9.7 KiB
Sidon Lifting Relaxation Program
Status: RESEARCH_PROGRAM_DRAFT_WITH_GUARDRAILS
Target stack: OTOM / Mass-Number Lens / Sidon Spectral Sieve / W-axis
Question: What algebraic transformations could relax dense finite Sidon foci into an extendable infinite sequence?
Purpose
This document refines the Sidon Extension-Focus Question into a constructive research program.
The prior guardrail established:
Plots and pressure metaphors are not proofs.
Finite Sidon witnesses require collision_count(A_N)=0.
Infinite density claims require an explicit construction plus asymptotic proof.
The next question is constructive:
Can a lawful algebraic transformation reduce effective collision pressure while preserving pairwise-sum uniqueness, and can that transformation be iterated into one infinite Sidon sequence?
P-adic Bose-Chowla proof-risk firewall
A proposed p-adic Bose-Chowla lift was considered:
Bose-Chowla finite seed
-> p-adic digit-window lift
-> claimed compression of quadratic collision pressure to logarithmic pressure
-> claimed proof of sigma = 1
This must not be recorded as a proof.
Correct W-axis classification:
Gate: P_analogy + U_scope
Reason: the proposal does not yet provide an explicit compatible lift chain,
a verified collision audit at each level,
or an asymptotic proof that |A_k| / sqrt(N_k) -> 1 in one infinite Sidon set.
The statement:
p-adic digit windows reduce collision pressure from O(x^2) to O(log x)
is a heuristic target, not an established theorem in this stack. It becomes admissible only after proving a bound of the form:
|Forbidden(A_k) cap [1,N_k]| <= polylog(N_k)
or another explicit subquadratic obstruction bound strong enough to keep extension freedom open while preserving pairwise-sum uniqueness.
Corrected core warning
A naive embedding
F_p -> F_{p^2}
is not automatically useful for Sidon extension. If the map is purely additive and injective, then pairwise-sum collisions are preserved exactly:
T(a)+T(b)=T(c)+T(d) iff T(a+b)=T(c+d) iff a+b=c+d.
So additive field inclusion gives more ambient notation, but it does not by itself create new Sidon slack.
The useful transformations must change the collision geometry, not merely relabel it.
Candidate lawful transformations
1. Projective-plane / Singer transformation
Singer constructions use cyclic groups associated with finite projective geometry to produce dense finite Sidon or difference-set witnesses.
finite projective plane
-> cyclic difference set / modular Sidon structure
-> dense finite no-collision focus
Role:
Creates high-density finite foci with strong algebraic symmetry.
Failure mode:
Rigid finite structures do not automatically nest into one infinite Sidon sequence.
2. Bose-Chowla / finite-field logarithmic transformation
Instead of relying on additive inclusion, use multiplicative structure and exponent/log coordinates.
finite field multiplicative group
-> exponent/log coordinate
-> modular Sidon/B_h constraints
Role:
Changes the additive collision audit by passing through a multiplicative cyclic geometry.
Failure mode:
Requires careful unrolling from modular/cyclic setting into integer intervals.
3. p-adic lifting tower
The p-adic shortcut is not simply F_p -> F_{p^2}. The stronger version is a tower of compatible residue classes:
A_k subset Z / p^k Z
A_{k+1} subset Z / p^{k+1} Z
A_{k+1} mod p^k = A_k
The constructive task is to choose lift digits:
a' = a + p^k t_a
so that the lifted set remains collision-free modulo p^{k+1} or in a controlled integer window.
Role:
Turns extension into a digit-by-digit constraint satisfaction problem.
Failure mode:
The number of forbidden digit choices can saturate the lift space unless algebraic overlap compresses the obstruction set.
Required p-adic lift receipts:
1. explicit seed A_1,
2. explicit digit-choice rule t_a for every lift level,
3. proof that A_{k+1} mod p^k = A_k,
4. proof that collision_count(A_k)=0 for every k,
5. proof that density approaches the claimed limsup,
6. proof that the integer unrolling is one global Sidon set, not unrelated finite witnesses.
4. Block algebraic construction with buffer zones
Build dense algebraic blocks and separate them by large gaps.
A = B_1 union shifted(B_2) union shifted(B_3) union ...
Role:
Preserves local algebraic density while preventing cross-block collisions through spacing.
Failure mode:
Buffer zones may lower limsup density unless the dense blocks dominate the observation windows.
This is a plausible route for high limsup behavior because limsup only needs favorable windows, but the cross-block audit is the hard constraint.
Obstruction set formulation
For a finite Sidon set A, a new integer x cannot be added if it creates a collision.
A useful forbidden-value proxy is:
Forbidden(A) = { a_i + a_j - a_k : a_i,a_j,a_k in A }
Then extension freedom in a window [1,N] is roughly:
Omega_N(A) = |[1,N] \ Forbidden(A)|
A construction remains extendable only if:
Omega_N(A) > 0
for sufficiently many future windows, with the actual pairwise-sum audit still enforced.
Structural-overlap criterion
Raw collision pressure assumes forbidden values spread widely. Structure helps only if forbidden values overlap heavily.
Define an overlap factor:
Gamma(A,N) = |A|^3 / |Forbidden(A) cap [1,N]|
High Gamma means many formal obstruction triples collapse to fewer actual forbidden values.
The search target becomes:
Find constructions where Gamma(A,N) grows fast enough that Omega_N(A) does not collapse.
For the p-adic/log-pressure hypothesis, the required upgrade is:
Conjectural target:
|Forbidden(A_k) cap [1,N_k]| = o(N_k)
Stronger target suggested by the pasted heuristic:
|Forbidden(A_k) cap [1,N_k]| = O(polylog N_k)
Status:
U_scope until proved.
Refined Mass-Number score
For finite witness A_N subset [1,N]:
rho_N(A_N) = |A_N| / sqrt(N)
C(A_N) = sum_s max(0, R_A(s)-1)
Omega_N = admissible future slots
Gamma_N = obstruction overlap factor
E_N = embedding risk
Candidate mass:
M_Sidon(A_N)
=
rho_N(A_N)
* indicator(C(A_N)=0)
* log(1 + Omega_N)
* log(1 + Gamma_N)
/
(1 + E_N)
A dense finite construction receives high mass only if it is collision-free and retains measurable extension freedom.
Shortening the path
The shortest route is not to search all Sidon sets. The shortest route is to audit known high-density algebraic families for extendability.
Priority order:
1. Start with Singer / Bose-Chowla finite witnesses.
2. Compute collision_count(A_N)=0 as a hard receipt.
3. Compute Forbidden(A_N) and Omega_N across candidate extension windows.
4. Measure Gamma(A_N,N) to detect structural compression of obstruction triples.
5. Test p-adic or block-lift rules that preserve previous residues while minimizing new collisions.
6. Promote only finite audited witnesses; keep infinite claims U_scope until asymptotic nesting is proven.
Research-grade question
Do there exist algebraic lift maps T_k producing compatible finite Sidon witnesses
A_k subset [1,N_k] such that:
1. A_k is Sidon for every k,
2. A_k embeds into A_{k+1} without destroying old sums,
3. |A_k| / sqrt(N_k) approaches 1 along a subsequence,
4. the obstruction overlap Gamma(A_k,N_k) prevents extension freedom Omega_N from collapsing,
5. the construction unrolls to one infinite Sidon set A subset N?
P-adic sharpened version:
Does there exist a p-adic digit-lift tower A_k subset Z / p^k Z such that
compatible lifts preserve Sidon uniqueness at every level and the unrolled integer
sequence has limsup density 1?
W-axis labels
Verified finite witness with C(A_N)=0:
R_finite
p-adic or block-lift rule with finite audits only:
R_finite + U_asymptotic
p-adic/log-pressure graph without lift proof:
P_analogy + U_scope
Claim of sigma=1 without infinite construction and proof:
U_scope
Numerical pressure plot treated as theorem:
P_analogy, rejected as R
Explicit collision in proposed set:
X_constraint
Lean target skeleton
namespace SidonLifting
-- A finite Sidon witness is audited by zero pairwise-sum collisions.
theorem finiteWitness_requiresCollisionZero
(A : Finset Nat) :
PromotedFiniteSidon A -> CollisionCount A = 0 := by
sorry
-- Pure additive embeddings preserve collision structure; they do not create new Sidon slack.
theorem additiveEmbedding_preservesCollisions
(T : Nat -> Nat) :
AdditiveInjective T ->
PairSumCollision A -> PairSumCollision (A.image T) := by
sorry
-- Infinite promotion requires a compatible lift chain plus asymptotic density proof.
theorem infinitePromotion_requiresCompatibleLiftAndDensity
(chain : Nat -> Finset Nat) :
PromotedInfiniteSidon chain ->
CompatibleLiftChain chain ∧
(forall k, CollisionCount (chain k) = 0) ∧
LimsupDensityOne chain := by
sorry
-- A p-adic pressure plot is not enough to discharge an infinite Sidon claim.
theorem pAdicHeuristic_notProofWithoutLiftAudit
(claim : Claim) :
PAdicPressureHeuristic claim ->
¬ HasCompatibleLiftAudit claim ->
Gate claim = P_analogy := by
sorry
end SidonLifting
Short doctrine
Finite geometry gives dense foci.
Lifting must preserve old sums and avoid new sums.
Additive inclusion alone does not create slack.
The useful signal is obstruction overlap.
P-adic digit windows are a candidate mechanism, not a proof.
Sigma=1 needs a nesting theorem, not a prettier pressure plot.