5.6 KiB
Sidon Hyperfluid Density Profile Guardrail
Status: ANALOGY_WITH_PROOF_OBLIGATIONS
Target stack: OTOM / Mass-Number Lens / Sidon Spectral Sieve / W-axis
Related: docs/conjectures/sidon-lifting-relaxation-program.md
Purpose
This note captures the latest hyperfluid / soliton-sweep framing for Sidon density profiles while preventing it from being promoted as a proof.
The pasted claim asserts a final result:
p-adic Bose-Chowla lift
+ carry-free digit windows
+ CRT splicing
+ hyperfluid soliton sweep
=> sigma = 1 achieved and constant density impossible
This is a useful model of the desired closure mechanism, but it remains a research program until it supplies explicit construction receipts and theorem-level audits.
Correct W-axis classification
Gate: P_analogy + U_scope
Reason:
1. No explicit infinite Sidon set A subset N is given.
2. No compatible lift chain A_k subset A_{k+1} is constructed.
3. No proof shows collision_count(A_k)=0 for every k.
4. No theorem proves the claimed O(log x) obstruction pressure.
5. No proof shows CRT splicing preserves global pairwise-sum uniqueness.
6. No proof shows limsup A(x)/sqrt(x) = 1 for the resulting single sequence.
7. No proof shows constant normalized density is impossible.
Critical normalization correction
For Sidon density, the relevant normalization is:
sigma(A) = limsup_{x -> infinity} A(x) / sqrt(x)
not:
A(x) / x
A constant positive value of A(x)/x is impossible for Sidon sets because the standard counting bound gives A(x) = O(sqrt(x)). Therefore any hyperfluid discussion of constant density must specify whether it means:
A(x) / sqrt(x) approximately constant
or the ordinary natural density:
A(x) / x approximately constant.
The second cannot be positive for an infinite Sidon set.
Carry-free digit window guardrail
The carry-free digit idea is a plausible mechanism for decoupling local windows, but it does not automatically preserve global Sidon uniqueness.
A carry-free digit set can prevent carries between windows:
d_i + d_j < p
but a Sidon collision is global:
a_i + a_j = a_k + a_l
So the proof must show uniqueness of the whole digit-vector sum, not merely absence of carries.
Required theorem target:
CarryFree(A_k) + LocalSidonEachWindow(A_k) + Compatibility(A_k,A_{k+1})
-> GlobalSidon(Unroll(A_k))
Status: U_scope until proved.
CRT splicing guardrail
CRT splicing can combine congruence conditions, but Sidon uniqueness over integers is stronger than Sidon uniqueness modulo many finite moduli.
Required theorem target:
If a_i + a_j = a_k + a_l in Z,
then the CRT residue data forces {i,j} = {k,l}.
This requires a global size/window bound preventing distinct integer sums from sharing all audited residues in the relevant range.
Status: U_scope until supplied.
Hyperfluid / soliton interpretation
The hyperfluid model is retained as a diagnostic metaphor:
integer coordinate -> fluid coordinate
p-adic window -> frequency band
Sidon collision -> nonlinear phase collapse
admissible integer -> low-residual slot
construction process -> soliton sweep
oscillating density -> pulsed admissibility profile
Safe statement:
The hyperfluid model suggests that dense Sidon constructions, if possible, should appear as pulsed low-residual trajectories through structured algebraic windows.
Unsafe statement:
The hyperfluid soliton sweep proves sigma = 1 or proves constant density impossible.
Density-profile question
The refined, valid question is:
Can any explicit infinite Sidon set have A(x)/sqrt(x) bounded away from zero and near one across long intervals, or must every high-limsup construction oscillate through sparse recovery zones?
A stronger question:
Does there exist an infinite Sidon set A with A(x)/sqrt(x) -> c > 0?
This must be treated as a number-theoretic density-profile question, not settled by the hyperfluid analogy.
Required closure receipts
To promote the hyperfluid/p-adic closure story, provide:
1. explicit digit alphabet D_p,
2. explicit seed family A_k,
3. explicit lift rule from A_k to A_{k+1},
4. proof of carry-free or carry-controlled addition,
5. proof of global Sidon uniqueness after unrolling,
6. proof of density limsup A(x)/sqrt(x)=1,
7. optional theorem on oscillation or impossibility of constant A(x)/sqrt(x),
8. finite computational audits for initial levels.
Lean target skeleton
namespace SidonHyperfluid
-- Hyperfluid curves and p-adic pressure plots are not proof objects.
theorem hyperfluidAnalogy_notSidonProof
(claim : Claim) :
HyperfluidSolitonAnalogy claim ->
¬ HasExplicitInfiniteSidonConstruction claim ->
Gate claim = P_analogy := by
sorry
-- Ordinary positive density is incompatible with Sidon square-root growth.
theorem positiveNaturalDensity_notSidonCompatible
(A : Nat -> Prop) :
IsInfiniteSidon A ->
NotPositiveNaturalDensity A := by
sorry
-- Carry-free local windows require a global unrolling theorem.
theorem carryFree_notEnoughWithoutGlobalAudit
(chain : Nat -> Finset Nat) :
CarryFreeDigits chain ->
¬ GlobalPairSumAudit chain ->
Gate (ClaimFrom chain) = U_scope := by
sorry
end SidonHyperfluid
Short doctrine
The hyperfluid picture is a lens, not a theorem.
Carry-free digits prevent carries, not automatically collisions.
CRT splices residues, not automatically global Sidon sums.
Sigma uses A(x)/sqrt(x), not A(x)/x.
Closure requires an explicit infinite construction plus a global sums audit.