# SWC positive-mass closure attempt

Date: 2026-09-04. Status: sufficient mass bound derived and tested;
universal Erdős–Straus conclusion remains unproved.

This note applies the supplied SWC manuscript's exact positive-mass,
second-moment, conditioning, and pushforward rules. It makes no novelty claim
for the underlying probability inequalities.

## Probability space for each fixed prime

Fix a prime p congruent to 1 modulo 24. Choose h congruent to 3 modulo 4,
with 0<h< p, and put C=(p+h)/4. Factor C=product r_i^e_i. Independently
choose each integer z_i uniformly from [-e_i,e_i], and form

    R = product r_i^z_i mod h.

All r_i are units modulo h. These independent choices define an actual
finite probability space, with N=product(2e_i+1) equally weighted outcomes.
The map d=C R corresponds to choosing a divisor of C^2 before reduction
modulo h. Thus the manuscript's two divisor tests become

    R = -1 mod h, or R = -p^{-1} mod h.

Each accepted seed reconstructs positive integer denominators by the
manuscript's rectangle identity. Positive accepted probability is therefore
an exact SWC existence certificate for this p.

Inversion preserves the exponent distribution. The additional target -p is
also safe: invert its exponent vector to obtain the second target. Duplicate
targets must be counted only once.

## Collision bound: a sufficient SWC second-moment certificate

Let G be a finite group containing every outcome of R; let m=|G|. This can
be the full unit group or the subgroup generated by the prime factors of C.
Let T be the set of valid targets in G and k=|T|>0. Let n_g be the number of
exponent vectors producing g, and define

    S = sum_g n_g^2,
    P_col = S/N^2 = Pr(R=R'),

where R' is an independent copy of R. If k=m, success has probability one.
Otherwise a sufficient condition for positive accepted mass is

    (m-k) S < N^2.

An explicit rational lower bound is

    Pr(R in T) >= max(0, (N^2-(m-k)S)/(2N^2)).

Proof: write rho_g=n_g/N. Cauchy–Schwarz on the non-target residues gives

    rho(G minus T)^2 <= (m-k) sum_{g outside T} rho_g^2
                     <= (m-k) P_col.

Thus rho(T)>=1-sqrt((m-k)P_col). When the displayed sufficient condition
holds, put z=(m-k)P_col<1 and use 1-sqrt(z)>=(1-z)/2. Substitution gives
the rational bound above. If the condition fails, the lower bound zero is
still valid. Equivalently, a distribution avoiding k target residues has
collision probability at least 1/(m-k); the certificate rules that out.

This proves existence without counting the target residue directly. Computing
S can still be difficult; neither its exact computability nor the inequality
for each tested p proves that some h works for every p.

## Character-sum version of the missing analytic estimate

For the finite abelian group G, character orthogonality gives

    Pr(R=t) = (1/m) sum_chi conjugate(chi(t)) product_i A_i(chi),
    A_i(chi) = [sum_{j=-e_i}^{e_i} chi(r_i)^j]/(2e_i+1).

Consequently, proving sum_{chi != 1} |product_i A_i(chi)| < 1 would
certify positive mass at every residue in G, including a valid target.
This is an analytic route to SWC closure. The bound is sufficient and can
fail even when witnesses exist. No uniform estimate across the shifted
factorizations (p+h)/4 has been established here.

## Exact experiments

The residue convolution was compared against direct divisor enumeration in
1,468 anchor/modulus cases. Collision bounds were checked using exact
rational arithmetic. The full-unit-group two-target test yields:

| p | h | SWC lower bound | Exact target mass |
|---:|---:|---:|---:|
| 73 | 7 | 53/450 | 2/5 |
| 409 | 7 | 101/882 | 8/21 |
| 118801 | 63 | 5/486 | 7/135 |
| 496609 | 63 | 1007/71442 | 1/21 |
| 532249 | 55 | 11/4050 | 1/27 |

Conditioning on the generated group certifies p=22605361 at h=91:
N=9, m=6, k=1, S=15, giving lower bound 1/27 and exact mass 2/9.

For p=806521, 2458369, and 8803369, even the subgroup version with the
extra inverse target does not give a positive bound at any tested h<=4095.
They do have exact witnesses. For example, p=8803369 has accepted mass
2/225 at h=107, while the coarse collision estimate is zero. Therefore this
bound's failure is a limitation of the bound, not evidence against the conjecture.

## Why repeated random-walk mixing does not close the proof

Take the hard-class prime p=73, h=27, C=25=5^2. Valid signed exponents
range only from -2 to 2. Their residues are {13,11,1,5,25}. Neither target
17=-p^{-1} mod 27 nor 26=-1 mod 27 appears.

An unconstrained random walk on the generated group eventually reaches both
targets: 5^3=17 and 5^9=26 modulo 27. But exponents 3 and 9 exceed the
available exponent 2. They would not encode divisors of C^2. A mixing theorem
for this extended walk cannot be pushed through the original sound verifier.
SWC support preservation means preserving valid candidate semantics, as well
as assigning positive weight to candidates.

## Induction with probability measures

The earlier Type-II transformer gives a sound pushforward once a compatible
parent certificate exists. For target p and parent q, compatibility requires
the sampled first denominator x to satisfy p=q+4(L-1)x, plus the necessary
denominator divisibilities. If A_q denotes valid parent witnesses and B_{p,q}
denotes this alignment event, the needed premise is

    mu_q(A_q intersect B_{p,q}) > 0.

The induction premise mu_q(A_q)>0 alone does not imply it. Conditioning on
B_{p,q} requires positive conditioning mass and may remove every accepted
parent witness. Hence probabilistic induction still needs a theorem supplying
positive aligned mass for every target.

## What would finish the universal SWC proof

One sufficient theorem is: for every hard-class prime p, some admissible h
has at least one target in its generated group and satisfies (m-k)S<N^2.
Alternatively, a sharper target-sensitive character bound, or a positive
aligned-parent-mass theorem, would work. A fixed universal numeric lower bound
is unnecessary: a rigorously positive bound depending on p is enough.

The present work does not establish any of these universal premises. It does
establish exact probabilistic certificates on the listed instances and
identifies which analytic estimate is missing.

Sources: the supplied v3 SWC manuscript, its second-moment and pushforward
rules; [Elsholtz–Tao](https://arxiv.org/abs/1107.1010) for average solution
counts, which do not assert pointwise positivity; and
[Dahan](https://arxiv.org/abs/2608.24035) for divisor criteria and exceptional
sets. The latter explicitly retains the conjecture as open.

Reproduce with:

    python3 -B research/swc_residue_mass.py --out research/evidence/swc-residue-mass.json
