Width-four site sources and exact rank conditioning (do not merge)

[LightChainr]

What this adds

Additive on #708's finite automaton. Independent of unmerged #710 and #716, and of the previous root/TL bundle.

• Source-compatible exact rank quotients. All seven D4 types of column-probability groupings are classified exactly on PR708's automaton. Common strong lumpings have 94/303/179/262/509/303/509 states respectively; every one equals the corresponding colour-preserving D4 orbit partition blockwise. Two independently addressed adjacent columns already require all 509 deterministic rank classes within this lumping class. No further width-four state enumeration is needed.
• Physical two-row site source. A genuine square-site two-row dipole source has zero linear response but nonzero mixed response; the 4x4 coefficients are 327/1024 and 633/2048 at p=1/2, and the finite root's mixed shifts carry strict rational sign certificates. Four-sign extraction of M is exact at finite amplitudes, by separate degree<=2 multiaffinity. This does not inherit P398's continuous-time response formula.
• Rank-conditioned exact backward sampler. An exact integer backward sampler conditions on any final rank. The complete nonuniform 4x3 conditional law is verified configurationwise; the 4x128 rare case has exact normalizers and fixed-seed algorithm controls. These rare paths are algorithm controls, not new probability estimates.
• Dictionary correction for [Grok retrieval] Cylinder/TM literature vs width-4 all-p matching identity #711/Lit retrieval: cylinder/TM literature vs width-4 all-p matching identity (#711, do not merge) #717. Withdraws Lit retrieval: cylinder/TM literature vs width-4 all-p matching identity (#711, do not merge) #717 §2.3's categorical distinction between critical-polynomial 2D/0D and the same finite site ensemble's rank2/rank0; see notes/finite-critical-polynomial-dictionary-20260912.md for the exact periodic-lift and site-weight dictionary, including the rank-one spiral trap. The correct boundary is preserved: no all-width B5/B15/B16 matrix intertwiner or closure-weight theorem follows.

Dependency

Requires PR708's results/research-control-20260912/width4-rank-closure-certificate.json (blob 50b7297deefe7c50215aea2ed534ca5810461af3), which exists on this base branch. It is deliberately not duplicated here. No new Monte Carlo, no GPU, no compiler; Python 3.10+ standard library only.

Verification boundary

• 14 local tests pass on this tree (6 + 4 + 4), run against the committed certificate.
• All 17 added files match the SHA-256 manifest recorded in the delivery archive's VALIDATION.json; the patch applied byte-identically.
• The repository's complete CI suite was NOT executed for these new files. Past successful CI on Width-four exact rank closure: 509 futures and 15-mode scalar (do not merge) #708/All-p width-four spectrum and diamond 3L−1 onset (stacked on #708; do not merge) #710 is not a substitute.
• Fixed-width oracle caveat: do not commission billions of direct snapshots merely to see rare sectors that this oracle can condition on; conversely, do not claim the fixed-width oracle scales cheaply to large circumference.

Non-claims

Readouts remain unmarked rank. No generic-q derivative, continuum field, original-U candidate identification, or new production claim is made. All-width structural closure weights and width-uniform spectral estimates remain separate, uncompleted questions. No merge, hardware allocation, or STATUS promotion is requested.

[LightChainr]

Prior-art sweep posted as PR #734 (notes/literature-sweep-torus-rank-observables-20260912.md, base main, do not merge). One finding belongs on this PR.

The observable itself has prior art at the definition level. r = rank im[H_1(K(omega)) -> H_1(T^2)] is the inclusion-induced map studied by Langlands-Pouliot-Saint-Aubin (1994), Pinson (J. Stat. Phys. 75, 1167, 1994), Arguin (arXiv:hep-th/0111193, Q=1 is our case), Morin-Duchesne-Saint-Aubin (arXiv:0812.2925) and Duncan-Kahle-Schweinhart (arXiv:2011.11903 / AIHP 61, 2235, 2025), who call the image elements giant cycles and prove a sharp transition. The implied dictionary:

• P0 <-> pi({0}); P1 <-> sum over coprime (a,b) of pi({a,b}); P2 <-> pi(Z x Z)
• so A_top = P2 - P0 and E_top = P2 + P0 are sums/differences of subgroup probabilities that have closed forms in Jacobi theta functions.

Two consequences for this PR:

1. The rank observable should not be presented as new. The finite-width exact rank decomposition, the source response and the conditioned oracle are the contributions; Pinson/Arguin/DKS are the definition's origin and should be cited up front.
2. This gives a free numerical benchmark: evaluate the Q=1 closed forms at our aspect ratios and report our exact width-4 P0/P1/P2 as the finite-size deviation from them. That is a much stronger sentence than an internal consistency check.

Also recorded in #734: Akhunzhanov-Eserkepov-Tarasevich (arXiv:2204.01517) reach exact L x L torus polynomials to L=12 with direction-based events (competitive frontier, and the direction-vs-rank dictionary is still unmade); Bobrowski-Skraba (PRE 101, 032304) on Euler-characteristic zeros approximating homological thresholds; May-Wierman (2005) and Jacobsen (arXiv:1401.7847) as the nearest relatives of the D4/lumping reduction. Search boundary is stated in the note: no Scholar/MathSciNet citation-graph traversal, several entries known only via secondary quoting.

[LightChainr]

Completed continuation in the current owner handoff, not yet committed to this branch; no new production request.

1. Full 4x4 spatial source Hessian. Translation invariance gives q_i=−1/N and q_ij=−M_ij/M' + M''/(N² M') for the finite root under additive probability fields. All zero-mean fields have zero first-order root response, without needing a reflection-odd hypothesis. The complete 16-mode Hessian has inertia (9 positive, 6 negative, 1 uniform-gauge zero), with rational sign enclosures. Unit-RMS stripes (−1)^x give q''=−1.9320342433611963; checkerboard (−1)^(x+y) gives +2.519895989268161. Logit fields have a different contact term; they are not silently equated with p±epsilon fields.

2. In logit coordinates the zero-mean root response is exactly −[Var(H|r=2)−Var(H|r=0)]/[E(K|r=2)−E(K|r=0)], H=sum h_i n_i. This is a conditional spatial structure-factor difference, not an unmarked K moment.

3. Strong finite nonidentifiability control: strictly positive translation/D4-invariant marks on rank0,K4 square/L/T tetromino orbits preserve the ENTIRE homogeneous-p joint law of (r,K), hence every homogeneous thermal/unmarked-rank-source jet, but change two independent spatial responses. Stripe/checker versus L/T integer response matrix [[−512,−128],[0,−256]] has determinant 131072. These are marked dependent countermodels, NOT claims that the known Bernoulli law is ambiguous or replacements for [P1 theory-blocked flagship] Original U: two same-source forward maps before further acquisition #275's actual candidate columns.

4. Independent mean-zero site-probability disorder leaves the annealed M exactly unchanged by multiaffinity, but averaging finite environment roots gives a second-order bias. The 4x4 all-site independent unit-variance coefficient is M''/(2N M')=0.020558979708451625. Four exact two-site environments verify the distinction; no disorder Monte Carlo was run.

All 65,536 physical ranks were checked against the unchanged PR708 automaton; 45 exact likelihood-score Hessian checks, finite-amplitude bivariate controls, and 27 targeted tests pass. Full repository CI has not been run. Separate notes also finish a bounded-source zero-free strip and geometrically anchored conditional-odds integration.

For the preceding literature comment: keep the prior-art attribution, but #734 section 1a needs the continuum-mesh versus fixed-lattice-width correction now posted there. A continuum value can be a properly labelled benchmark at the same physical critical point; it is not an exact finite-width oracle, and q4 is not the infinite square-site pc. No merge/STATUS change is requested.

[LightChainr]

Completed further analysis in the current owner handoff (new additive patch, not committed to this branch). No new production or duplicate issue is requested.

1. A strictly stronger local-information countercontrol than the previous uniform-jet example. On the 4x4 rank0,K4 cell there are 1812 configurations / 32 translation-D4 orbits. The exact incidence ranks through local orders d=0..4 are 1,1,5,10,32 (blind dimensions 31,31,27,22,0). With O(m) the full translation/D4 orbit of bitmask m, f_A=-21_O(30)+1_O(53) is invisible to EVERY finite probability change on at most two addressed sites. f_B=1_O(27)-1_O(30)-21_O(51)+1_O(60) is invisible on at most three. W_±=1±f/4 are strictly positive in [1/2,3/2], and preserve the normalized entire (r,K) law in those regimes. Their first omitted inclusion contrasts are -4 on sites {0,1,2} and +1 on {0,1,2,5}. These are dependent marked countermodels, not ambiguity in the known Bernoulli definition or replacements for [P1 theory-blocked flagship] Original U: two same-source forward maps before further acquisition #275 candidates.

2. General finite-order existence proof, no larger census: for any fixed d, take K=d+1<L. Every such configuration is rank0. There are at least ceil[binom(N,d+1)/(8N)] orbit variables and at most 1+sum_{j=1}^d binom(N-1,j-1) inclusion constraints. For large N the nullspace is nonzero; the first detectable order is exactly d+1. Important limit: fixed-K mass is exponentially small at interior p. This does NOT establish a leading critical-scaling ambiguity or justify an unlimited higher-order source campaign.

3. Actual unmodified site model, not a marked countermodel: h_±(x,y)=cos(pix/2)+cos(piy/2)±cos(pi*(x+y)/2) have equal power spectra / quadratic response, mean zero, and mean-square strength 3/2. h_-=-h_+(x+2,y+2), hence q_-(e)=q_+(-e). The cubic response is nonzero: q''=-0.67360682252439048..., q_+'''=13.657437539796486..., q_-'''=-q_+'''. Full all-p jets through order four retain the moving-root contact terms. Independent Walsh-Hadamard arithmetic gives M_hhh(1/2)=-14997/256. Root values were rechecked at 100-digit precision.

4. Approximate rank bridges now have a nonasymptotic variance certificate, not an unsupported 'exact conditional sample' label. With exact feasible support and alpha_j<=g_j/h_j<=beta_j, weighted independent paths obey CV^2<=product_j[(alpha_j+beta_j)^2/(4 alpha_j beta_j)]-1. Keeping b significant binary bits gives [1+4^(-b)/(1-2^(1-b))]^m-1. On p=3/5,4x128,rank2 (probability 1.62924e-9), the 8-bit proposal CV^2 is enclosed near 0.000270681251327 by an outward-rounded rational DP interval of width 3261/2^80; the universal bound is 0.00197042755. Full 4x3 laws independently verify proposal normalization, exact unbiasedness and variance. The guide was obtained by truncating an already exact oracle: no cheap large-width guide or speed advantage is claimed. Approximate paths MUST be reweighted.

20 local tests passed; patch applied in a minimal Git tree with the unchanged PR708 certificate; all 15 new files byte-identical; all three result files regenerated identically except runtime. Full Matching-One CI was not run. General homometry/k-deck and approximate-Doob importance-sampling methods have prior art (Grimm-Baake 0808.0094; Jaming-Kolountzakis math/0207051; Hartmann-Richter 2102.09606; Zhang-Sahai-Marzouk 2101.07330, primary HTML read). No novelty, merge, STATUS or acquisition request.