Add Ahmed Integral scalar witness gate documentation

6-Documentation/famm/AHMED_INTEGRAL_SCALAR_WITNESS_GATE.md

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+# Ahmed Integral Scalar Witness Gate
+
+## Purpose
+
+Add Ahmed's Integral as a definite-integral scalar witness gate.
+
+The project-useful shape is:
+
+```text
+complicated analytic integrand
+→ definite-integration projection
+→ exact scalar witness
+→ residual receipt
+→ branch / precision / transform verification
+```
+
+## Source identity
+
+Original paper metadata:
+
+```text
+Ahmed, Z.; Dale, Knut; Lamb, George.
+Definitely an Integral: 10884.
+The American Mathematical Monthly, 2002.
+DOI: 10.2307/3072448.
+```
+
+Common identity:
+
+```math
+\int_{0}^{1}
+\frac{\arctan\left(\sqrt{2+x^{2}}\right)}
+{(1+x^{2})\sqrt{2+x^{2}}}
+\,dx
+=
+\frac{5\pi^{2}}{96}
+```
+
+## Project name
+
+```text
+AHMED_INTEGRAL_SCALAR_WITNESS_GATE
+```
+
+## Universal Shortcut Center packet
+
+```math
+\Gamma_{\mathrm{Ahmed}}
+=
+(
+X_f,
+\pi_{\int},
+W_{5\pi^2/96},
+R_{\mathrm{verify}},
+I_{\mathrm{area}},
+G_{\mathrm{integrable}},
+K,
+\epsilon
+)
+```
+
+| Packet term | Meaning |
+|---|---|
+| `X_f` | full arctangent/radical integrand manifold |
+| `pi_int` | definite integration projection over `[0,1]` |
+| `W_5pi2_96` | exact scalar witness `5*pi^2/96` |
+| `R_verify` | analytic proof, high-precision quadrature, or symbolic reduction receipt |
+| `I_area` | preserved accumulation / area invariant |
+| `G_integrable` | branch, endpoint, convergence, and real-valuedness guards |
+| `K` | symbolic/numeric/proof cost |
+| `epsilon` | residual between evaluated and exact witness |
+
+## Residual receipt
+
+```math
+R_{\mathrm{Ahmed}}
+=
+\left|
+\int_{0}^{1}
+\frac{\arctan\left(\sqrt{2+x^{2}}\right)}
+{(1+x^{2})\sqrt{2+x^{2}}}
+\,dx
+-
+\frac{5\pi^{2}}{96}
+\right|
+```
+
+Pass condition:
+
+```math
+R_{\mathrm{Ahmed}}\le \Theta_{\mathrm{tol}}
+```
+
+## Why this differs from the polynomial-exponential ladder gate
+
+The polynomial-exponential integral gate is a finite ladder collapse:
+
+```text
+repeated integration by parts
+→ finite summation
+→ derivative receipt
+```
+
+Ahmed's Integral is a hidden transform/scalar witness gate:
+
+```text
+arctangent + radical definite integral
+→ special reduction / quadrature / probability-integral route
+→ exact pi^2 scalar receipt
+```
+
+## Adversarial verification checks
+
+The Warden should check:
+
+```text
+branch choice of arctan and square root
+endpoint behavior on [0,1]
+numeric precision / quadrature stability
+symbolic transformation validity
+hidden integrand perturbation that changes the scalar receipt
+```
+
+Formal hidden-perturbation check:
+
+```math
+h\in\ker \Pi_{\mathrm{metadata}}
+\quad\text{but}\quad
+\int_0^1 h(x)\,dx\ne0
+```
+
+Meaning:
+
+```text
+a perturbation is invisible to the current symbolic description,
+but it changes the definite-integral scalar witness.
+```
+
+## FAMM object
+
+```math
+\mathfrak C_{\mathrm{AhmedIntegral}}
+=
+A_{16}(u_{\mathrm{ahmed}})
+\otimes
+[
+\Sigma_f
++
+\Sigma_{\int_0^1}
++
+\Sigma_{\arctan}
++
+\Sigma_{\sqrt{2+x^2}}
++
+\Sigma_{5\pi^2/96}
++
+\Sigma_{\epsilon}
++
+\Sigma_{\mathrm{branch}}
++
+\Sigma_{\mathrm{receipt}}
+]
+```
+
+## Stack placement
+
+```text
+AHMED_INTEGRAL_SCALAR_WITNESS_GATE
+→ Universal Shortcut Center Manifold
+→ Builder-Judge-Warden
+→ adversarial branch / precision checks
+→ NUVMAP scalar witness node
+→ exact / high-precision quadrature receipt
+```
+
+## Warden boundary
+
+This gate records a definite-integral scalar witness and its project use. It does not claim every hard integral has the same structure or that numeric agreement alone is proof.
+
+Allowed claim:
+
+```text
+Ahmed's Integral is a clean scalar-witness benchmark for checking whether a proposed analytic shortcut preserves the exact definite-integral value.
+```
+
+Disallowed claim:
+
+```text
+A numerical match to 5*pi^2/96 proves a new integral theorem without branch, endpoint, and proof receipts.
+```
+
+## Project sentence
+
+Ahmed's Integral is a scalar witness gate: a complicated arctangent-radical integral collapses to `5*pi^2/96`, giving the stack a clean exact-receipt target where Builder proposes reductions, Judge verifies the scalar invariant, and the Warden checks hidden branch, precision, and transformation failures.