Thread by @John_A_Ridge

So what is S-500? S-500 is a long-range air and missile defense system operated by the VKS (Russian Aerospace Forces).

It is a direct derivative of S-300V, the principal long-range maneuver SAM operated by the SV (Russian Ground Forces).
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I won't detail the complete program history of S-300P/F/V as @ChungTzuW has already written excellent threads about them which I strongly recommend reading for context.

In short, all three systems emerged from a joint VKS/VMF/SV procurement program.
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VKS/VMF requirements were focused on ABTs (air-breathing threats) while SV requirements were focused on TBMs (tactical ballistic missiles). This led to a divergence into S-300P/F and S-300V using radically different missiles and radars.
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S-300P received mild upgrades beginning in the mid-1990s and was redesignated S-400, originally S-300PM(U)3.

The SV pushed continual upgrades for S-300V through the early-2010s, terminating in all Russian fire units being upgraded to the S-300V4 standard by 2015.
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Concurrently, the VKS initiated a development program for an overhauled S-300V derivative, now S-500, to meet their own requirements in the late 2000s.

The first production S-500 regiment was accepted into VKS service in 2021.
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The VKS wanted a mobile ABM system to supplement the siloed A-135/235 complex, which provides terminal defense against ICBMs around Moscow, in addition to filling a mobile theater missile defense requirement.
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With the history out of the way, what is S-500 actually?

One of the VKS' primary requirements was for an expanded defended footprint against TBMs and capability against a wider range (faster) of TBMs relative to S-300V.
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Against ABTs, the radius of your defended footprint is effectively a function of engagement range.

However against TBMs, defended footprint is primarily a function of engagement altitude as engaging a TBM at greater range means engaging it at greater altitude.
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One of the primary design challenges facing TBM-optimized interceptors is maneuverability and control authority.

Most missiles across the spectrum use aerodynamic control surfaces to generate control authority.
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However, for extended range TBM engagements at high altitude, the atmosphere is extremely thin and control surfaces generate dramatically less control authority.

Atmospheric density at 50 km altitude is <0.1% atmospheric density at sea-level.
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Below is the maneuverability diagram for 5V55, the initial interceptor deployed with S-300P when it entered service. 5V55's maneuverability shrinks dramatically as it approaches apogee due to loss in control authority.
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There are really two paths to solve this problem.

The first is to generate control authority non-aerodynamically, typically via an Attitude Control System. The second is to increase the efficacy of the control surfaces.
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ACS has been employed with enormous success by PAC-3, Aster, and all exoatmospheric interceptors (THAAD, SM-3...) by necessity.

An ACS consists of solid- or liquid-fueled Attitude Control Motors that enable the missile to maneuver without aerodynamic control authority.
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The second approach is to generate additional aerodynamic control authority by increasing airflow over the control surfaces. In other words, make the missile faster.

As can be seen with 5V55, the missile's G value increases for a fixed altitude if velocity increases.
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This is the approach taken by the Russians for 77N6-N/N1, the interceptors for S-500.

Fundamentally, 77N6-N is just a 9M82M from S-300V4 with an enlarged and overhauled booster section to increase it's cutoff velocity by ~1,000 m/s.
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As referenced by Russian arms commentators, the additional velocity seems to have roughly doubled 77N6-N's engagement ceiling to ~60 km from 9M82M's 30 km ceiling.

I very roughly estimate, albeit without modeling, this likely increased its defended footprint to 60-80 km.
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Now what is funny about this?

77N6-N is an comically enormous missile for a mobile theater missile defense platform. Two 77N6-N rounds on a 77P6 launcher have a larger form factor than 8 MIM-401 Talon rounds on a THAAD launcher.
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So against TBMs, 77N6-N offers ~25% of the engagement ceiling, <50% of the defended footprint, 25% the magazine depth, and worse endgame maneuverability than Talon.

And really the only thing to show for it is that 77N6-N can engage ABTs while Talon cannot.
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The Russians could've achieved their TBM defeat requirements with dramatically lighter and smaller missiles had they pursued ACS, especially paired with a Hit-To-Kill warhead.

9M82/77N6-N's 150 kg warhead greatly contributes to overall missile volume and mass.
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However prior Russian ACS programs, notably 9M96, have demonstrated a consistent lack of technical maturity and have only been deployed in fairly limited service.
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Against even relatively moderate numbers of shorter range TBMs such as ATACMS, 77N6-N is particularly poorly optimized.

Assuming ripple fire of two interceptors per TBM, each launcher can only engage a single threat before exhausting its magazine.
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Aside from struggling to cost-effectively engineer HTK missiles with ACS, its worth considering Russian doctrine.

@MatejRisko and @GrangerE04117 can elaborate in far more detail, but the Russians likely compromised S-500's TBM requirements with its ABT requirements.
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Now usual disclaimers apply here. I notably did not touch on S-500's sensors or the rest of the kill chain which is critical to successfully completing an engagement.

But from the kinematics alone, S-500 is a comically unoptimized platform for its TBM defeat requirements.
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