WS-CAND-003: Erdős-Straus Type A/B witness depth and congruence shadow structure

\frac{4}{n}=\frac1x+\frac1y+\frac1z.

p\equiv -4d \pmod{4dn-1}

p\equiv -n \pmod{4dn-1}.

C_{AB}(p),

m=4dn-1,

p\equiv-4d\pmod m,

p\equiv-n\pmod m.

H=\{1,121,169,289,361,529\}.

k=dn,\qquad m=4k-1=4dn-1.

p+n=qm.

p\equiv-n\pmod m.

X=dnq=kq,

Y=dqp,

Z=dnp=kp,

\frac4p=\frac1X+\frac1Y+\frac1Z.

p+4d=qm.

p\equiv-4d\pmod m.

u=nq-1.

X=du=d(nq-1),

Y=dnp,

Z=dnp(nq-1),

\frac4p=\frac1X+\frac1Y+\frac1Z.

m_k=4k-1.

T_k=\{-e,-4e\pmod{m_k}:e\mid k\}.

\boxed{ C_{AB}(p)=\min\{k\ge1:p\bmod(4k-1)\in T_k\}. }

\boxed{ C_{AB}(p)= \min\left\{ k\ge1:\exists e\mid k, p\equiv-e\text{ or }p\equiv-4e\pmod{4k-1} \right\}. }

3, 8, 22, 25, 28, 45, 70, 170, 245, 1050, 1403, 1435, 2622

Type B
d = 69
n = 38
k = 2622
m = 10487
q = 921
p + n = 9658527 = 10487 * 921

55bf4226937122b1021f7be1d25e8ab8cd411e64fb4200a2392a23b9de37cf10

\boxed{ \frac4{9658489} = \frac1{2414862} + \frac1{613787317461} + \frac1{25324558158}. }

centl 'assert(4/9658489 = 1/2414862 + 1/613787317461 + 1/25324558158)'

verdict: verified (closed_exact_rational via closed_rational_comparison); comparison=equal

P(t)=10487t-38,

X(t)=2622t,

Y(t)=69t(10487t-38),

Z(t)=2622(10487t-38).

4XYZ=PYZ+PXZ+PXY

verdict: verified (univariate_rational_polynomial via polynomial_zero_difference); comparison=equal

C_{AB}(p)>K.

L_K=\operatorname{lcm}\left(840,\;4k-1:1\le k\le K\right).

p\equiv1\pmod{L_K},

p\equiv1\pmod{4k-1}.

1\equiv-e\pmod{4k-1}

1\equiv-4e\pmod{4k-1}.

0<e+1\le k+1<4k-1.

0<4e+1\le4k+1=(4k-1)+2,

4e+1=4k-1,

C_{AB}(p)>K.

T_k=\{-d,-4d\pmod{4k-1}:d\mid k\}.

\boxed{

|T_k|=2\tau(k)-1-\mathbf1_{4\mid k}\tau(k/4). }</div>

Proof

The residues -d are distinct as d ranges over divisors of k, because all representatives lie in an interval shorter than the modulus 4k-1. The same is true for the residues -4d. Before cross-collisions there are therefore 2 tau(k) entries.

A cross-collision occurs when

for divisors a,b | k. Since 1 <= a <= k and 1 <= b <= k, the quantity 4b-a lies strictly above -(4k-1) and at most 4k-1. Thus a congruence collision can occur only when

or

The second has the unique solution (a,b)=(1,k), giving the universal collision

The first condition is a=4b. Such collisions exist exactly when 4 | k; then they are in one-to-one correspondence with divisors b | k/4. Hence there are tau(k/4) additional cross-collisions when 4 | k.

Subtracting these from 2 tau(k) proves the formula. QED.

Checks from the experiment

For

tau(2622)=16 and 4 does not divide 2622, so

The independently generated fingerprint also counted exactly 31 distinct trap residues.

For k=1050, tau(1050)=24, giving |T_1050|=47, again matching the computation.

For k=8, the correction term matters:

11. Hazard analysis and why the naive model failed

The finite sieve records a conditional kill fraction at each layer:

A first naive comparison used

This was wrong as a probabilistic baseline because the population consists of primes already restricted to the six hard classes modulo 840. Such primes do not occupy arbitrary residue classes modulo m_k.

11.1 Prime- and hard-class-conditioned null

For a hard class h in H, put

and define the compatible unit residues

Then

and the class-specific Type A/B trap density is

Weighting these probabilities by the current number of survivors in each hard class gives a more appropriate local null model.

This correction explained several large apparent enrichments. For example, at k=10, the raw density was about 17.95%, while the prime-compatible conditioned density is 33.333%, close to the observed 34.2445%.

11.2 Residual anomalies after conditioning

Even after prime/hard-class conditioning, some layers remained highly anomalous. Examples in the high-population regime included:

k	survivors before	observed	conditioned expected	enrichment	standardized residual
25	1437	152	95.800	1.5866	+5.943
12	4044	870	791.217	1.0996	+3.123
51	373	27	17.286	1.5620	+2.425
14	3174	0	478.300	0	-23.966
24	1531	94	211.556	0.4443	-8.734
29	973	0	67.500	0	-8.551
39	609	0	31.200	0	-5.751
52	346	0	20.970	0	-4.725

The extreme negative layers suggested that conditioning on primality and the hard residue classes was still insufficient. The missing information was survival history: a prime present at layer k has already escaped every earlier layer.

That observation led to the shadow relation.

Late-stage giant enrichment or z values with only one, two, three, or four survivors are not treated as meaningful statistical evidence. They are dominated by tiny sample size and selection effects.

12. Direct congruence shadowing

12.1 Admissible candidate classes

For layer k, modulus m_k=4k-1, hard class h in H, and trap t in T_k, call (h,t) admissible when

and

The pair determines a simultaneous congruence class

By CRT this is a class

where

12.2 Shadow criterion

Fix an earlier layer j<k, with modulus m_j=4j-1, and put

As x runs through the candidate progression r mod L, the possible residues modulo m_j are exactly the fibre

Therefore the candidate class (h,t) at level k is directly shadowed by level j if and only if

This is an exact finite criterion, not a statistical one.

12.3 Proof of the fibre statement

The simultaneous system

is solvable exactly when

Thus the listed fibre is precisely the set of residues modulo m_j attained by members of the candidate progression. If every such residue belongs to T_j, then every integer in the candidate class is already a Type A/B hit at j. Conversely, if some residue in the fibre is not in T_j, the CRT supplies integers in the candidate class that evade j.

12.4 Layer-level notions

For a layer k, let A_k be the set of admissible (h,t) pairs.

A pair is directly shadowed when some single earlier level j shadows it by the criterion above.

Define the direct novelty fraction

Interpretation:

• nu(k)=0: every admissible pair is directly shadowed by at least one earlier layer;
• 0<nu(k)<1: the layer is partially directly redundant;
• nu(k)=1: no admissible pair is wholly shadowed by any single earlier layer.

Important limitation: nu(k)=1 does not yet prove that earlier layers cannot collectively cover a candidate class by a union of partial fibres. This experiment has established a single-earlier-layer shadow relation, not complete union irredundancy. The specific class containing an actual prime with verified C_AB(p)=k is stronger evidence of noncoverage, because that prime is itself a witness that the union of all earlier traps did not cover that point.

13. Exact example: layer 14 is redundant

For k=14,

After hard-class and prime compatibility are imposed, the relevant trap residues reduce to the residues 41, 51, and 54 modulo 55 across the admissible pairs. Modulo 11,

At k=3,

Thus every admissible hard-prime hit at layer 14 is already a hit at layer 3.

The shadow-map computation reported:

k=14  m=55  classes=9  shadowed=9  novel=0  sources=[3]

Hence, for the selected hard-prime population,

This is an infinite congruence statement, not merely the observation that no example appeared below 10^7.

The huge negative hazard residual at k=14 was therefore not a random statistical anomaly. The correct conditional expectation after incorporating survival history is exactly zero.

14. Shadow map through k=3000

The optimized shadow computation analyzed every layer through 3000. It found many completely directly shadowed layers. Early examples include:

14 <- 3
29 <- 6
39 <- 8
47 <- 3,14
52 <- 6
59 <- 12
62 <- 5,10,24
69 <- 3,14
74 <- 15
75 <- 6,10,29
89 <- 18
100 <- 5,24
101 <- 8,39
106 <- 12
113 <- 3,14
119 <- 5,24,100
129 <- 26
134 <- 27
146 <- 3,14,40
159 <- 32
166 <- 10
167 <- 6,29
178 <- 20
179 <- 3,14,36
194 <- 8,39
201 <- 3,14
205 <- 10
213 <- 6,29
218 <- 10,17,84
239 <- 48
249 <- 50
251 <- 15,74
256 <- 8,39
257 <- 20,99
267 <- 3,14
269 <- 11,54
278 <- 3,14,76
282 <- 6,29
299 <- 60

The complete program output continues with hundreds of directly redundant layers through k=3000. The important mathematical fact is not the finite count by itself, but that each entry carries a finite CRT shadow certificate that can in principle be independently checked.

14.1 Selected novelty measurements

k	m	admissible classes	directly shadowed	directly novel	nu(k)
14	55	9	9	0	0.0000
24	95	15	9	6	0.4000
25	99	12	0	12	1.0000
29	115	9	9	0	0.0000
36	143	84	60	24	0.2857
39	155	9	9	0	0.0000
40	159	36	0	36	1.0000
51	203	8	0	8	1.0000
54	215	21	9	12	0.5714
62	247	42	42	0	0.0000
65	259	8	0	8	1.0000
70	279	48	24	24	0.5000
75	299	66	66	0	0.0000
83	331	18	0	18	1.0000
90	359	138	0	138	1.0000
104	415	15	9	6	0.4000
114	455	4	0	4	1.0000
170	679	16	0	16	1.0000
245	979	66	18	48	0.7273
1050	4199	282	216	66	0.2340
1403	5611	42	18	24	0.5714
1435	5739	48	0	48	1.0000
2622	10487	186	0	186	1.0000

14.2 Layer 25

Layer 25 was particularly informative because the conditioned hazard analysis had identified it as a strong positive residual:

N=1437
observed first hits=152
conditioned expected=95.800
E*=1.5866
z=5.943

The shadow map then found all 12 admissible (h,t) pairs directly fresh under the single-layer criterion. The trap residues are 79 and 94 modulo 99 for every hard class. Their residues modulo 11 are 2 and 6, neither of which lies in T_3={7,8,10}.

This does not establish a new probabilistic law, especially because layers are dependent and multiple layers were inspected. It does show that the strong positive residual at k=25 is not explained by the same direct-shadow mechanism that forces the zero at k=14.

14.3 Layers 1435 and 2622

The shadow map found:

k=1435  m=5739   classes=48   shadowed=0  novel=48   nu=1.0000
k=2622  m=10487  classes=186  shadowed=0  novel=186  nu=1.0000

Again, nu=1 means no single earlier level completely shadows an admissible class. It does not yet prove absence of union shadowing.

For the particular Type B class containing p=9,658,489, however, the independent minimality certificate proves that this actual prime evades every level 1..2621. Thus the union of all prior Type A/B traps does not cover that point.

15. The emerging redundancy graph

The direct shadow relation naturally defines a directed graph on congruence layers or, more finely, on admissible (k,h,t) classes.

An edge

j -> (k,h,t)

means the earlier layer j covers the entire CRT progression represented by (k,h,t).

At layer level, one can record the set of earlier sources responsible for shadowing every admissible class. This produces a hierarchy of inherited congruence work.

Examples:

3 -> 14
6 -> 29
8 -> 39
12 -> 59
5,10,24 -> 62
6,10,29 -> 75

This suggests replacing the nominal sequence of all integer layers with an irredundant Type A/B sieve whose active objects are only congruence classes that add information not already forced by earlier layers.

That irredundant object has not yet been completely constructed because direct shadowing is only the first notion of redundancy. Union shadowing remains to be computed and formalized.

16. Candidate novelty, stated precisely

The novelty claim under investigation is not:

• the Erdős-Straus conjecture;
• the Type A or Type B solution forms;
• the congruences p == -4d or p == -n mod (4dn-1);
• the six difficult classes modulo 840;
• or a new record for brute-force verification of Erdős-Straus.

The candidate novelty is the following structural package built on López's Type A/B framework:

Candidate contribution A: minimal witness-depth invariant

The explicit use of

as a Type A/B complexity/depth invariant, together with certified record behavior.

Candidate contribution B: exact trap cardinality

The formula

and its elementary collision proof.

Candidate contribution C: unboundedness of Type A/B depth

The Dirichlet construction proving that for every fixed K, infinitely many primes in 1 mod 840 have C_AB(p)>K.

Candidate contribution D: direct congruence-layer shadowing

The CRT fibre criterion that determines when an admissible later Type A/B congruence class is completely implied by a single earlier layer.

Candidate contribution E: irredundant Type A/B sieve

The resulting distinction between fully shadowed, partially shadowed, and directly fresh layers, and the associated redundancy graph/poset as an object of study.

Candidate contribution F: certified computational frontier

A reproducible minimal-depth frontier on the six hard residue classes through 10^7, capped by

with an independent exhaustive minimality verifier and exact CENTL verification of the final decomposition and its parameterized polynomial family.

These items are strongly connected: the depth invariant motivates the layer sieve, the trap cardinality measures each layer, the unboundedness theorem rules out a finite constant-depth strategy, and shadowing explains why many nominal layers add no new first-hit information.

Current status: targeted searches have not found this exact package in the primary literature checked, but FCF does not claim priority until a broader literature review and independent expert scrutiny are completed.

17. What would falsify, narrow, or demote the candidate

This candidate should be narrowed or retired if any of the following occurs:

1. prior literature is found that already defines an equivalent minimal Type A/B depth invariant and substantially the same shadow hierarchy;
2. the claimed trap-cardinality formula is shown false for some k;
3. the CRT shadow criterion is shown to misclassify an admissible class;
4. an independent implementation finds a smaller Type A/B witness for one of the certified frontier primes;
5. the computation is shown to have omitted valid López Type A/B witnesses under the stated definition;
6. the distinction between direct shadowing and union shadowing proves too weak to support the proposed irredundant-sieve program.

Discovery of prior art would not invalidate the computations, but it could remove or substantially narrow the novelty claim.

18. Reproducibility architecture

The work intentionally separated discovery from checking:

Type A/B discovery miner
        |
        v
minimal-depth k-sieve
        |
        v
independent frontier verifier
        |
        +--> JSON minimality certificates + SHA-256
        |
        v
CENTL exact rational / polynomial verification
        |
        v
arithmetic fingerprint
        |
        v
prime-conditioned hazard analysis
        |
        v
CRT shadow-map computation

The major experimental scripts created during the session were:

esc-miner.py
esc-k-sieve.py
esc-frontier-verify.py
esc-frontier-fingerprint.py
esc-hazard-profile.py
esc-conditioned-hazard.py
esc-shadow-map.py
esc-shadow-map-fast.py

The first shadow-map implementation was intentionally replaced after it was observed consuming a full CPU core for a long period. The optimized implementation precomputed trap sets and replaced repeated explicit fibre construction with residue-bucket counts. The optimized program implements the same single-layer shadow criterion and completed through k=3000.

Future archival work should commit normalized versions of the scripts and machine-readable certificates, with relative paths and deterministic manifests, rather than relying on workstation-local paths.

19. Open problems created by the finding

This candidate opens several distinct research streams.

19.1 Union shadowing

Determine whether a candidate class not shadowed by any single earlier level may nevertheless be completely covered by the union of multiple earlier levels. Define and compute a stronger irredundancy invariant.

19.2 Classification of fully redundant depths

Characterize the integers k for which every admissible hard-prime Type A/B class is already covered earlier. A closed arithmetic criterion would convert a large empirical shadow list into a theorem family.

19.3 Irredundant core and graph structure

Construct the transitive reduction or another canonical representation of the shadow graph. Determine whether recurring source patterns such as 3 -> 14, 6 -> 29, and 8 -> 39 extend to infinite families.

19.4 Growth of C_AB

Study

and the least prime whose depth exceeds a given threshold. The unboundedness theorem guarantees arbitrarily deep survivors exist, but does not control their least size.

19.5 Relation to quadratic characters and other hardness measures

Several frontier primes overlap with record primes arising in other, different Erdős-Straus parametrization searches. This may be coincidence or may reflect shared arithmetic obstructions. Quadratic characters, factorizations of p-1, p+1, k, m, and multiplicative-order data are candidates for further study, but no theorem is claimed from the current small frontier sample.

19.6 Formal verification

CENTL already verifies the supported algebraic identities exactly. The external modular lemmas, trap-cardinality theorem, and shadow theorem could be formalized in a proof assistant or added to a future verified arithmetic layer. Until then, CENTL receipts and the number-theoretic arguments have separate assurance boundaries.

19.7 López Type A/B coverage

The deepest open target remains proving that

for every prime. This is essentially the Type A/B coverage conjecture studied by López and would imply Erdős-Straus. Nothing in the present work establishes that global finiteness claim.

20. Publication discipline

Any public summary of this work should include the following sentence or equivalent:

FCF has identified a candidate structural contribution concerning minimal Type A/B witness depth and congruence-layer redundancy in the Erdős-Straus problem. The computations and elementary lemmas are reproducible, but novelty and priority remain under review, and the Erdős-Straus conjecture remains open.

Avoid phrases such as:

FCF solved Erdős-Straus.
CENTL proved Erdős-Straus.
FCF verified Erdős-Straus farther than previous work.
C_AB <= 2622 for all primes.
Layer 2622 is globally irredundant in every admissible class.

None of those statements is established by the current evidence.

21. References

1. Miguel Angel López, A Complete Congruence System for the Erdos-Straus Conjecture, arXiv:2404.01508, 2024. <https://arxiv.org/abs/2404.01508>
2. S. Salez, computational work on the Erdős-Straus conjecture through 10^17, arXiv:1406.6307. <https://arxiv.org/abs/1406.6307>
3. M. Bello-Hernández, M. Benito, E. Fernández, A Divisor Parametrization for the Erdős--Straus Conjecture, arXiv:2606.10922, 2026. <https://arxiv.org/abs/2606.10922>
4. Current FCF Wellspring policy: ../FCF-WELLSPRING.md.

22. Current assessment

The evidence is strong enough to preserve this work as a Wellspring Candidate because it has generated multiple independent research directions and includes exact arguments, independent computation, and CENTL verification.

It is not yet designated an FCF Wellspring. The remaining barriers are principally prior-art review, independent mathematical review, union-shadow analysis, and archival reproduction from repository-controlled scripts and certificates.

The central research question created by the experiment is now:

That question is narrower than solving Erdős-Straus directly, but it is precise, testable, and structurally richer than the brute-force experiment from which it emerged.