# OP0a Scalar / 1D Multiwell Anti-Uniqueness Formal Theorem Note

Status: accepted scalar theorem-language note after post-specialist formal review route `op0a_scalar_theorem_ready_for_accepted_language_after_formal_review`, 2026-05-26. This applies only to the scalar / one-dimensional OP0a theorem statement below. It is not global OP0a theorem status, and the high-dimensional `R^m` target remains separately gated.

Claim class: formal-review-cleared scalar theorem object. Its role is specificity-boundary mathematics for OP0; Standard Model, QFT representation, Layer-3, Tau-X, hardware, material, mission, and capability interpretations remain separate promotion-gated questions.

Supporting artifacts:

- `open-theorem-working-notes/op0a-multiwell-anti-uniqueness-theorem-note-2026-05-22.md`
- `open-theorem-working-notes/op0a-scalar-1d-theorem-development-2026-05-25.md`
- `cct-public-replication/theorem_verifiers/verify_op0a_multiwell.py`
- `cct-public-replication/examples/op0a_multiwell_cases.csv`
- `external-reviews/op0a-dynamical-systems-postinterim-review-2026-05-25.md`
- `external-reviews/op0a-scalar-theorem-specialist-review-2026-05-26.md`
- `external-reviews/op0a-scalar-post-specialist-formal-review-2026-05-26.md`

## Role

OP0a is a specificity-boundary result. It shows that smooth scalar stability potentials can realize arbitrary finite basin counts and prescribed local point Hessians. Therefore basin count, local curvature matching, or generation-like hierarchy shape is not specificity evidence by itself.

This result supports OP0b by explaining why additional filters are needed before any rule-space object can be read as physically specific.

## Accepted Scalar Theorem Statement: Scalar Multiwell Anti-Uniqueness

Let `n >= 2`. Let

```text
x_1 < x_2 < ... < x_n
```

be prescribed centers in `R`. Let

```text
h_1, ..., h_n > 0
```

be prescribed point Hessians, and let `beta > 0` be a prescribed adjacent two-sided barrier lower bound.

For any interleaving points

```text
x_1 < y_1 < x_2 < y_2 < ... < y_{n-1} < x_n,
```

there exists a smooth proper Morse function

```text
S: R -> R
```

such that:

1. The critical set is exactly

   ```text
   Crit(S) = {x_1, ..., x_n, y_1, ..., y_{n-1}}.
   ```

2. Each `x_k` is a strict local minimum and

   ```text
   S''(x_k) = h_k.
   ```

3. Each `y_i` is a strict local maximum.
4. `S(x) -> +infinity` as `|x| -> infinity`.
5. The adjacent two-sided barrier convention is frozen as

   ```text
   S(y_i) - S(x_i) >= beta
   S(y_i) - S(x_{i+1}) >= beta
   ```

   for every `i = 1, ..., n-1`. When this note says “communication height” in the scalar theorem, it means this adjacent two-sided barrier convention, not a separate stochastic or high-dimensional quasipotential.
6. For any smooth positive mobility `M_mob: R -> (0,infinity)` that is locally bounded above and below on compact intervals, the gradient-like flow

   ```text
   dX/dt = -M_mob(X) S'(X)
   ```

   has exactly `n` attracting basins, separated by the maxima `y_i`. The basins are the open intervals

   ```text
   B_1 = (-infinity, y_1)
   B_k = (y_{k-1}, y_k), 2 <= k <= n-1
   B_n = (y_{n-1}, infinity),
   ```

   with the maxima `y_i` excluded as unstable equilibria.

## Construction

Define

```text
P(x) = prod_{k=1}^n (x - x_k) * prod_{i=1}^{n-1} (x - y_i).
```

The roots are ordered as

```text
x_1, y_1, x_2, y_2, ..., y_{n-1}, x_n.
```

There are `2n-1` simple roots.

## Lemma 1: Ordered-Root Sign Alternation

Let the ordered roots be

```text
r_1 = x_1, r_2 = y_1, r_3 = x_2, ..., r_{2n-1} = x_n.
```

For

```text
P(x) = prod_{j=1}^{2n-1} (x-r_j),
```

the derivative at a root is

```text
P'(r_j) = prod_{ell != j} (r_j-r_ell).
```

There are `j-1` positive factors and `2n-1-j` negative factors, so

```text
sign P'(r_j) = (-1)^(2n-1-j).
```

Because each `x_k` is an odd-indexed root and each `y_i` is an even-indexed root:

```text
P'(x_k) > 0
P'(y_i) < 0.
```

The construction will use

```text
S'(x) = q(x) P(x)
S(x) = integral_0^x q(t) P(t) dt + C
```

with `q in C^\infty(R)`, `q(x) > 0`, and `q(x) = 1` outside a compact interval after the barrier-tuning factors are applied.

## Lemma 2: Positive Multiplier Preserves Exact Critical Set

If `q(x) > 0` for all `x`, then the zeros of `S'` are exactly the zeros of `P`.

At any root `r` of `P`,

```text
S''(r) = q(r) P'(r),
```

because `P(r)=0`. Since `q(r)>0`, the sign of `S''(r)` is the sign of `P'(r)`.

Therefore each `x_k` is a nondegenerate local minimum and each `y_i` is a nondegenerate local maximum. The critical set is exactly the declared root set. No additional critical points can be introduced by shaping `q` as long as `q` remains positive. The exact second derivatives at the maxima are not prescribed by this theorem; only nondegeneracy and maximum type are required.

## Lemma 3: Point-Hessian Prescription

Set

```text
a_k = h_k / P'(x_k).
```

Because `P'(x_k)>0`, each `a_k>0`.

Choose pairwise disjoint compact intervals around the centers `x_k`, avoiding all other roots, and choose smooth bump functions `psi_k` supported in those intervals with `psi_k(x_k)=1`. Define

```text
q_0(x) = exp(sum_k log(a_k) psi_k(x)).
```

Then:

```text
q_0(x) > 0
q_0(x_k) = a_k
q_0(x) = 1 outside a compact set.
```

Thus the function defined by `S'_0=q_0 P` satisfies

```text
S''_0(x_k) = q_0(x_k) P'(x_k) = h_k.
```

This theorem prescribes point Hessians. Exact local quadratic germs are a stronger variant and require an added local-jet condition.

## Lemma 4: Two-Sided Adjacent Barrier Tuning

Use the two-sided adjacent barrier convention:

```text
S(y_i)-S(x_i) >= beta
S(y_i)-S(x_{i+1}) >= beta.
```

Choose the Hessian-prescription neighborhoods around the `x_k` small enough that each open interval `(x_i,y_i)` and `(y_i,x_{i+1})` still contains a nonempty compact subinterval away from all roots and away from those neighborhoods.

For each `i`, choose compact subintervals

```text
I_i^L subset (x_i, y_i)
I_i^R subset (y_i, x_{i+1})
```

that avoid all roots and avoid the Hessian-prescription neighborhoods around the `x_k`. On `(x_i,y_i)`, `P` has the sign that makes

```text
S(y_i)-S(x_i) = integral_{x_i}^{y_i} q(t)P(t)dt
```

positive. On `(y_i,x_{i+1})`, `P` has the opposite sign, so

```text
S(y_i)-S(x_{i+1}) = - integral_{y_i}^{x_{i+1}} q(t)P(t)dt
```

is positive.

Let `eta_i^L` and `eta_i^R` be nonnegative smooth bump functions supported in `I_i^L` and `I_i^R`, respectively, and positive on some subinterval. Define

```text
q_lambda(x) = q_0(x) * exp(sum_i lambda_i^L eta_i^L(x) + lambda_i^R eta_i^R(x)).
```

For finite nonnegative `lambda`, `q_lambda>0`, so the critical set remains the declared root set.

Because each `eta` support is disjoint from every `x_k`, we have

```text
q_lambda(x_k) = q_0(x_k)
```

and the prescribed point Hessians survive barrier tuning.

As `lambda_i^L -> infinity`, the integral mass over `I_i^L` tends to `+infinity`, so `S(y_i)-S(x_i)` can be made at least `beta`. Quantitatively, on any compact subinterval `J` where `eta >= c > 0` and `q_0 |P| >= m > 0`,

```text
integral_J q_0 exp(lambda eta) |P| dt >= m |J| exp(lambda c) -> +infinity.
```

The same bound applies to the signed integral over `I_i^R`, so `S(y_i)-S(x_{i+1})` can be made at least `beta`. For each side, finite sufficiently large `lambda` values therefore meet the declared lower bound. Since the supports can be selected disjointly for each side and away from roots, all adjacent two-sided barriers can be met without moving roots or changing prescribed point Hessians.

For non-adjacent minima `x_i < x_j`, every path in `R` crosses the endpoint-adjacent maxima `y_i` and `y_{j-1}`. The two-sided adjacent bound gives a barrier at least `beta` above `S(x_i)` at `y_i` and at least `beta` above `S(x_j)` at `y_{j-1}`. Thus the scalar non-adjacent communication statement follows from the adjacent convention.

## Lemma 5: Exterior Collar Implies Properness

Require the final multiplier to satisfy an exterior lower-bound/collar condition. The construction above chooses supports compactly, so after selecting a compact interval containing all roots and all barrier-tuning supports:

```text
q_lambda(x) = 1
```

outside that interval.

The polynomial `P` has odd degree `2n-1` and positive leading coefficient. Therefore

```text
P(x) -> +infinity as x -> +infinity
P(x) -> -infinity as x -> -infinity.
```

Since `q_lambda=1` outside a compact set, `S'(x)=q_lambda(x)P(x)` has the same exterior sign and growth. Hence

```text
S(x) -> +infinity as |x| -> infinity.
```

At the left end, for large negative `x`,

```text
S(x) = integral_0^x P(t)dt = - integral_x^0 P(t)dt -> +infinity
```

because `P(t)<0` with polynomial growth on the negative exterior.

This exterior condition is necessary. A positive multiplier alone is not enough: if `q` decays too quickly at infinity, the integral of `qP` can remain bounded. The companion verifier includes a computed expected-failure fixture for this case.

## Lemma 6: Positive-Mobility Basin Convergence

Let `M_mob` be smooth and positive, locally bounded above and below on compact intervals. For

```text
dX/dt = -M_mob(X) S'(X),
```

properness of `S` confines trajectories to compact sublevel intervals because

```text
dS/dt = -M_mob(X) (S'(X))^2 <= 0.
```

The maxima `y_i` are equilibrium points and separate the line into basin intervals:

```text
B_1 = (-infinity, y_1)
B_k = (y_{k-1}, y_k), 2 <= k <= n-1
B_n = (y_{n-1}, +infinity).
```

Each such interval contains exactly one minimum `x_k` and no other critical point. The vector field points toward that minimum on both sides inside the interval. A trajectory starting inside the interval cannot cross a separating maximum because uniqueness for the smooth ODE prevents crossing an equilibrium point. The trajectory is globally defined inside a compact sublevel interval. In one dimension it is monotone toward the unique equilibrium in its interval, and its omega-limit must be a critical point. Since there is only one critical point in the interval, the trajectory converges to `x_k`.

If `X(0)=x_k`, the trajectory is constant at the attracting equilibrium. If `X(0)=y_i`, the trajectory is constant at the excluded separating equilibrium. Otherwise, trajectories inside a basin interval converge monotonically to the unique `x_k`.

Thus the attracting basins are exactly the `n` intervals separated by the maxima, with the maxima themselves as the excluded separating equilibria.

## Anti-Uniqueness Consequence

For every finite `n >= 2`, the same scalar construction realizes an `n`-well stability potential with prescribed point Hessians and declared adjacent barriers.

Therefore:

- `n=3` is compatible but not selected;
- local curvature hierarchy can be fitted;
- basin count and Hessian matching are underdetermined without additional filters;
- OP0b-style specificity filters are required before any rule-space object can carry physical specificity.

## Verifier Relationship

The companion verifier remains a public-safe scalar synthetic route companion, not a proof engine.

It currently checks:

- scalar multiwell examples for `n=2,3,4,5,8`;
- an explicit scalar derivative-construction companion with nonuniform centers and arbitrary prescribed point Hessians;
- `n=3` anti-specialness;
- global convex-Hessian / metric conflation expected failure;
- positive-quadratic plus positive-quartic expected failure;
- computed naive additive-bump / gluing expected failure;
- high-dimensional product-potential sentinel;
- barrier-rescaling / exact-Hessian conflict;
- remote-minimum sentinel;
- stochastic-quasipotential sentinel;
- computed positive-`q` nonproperness expected failure.

The verifier's positive scalar derivative fixture is a companion to the theorem construction: it exercises `S'=qP`, positive `q`, nonuniform centers, arbitrary point Hessians, and compact exterior behavior. Its numerical barrier fixture is not a proof of the theorem. Accepted theorem status comes from proof review, not from route checks or numerical scans.

## Small-Noise Boundary

Small-noise retention is not part of the scalar theorem.

A later separate corollary may ask whether, for fixed finite `T` and compact `K` inside one deterministic basin away from the separating maxima,

```text
P_x[exit before T] -> 0 as D -> 0
```

under declared drift/noise regularity. Metastable exponential claims require additional reversible Langevin or quasipotential assumptions and are outside this theorem.

## Accepted Scalar Status

Current review route:

```text
op0a_scalar_theorem_ready_for_accepted_language_after_formal_review
```

The scalar note has passed post-specialist formal review for accepted scalar theorem language. The accepted-language scope is exactly the scalar / one-dimensional theorem stated above. Global OP0a language, high-dimensional `R^m` theorem language, OP0b specificity, Standard Model, Layer-3, hardware, material, Tau-X, and mission interpretations remain separate review-gated targets.

The high-dimensional `R^m` theorem remains behind a separate relative Morse / handle-construction lemma.
