RENAMED 2026-08-20 — replaced, not deleted. This heading read
“A certified bifurcation bracket for rational attention” from 2026-07-31 until
today. The page's own proof-status ledger, unchanged below, had already retired every bracket the
path ever carried: the Newton-from-8k bracket is demoted (T8), the
(2.5, 2.7) linear-stability loss is demoted as an 80k-step cutoff (T9), and the tight
omega-limit β* for seed +0 is open and not claimed. A heading that
promises a bracket over a body that retired all of them overclaims against this page's own
evidence, in the direction this page exists to refuse — so the heading now names what
survived, and the wording it replaced is kept here rather than erased.
What survived, and what the new heading points at, is in the ledger
below at measured: the 1-cluster spectrum
{0, −1} independent of β, and exact residual zero on the
closed-form 2-cluster configurations in exact rational arithmetic. The three demotion records in
Corrections are untouched, and so is everything else on this page: the draft chrome, the
never row that forbids reporting any of this as softmax, and the statement that
no literature gate has run.
Transformers are an interacting particle system, and their tokens cluster.
Softmax needs exp, which this tree does not have — so replace the kernel
with a rational one and check, first, whether the phenomenon comes with it. The PATH 12 rule:
when the arithmetic cannot represent your model, change the model, do not approximate the function.
Softmax attention needs exp. The PATH 12 move
is to substitute a rational kernel rather than approximate the exponential. The deciding
question: does the clustering phenomenon survive the swap? If not, the certificate would
describe a system nobody cares about.
It survives, and it survives structurally. Measured on 12 tokens on S²,
fixed seed (attention-substitution.py):
| kernel | low sharpness | mid | high |
|---|---|---|---|
| softmax — blocked | 1 cluster | 1 cluster | 3 clusters |
polynomial (1+⟨xi,xj⟩)p | 1 cluster | 1 cluster | 2 clusters |
Both undergo a cluster-count change as the interaction sharpens. Green control: if softmax ever fails to cluster the script exits 2.
python3 research/probes/attention-substitution.py
An integer exponent is a poor bifurcation parameter. The form that stays rational in the state and smooth in the knob:
wij ∝ (1 + β⟨xi, xj⟩)p
with integer p and continuous β —
only + − × ÷ and integer pow.
The projected residual is exactly zero on closed-form
coincident-cluster configurations, decided by exact rational arithmetic.
Antipodal 2-clusters and ⟨u,v⟩ = −1/β 2-clusters
(when β ≥ 1) are equilibria for every such β.
Existence of multi-cluster equilibria is not what any float β* measured.
What would falsify the enclosure. A nonzero exact residual on the closed-form configurations (T1–T3). A zero residual on a deliberately wrong angle (T4 must stay red). Krawczyk closing on a non-isolated coincident root (T5 must refuse).
Locator transient. attention-transition.py reported
β* ∈ [1.601593, 1.601624] from 4000-step terminals.
That bracket collapses under Newton / 20k steps. Gated in
attention-equilibrium.py T5. Same species as PATH 12's first Huggett test.
Newton-from-8k is not an omega-limit.
attention-equilibrium.py locked β* ∈ [1.615215, 1.615220]
by short flow + Newton. Near criticality that is metastability.
Gated as T8 in attention-basin.py: 100k-step flow from seed +0 at the old
βhi reaches the 1-cluster.
(2.5, 2.7) was an 80k-step cutoff, not a linear crossing.
The old basin T6/T7 reported the exact 2-cluster “loses local stability in (2.5, 2.7)”
because β=2.5 still showed 2 clusters at 80k steps and β=2.7 showed 1. Extending the
budget: the same seed/eps collapses at ~84.5k for β=2.5 (gated T9). Exact rationals
(attention-stability.js) decide the reduced perfect-cluster law has a
double zero at c*=−1/β and ċ>0 off equilibrium
for every tested β — one-sided semi-stable, no linear sign change in β.
attention-enclosure.js — exact rational
arithmetic (core/interval/rational.js) decides the residual is all zero on:
| configuration | β | residual |
|---|---|---|
2-cluster, ⟨u,v⟩=−3/5, Pythagorean v=(−3/5,4/5,0) |
5/3 | exactly 0 |
| antipodal 2-cluster | 1/2 (below any float bracket) |
exactly 0 |
| 1-cluster | 5/3 | exactly 0 |
Claim revision, gated by the antipodal β=1/2 case:
multi-cluster equilibria exist far below any float bracket. Krawczyk on an ungaged coincident
root refuses (geometric non-isolation).
node research/probes/attention-enclosure.js
attention-spectrum.py. At any 1-cluster
equilibrium, every sphere-tangent perturbation satisfies
(Jv)i = mean(v) − vi
— spectrum {0 (gauge), −1 (relative)},
independent of β. At consensus the weights are uniform and their
first variation vanishes.
Therefore no float β* on this path is a linear bifurcation
of the 1-cluster. The 1-cluster stays linearly attracting for all β.
python3 research/probes/attention-spectrum.py
attention-basin.py. At rational
β = 5/3, both closed-form equilibria are locally stable under the
normalized flow (50k steps after a 10−3 perturbation):
| start | after perturbation + flow |
|---|---|
| 1-cluster | 1 cluster |
exact 2-cluster ⟨u,v⟩ = −3/5 | 2 clusters |
From seed +0, 100k-step flow selects 1 cluster at β = 3/2
and 2 clusters at β = 2. Perfect-cluster kicks: above c* merges,
below returns — at both β=2.5 and 2.7.
A tight certified omega-limit β* for seed +0 is not claimed
(critical slowing).
python3 research/probes/attention-basin.py
attention-stability.js — exact
rationals decide the perfect-cluster reduced law (equal sizes, p=2):
ċ = 2(1+βc)²(1−c²) / [(1+β)² + (1+βc)²]
Double zero at c*=−1/β (so f=f′=0);
ċ>0 off equilibrium on every tested rational point —
one-sided semi-stable for every β, not a linear crossing in (2.5, 2.7).
Cross-weights are flat to first order at p=2, so the tangent Jacobian
block-decouples into two 1-cluster copies with spectrum {0,−1}.
node research/probes/attention-stability.js
| claim | status | what backs it |
|---|---|---|
| rational kernel still clusters | measured | attention-substitution.py, exit 0 |
| exact residual zero on closed forms | measured | attention-enclosure.js, T1–T3 exact rationals |
| 1-cluster spectrum {0, −1}, β-free | measured | attention-spectrum.py, analytical J = mean − Id |
| bistability at β=5/3 | measured | attention-basin.py T1–T3 |
| reduced 2c angle: one-sided semi-stable | measured | attention-stability.js T1–T7 exact rationals |
| tight omega-limit β* for seed +0 | open | not claimed; Newton bracket demoted (T8) |
| (2.5, 2.7) linear stability loss | demoted | 80k cutoff; T9 + double-zero law |
| Transformer / softmax threshold | never | rational system only; must not be reported as softmax |
Second independent instance of the PATH 12 rule. PATH 11 died approximating; PATH 12 worked by substituting in economics; this substitutes in machine learning. Two confirmations and one refutation is what turns a trick into a method.
It takes attention-bifurcation (CONFIRMED in
SCORING.json) out from behind interval-transcendentals.
PATH 07's consumer count drops again.
Harder swap than PATH 12. Huggett is still a canonical model. Linear / polynomial attention is a different attention; Geshkovski et al. is about softmax. The certified number would be a fact about the rational system and must never be reported as a fact about Transformers.
Evidence is still thin, and under-iteration already bit thrice.
Bistability is gated at one rational β; seed +0 selection at two
safe distances. Never trust a cluster count that has not been long-integrated or
Newton-polished to a small algebraic residual.
Occupancy, ungated. Geshkovski, Letrouit, Polyanskiy and Rigollet are named from memory; Lessard–Pugliese likewise. N is near zero and the artifact is the claim — no literature gate has run.
N: near ZERO. Interacting-particle views of
attention are published; certified bifurcation brackets are standard machinery.
A: HIGH and unverified — a re-runnable, zero-dependency artifact for an
attention system does not appear to exist (same A-axis claim as
attention-bifurcation in SCORING.json).
Citation state: Geshkovski et al. and Lessard–Pugliese are named from memory and unverified at source. The measurements above were produced on 2026-07-31 by the scripts named, and are reproducible with a fixed seed.