FINAL PROJECTED-BAND REPAIR: INDEPENDENT REVIEW ADDENDUM
18 October 2026. Internal AI review, research only.

FINAL SOURCE
revised_stopping_theorem.txt
SHA256 7d2ed0072718a9fbb84581ed4c67fe75d4781468592bafa3198e163bbcd12edd

PRESERVED IMMEDIATE PREDECESSOR
reviews/source_before_projected_band_clarification.txt
SHA256 f5406cc1ccedf46b7bbcf0d7d0544f7ea84891632f7b6dbb48c91a6bc1675089

DISPOSITION
PASS for the repaired projected-band concentration argument. It is a valid
and sufficient proof repair, with unchanged theorem statements and scope.
This source-specific addendum supersedes final-source binding in my earlier
independent_code_and_final_binding.txt, which accurately describes the
preceding f5406c... source and its diagnostic evidence. The preserved older
full proof review remains a review of its identified earlier source; this
addendum does not silently relabel that report as a fresh full-source audit.

1. EXACT CHANGE AND WHY THE REPAIR MATTERS
Compared both complete files and their complete unified diff. The only
change from f5406c... to 7d2ed0... is the high spectral-band variance
paragraph in Section 7. No theorem, hypothesis, stopping definition,
endpoint, inference statement, code, constant or originality claim changed.

The prior paragraph invoked global bound (21) for the high projected band.
Its coarse diagonal envelope epsilon^(-d/2)4^(-J) alone does not, for every
arbitrary lower band beta_L<1/2, follow from just the stated narrow-band
width condition. The replacement sharpens that envelope on the actual
projected vector. This supplies the missing quantitative justification;
the preserved earlier assessments should be read together with this repair.

2. INDEPENDENT DERIVATION OF BOTH VARIANCE TERMS
Let Pi_H be the deterministic spectral projection for C onto the fixed
band beta=(1-lambda)/2 in [beta_H,beta_H2], with
  0<beta_L<beta_H<beta_H2<1/2.
Since beta_H2<1/2, its lambda values are bounded below by the positive
fixed constant 1-2beta_H2. It has order epsilon^(-d/2) eigenvalues by the
fixed interior-band count already established. Put
  Z_H=(1/2) Pi_H C^(1/2) B^J [a(X)k_X-v].
This vector is centered. Its covariance is
  Sigma_H=(1/4) Pi_H C^2 B^(2J),
because the stationary rank-one term is annihilated for J>=1. Define
  m_H=E||Z_H||^2=tr Sigma_H.
Fixed-band eigenvalue bounds and multiplicity give
  m_H >= c epsilon^(-d/2) beta_H^(2J).

The deterministic feature envelope is
  ||a(X)k_X-v||^2 <= C epsilon^(-d/2).
On the projected band, ||Pi_H C^(1/2)B^J||^2 is at most beta_H2^(2J).
Therefore
  ||Z_H||^2 <= M_H:=C epsilon^(-d/2) beta_H2^(2J).
Also ||Sigma_H||op <= C beta_H2^(2J). These bounds apply to the actual
projected vector, so they replace the coarser global 4^(-J) envelope.

The variance identity for the sample-mean squared norm gives two relative
contributions. For the off-diagonal contribution,
  tr(Sigma_H^2)/m_H^2
    <= ||Sigma_H||op/m_H
    <= C epsilon^(d/2) (beta_H2/beta_H)^(2J).
For the diagonal contribution,
  E||Z_H||^4/(n m_H^2)
    <= M_H/(n m_H)
    <= C(1/n)(beta_H2/beta_H)^(2J).
Dropping the negative diagonal subtraction only enlarges the upper bound.
Thus the final displayed relative-variance bound is correct:
  C[epsilon^(d/2)+1/n](beta_H2/beta_H)^(2J).

With J=floor[u_- log(1/epsilon)/log 2], let
  a_H=2u_- log(beta_H2/beta_H)/log 2.
The positive width can be chosen to make a_H<d/2. Then the first term
is O(epsilon^(d/2-a_H))->0. Since n epsilon^(3d/2)->infinity,
the second is epsilon^(3d/2-a_H)/(n epsilon^(3d/2))->0 as well.
Integer rounding changes these bounds only by a fixed factor.

The low/high expected-energy ratio is at most
  C(beta_L/beta_H)^(2J)->0.
Markov for the low part and concentration for the high part imply the
low component is o_p of the total Q_J without any independence between
spectral components. A positive spectral sum then gives
  q_(J+1)/q_J >= beta_L-o_p(1).
Taking beta_L arbitrarily near 1/2 and using nondecreasing successive
ratios bounded above by 1/2 proves the claimed uniform later-ratio limit.
The already established threshold-relative norm approximation transfers
this to the attained residual lower bound for every fixed c<1/2.

3. BINDING AND LIMITS OF THIS ADDENDUM
The source comparison and the foregoing independent calculation establish
that this edit repairs the identified Section 7 issue without changing
the result. The code/provenance and analytic endpoint-constant conclusions
in independent_code_and_final_binding.txt remain applicable: none of those
files, definitions or constants is affected by the repair. The literature
comparison, specialist-value assessment and stated application limitations
also retain their mathematical target; I have not repeated external
literature research or every estimate of the full proof in this addendum.

My earlier report establishes the notation-only equivalence between the
22d426... reviewed source and f5406c..., not a new fresh full mathematical
review of the present 7d2ed0... source. Current overall proof status should
combine the preserved initial review with fresh final-source cross-review
and this explicit independently verified repair, rather than substituting
a new hash into an old review and claiming nothing substantive changed.

No existing source, code, proof, evidence, report or prior edition was
edited. Only this new addendum was written. This is internal AI review,
not external human peer review or publication authorization.
