Arbitrary-period balance-root consistency and Jordan visibility (do not merge)

[LightChainr]

What this adds

Additive and standalone. No dependency on unmerged #708/#710/#716/#718 scripts
or JSONs. Python 3.10+ with SymPy; no compiler, no GPU, no new Monte Carlo.
New files only: 6 notes, 4 result JSONs, 4 scripts, 3 tests.

1. Balance roots no longer require axial rectangles. For any full-rank
integer period lattice Lambda <= Z^2 with index N and shortest period
ell(Lambda), the square-site balance root p_Lambda (the zero of
M = P_2 - P_0) satisfies

lim_{L->inf} sup_{Lambda : ell >= L} |p_Lambda - p_c^site(Z^2)| = 0

with no aspect-ratio bound, no area-versus-systole growth condition, and no
primitivity requirement on a Gaussian representative. The condition is on the
genuine shortest period, not on whichever basis vector happens to look long in
an input file. The construction takes the shortest period u, completes a
reduced basis, uses q(x) = det(u,x) mod N as a transverse coordinate (a choice
of period coordinates — it does not rotate the physical NN interaction onto
another lattice), and builds k = floor(8N/ell^2) vertex-disjoint oblique
bands. Positive association is used for the overlapping local arm events;
independence is used only after the band vertices are disjoint. The fixed-p
subcritical bound is P_2/P_0 <= exp[-kappa(p) N/ell], with the reverse bound
above p_c on the matching complement.

Site sharpness adaptation is kept explicit (Duminil-Copin–Tassion state the
main theorem in bond language; the site adaptation is §1.2 and is not skipped),
and the matching side follows Grimmett–Li.

2. Which sources move the finite root without moving the limit. For a
positive p-independent weight W with log oscillation Omega = osc(log W),
the two-sector odds ratio is controlled to exp(±Omega), so Omega = o(N/ell)
implies all balance zeros tend to p_c. Product log-odds fields
(Omega = sum_v |eta_v|) satisfy this, keep an independent site measure after
normalisation, and keep a unique root. Arbitrary positive marks do not keep
finite uniqueness: a completed 4x4 example (7-point rank-2 cross and 9-point
rank-0 block, both reweighted by 1e12) has at least three distinct finite
balance roots, verified against all 65,536 configurations. The theorem must
therefore say "all zeros tend to p_c", not "the zero".

3. Two algebraic corrections, now with exact counterexamples. The useful
part of #724 survives: an m lambda^m factor in a thermal derivative does not
by itself identify a physical Jordan block. Two over-strong inferences are
corrected by explicit matrices:

• A(t) = [[1+t, 1], [0, 1-t]] has eigenvalues 1±t but A(0) is a genuine
  order-2 Jordan block — linear splitting does not make the crossing semisimple.
• C(t) = [[0,t,0],[0,0,t],[t,0,1]] with det(xI - C(t)) = x^2(x-1) - t^3 is
  diagonalisable at t=0 and has distinct eigenvalues for small nonzero t,
  yet the near-zero eigenvalues cannot be analytic: any integer vanishing order
  k would force 2k = 3.

Bamieh §2 takes analytic eigenvectors and eigenvalues as hypotheses; the
correction is to how the retrieval used those hypotheses, not an accusation that
the source proved a false theorem.

4. A stronger positive control for Jordan invisibility. Two families of
3-state irreducible Markov generators, pushed through T = I + G/2, give
stochastic matrices with strictly positive entries on |t| <= 1/4, a common
uniform stationary distribution, and

tr T_D(t)^m = tr T_J(t)^m = 1 + ((1+t)/2)^m + ((1-t)/2)^m   for all m, t

so all ordinary traces and all parameter derivatives of them coincide — yet at
t = 0 one family is semisimple and the other has a nontrivial Jordan block.
Positivity, irreducibility and a common stationary normalisation are therefore
not enough. With a specified source and readout (start in state 1, read whether in state 3) they separate: minimal constant-coefficient recurrence order 2
versus 3, with exact Hankel determinants -1/36 and -1/6912. The question is
not whether Jordan structure is ever visible, but whether it enters the
specified source/readout response: the coefficients that matter are
C N_lambda^k P_lambda B, k >= 1, which ordinary traces cancel.

Also flagged: the thermal jet lift J_1 = [[A, A'], [0, A]] carries its own
nilpotent part even when the physical A is semisimple, so reconstructing a
repeated pole in a derivative response is not automatically reconstructing a
Jordan block of the physical operator.

Verification boundary

• 17 local tests pass on this tree (8 + 6 + 3); all 17 files match the delivery
  sha256 manifest; the patch applied byte-for-byte after git apply --check.
• Executed controls: 2,080 HNF period bases reduced and checked for shortest-
  ness; 215,040 small oblique-torus graph/configuration checks with 107,520
  complement pairs; 80 larger fixed-structure controls including genuinely
  sheared Gaussian ideals; 65,536 configurations for the multi-root example.
• Full repository CI was not executed for these files.
• Four review comments were already posted by this delivery to [Grok retrieval] Thermal m λ^m vs Jordan/LCFT diagnostics — literature note (#714) #724, [Round B peer check] Cross-check of #724: thermal m λ^m vs Jordan/LCFT diagnostics (#727) #731, #720 Round B peer check of #717: Jacobsen 2015 / Mertens–Ziff 2016 cylinder claims #728
  and Aspect-uniform matching roots; P398 is periodic IC TL (do not merge) #718 (IDs in REMOTE_ACTIONS.json of the source archive). No branch was
  pushed and no PR or issue was merged or closed by that step.

Non-claims

Not a full threshold-distribution concentration theorem: on extremely long thin
sequences the birth-time mixing law can keep splitting toward 0 and 1 while the
balance root still tends to p_c. No quantitative near-critical rate, no
L^{-4} law, no new numerical p_c, no finite-size exponent. The arm constants
needed for a numerical rate are not supplied. No solution to #275, and no
authorisation to change the existing source, normaliser or moving-root contract.
No novelty claim is made — see the separate literature sweep.

[LightChainr]

Focus reset under the owner's latest request: #650 now selects the PROBABILITY part of this PR as the single active consolidation/audit deliverable. This is not a merge request or a claim promotion. The original #276 task remains completed; do not reissue it. Jordan counterexamples and marked-source extensions remain retained assets, but should not be dependencies or sections of the core matching-root manuscript.

I reread the complete arbitrary-period-balance-consistency-20260912.md at 9d29d01 and its main-branch digital-Alexander input. The next independent check is the oblique-band lemma, not another finite-width census: first-exit support, disjoint slab site sets, the last-entry/first-exit extraction for backtracking cycles, arbitrary-tilt entry counting (including u nonprimitive in ambient Z^2), matching diagonal jumps, and constants uniform in area/shear. The existing #732 check covers the rectangular predecessor, not this full claim.

One algebraic simplification of the SAME proof is now complete. Set S=ell^2 (an integer), B=4 ceil(sqrt(S)), ell>=64, and let abar<1 be a rigorously justified upper bound on the local arm probability at radius ell/64. If

(B*abar)^4 < (1-abar)^S,

then B*abar<1 and (A)–(C) of the proof give

P2/P0 <= [ (B*abar)^4 / (1-abar)^S ]^(N/S) < 1.

For rational abar this is an exact rational finite-root sign check with no tiny-probability division. A matching-side check at 1-p gives the opposite sign for the primal. This is NOT a numerical enclosure of the infinite pc, nor proof that an arbitrary finite root bounds pc; a quantitative convergence modulus or a separate infinite-volume finite-size criterion would be needed. No near-critical arm calculation or new compute is authorized by this restatement. I checked the arithmetic using the exact radius-one arm a=p[1-(1-p)^d] at S=4096: p=1/100 passes for d=4 and 8, while d=4,p=1/2 fails. These are elementary controls, not competitive threshold estimates.

Acceptance target: one self-contained probability manuscript, a lemma-indexed independent proof audit, and a closest-prior-theorem hypothesis/conclusion comparison. Do not substitute repeated script execution, source quotation checks, or 'no search hit' for proof validity or novelty. If the all-period lemma fails, identify the exact implication and retain the narrower valid scope. No L^-4, original-U/Jordan identification, new pc decimal, all-width spectral-weight theorem or full birth-law concentration is added to this deliverable.

[LightChainr]

The single probability manuscript is now submitted as draft #739, head 758800f92fd84ee036e9b10e91a7facd54d3712c, on current main d31fa5fe. It consolidates the probability part of this PR and #736, and proves the previously missing arbitrary-direction full-law necessity, rather than making that lemma another task.

For every sequence of honest integer-period square-site tori with N→∞, the whole birth law concentrates at pc (equivalently all fixed interior quantiles converge) iff log N / ell→0. This is now for all integer period shapes, not only axial rectangles. Your existing root theorem needs only ell→∞ and is reproduced in the same text; its systole condition is not claimed necessary for the median.

New construction: integer staircase from 0 to the actual shortest u; axis-aligned hub/connector crossing rectangles of side 2s--3s; the final hub is exactly Q0+u, so the connected lift closes from z to z+u. Critical square-site RSW and finite-product continuity give a fixed p_eta<pc with event probability ≥exp(-eta ell). The support lies in a 4s tube, so |S−S|≤64s ell. The elementary maximal-translate packing lemma gives at least N/(64s ell) disjoint supports in Z²/Lambda, hence

P0(p_eta) ≤ exp[-N/(64s ell) exp(-eta ell)].

This handles orientation, shear and ambient nonprimitivity without rotating the physical interaction or treating overlapping rectangles as independent. A simpler forced-path version also gives endpoint splitting for log N/ell→∞ on arbitrary period shapes.

One new manuscript directory, one standalone script, one test file and one result JSON; no source/Jordan/width-four-stack dependency. Five local tests passed; 135,168 tiny configurations check the deterministic implication; six larger reduced oblique bases and exact packing controls pass. Full repository CI was not run. This is the author's proof and bounded primary-source comparison, not an independent acceptance or originality certificate. No other PR merge, new issue, production request, or automatic parallel review round is requested. Existing evidence and the #738 navigation reset are unchanged.