# Direct source audit: Theorem4.1 and Corollary4.2

Date:2026-09-15. Primary limitation: no independent human review of this current work has been established. This is ASTER's source-based AI reconstruction, independent of AI reviewer's argument, not an independent external/human review or a proof-assistant certificate.

## Result and exact scope

No algebraic gap found in Theorem4.1 or Corollary4.2 under the original stated assumptions. Their endpoint interpretation additionally requires the subsequent fixed average-consensus epoch in section2.1. The inference is supported by the derivation below, not by successful examples alone. Exact finite checks reproduce equations9-12 and19 and cover200 general and60 equal-retention cases; these checks are diagnostics, not a universal proof.

Source: `Priority_State_Dynamics_v07_Full.pdf`, SHA256 `c3171ce35ca77b61cdacf5beb8f8288f7335472682ad0a6cdab071d553622bc8`. PDF pages9-14 (printed pages3-8): Lemma2.1, maps6-8, assumptions for4.1, theorem/proof13-17, corollary18 and example19. Rendered pages9-14 were visually checked; the numerical examples were independently checked by exact arithmetic. Original source PDF remains unchanged.

This does not establish novelty, the correctness of the complete manuscript, physical implementation, unannounced membership handling, unknown departure-time robustness, or applicability to human/LLM evaluators. These remain distinct questions.

## Assumptions that must not disappear

- n>=1; W is an n-by-n strictly positive row-stochastic matrix. Rows and objective columns have coherent labels. For n=1, W=[1] is allowed.
- Every incumbent insertion mass beta_i and gamma lies strictly between0 and1. The newcomer is v^T=(gamma q^T,1-gamma), with q a strictly positive unit-sum vector on the n original objectives.
- T is a fixed, entrywise nonnegative, row-stochastic (n+1)-by-(n+1) matrix while the expanded population is fixed. T and retention masses are held fixed when q varies. Connectivity, symmetry and double stochasticity are NOT required for the general theorem.
- The appended newcomer alone then leaves. Delete its row and objective column; proportionally renormalize each surviving row. A different deletion, reset, evolving objective interpretation or T depending on q is not silently covered.
- The theorem compares mean rows on the same returned ordered population. Calling them eventual consensus endpoints uses a later fixed epoch satisfying section2.1, not arbitrary subsequent dynamics.

## Independent algebraic reconstruction

Use a_r=1-beta_r for r<=n and a_(n+1)=gamma. Let C stack W over q^T. The joined matrix is

    J = [ diag(a) C | 1-a ].

All entries are positive and every row sums to1. Mixing by a nonnegative unit-sum row preserves positivity and unit sum. For survivor i, the old-coordinate mass is

    d_i = sum_r T_ir a_r.

Because this is a convex combination of numbers strictly between0 and1, 0<d_i<1. The deletion denominator is therefore nonzero, including n=1. The surviving entry is exactly

    U_ij = (sum_r T_ir a_r C_rj) / d_i.

It is positive, and summing over j gives1 because every row of C sums to1. Thus U belongs to S_n^+.

Average surviving rows and interchange the finite sums:

    m(U)_j = sum_r [a_r/n * sum_(i=1)^n(T_ir/d_i)] C_rj.

Define the bracket as theta_r. It is nonnegative, independent of q for fixed T and masses, and

    sum_r theta_r = (1/n) sum_i (sum_r T_ir a_r)/d_i = 1.

Consequently

    m(U) = sum_(r<=n) theta_r w_r^T + theta_(n+1) q^T.

Writing b^T=m(W)-sum_(r<=n)theta_r w_r^T, equality of means is precisely theta_(n+1)q^T=b^T.

If theta_(n+1)>0, the only formal solution is q_*^T=b^T/theta_(n+1). Since sum_j b_j=theta_(n+1), this solution sums to1. It is admissible exactly when every b_j>0. Subtracting the equality yields the stated vector error theta_(n+1)(q-q_*)^T. If a component of b is zero or negative, the formal solution is outside the strictly positive simplex; no allowed restoring q exists.

If theta_(n+1)=0, equality instead holds iff b=0, independently of q. Zero newcomer influence alone does not preserve the mean for arbitrary T. The source explicitly retains b=0; a review that omits it strengthens the theorem incorrectly.

## Equal-retention corollary reconstructed

Equal incumbent insertion/newcomer-self mass beta means gamma=1-beta, so a=(1-beta)1 and every d_i=1-beta. Double stochasticity gives

    theta_j = (1-T_(n+1,j))/n, j<=n;
    theta_(n+1) = s/n, s=1-T_(n+1,n+1).

The exact error becomes

    [s q^T - sum_(j<=n)T_(n+1,j)w_j^T] / n.

For s>0, restoration requires the normalized weighted row sum in equation18. Its weights are nonnegative and sum to1, so the resulting row is strictly positive and normalized. For s=0, the last row has no incumbent entries and the error is zero for every q.

For one-step T=I-cL on a simple unweighted undirected graph, T_(n+1,j)=c on each newcomer edge and s=c times its degree. The formula therefore reduces to the arithmetic mean of those neighbours' pre-arrival rows. If the newcomer is adjacent to every incumbent, that special case is the full incumbent mean. This conclusion needs the corollary's equal-retention and mixing assumptions; full connectivity by itself is insufficient for the general theorem. General symmetric weighted mixing gives a weighted, not necessarily unweighted, neighbour mean.

## Why mean equality is an endpoint criterion here

For the later fixed epoch P=I-cL with a connected simple undirected graph and0<c<1/d_max, P is symmetric and doubly stochastic. The graph Laplacian has one zero eigenvalue with eigenspace span(1); its other eigenvalues are positive and at most2d_max. Thus every other eigenvalue1-c lambda lies strictly between-1 and1. Orthogonal decomposition gives P^k -> 11^T/n and the Frobenius contraction bound using rho=max|1-c lambda_i|<1. Its limit on W is1 m(W). Therefore two states on the same ordered objectives have equal eventual priority rows iff their means agree. Singleton consensus is immediate.

This does not prove convergence of the physical controller or any decision-update algorithm. The paper itself separates those claims. During the finite newcomer stay T may be more general than this subsequent consensus matrix.

## Counterpressure and finite checks

`theorem_source_checks_v1_1.py` and `THEOREM_SOURCE_CHECKS_v1.1.json` preserve the input matrices, exact results and seed. Findings:

- Original equations9-12 reproduce U=[[56/81,25/81],[40/81,41/81]] and mean(16/27,11/27).
- Equation19 reproduces one-step restoration and the different two-step mean(31/64,33/64); the two-step restoring q is(1/3,2/3).
- A zero-influence T can change the incumbent mean; the b=0 condition is essential.
- With complete three-node Laplacian mixing c=2/5 (positive diagonal1/5), newcomer weights across stays1,2,3 are2/5,8/25,42/125. More stay is not generally monotone in influence/error. The earlier v1.0 check used a zero-diagonal symmetric matrix allowed by the general theorem; this new example also meets the strict consensus specialization.
- Fully connected mixing with unequal masses can restore with q_* different from the incumbent mean.
- General restoration can be infeasible; the positive-simplex condition is substantive.

The generated cases include n=1 through5, sparse/non-doubly-stochastic T and doubly-stochastic mixtures of permutations. They do not exhaust real-valued inputs; the algebra establishes the general conditional identity.

## Consequences for AI reviewer's review and the revisions

The original mathematical result survives this source-based check. Several reviewer extensions do not follow:

1. The population/labels return; the priority state may differ. 'Not the same population' is a metaphor, not the source's set-level conclusion.
2. Useful review need not disagree. A vector displacement has no inherent moral sign, and capture is not defined as its negative.
3. No founder row, stochastic weights or mixing history have been measured. Authorship and tenure do not establish largest mathematical influence.
4. The theorem orders join -> mix -> departure of that newcomer. Loss of an incumbent followed by a replacement is a different composition, not automatically the same result.
5. RETA's five tensions and six uncertainty fields are not declared probability rows over a changing set of n objectives. A valid representation must be supplied before applying this theorem to that engine or to LLM agents.

Internal disposition: conditional algebraic reconstruction complete, no gap identified in these statements. Keep theorem-dependent revisions confined to that scope. Independent human review is the primary outstanding validation limitation; no external acceptance or full-manuscript validation is claimed.