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Equation 4 · AI Datacenter Interconnects in 2035: Scenarios, Signals, and Falsifiable Predictions

What does this equation mean?

R(t)=R0⋅2(t−t0)/TR(t) = R_0 \cdot 2^{(t - t_0)/T}

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Inputs and operationsR_0 × 2^(t - t_0)/T
Result or conditionR(t)
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This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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RR

Symbol R

R is part of the quantity the equation computes from the expression on the right.

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tt

Symbol t

t is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.

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R0R_0

Symbol R_0

R0R_0 is one of the signed contributions combined to compute the quantity on the left.

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t0t_0

Symbol t_0

the time or time index used in this relationship.

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TT

Symbol T

the assumed doubling period: [displayed formula].

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=

=

The expressions on both sides represent the same quantity under the stated assumptions.

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multiplication

multiplication

Multiply the quantities on either side.

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subtraction

subtraction

Subtract the following term or group from the preceding one. A leading minus marks a negative quantity.

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subscript

subscript

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

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superscript

superscript

A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.

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How to interpret it

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What the article says around this equation

Analysis. A naive extrapolation makes the compressing cadence explicit rather than hiding it. Writing per-lane rate as a function of time, with R0R_0 the most recent standardised rate at time t0t_0 and T the assumed doubling period: R(t)=R0⋅2(t−t0)/TR(t) = R_0 \cdot 2^{(t - t_0)/T}. Anchoring at R0R_0 = 200 Gb/s per lane, t0t_0 = 2026 , and holding the most recently observed four-year doubling period fixed gives R(2035) = 200 ⋅\cdot 2^{9/4} ≈\approx 950 Gb/s per lane by 2035 — essentially a 1 Tb/s single electrical lane. That number is offered to be doubted, not believed. The assumption it encodes — that the doubling period keeps compressing on a fixed clock rather than lengthening as channel loss and signal-to-noise margin close…
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Analysis. A naive extrapolation makes the compressing cadence explicit rather than hiding it. Writing per-lane rate as a function of time, with R0R_0 the most recent standardised rate at time t0t_0 and T the assumed doubling period: R(t)=R0⋅2(t−t0)/TR(t) = R_0 \cdot 2^{(t - t_0)/T}. Anchoring at R0R_0 = 200 Gb/s per lane, t0t_0 = 2026 , and holding the most recently observed four-year doubling period fixed gives R(2035) = 200 ⋅\cdot 2^{9/4} ≈\approx 950 Gb/s per lane by 2035 — essentially a 1 Tb/s single electrical lane. That number is offered to be doubted, not believed. The assumption it encodes — that the doubling period keeps compressing on a fixed clock rather than lengthening as channel loss and signal-to-noise margin close in — is exactly the part the standards process itself is already hedging against: P802.3dj did not simply push PAM-4 harder, it added an 800 Gb/s coherent-signalling option specifically for the reaches where intensity-modulated PAM-4 stops being the cheapest way to add bandwidth [ 5 ] . That is a documented fact about how the last doubling was actually achieved — partly through a different modulation scheme, not only through a faster clock — and it is a reason to expect the next doubling to arrive through some mixture of higher-order modulation, coherent optics, and more parallel lanes rather than through PAM-4 alone continuing to double on schedule.

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