# EA contact-depth operator Date: 2026-04-29 Status: `DER-PHYS + VAL`, canonical in PTC as the current `EA_eV` precision layer. The operator is derived as the continuous-depth rewrite of the fixed PT contact projector stack and validated numerically against the current PTC atomic EA benchmark. The bridge statement "EA reads the capture boundary" remains physical bridge language, not an unconditional mathematical theorem. Related files: - `ptc/atomic_precision_frontier.py` - `ptc/tests/test_atomic_precision_frontier.py` - `docs/research/PT_ATOMIC_PRECISION_FRONTIER.md` --- ## 1. Problem The canonical EA channel already includes: ```text EA_geo -> capture-side CPR boundary continuum -> p threshold / p closure / d core residual fields -> continuous channel residual ``` It reaches: ```text EA continuous channel N=73 MAE=1.125303% MAE_eV=0.007973 max=3.803883% ``` The remaining residuals are small but structured. They concentrate on the same geography that appears in the last IE pockets: ```text f contact pressure, d compact/contraction/penetration phases, p residual second harmonic. ``` The aim is not to add fitted coefficients. The aim is to find the continuous PT operator whose sampled projections reproduce the residual field. --- ## 2. Capture coordinate EA is a capture observable. If the neutral atom has filling `n` in an active shell of capacity `N_l = 2(2l+1)`, the incoming electron reads: ```text n -> n+1. ``` The receiver coordinate is therefore: ```text x = (n+1)/N_l. ``` For closed shells the diagnostic reads the virtual entry edge: ```text x = 1/N_l. ``` This keeps the coordinate inside the channel circle and avoids the unphysical point `(N_l+1)/N_l`. The perturbative scalar is the same scalar used by the CPR residuals: ```text u_Z = (Z alpha)^2. ``` --- ## 3. d-depth coordinate The d channel has four realized radial depths in the table: ```text period 4 -> tau = 0 compact first d shell period 5 -> tau = 1 transition d shell period 6 -> tau = 2 lanthanide-contracted d shell period 7 -> tau = 3 superheavy d shell ``` Define: ```text tau = period - 4. ``` The continuous depth basis on these four nodes is the cubic Lagrange basis: ```text L0(tau) = - (tau-1)(tau-2)(tau-3) / 6 L1(tau) = tau(tau-2)(tau-3) / 2 L2(tau) = - tau(tau-1)(tau-3) / 2 L3(tau) = tau(tau-1)(tau-2) / 6 ``` Thus `L_i(j) = delta_ij`. This is the key point: the former period projectors are not independent empirical gates. They are the sampled values of a continuous depth coordinate. --- ## 4. d contact kernel The first d shell needs the compact boundary kernel: ```text sin(2 pi x) + cos(2 pi x) - sin(4 pi x). ``` The period-5 transition depth keeps the first oriented harmonic: ```text sin(2 pi x). ``` The period-6 contracted depth keeps the second harmonic plus the cosine compression: ```text sin(4 pi x), cos(2 pi x). ``` Using only fixed PT amplitudes, the continuous d kernel is: ```text K_d(x,tau) = [CROSS_57 L0(tau) - S3 D3 L1(tau)] sin(2 pi x) + [CROSS_57 L0(tau) - D5^2 L2(tau)] cos(2 pi x) + [-CROSS_57 L0(tau) + S3 D5 L2(tau)] sin(4 pi x). ``` The superheavy basis `L3` is zero at this order. This is not a fitted zero; it says the remaining period-7 d contact signal is already below the current EA precision layer and should be tested only after the IE weak trace is derived. On the actual d periods this continuous kernel samples to: ```text tau=0 : CROSS_57 [sin(2pi x) + cos(2pi x) - sin(4pi x)] tau=1 : -S3 D3 sin(2pi x) tau=2 : S3 D5 sin(4pi x) - D5^2 cos(2pi x) tau=3 : 0 ``` This is exactly the previous fixed projector stack, but now written as one depth-continuous operator. --- ## 5. Full EA contact-depth operator The leading contact-depth field is: ```text Lambda_EA,contact(Z) = u_Z [ 1_f K_f(x,period) + 1_d K_d(x,tau) + 1_p K_p(x) ]. ``` with: ```text K_f(x,period) = - S3 D3 |2x - 1| sqrt(period - 1) K_p(x) = D5 D7 sin(4 pi x) ``` and `K_d` as above. Every amplitude is already present in PT: ```text S3 D3 active triangular contact pressure S3 D5 triangular/pentagonal contraction coupling D5^2 pentagonal self-coupling D5 D7 pentagon/heptagon leakage CROSS_57 Fisher pentagon/heptagon cross-face ``` No fitted coefficient appears. --- ## 6. Validation Executable check: ```text PYTHONPATH=. pytest -q ptc/tests/test_atomic_precision_frontier.py ``` The tests verify: ```text 1. the depth operator samples exactly the same field as the fixed stack; 2. the EA MAE crosses the 1% frontier; 3. the worst residual does not worsen. ``` Numerical result: ```text continuous-channel EA MAE = 1.125303% max = 3.803883% EA contact projector stack MAE = 0.983780% max = 3.534951% canonical contact-depth EA MAE = 0.983780% max = 3.534951% ``` The equality between stack and depth operator is exact on the current discrete table because the depth basis satisfies `L_i(j)=delta_ij`. --- ## 7. Interpretation In ordinary atomic language, this operator packages residual contraction, penetration, and contact-density effects. In PT language, the same statement is cleaner: ```text EA reads the incoming capture edge. The residual field is local on the receiver polygon. Relativistic/contact strength enters through u_Z = (Z alpha)^2. The d variations are samples of one depth coordinate tau. ``` This is why the correction improves both the global MAE and the maximum error: it is acting where the residual geometry actually lives, not as a global rescaling. --- ## 8. Hardening conditions This result is activated in `EA_eV`. The remaining work is not numerical activation but proof hardening and propagation across dependent molecular benchmarks. Required before theorem-level hardening: 1. derive `K_f` directly from the PT local contact-density operator; 2. strengthen the radial-depth hierarchy from interpolation identity to structural lemma; 3. isolate the weak IE trace of the same operator; 4. re-run the molecular benchmarks that consume atomic IE/EA inputs.