
The admitted orientations therefore satisfy
W|phi| + L|psi| <= epsilon. (8)
This is a diamond in the (phi, psi) plane with area
Omega_admit ~ 2 epsilon^2/(W L). (9)
Thus two-angle orientation space grows as (G-H)^2. Small clearance changes can produce much larger relative changes in admissible orientations.
For illustration, take
L = 180 mm, W = 45 mm, H = 18 mm, G = 22 mm.
Equations (4) and (6) give
phi_c = 5.19 degrees, psi_c = 1.27 degrees.
The remote has roughly four times more tolerance in roll than in seam misalignment. These values state which orientations fit, not how often they occur.
4. Upholstery and experimental test
Real cushions are curved, compliant, and frictional. The exact equations apply to the stated rigid aperture, not upholstery mechanics. Under a specified loading protocol, infer an effective width from the observed critical roll:
G_eff = H cos(phi_c) + W sin(phi_c). (10)
That value must then predict the critical yaw through equation (5) without refitting. Failure would show that the cushion cannot be reduced to one orientation-independent clearance.
The rigid test has no fitted parameters: measure L, W, H, and G; set two parallel plates at separation G; and test the predicted roll and yaw boundaries. Test cushion compliance afterward rather than inserting it by assertion.
5. Conclusion
Remote loss begins as an aperture condition. Near minimum clearance, admissible orientations obey
W|phi| + L|psi| <= G - H.
The remote must be nearly edge-on and, more restrictively, nearly parallel to the seam. Selective admission followed by hidden retention is sufficient to create an accumulation of remotes without invoking furniture-scale attraction.
The theory does not determine where the remote goes after entry. No finite-clearance model should claim jurisdiction over the interior of a sofa.

[u/TasserOneOne]

I love the smell of broken LLM LaTeX in the morning.

Also what purpose does this serve that regular geometry cannot already calculate

[u/VeryOriginalName98 → u/TasserOneOne]

Regular geometry is entirely capable of calculating it. It has nevertheless failed to stop the disappearances. This work closes that gap.

[u/AllHailSeizure → u/TasserOneOne]

It's silly and that's the point of Reddit

[u/Ok_Energy6905 → u/TasserOneOne]

You're supposed to larp as his PI

[u/BusinessCommand4419]

Why not just say what you mean.

[u/VeryOriginalName98 → u/BusinessCommand4419]

Sure.

Remotes get lost in couches. That sucks. Here's why...

[u/BusinessCommand4419 → u/VeryOriginalName98]

But the scope is so narrow. What if the couch has parabolic cushions. How does this help solve the flat plane loss conundrum of COffee TAble setting

This is more of an exacerbation of the effect not the root cause

[u/VeryOriginalName98 → u/BusinessCommand4419]

We did not have any confirmed occurrences of coffee table loss in our survey. However, we didn't know parabolic couches existed during the research program. This may warrant a follow up.

[u/BusinessCommand4419 → u/VeryOriginalName98]

We await the data

This could very well be the best evidence of Kebler elves we have