PT chemistry / capture boundary

How PT computes an electron affinity

Electron affinity EA is the energy released when a neutral atom captures an electron. In PT, this capture is read as an entry at the edge of a discrete channel.

Short version: PT does not ask only whether a place is open. It asks where the next place sits, and in which geometry it opens.

The active shell has capacity $N = 2(2\ell+1)$ and contains $n$ electrons. The added electron is read on the receiver edge $x=(n+1)/N$, then PT threshold, core, channel, and contact-depth fields correct the capture energy.

Canonical score

0.984%

MAE over 73 positive atomic electron affinities.

The score compares the PT value to the experimental reference as a mean relative error. The table excludes zero or negative EAs, which belong to a separate classification test.

6.86 meV

absolute MAE

3.535%

max residual

free coefficients

Status: physical derivation plus numerical validation. The statement "EA reads the capture boundary" remains bridge language, not an unconditional mathematical theorem.

Plain

Understand the idea without heavy formulas

EA tells us whether an atom welcomes an extra electron. PT gives a simple picture: the added electron must find a stable entry point on the edge of the shell. Halogens are very good at this; some metals have a much more fragile entry.

Choose the shell

The active shell gives the number of available channel positions.

Read the edge

The added electron enters through the next receiver position.

Correct capture

Depth and closures adjust the released energy.

The edge picture

In this drawing, green positions are already occupied. The orange line connects the last occupied position to the next capture position. This edge position is what the PT engine turns into an energy. The more naturally the edge closes the channel, the stronger capture tends to be.

Why this matters in chemistry

Electron affinities control anion formation, halogen strength, charge transfer, and the reactivity of many families. A single geometric reading that follows metals, p blocks, and d/f blocks is therefore a demanding test.

Some checkpoints

Element	PT EA	Ref.	Error
H	0.749	0.754	-4.98 meV
C	1.246	1.262	-16.33 meV
O	1.464	1.461	+2.52 meV
Cl	3.617	3.613	+4.25 meV
Mo	0.770	0.748	+21.79 meV

Standard

Working view: stages, calculator, table

This level keeps the chemical vocabulary and numerical quantities: PT value, experimental reference, signed error, and channel position. The percentages below are the MAE after each model layer, from the geometric reading to the contact-depth correction.

EA_geo

1.367%

polygonal capture/ejection reading

the active channel fixes N = 2(2ℓ+1)

CPR surface

1.332%

surface transmission

the capture boundary is weighted by exp[-ℓ(ℓ−1)]

threshold + core

1.248%

p closures and compact d polarization

the first threshold residuals become fixed fields

continuous channel

1.125%

channel harmonics

d, f, and p are read on x = (n+1)/N

contact-depth

0.984%

radial hierarchy

τ = period − 4 replaces discrete d-block gates

PT EA calculator

Pick a positive reference. The calculator shows the canonical value, the experimental residual, and the channel reading.

Element

Z = 17 · p block · period 3

PT EA

3.617 eV

Reference

3.613 eV

Error

+4.25 meV

Relative error

+0.118%

Channel reading

n is the occupation already present, N is the active-channel capacity, and x=(n+1)/N is the position reached after capture.

1.000

Final stages

Full table: 73 positive affinities

Values in eV; signed error is PT − reference. Zero or negative EAs are not listed here.

Z	Element	Block	n/N	PT EA	Reference	Error meV	Error %
1	H	s	1/2	0.749	0.754	-4.98	-0.660
3	Li	s	1/2	0.628	0.618	+10.45	+1.690
5	B	p	1/6	0.277	0.277	+0.25	+0.091
6	C	p	2/6	1.246	1.262	-16.33	-1.294
8	O	p	4/6	1.464	1.461	+2.52	+0.173
9	F	p	5/6	3.426	3.401	+25.10	+0.738
11	Na	s	1/2	0.544	0.548	-4.30	-0.785
13	Al	p	1/6	0.425	0.433	-8.50	-1.962
14	Si	p	2/6	1.369	1.385	-15.79	-1.140
15	P	p	3/6	0.760	0.746	+14.05	+1.884
16	S	p	4/6	2.058	2.077	-18.88	-0.909
17	Cl	p	5/6	3.617	3.613	+4.25	+0.118
19	K	s	1/2	0.494	0.502	-7.87	-1.567
20	Ca	s	2/2	0.025	0.025	+0.04	+0.148
21	Sc	d	1/10	0.183	0.188	-4.61	-2.450
22	Ti	d	2/10	0.078	0.079	-0.85	-1.083
23	V	d	3/10	0.513	0.526	-12.94	-2.459
24	Cr	d	5/10	0.665	0.666	-0.84	-0.126
26	Fe	d	6/10	0.153	0.151	+2.19	+1.454
27	Co	d	7/10	0.656	0.661	-4.55	-0.689
28	Ni	d	8/10	1.135	1.156	-21.30	-1.843
29	Cu	d	10/10	1.217	1.228	-10.85	-0.884
31	Ga	p	1/6	0.427	0.430	-3.15	-0.733
32	Ge	p	2/6	1.236	1.233	+3.15	+0.256
33	As	p	3/6	0.792	0.804	-11.64	-1.447
34	Se	p	4/6	2.023	2.021	+1.93	+0.096
35	Br	p	5/6	3.385	3.365	+20.06	+0.596
37	Rb	s	1/2	0.490	0.486	+3.77	+0.776
38	Sr	s	2/2	0.051	0.052	-1.01	-1.940
39	Y	d	1/10	0.314	0.307	+6.61	+2.154
40	Zr	d	2/10	0.431	0.426	+5.25	+1.234
41	Nb	d	4/10	0.871	0.893	-21.91	-2.454
42	Mo	d	5/10	0.770	0.748	+21.79	+2.914
43	Tc	d	5/10	0.541	0.550	-9.26	-1.684
44	Ru	d	7/10	1.041	1.050	-8.77	-0.835
45	Rh	d	8/10	1.139	1.137	+2.12	+0.186
46	Pd	d	10/10	0.557	0.557	+0.14	+0.025
47	Ag	d	10/10	1.302	1.302	-0.26	-0.020
49	In	p	1/6	0.409	0.404	+5.50	+1.361
50	Sn	p	2/6	1.109	1.112	-2.93	-0.264
51	Sb	p	3/6	1.051	1.047	+3.75	+0.359
52	Te	p	4/6	1.969	1.971	-1.76	-0.089
53	I	p	5/6	3.099	3.059	+39.90	+1.304
55	Cs	s	1/2	0.470	0.472	-1.66	-0.351
56	Ba	s	2/2	0.147	0.145	+2.08	+1.433
57	La	d	0/10	0.479	0.470	+9.44	+2.009
58	Ce	f	2/14	0.653	0.650	+2.61	+0.401
59	Pr	f	3/14	0.994	0.962	+32.03	+3.330
60	Nd	f	4/14	1.900	1.916	-15.94	-0.832
61	Pm	f	5/14	0.128	0.129	-1.28	-0.994
62	Sm	f	6/14	0.162	0.162	+0.24	+0.147
63	Eu	f	7/14	0.858	0.864	-6.50	-0.752
64	Gd	f	8/14	0.134	0.131	+3.04	+2.319
65	Tb	f	9/14	1.167	1.165	+2.18	+0.187
66	Dy	f	10/14	0.353	0.352	+0.85	+0.242
67	Ho	f	11/14	0.335	0.338	-2.73	-0.807
68	Er	f	12/14	0.311	0.312	-1.33	-0.427
69	Tm	f	13/14	1.033	1.029	+3.95	+0.384
70	Yb	f	14/14	0.019	0.020	-0.71	-3.535
71	Lu	d	1/10	0.344	0.346	-1.62	-0.469
72	Hf	d	2/10	0.178	0.178	+0.22	+0.125
73	Ta	d	3/10	0.331	0.322	+8.86	+2.751
74	W	d	4/10	0.814	0.815	-1.46	-0.179
75	Re	d	5/10	0.146	0.150	-3.61	-2.409
76	Os	d	6/10	1.094	1.078	+16.09	+1.493
77	Ir	d	7/10	1.563	1.565	-2.06	-0.132
78	Pt	d	9/10	2.135	2.128	+7.12	+0.334
79	Au	d	10/10	2.309	2.309	-0.49	-0.021
81	Tl	p	1/6	0.380	0.377	+2.62	+0.695
82	Pb	p	2/6	0.364	0.364	+0.02	+0.006
83	Bi	p	3/6	0.946	0.946	-0.25	-0.026
84	Po	p	4/6	1.899	1.900	-0.57	-0.030
85	At	p	5/6	2.413	2.416	-3.06	-0.127

Technical

Canonical mechanism and residuals

The technical level exposes the action formula, constants, harmonics, and remaining residuals. The result is a physical bridge validation, with its domain of validity made explicit.

Operational formula

The calculation starts from a geometric value $EA_{\rm geo}$, then multiplies it by a correction exponential:

EA_PT(Z) = EA_CPR(Z) exp(Λ_threshold + Λ_core + Λ_cont + Λ_contact)

These terms use fixed PT constants; they are not adjusted on the EA benchmark.

Important point: the periodic table supplies discrete samples (element, block, period), but the calculation is not a list of cases. Once the channel is selected, the corrections are evaluated by continuous functions of the capture position x, the depth τ, and the relativistic scale Zα.

Layer	Amplitude	Role	Support
p threshold	−δ₃δ₅	fine p closure	p4/p5 edges
d core	sδ₅	compact d polarization	weighted by Z²α²
d continuous	−δ₅²	pentagon harmonic	sin(2πx)
f leakage	−CROSS₃₇	f entry/closure	incoming edge minus outgoing edge
p center	+CROSS₅₇	p central pressure	before double d closure
d depth	S₃δ₃, S₃δ₅, δ₅², CROSS₅₇	radial Kd kernel	L₀..L₃ basis on τ

Channel and radial depth

The channel names the shape occupied by the captured electron: s, p, d, or f. Radial depth names the same channel when it appears lower in the periodic table. A period-4 d electron and a period-6 d electron therefore live in the same polygonal channel, but not at the same depth.

1. Same channel

Sc, Ti, Zr, and Hf all belong to the d channel: capacity 10, same polygonal logic.

2. Different depth

The period tells how far radially the channel is realized: 4d, 5d, 6d, 7d.

3. One continuous coordinate

PT writes this depth as τ = period − 4. Former period effects become points on one curve, not separate exceptions.

The L₀..L₃ basis then acts as the reading rule: at each realized depth, it selects the matching sample of the continuous kernel. This discrete/continuous passage prevents each period from becoming an ad hoc coefficient.

Best locks

Pb	0.364021 eV	+0.0059%
Ag	1.301745 eV	-0.0196%
Au	2.308514 eV	-0.0211%
Pd	0.557139 eV	+0.0250%
Bi	0.945753 eV	-0.0261%
Po	1.899428 eV	-0.0301%

Largest remaining residuals

Yb	f	-3.535%
Pr	f	+3.330%
Mo	d	+2.914%
Ta	d	+2.751%
V	d	-2.459%
Nb	d	-2.454%

Attached sources

This page mirrors the canonical PTC engine and the contact-depth derivation note. The files served here are readable copies of the source.

ea_residual_fields.py ea_cpr_capture.py contact-depth proof

Halogen and metal: two polygon cases

These drawings are not spatial orbitals: they are discrete channel maps. Occupied positions are green, the next capture position remains white, and the orange segment marks the edge that turns electron entry into affinity energy. Geometry gives the base reading; the CPR, threshold, channel, and contact-depth layers then correct the amplitude.

Chlorine: p-channel closure

5 positions occupied out of 6; the added electron closes the p edge.

p5 → p6

In concrete terms, the electron is not merely entering an open slot: it completes the p channel. That closure strongly stabilizes the anion, giving chlorine a high EA and a clear tendency to form Cl⁻. In the PT reading, the captured position sits between two occupied neighbours: it receives a local two-sided clamp, which strengthens the lock further.

PT EA

3.617 eV

ref.

3.613 eV

gap

+4.25 meV

Titanium: still-open d edge

2 positions occupied out of 10; capture enters a weakly closed d channel.

d2 → d3

Here the added electron closes nothing: it extends a still-open d channel. It is not held between two occupied neighbours as in chlorine: it has one local support and still-open channel space. Isolated capture therefore releases little energy; titanium gets most of its chemistry from d reorganization, bonding, and coordination rather than from a simple atomic Ti⁻ lock.

PT EA

0.078 eV

ref.

0.079 eV

gap

-0.85 meV

The PT distinction is therefore qualitative: chlorine captures by closure, while titanium captures by extending an open edge. In the first case, the atom gains a very stable configuration; in the second, the extra electron remains a channel possibility, but not a strongly energizing closure.