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I asked my AI who know something about semantic attractor dynamics and prime number gaps.
https://discuss.huggingface.co/t/164607/5

And this is its reply, hope it help.

https://chatgpt.com/share/68d7c80f-7608-8010-be48-bc72196ad178 <= this link can see those formula better.

Yes: your square ↔ pronic alternation can be turned into a useful anchor–edge memory scaffold for LLMs.

  • Squares act as anchors (stable “1” with scale/magnitude).
  • Pronics act as edges/transitions (contextual “0” defined by its two neighboring anchors).
    This yields a multi-scale positional/memory scheme with ~O(N)O(\sqrt{N}) anchor slots for a length-NN sequence—handy for long-context retrieval and compression.

Why this is promising

Your recurrence

an+1=an+⌈an⌉,a1=1a_{n+1}=a_n+\lceil\sqrt{a_n}\rceil,\quad a_1=1

produces diffs Δa=1,2,2,3,3,4,4,…\Delta a = 1,2,2,3,3,4,4,\ldots (each integer repeats twice). That induces a very clean two-phase geometry:

  • Odd indices (perfect squares m2m^2) → anchors with intrinsic scale mm.
  • Even indices (pronics m(m+1)m(m+1)) → edges that bind consecutive anchors m2↔(m+1)2m^2 \leftrightarrow (m{+}1)^2.

Intuition: anchors are “stable lattice states”, pronics are “allowed transitions”. This is exactly the structure many long-context systems want: a sparse set of global reference points (anchors) plus local bridges (edges).

A concrete design: Nitya-PE (Positional Encoding) + Anchor-Edge Attention

Encoding for token position tt:

  • m=⌊t⌋m=\lfloor \sqrt{t}\rfloor (segment index / scale)
  • ϵ=t−m2\epsilon = t-m^2 (offset within the [m2,(m+1)2][m^2,(m+1)^2] segment)
  • Phase bit b=1[t is a square]b = \mathbb{1}[t \text{ is a square}]

Form a feature vector, then project to model dim:

  • u1=tu_1=\sqrt{t} (magnitude/scale)
  • u2=ϵ/(2m+1)u_2=\epsilon / (2m+1) (normalized in-segment offset)
  • u3=bu_3=b (anchor/edge phase)

Use PE(t)=W [u1,u2,u3]\text{PE}(t)=W,[u_1,u_2,u_3] (learned linear WW), optionally concatenated with your usual RoPE/ALiBi/relative biases.

Attention bias (Anchor-Edge mask):

  • Give a small learned bonus for queries to attend to:
    1. their nearest anchor m2m^2,
    2. the bridging edge m(m+1)m(m{+}1), and
    3. the next anchor (m+1)2(m{+}1)^2.
  • Within a segment (between m2m^2 and (m+1)2(m+1)^2), allow more order-insensitive aggregation; across segments, keep order-sensitive weighting.
    (Rationale: your diffs are flat inside segments and “jump” only at boundaries—those jumps are natural event markers.)

Memory compression:

  • Keep external memory only at anchors (and optionally edges).
  • Number of anchors up to position NN is ⌊N⌋\lfloor\sqrt{N}\rfloor → ~O(N)O(\sqrt{N}) memory footprints, while still indexing the full length NN.

How it’s different from plain binary

Your “1/0” isn’t free-floating; it’s binary with magnitude and context:

  • “1” = square carries the scale mm explicitly (via t\sqrt{t}).
  • “0” = pronic is defined by its neighbors; it encodes relational info (a transition), not a standalone state.

This makes it a natural fit for hierarchical or multi-scale reasoning, with built-in boundary detection (the moments when the diff increases).

Relation to known ideas (at a glance)

  • Positional encodings: complements RoPE/ALiBi/T5-relative by adding an explicit segment scale and phase.
  • Hierarchical/long-context attention: akin to global tokens or skip-lists, but with a deterministic arithmetic schedule for where the globals live (the squares) and how to bridge them (the pronics).
  • State-space / memory models: you can treat anchors as persistent states and pronics as update edges.

Minimal experiment you can run

  1. Tasks: long-range retrieval (needle-in-a-haystack), cross-chapter QA, code with far references, or any long-document summarization.
  2. Models: a small Transformer baseline (e.g., 100–300M).
  3. Variants:
  • Baseline PE (RoPE or ALiBi)
  • Baseline + Anchor-Edge attention bias
  • Baseline + Nitya-PE
  • Baseline + both (bias + Nitya-PE)
  1. Metrics: retrieval hit-rate @k, cross-segment F1 (can it follow references across square boundaries?), perplexity on long contexts, and memory/read latency if you add external memory at anchors.
  2. Ablations: remove the phase bit; replace u1u_1 with log⁡t\log t; jitter anchor locations slightly to check robustness.

Descending variant (reversible pass)

Your decreasing rule (e.g., 100→90→81→72→⋯→0100\to90\to81\to72\to\cdots\to0) is essentially a reverse traversal of the same anchor-edge graph. Two uses:

  • Reversible compression / replay: write forward (ascending), read back (descending) to reconstruct long chains with low drift.
  • Curriculum & stabilization: a “cool-down” pass that progressively collapses transitions back into anchors—useful for denoising or for iterative summarization that preserves anchor fidelity.

Pseudocode sketch (toy)

def nitya_features(t):
    import math
    m = int(math.sqrt(t))
    eps = t - m*m
    u1 = math.sqrt(t)
    u2 = eps / (2*m + 1) if (2*m + 1) > 0 else 0.0
    u3 = 1.0 if m*m == t else 0.0  # square = anchor
    return [u1, u2, u3]  # project with a learned linear layer

# Attention bias: for a query at t, bonus to nearest anchors/edge
def nitya_bias(t, s):
    import math
    m = int(math.sqrt(t))
    anchors = {m*m, (m+1)*(m+1)}
    edges   = {m*(m+1)}
    return 1.0 if s in anchors else (0.5 if s in edges else 0.0)

Open questions (would love community input)

  • Does anchor-edge bias improve fidelity on truly long contexts (>128k) without extra compute?
  • What’s the best way to combine Nitya-PE with RoPE (concat vs. gating vs. mixture-of-positional-experts)?
  • Can we push the idea into external memory: anchors as keys, pronics as linkers, for O(N)O(\sqrt{N}) retrieval hops?

Bottom line: your “recursive binary-with-magnitude” looks like a clean, testable multi-scale memory/indexing prior. If you’re up for it, I can help draft a small training script to benchmark Nitya-PE + anchor-edge attention on a public long-context task and share results.

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