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Synthesis · hosted from the CENTL repository

Research library · Synthesis

Synthesis

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Source in the repository

Date: 2026-08-15

Claim boundary: Erdős-Straus remains open. Universal López Type A/B coverage remains open. Universal DSC-0 and DSC-P are false. Universal prime strong/Type-II coverage is also unproved. The Mizony/Thépault divisor-square mechanism is classical prior art. The active FCF work is the structural synthesis, shadow theory, Kneser theory, and exact fixed-shift analysis described below.

1. The proof search now has two distinct lanes

Do not conflate these.

Lane A: exact prime Erdős-Straus

For prime

p\equiv1\pmod4

and an admissible shift

k\equiv3\pmod4, \qquad \gcd(k,p)=1,

put

C_k=\frac{p+k}{4}

and define

\boxed{ \mathcal R_k(C_k) = \left\{ \prod_i r_i^{z_i}\pmod k: -e_i\le z_i\le e_i, \quad C_k=\prod_i r_i^{e_i} \right\}.}

ES-TWO-TARGET-SIGNED-BOX-EQUIVALENCE.md proves

\boxed{ p\text{ satisfies Erdős-Straus} \iff \exists k: \{-p^{-1},-1\} \cap \mathcal R_k(C_k) \ne\varnothing.}

The targets are standard Type I and Type II respectively. Inversion symmetry adds the equivalent Type-I orientation -p.

This is an exact reformulation of prime ES.

Lane B: classical strong/Type-II route

For a layer index a, define

\boxed{ S_a = \{-4D\pmod{4a-1}:D\mid a^2\}.}

Universal prime coverage by the layers S_a would prove the classical strong/Type-II form and therefore Erdős-Straus, but it is logically stronger than original ES.

The condition

D\mid a^2, \qquad 4a-1\mid p+4D

belongs to the historical Mizony/Thépault/Rosati-Yamamoto lineage. See:

  • STRONG-ES-MIZONY-THEPAULT-PROVENANCE.md
  • SQUARE-COMPLETION-PRIOR-ART.md

No novelty claim should be made for the square-divisor criterion itself.

2. DSC is closed as the universal bridge

The explicit hosted counterexample proves

\boxed{\mathrm{DSC\!\!-0}\text{ is false}, \qquad \mathrm{DSC\!\!-P}\text{ is false}.}

The following remain valid supporting mathematics:

  • strong/weak/pointwise q=3 absorption;
  • exact reduced-parameter domain;
  • finite exact-depth certificates;
  • direct-shadow smoothness;
  • character and multiplicative quotient structure;
  • covering-core / hypergraph depth theory.

Do not spend the main ES effort trying to restore universal DSC.

3. López A/B are the two boundary orthants of the classical square layer

The ordinary López trap at layer a is

T_a = \{-e,-4e:e\mid a\} \pmod{4a-1}.

Write

a=\prod_i\ell_i^{E_i}, \qquad D=\prod_i\ell_i^{U_i}, \qquad 0\le U_i\le2E_i.

Then ES-SQUARE-COMPLETION-TRAP-GEOMETRY.md proves:

  • López Type A is the lower orthant U_i<=E_i for every i;
  • López Type B is the upper orthant U_i>=E_i for every i;
  • the additional standard Type-II parameters are the mixed cross-orthant points.

The exact mixed parameter count is

\boxed{ M(a)=\tau(a^2)-2\tau(a)+1.}

Thus

\boxed{M(a)=0\iff a\text{ is a prime power}.}

Prime-power layers are unchanged by completion:

\boxed{S_a=T_a\quad\text{if }\omega(a)=1.}

4. Central synthesis: the completed layer is a symmetric Kneser box

Center the square-divisor exponents:

z_i=U_i-E_i.

Since

4a\equiv1\pmod{4a-1},

one gets

-4D \equiv -\prod_i\ell_i^{z_i} \pmod{4a-1}.

Therefore

\boxed{ S_a =-\mathcal R_{4a-1}(a).}

This is the main structural merger.

The old program studied cross-layer congruence shadowing. The newer program studied Kneser expansion and stabilizers of symmetric product boxes. They now act on the same classical strong/Type-II object.

Divisor complement

D\mapsto a^2/D

is exactly

z\mapsto-z

and therefore residue inversion.

The López A/B mutual-inverse relation is the boundary restriction of this global symmetry.

See:

  • ES-SQUARE-TRAP-SIGNED-BOX-IDENTITY.md
  • ES-SQUARE-TRAP-COMPLEMENT.md

5. The old character shields survive completion

Let

H_a = \langle \ell\bmod(4a-1):\ell\mid a\rangle.

The complete strong layer satisfies

\boxed{ T_a \subseteq S_a \subseteq -H_a \subseteq \{x:(x/(4a-1))=-1\}.}

Thus square completion changes exact occupancy inside the old multiplicative coset, but does not weaken the coarse multiplicative or Jacobi shields.

See ES-SQUARE-COMPLETION-COSET-SHIELD.md.

6. Root geometry

Every square-divisor parameter may be written

D=sb^2, \qquad \frac{a^2}{D}=sc^2, \qquad a=sbc,

with s squarefree.

The Type-II equations become

\boxed{ p+q=4sbt, \qquad b+t=cq.}

López comparability is exact:

\boxed{ \begin{array}{ccl} \text{Type A}&\iff&b\mid c,\\ \text{Type B}&\iff&c\mid b,\\ \text{mixed strong Type II}&\iff&b\nmid c\text{ and }c\nmid b. \end{array}}

See ES-TYPEII-ROOT-GEOMETRY.md.

7. Completed depth is unbounded, but finite compression is strong

Two elementary facts hold for every layer:

\boxed{1\notin S_a,} \qquad \boxed{-1\in S_a.}

The prime-modulus CRT/Dirichlet backbone therefore survives completion.

Whenever

4a-1>7

is prime, infinitely many Mordell-hard primes have exact strong/Type-II first depth a.

Hence completed first-hit depth is unbounded.

A reproducible finite census through

p\le50,000,000

contains 93,457 Mordell-hard primes, all captured by completed layers with

\boxed{a\le624.}

The unique deepest observed prime is

\boxed{p=2,031,121}

with mixed witness

\boxed{a=624, \quad D=576, \quad 4a-1=2495, \quad q=815.}

Its López A/B first depth is 1403.

This is finite compression only, not a universal ceiling.

See:

  • ES-SQUARE-COMPLETION-BACKBONE.md
  • SQUARE-COMPLETION-FINITE-CENSUS.md
  • square_completion_probe.py

8. Exact prime-index spectrum

If the layer index a is prime, then

S_a=\{-4,-1,-a\}.

Every ancestry edge into such a layer is a complete shadow. Consequently:

\boxed{ a\text{ prime is an exact completed depth} \iff 4a-1\text{ is prime}.}

The positive case is realized infinitely often by Mordell-hard primes.

See ES-SQUARE-PRIME-INDEX-SPECTRUM.md.

9. Prime-power and squarefree-semiprime dichotomy

Prime powers

If a is a prime power,

S_a=T_a.

In particular the power-of-two Mersenne shadow lattice carries over unchanged, including its infinite structural-gap families.

Squarefree semiprimes

If

a=uv

with distinct primes u<v, the only mixed square divisors are

u^2,\qquad v^2.

Their signed ratios

u/v,\qquad v/u

lie outside every López boundary residue.

Therefore

\boxed{T_{uv}\subsetneq S_{uv}}

for every squarefree semiprime layer.

See ES-SQUAREFREE-SEMIPRIME-MIXED-RESIDUES.md.

10. Multiplicative ancestry is completely classified

Fix an ancestor j and let

R_j=\mathcal R_{4j-1}(j), \qquad H_j=\operatorname{Stab}(R_j).

Take a multiplicative descendant

k=jB

on an ancestry edge. The ancestry condition is

B\equiv1\pmod{4j-1}.

Then

\boxed{ S_{jB}\bmod(4j-1)\subseteq S_j \iff r\bmod(4j-1)\in H_j \text{ for every prime }r\mid B.}

Any containment is automatically equality.

This is an exact iff classification of direct shadows on multiplicative ancestry edges.

See ES-SQUARE-MULTIPLICATIVE-SHADOW-IFF.md.

11. Internal stabilizers manufacture infinite cross-layer gaps

If every prime factor of an extension B lies in H_j and

B\equiv1\pmod{4j-1},

then

S_{jB}\bmod(4j-1)=S_j.

For any h in H_j, Dirichlet gives primes

r\equiv h, \qquad s\equiv h^{-1} \pmod{4j-1}.

Then B=rs gives an infinite structural-gap cone above j.

This is the first exact theorem where an internal Kneser stabilizer generates cross-layer shadow edges.

See ES-SQUARE-STABILIZER-EXTENSION-SHADOW.md.

12. Nonmultiplicative ancestry also has infinite exact structure

Squarefree factor-lift theorem

If the later index is squarefree

k=r_1\cdots r_t

and the ancestor factors as

j=A_1\cdots A_t

with

r_i\equiv A_i\pmod{4j-1},

then

\boxed{S_k\bmod(4j-1)\subseteq S_j.}

Every ancestry quotient has infinite factor-lift families

For any

Q=4s+1,

choose a squarefree divisor A|s, put t=s/A, choose a prime

r\equiv-t\pmod Q,

and define

B=\frac{r+t}{Q}, \qquad j=AB, \qquad k=Ar.

Then

\boxed{4k-1=Q(4j-1)}

and the later layer is completely shadowed by j.

Dirichlet supplies infinitely many such r.

Therefore:

\boxed{ \text{every allowed ancestry quotient supports infinitely many exact completed structural gaps}.}

See:

  • ES-SQUARE-SQUAREFREE-FACTOR-LIFT.md
  • ES-SQUARE-ALL-QUOTIENT-FACTOR-LIFT.md

13. Exponent-lattice normal form for arbitrary ancestry

For

j=\prod_i p_i^{E_i},

define

\phi_j:\mathbb Z^d\to(\mathbb Z/(4j-1)\mathbb Z)^\times, \qquad z\mapsto\prod_i p_i^{z_i},

and

L_j=\ker\phi_j.

The ancestor completed box is the image of

B_j=\prod_i[-E_i,E_i]_{\mathbb Z}.

For a later ancestry layer

k=\prod_\nu r_\nu^{F_\nu},

choose exponent lifts v_nu with

\phi_j(v_\nu)=r_\nu.

Define the later discrete zonotope

Z(k\to j) = \left\{ \sum_\nu z_\nu v_\nu: -F_\nu\le z_\nu\le F_\nu \right\}.

Then direct shadow is exactly

\boxed{ Z(k\to j) \subseteq B_j+L_j.}

A strong sufficient coordinate-budget test is

\boxed{ \sum_\nu F_\nu |(v_\nu)_i|\le E_i \quad\forall i.}

This turns residual nonmultiplicative shadowing into a finite zonotope-in-lattice-cover problem.

See ES-SQUARE-EXPONENT-LATTICE-SHADOW.md.

14. Effective dimension can be much smaller than prime support

For

j=2^e p

with p odd prime,

4j-1=2^{e+2}p-1

gives

p\equiv2^{-(e+2)}.

Therefore the raw two-dimensional completed box collapses exactly to

\boxed{ \mathcal R_{4j-1}(j) = \{2^z:-2e-2\le z\le2e+2\}.}

For j=2p, this is the nine-step interval [-4,4] in powers of 2.

This explains non-factor-lift shadows such as the finite edge j=10 -> k=8083=59*137.

See ES-SQUARE-BINARY-PRIME-INTERVAL.md.

The next internal invariant should be effective signed-box dimension modulo 4j-1, not merely omega(j).

15. Completed depth spectrum is quantitatively infinite and coinfinite

The prime-modulus backbone gives

\boxed{

|\mathcal D_{\rm sq}\cap[1,K]| \ge (1+o(1))\frac{2K}{\log(4K)}.}</div>

The quotient-nine factor-lift family

k=2r, \qquad r\equiv8\pmod9

gives structural gaps with

\boxed{

|\mathcal G_{\rm sq}\cap[1,K]| \ge (1+o(1))\frac{K}{12\log K}.}</div>

Thus the completed strong/Type-II depth spectrum is provably infinite and coinfinite, with explicit backbone and anti-backbone subfamilies both of prime-counting order.

See ES-SQUARE-SPECTRUM-INFINITE-COINFINITE.md.

16. Fixed-shift strong/Type-II search lives in a finite corridor

A successful Type-II shift q satisfies

\boxed{3q\le p+4.}

Writing

A=\frac{p+3}{4},

the shifts

q_h=4h+3

correspond to consecutive integers

C_h=A+h.

Only

0\le h\le\left\lfloor\frac{p-5}{12}\right\rfloor

can support a Type-II solution.

Thus a hypothetical strong counterexample requires simultaneous signed-box defects across a long finite corridor of consecutive integers.

See STRONG-ES-FINITE-SHIFT-CORRIDOR.md.

17. Exact small-shift factor filters

For Mordell-hard primes:

q = 3

\boxed{ q=3\text{ misses} \iff \text{every prime factor of }\frac{p+3}{4} \text{ is }1\pmod3.}

q = 7

\boxed{ q=7\text{ misses} \iff \text{every prime factor of }\frac{p+7}{4} \text{ is a quadratic residue mod }7.}

q = 11

A miss is either:

  1. pure quadratic splitting modulo 11, or
  2. a thin defect with v_3=1, all other QR factors 1 mod11, only primitive NR classes 2,6, and total such valuation at most 2.

q = 23

Because 6|(p+23)/4, the forced factors 2,3 generate almost the entire QR subgroup. A miss is either:

  1. pure quadratic splitting modulo 23, or
  2. a thin defect with v_2=v_3=1, all other QR factors 1 mod23, only NR classes 5,14, and total such valuation at most 2.

See:

  • FAB-HARD-FIRST-FILTERS.md
  • STRONG-ES-Q7-EXACT-FILTER.md
  • STRONG-ES-Q11-EXACT-FILTER.md
  • STRONG-ES-Q23-EXACT-FILTER.md

18. The first four prime shifts already give a dimension-three sieve

Classical upper-bound sieve theory applied to the exact corridor filters gives:

\boxed{ \#\{p\le X:\ q=3,7\text{ both miss}\} \ll \frac{X}{(\log X)^2}.}

Adding q=11 gives

\boxed{ \#\{p\le X:\ q=3,7,11\text{ all miss}\} \ll \frac{X}{(\log X)^{5/2}}.}

Adding q=23 gives

\boxed{ \#\{p\le X:\ q=3,7,11,23\text{ all miss}\} \ll \frac{X}{(\log X)^3}.}

The last estimate is a relative-prime exceptional proportion

\boxed{O((\log X)^{-2}).}

These are specific applications of classical Selberg/Brun sieve ideas. Classical full-ES exceptional-set theorems are much stronger.

See:

  • STRONG-ES-Q3-Q7-SIEVE.md
  • STRONG-ES-Q3-Q7-Q11-SIEVE.md
  • STRONG-ES-Q3-Q7-Q11-Q23-SIEVE.md

19. Exact-ES external-shift Kneser obstruction theory remains active

For external prime shifts

q\equiv3\pmod4, \qquad (q/p)=-1,

a combined Type-I/Type-II failure has even stabilizer index

\boxed{n\ge6}

and symmetric defect budget

\boxed{ \sum_i \left( \min(2e_i+1,\operatorname{ord}(r_iH))-1 \right) \le n-4.}

At index six the failure reduces to one simple primitive sextic factor and a forced external-nonresidue edge.

Consecutive primitive defects can occur, with exact recurrence

\boxed{ q_{i-1} \equiv q_{i+1}^{\pm2}u_i^6 \pmod{q_i}.}

A hypothetical ES counterexample would require unbounded full-stabilizer quotient complexity as the auxiliary shift varies, and the least odd prime divisor of that defect index can be forced arbitrarily large.

Thus no finite classification of low Kneser defect indices can finish exact ES.

20. Two-target corridor companions (2026-08-15)

The exact two-target reformulation now has four additional corridor theorems on the original-ES side, not merely the strong/Type-II side.

Linear form 2p+1

TWO-P-PLUS-ONE-FILTER.md proves that a Mordell-hard prime is solved as soon as 2p+1 has a divisor 7\bmod8. A counterexample must place 2p+1 in the same {1,3}\bmod8 semigroup already forced on p+2.

q=3 and q=7 have no Type-I surplus

At q=3 the two targets coincide for hard primes. At q=7, HARD-Q7-TYPE-I-NO-RESCUE.md proves they fail together: the forced factor 2 fills the whole quadratic-residue subgroup, and every hard class is itself a residue modulo 7. Combined failure equals the existing Type-II miss.

q=11 has a genuine Type-I companion

Q11-TYPE-I-COMPANION.md classifies the rescues. After a Type-II miss, Type I still hits on explicit residue classes modulo 11 once the QR box is full, and on the thinner set {7,8,10} when the box is only {1,3,4}. Through 2\cdot10^6 this companion solves 13 hard primes that Type II missed at the same shift.

Composite shift k=15

K15-TWO-TARGET-FILTER.md identifies the two-primary subgroup

H=\langle2\rangle=\{1,2,4,8\}\subset(\mathbb Z/15\mathbb Z)^\times.

Both hard-class Type-I targets and the Type-II target lie outside H. Combined failure occurs exactly on the H-trap (every prime factor of (p+15)/4 is 1,2,4,8\bmod15) or on one thin 11-packet. Any prime factor 7,13,14\bmod15, or 11 with v_2\ge2, is an immediate Type-II hit.

Finite residual after 3,7,11

Through 2{,}000{,}000 there are 4519 Mordell-hard primes. Combined two-target failure at 3,7,11 leaves 711 primes, all solved by some later shift in

\{15,19,23,27,31,35,39,43,47,51,55,59\}.

The next exact target after k=19 is the Type-I companion to the existing q=23 Type-II theorem.

These are corridor theorems, not a bounded-window existence proof.

Independent covering obstruction

A separate attack asked whether a fixed multiplier M can force M\mid(p+k)/4 at the aligned shift k\equiv-p\pmod{4M} and hit Type II from the signed box of M alone. HARD-SMOOTH-TYPEII-OBSTRUCTION.md proves this is impossible whenever M is {2,3,5,7}-smooth: the whole forced box is Jacobi-positive, while -1 is Jacobi-negative. Any uniform Type-II arithmetic-progression cover of a hard class must import an external prime ℓ≥11. Including 13 produces at least one explicit infinite family,

p=10920t+10369, \qquad \frac4p=\frac1{546Tp}+\frac1{2730T}+\frac1{5Tp},\quad T=t+1.

That family sits inside the single hard class 289\bmod840 and does not cover the class. Original Erdős--Straus remains open.

21. Current highest-priority proof targets

A. Strong/Type-II cross-layer ancestry

Use the exact object

S_a=-\mathcal R_{4a-1}(a)

and classify the residual nonmultiplicative ancestry edges after removing:

  1. multiplicative stabilizer extensions, already solved exactly;
  2. factor-lift families, already infinite at every quotient;
  3. low effective-dimension folds such as 2^e p.

The exponent-lattice criterion is now the primary language.

B. Strong/Type-II finite corridor

Continue exact fixed-shift classifications at useful small primes q and track the added sieve dimension.

The immediate questions are:

  1. identify more shifts where hard congruences force a large QR product subset;
  2. quantify all exceptional low-entropy branches;
  3. determine whether a useful uniform family of corridor shifts exists.

C. Exact ES two-target lane

Continue the corridor from the new k=19 combined filter. The immediate exact target is the Type-I companion to the already-classified Type-II shift k=23.

The proven unbounded-defect forcing theorem still means no finite list of Kneser indices can finish the external-nonresidue lane. The corridor lane is a different finite-for-each-prime search and is not forbidden by that theorem.

D. Prior-art review

Continue tracing Thépault, Mizony, Rosati-Yamamoto, Mordell, Bradford, López, Chamberland, and BHB-F so all structural novelty claims remain conservative and publication-safe.

E. Public hunt

The operator-facing attack is now a public infinite hunt: ES-HUNT.md, kernels bb.kernel and CC.kernel, findings under findings/. The recurrence is the windows \((s,s+\Delta]\) with no last interval. Letters are collected; the engine stops when the operator stops it. Letter numbers are the first 128 bits of SHA-256 of ES-LETTER-v1 and do not depend on the start factor. A cleared window is not a proof.

22. One-line status

The main new research object is now clear:

\boxed{ \text{classical Mizony/Thépault strong layer} = \text{square-completed López layer} = -\text{symmetric signed divisor box}.}

Its internal Kneser geometry and cross-layer shadow geometry are now partially unified, multiplicative ancestry is completely classified, every ancestry quotient has infinite exact gap families, the residual shadow problem has an exact exponent-lattice form, and four tiny fixed Type-II shifts already leave only an O(X/(log X)^3) prime survivor set. On the original-ES two-target corridor the first three prime shifts are now combined-exact, 2p+1 is an additional linear-form filter, k=15 is a complete two-target theorem, and k=19 is now a complete two-target theorem (QR-trap, class-121 filling of Q, and a one-pair Type-II companion table). Original Erdős-Straus remains open.