[BUG] Alpha Injection Docs Still Use Legacy Price-Based Δτ Formula (Conflict With New Flow-Based Emissions Model)

[chrismaresca]

Bug Overview

This is a docs bug:
The “Alpha reserve injection” section still uses the legacy price-based TAO injection formula when deriving Δαᵢ, even though the earlier section has switched entirely to the new flow-based TAO injection model (Taoflow).

This causes the derivation to appear inconsistent and mathematically incorrect when read in the context of the updated flow-based Δτᵢ.

Bug Description

In the docs, under Alpha reserve injection, the following derivation appears:

Δαᵢ = Δτᵢ / pᵢ

When we fill in this equation with the previous formula for Δτᵢ,
the price pᵢ cancels out, giving:

Δαᵢ = Δτ̄ / Σⱼ pⱼ

However, this “pᵢ cancels out” step is only valid if the “previous formula for Δτᵢ” is price-based, not flow-based. But current protocol behavior as of November 2025) Δτᵢ is defined by the new flow-based emission model:

Δτᵢ = T emitted per block * zᵢ (**flow, EMA-smoothed).

which depends on:

• EMA of net TAO flows
• offset + clipping
• power normalization
• and has no dependence on price pᵢ

Therefore, you cannot substitute the flow-based Δτᵢ into
Δαᵢ = Δτᵢ / pᵢ
and get the cancellation shown in the docs.

This creates a contradiction:

1. The alpha injection section still assumes Δτᵢ ∝ pᵢ
2. But the TAO injection section (above it) has already replaced Δτᵢ with the flow-based version

This makes readers think either:

1. Alpha injection should use Δτᵢ(flow): in which case price does not cancel, or
2. A synthetic / price-based Δτᵢ is intended for α injection, separate from the flow-based Δτᵢ(real).

Currently the docs mix both models, which is confusing and mathematically inconsistent.

Suggested Fixes and Relevant Docs

Update the derivation to reflect the flow-based Δτᵢ if that is the intended behavior.

1. Flow-Based Emissions (Taoflow)
2. Alpha Reserve Injection (problematic derivation)