Explain the Lean theorem `compensatory_nonneg_of_sqNorm_le_one` in module `IndisputableMonolith.Cost.Ndim.Bridge`. Write for an educated reader who knows science and programming but may not know Lean. Cover: (1) what the declaration says in plain English, (2) why it matters in Recognition Science, (3) how to read the formal statement, (4) visible dependencies or certificates in the supplied source, and (5) what this declaration does not prove. Cite only declarations present in the supplied Recognition source.

(1) What the Declaration Says in Plain English

The theorem compensatory_nonneg_of_sqNorm_le_one establishes a fundamental inequality for quadratic costs in $N$ dimensions. It states that if you have a vector of weights $\alpha$ whose squared magnitude is at most $1$, the "additive" quadratic cost of any state vector $\varepsilon$ will always be greater than or equal to its "multiplicative" quadratic cost. Consequently, the difference between them—defined as the "compensatory" residual—is always non-negative.

(2) Why it Matters in Recognition Science

In Recognition Science, physics is forced by cost (a THEOREM of the framework). When generalizing cost processing from a single scalar to an $N$-dimensional vector of states $\varepsilon$, there are two canonical ways to approximate the cost quadratically:

• Additive: Process the cost of each component independently and sum them.
• Multiplicative: Aggregate the components using weights $\alpha$, then compute the combined cost.

This theorem guarantees that under normalized weights, the additive configuration always bounds the multiplicative configuration from above. The non-negative residual (the compensatory term) represents the structural gap between independent processing and aggregated processing. It ensures that no negative costs spontaneously appear in multi-dimensional generalizations, preserving the strict positivity required by the underlying Law of Logic.

(3) How to Read the Formal Statement

theorem compensatory_nonneg_of_sqNorm_le_one {n : ℕ}
    (α ε : Vec n) (hα : dot α α ≤ 1) :
    0 ≤ compensatoryQuadratic α ε

• {n : ℕ}: The theorem applies generically to any natural number dimension $n$.
• (α ε : Vec n): It accepts two $n$-dimensional real vectors: a weight vector $\alpha$ and a state/error vector $\varepsilon$.
• (hα : dot α α ≤ 1): This is the prerequisite hypothesis. The dot product of $\alpha$ with itself (its squared $L_2$ norm, $|\alpha|^2$) must be $\le 1$.
• 0 ≤ compensatoryQuadratic α ε: The conclusion asserts that the compensatory quadratic term evaluated at $\alpha$ and $\varepsilon$ is greater than or equal to zero.

(4) Visible Dependencies in the Source

The proof is extremely tight and routes through three explicit steps within the same module:

1. It unfolds the definition of compensatoryQuadratic, replacing it with additiveQuadratic ε - multiplicativeQuadratic α ε.
2. It invokes the immediately preceding theorem multiplicative_le_additive_of_sqNorm_le_one, which proves the multiplicative cost is bounded by the additive cost. That underlying theorem relies on a squared Cauchy-Schwarz bound provided by dot_sq_le_sqNorm_mul.
3. It uses the linarith tactic to close the proof algebraically (since if $A \ge B$, then $A - B \ge 0$).

(5) What this Declaration Does Not Prove

• The Exact RS J-Cost Function: This module establishes bounds for purely quadratic approximations ($x^2$). It does not prove the exact reciprocal-symmetric J-cost functional equation $J(x) = (x+x^{-1})/2 - 1$ unique to Recognition Science.
• Weight Normalization: It does not prove that the weight vector $\alpha$ must have a norm $\le 1$; the bound is strictly conditional on the external hypothesis hα being true.
• Equality: It does not compute the exact value of the compensatory term, merely that it cannot fall below zero.