Independent comparison

---

The isolated input was treated as unverified, then compared with all topical notes linked by the research index. Its assertions of prior checking were not accepted as evidence. The checks below used the displayed arguments, independent reconstructions, and the relevant primary-source passages. The original input is retained as an archive, not as an alternative entry point into current progress.

The full problem remains unresolved. None of the verified additions excludes the surviving least-budget minimal-ratio core or the all-deficient two-supported core.

The later quadratic completion improves both the span and equality thresholds to $r\ge64a^2$. The fourth-power comparisons below describe the earlier audited state.

Verified additions

The independent maximum-degree induction proves exactly the finite partition theorem used in Hereditary span. It is written there, making that part of the existing proof self-contained without changing its $64a^4$ threshold.

The nonpartite Turán gap in equation (6) is true, but its claimed linked proof is absent from the input bundle. A complete single-vertex-cloning proof was reconstructed and is in Packing tools. It agrees with the applicable branch of Kang--Pikhurko Theorem 1, checked in the supplied full source after reading its digest. The same note includes the stronger matching-transversal classification: the failed system on $k$ parts of size $k-1$ is $k-1$ disjoint transversal $K_k$'s, not just a collection of pairwise perfect matchings.

The exact equality classification in Appendix E is valid for unequal nonempty parts as well as equal ones. It is preserved with a full proof and conditional clique-merging consequence in Transversal equality. Neither this result nor the present global stability bounds prove a near-equality completion theorem. The statement that the relevant weights are “near” one is not a quantified conclusion of those bounds.

Appendix C provides a useful complementary relaxation counterexample: all hereditary caps and $\alpha=\omega=r$ can hold while the least-color budget fails. Its proof was checked and extends from primes to prime powers. The construction and exact excess $r(r-1)(r-2)/2$ are recorded in Local-cap obstructions.

The affine edit-distance bounds in Section 5 also survive checking. Their missing equality classification was reconstructed in full in Affine edit obstruction. An extension would need at least $2(r-1)^2$ old-edge recolorings, or $(r-1)(2r-1)$ for $r\ge5$. These improve the requirement to alter every singleton-direction class. They give no reduction of arbitrary colorings to affine constructions.

Valid arguments already matched or surpassed

Section 1 repeats Reductions. The two small-independence span claims in Section 2 follow from the reconstructed gap, with excess

$$\frac{-a^2+3a+2}{2}r-2a.$$

They give $a=2$ for $r\ge3$ and $a=3$ for $r\ge7$ by an elementary color-coupling argument, recorded in Cross-color cores. They do not improve that note's existing localized-inequality exclusions. Nor do they replace the exact-order structure in the two-row and three-row notes. The gap argument uses clique restrictions in the other colors, not just one-color caps.

The quantitative proofs in Section 3 and Appendix A check, including the extracted clique sizes, exceptional-vertex counts, the dependency degree in the finite probability lemma, and the final constants. Under full hereditary caps, both $r<256a^6$ and $r<4096a^5$ are weaker than the existing $r<64a^4$ condition. The sixth-power theorem nevertheless retains a distinct scope: it uses only

$$|H|=ar+1,\quad \alpha(H)\le a,\quad e(H)\le D_r(ar+1),$$

and the caps on $(r+1)$-sets. Its proof does not need caps on all intermediate subsets. The fifth-power argument uses the two-clique cap to bound their cross edges by $2r$; the present fourth-power proof instead includes the exceptional vertex in the capped set, obtaining the stronger estimate $c_{ij}\le m_i+m_j+d_id_j$. This explains the improved threshold.

Section 4's full-resolution exclusion is already in Packing tools. Appendix B's nonuniform pentagon blow-up supplies the same local-only obstruction as the first family in Local-cap obstructions, at parameters $r=5k$ rather than $r=5k+2$. Its exceptional component violates the full cap by

$$e(H)-D_r(2r+1)=5k^2+k-2>0.$$

Its claimed local minimum $2k^2$ is attained by occupancies $(2k+1,2k,0,0,k)$ in the input's cyclic labeling; the phrase “next small class” should not be read as selecting bag 2 instead of bag 4. No new coloring counterexample results.

Appendix D's private-clique construction, pairwise-disjoint exception classes, full-clique absorption, and support-component independence bounds are correct and already present in Two-supported private cliques. Its concern about iterating after absorption is legitimate, but the established Full-row deletion handles the initial independence drop when at least three full rows occur, and then justifies subsequent deletion and absorption. Its “another color has a full clique” conclusion is weaker than the existing $\Omega(\sqrt r)$-color conclusion. None of these arguments removes the all-deficient remainder.

Disagreements and failed inferences

The averaged independence bound left undecided in Appendix F is already an established input in Packing tools. Rechecking Abiad--Kumar--Pragada, Theorems 2.2 and 3.1 confirms the localized inequality; Jensen gives the averaged version. This uses that theorem, not an unjustified replacement of maximum degree by average degree.

Appendix E's warning about inheriting the original minority budget is correct. The present potential reduction does not inherit an average-degree upper bound of $r$; it proves $2e(H)\le r|H|+r(r-a)$. A concrete nonregular minimal core inside a graph satisfying the global budget, but failing the local cap, is recorded there to isolate the failed inference.

Appendix F's additive exception charge has no injective charging rule: several exceptions can share the same obstruction edges. The limitation is retained in Transversal equality. Optimizing a stability partition by local moves is still only a suggestion; no additional completion result follows from the input.

The support-component deletion in Appendix D preserves the one-color independence bound, but should not be conflated with a simultaneous component decomposition in all colors. More generally, actual color components and edge-disjoint spanning witnesses with leftover edges are different hypotheses.

Claims without enough supplied proof

Section 6 lists further assertions whose linked topical files are not included in the independent input. Its no-packing claim for $r$ spanning witnesses, each made of $r-2$ cliques of order $r$ and a $(2r+1)$-vertex independence-two component, does follow from the checked small-independence exclusion: color each witness with its own color and assign leftover edges arbitrarily. Every color still has independence at most $r$, while the exceptional component still has target independence at most two. The color-coupling span bound gives a contradiction.

The more specific exact-component inequality

$$r^2-(a^2-3a+1)r-4a+2\le0$$

was not reconstructed in the first audit. The second audit below supplies its proof in Cross-color cores. The unspecified longer bounded-template assertions still have no supplied proof and remain unverified.

Second independent input: checked comparison

The second isolated input was read as unverified mathematical input and compared with every topical note linked by the root index. Its named supporting companion files are not retained. The conclusions recorded here come from checking its displayed proofs or reconstructing those missing arguments, not accepting its claims of prior checking. The archived body is retained unchanged.

Additions and stronger deductions

Affine repairs. Section 5's local perturbation proof is valid: for an arbitrary old partition into $r$ sets of order $r$, independence at most $r$ and the local $(r+1)$-set cap force $t\ge r^{4/3}/4-r/2$ crossing edges, with asymptotic constant $4^{-2/3}$. Its pentagon example, local cap, old partition, exact edit counts, and change of distinguished vertex were reconstructed and checked. They are recorded in Local-cap obstructions. The example violates the full hereditary cap, so its claimed sharp exponent applies only to the weaker one-graph hypotheses.

The decisive comparison is to count defective rows, not high cross-degree vertices. There are at most $2rt/(r^2-t)$ defective rows. Matching-transversal rigidity controls the intact rows, and the established quadratic hereditary span theorem excludes the remaining small-independence core. The complete checked proof in Affine edit obstruction gives

$$t>\frac{r^2}{16\sqrt r+1},\qquad M>\frac{(r-1)r^2}{16\sqrt r+1} =\left(\frac1{16}-o(1)\right)r^{5/2}.$$

This improves both the input's $r^{7/3}$ repair bound and the previous quadratic bounds asymptotically, not necessarily at every small parameter. Its separate finite quadratic assertion is already weaker than the earlier checked bound and was not needed.

Incident palettes. Section 6's minimum-degree block accounting and square-root palette estimate check. Applying that accounting with all other colors' independence bounds, rather than only the local cap, gives a stronger result. If $q$ colors are absent at a vertex, then

$$q(5r^2-3r+4-2q)\le r^2(r-1), \qquad q<r/5.$$

Thus every vertex sees more than $4r/5$ colors. The convex block-cost proof, including its harmless $r=3$ rounding exception, is in Palette and cut bounds. That note also proves both supplied cut inequalities and records an additional parity-rounding improvement.

Packing and components. Section 3's transversal classification is already in Packing tools, but its forced independent-set-to-block charge was missing from one existing packing bound. It can be added simultaneously with the residual's Turán edge lower bound. The same note records the resulting formula and the checked exclusion of $r-1$ disjoint full cliques in a minority graph for $r\ge5$.

Section 4's component and leakage identities are correct, including the correction for uncolored pairs. They are in Cross-color cores. Applying the established nonpartite Turán gap to foreign colors on every independence-$s$ component residue gives the stronger necessary inequality

$$r^2-(s^2-3s+1)r-4s+2\le0.$$

This supplies a full proof of the unverified formula from the first independent input. It still requires the stated simultaneous component decompositions; pruning colors independently does not supply them while retaining edge coverage.

Failures of local-only stability. Section 7.1 validly extends the existing equal-bag pentagon family to every $r\ge11$. The more substantial Section 7.2 gives $r-\Theta(\sqrt r)$ full cliques, not $r-O(1)$, even with the minority budget, minimum degree $r-1$, local caps, and independence-criticality. Its full capped-simplex proof is in Local-cap obstructions. Every pentagon component already violates a full hereditary cap, with explicit positive excess. These are rigorous failures of weaker proposed lemmas, not coloring counterexamples.

Superseded claims and precise limits

Section 2's singleton-clique count, elementary high-minimum-degree partiteness lemma, deletion argument, and numerical threshold all check. The random-completion expectation should be described as at most $M/\min_i|N(v)\cap A_i|^2$; this is strictly less than one when $M>0$, while $M=0$ gives expectation zero directly. This is a wording correction, not a substantive gap.

In an actual coloring, that theorem is superseded by Hereditary span: the same $sr+1$ residue is impossible for $r\ge64s^2$, without an external monochromatic $K_r$. Its fixed-$s$ clique-packing consequence can likewise use this stronger threshold.

The independent statement that existence of a suitable minority $K_r$ is unproved describes only its isolated investigation. Packing tools already forces full cliques in a least color using the checked localized independence bound. It does not force a clique-rich near-resolution, which remains the relevant gap.

Section 7.3's critical-edge witness is correct; its disconnected independence-$r$ consequence is made explicit in Minimal-core structure. It does not exclude connected independence-$r$ graphs, and the surviving minimal-ratio core is already $K_r$-free without that argument.

Section 8's color-preserving transitive exclusion is weaker than Symmetry obstructions, which permits the action to permute colors. Its tensor failure and warnings about simultaneous repairs agree with the retained construction notes. The cut statement about balanced sets means a balanced bipartition of the whole vertex set, not any two small equal sets.

The unrestricted problem remains unresolved. No result from this comparison excludes either surviving core in its full parameter range or constructs a counterexample.

Third independent input: checked comparison

The third isolated input was read as unverified and compared with all topical notes linked by the root index. Its supporting notes/ files are absent. Claims of prior auditing were not treated as evidence; the useful proofs were checked from the displayed arguments or reconstructed independently. The archived body is preserved, including its explicit unresolved status.

Additions and stronger deductions

Small-independence graphs. Section 3's two-slot argument works under the one-graph hereditary caps, not only in an actual coloring. It gives capacity $|H|\le2r$ for $r\ge3$, improving the earlier cap-only threshold $r\ge4$. Its shortest-odd-cycle degree argument gives a much shorter proof of the existing two-row equality theorem for $r\ge6$. Both are in Two-row structure.

The independence-three proof also checks and can be shortened further: an overloaded $3r+1$-set cannot contain $K_r$, because deleting it would leave an overloaded independence-two set. Every nonneighborhood then has at most $2r-1$ vertices by two-row equality. The complement-degree upper bound contradicts the inherited lower edge bound by $(r-5)/2$. This gives cap-only span for $r\ge6$, recorded in Three-row structure, improving its previous threshold $12$ for span alone. Equality rigidity at order $3r$ remains proved only for $r\ge12$. These improvements do not establish new cases of the original coloring problem.

Packing in every color. Sections 5.3--5.5 survive checking. In particular the triangle-support lemma is valid: symmetrize vertices outside the triangle support using maximum total degree, and handle an empty first neighbor part separately. The resulting bound is sharp. The matching-transversal attachments, duplicate-color propagation, and both budget polynomials check. The omitted $r=3,4$ cases of $p\le r-2$ were reconstructed, rather than accepted from the unavailable scratch link.

There is a stronger conclusion after comparison. The established private-row excess in Cross-color cores changes the fiber bound to

$$t_i(A)\le(r-1)(p-1-e(r))+1, \qquad e(r)=\left\lceil\frac{\sqrt{8r-7}-5}{2}\right\rceil.$$

If $p=r-a$ and the residue contains an independent $(a+1)$-set, transversal rigidity gives $t_i(A)\ge(p-1)^2+a+1$. The exact difference of these bounds is $a(p-2)-(r-1)e(r)$. Combining this with the checked small-independence and quadratic span theorems yields

$$\boxed{p\le r-\max\{3,\lfloor\sqrt r/8\rfloor\}-1\quad(r\ge8).}$$

The complete elementary input proof and the stronger deduction are in Global clique packing. This is an upper bound on disjoint full cliques in every color, not a bound on their total number and not a lower bound supplying a contradiction. It does not validate the separate unaudited total-clique-count draft in Private-clique approach.

Concentrated affine repairs. Section 6's vertex-cover bound is valid: with $q=\min\{d\ge1:\binom{d+1}{2}\ge r-1\}$, replacing $k$ old points by $k+1$ arbitrary vertices requires

$$k\ge\left\lceil\frac{(r-1)q^2}{r+q-2}\right\rceil =2r-O(\sqrt r).$$

The proof, including empty residual lines, is in Affine edit obstruction. Applied to a vertex cover of the changed-old-edge graph, it bounds edit concentration. The existing order-$r^{5/2}$ edit-count bound alone gives only an order-$\sqrt r$ cover bound, so this addition has distinct content. The weaker assertion $M\ge(r-1)^2+1$ is already superseded even by the earlier quadratic bound $M\ge2(r-1)^2$; its longer equality analysis was not needed or incorporated.

A growing-independence barrier. Section 7.2's cycle blowup checks, including its exact local nonedge minimum, criticality, proper-induced minimality, and surplus formula. It gives more than the input states: heavy-bag anchors furnish two-support with $b<r$ and satisfy the desired light-set bound. Thus the least-core budget, local cap, two-support, bounded exceptions, criticality, and induced minimality can hold simultaneously while the overloaded independence number is unbounded.

The full proof is in Growing-independence barrier. The same note gives a simpler exact rejection than the input's asymptotic warning: a light--heavy--light union has only $t^2$ nonedges, less than the required $(r-1)(t-g)$, for every stated parameter. This is a failure of weaker proposed arguments, not a graph meeting the full core hypotheses and not a compatible family of color graphs.

Superseded results and useful limitations

Sections 1--2 agree with the existing reductions and stability induction. The $r\ge256a^6$ proof checks, including empty cores, full-part random choices, and the final constants. Under all hereditary caps it is weaker than $r\ge64a^2$. As in the first comparison, its narrower assumptions remain distinct: whole-set density and $(r+1)$-set caps suffice for that sixth-power proof. It does not require every intermediate cap.

Section 4 gives a minimal overloaded induced subgraph in every color. The current least-color potential reduction also retains the budget, and its structure gives further restrictions. Those stronger budget conclusions must not be assigned to every color's independently extracted core. Neither version proves that the core lies in the small-independence range.

Section 5.1's lower bound $e(G_i)\ge T+r-1$ is already established by both the nonpartite gap and Palette and cut bounds. Its batchwise local-cap refinement is weaker than the latter note's full-cap charge $(r-1)p_r(b)$. The fiber construction is the existing two-supported private-clique mechanism; it does not close the clique-safe assignment, deletion, or all-deficient normalization gaps. The matching-transversal extremizer also differs from the weight-one extremizer in Transversal equality; it supplies no missing near-equality completion theorem.

Section 7.1's pentagon constructions reiterate the distinction between local and hereditary caps in Local-cap obstructions. The weighted family's displayed counts and local minimum check, but the claimed model-specific sharp lower bound has no supplied proof and is not adopted. Its $r^{4/3}$ scale already occurs in the retained local-only examples. All the cycle blowups are line graphs of multicycles, so the established line-graph span and subdivision-network results reject them under full caps, without any bounded-independence assumption. The final provisional weighted-cycle calculation is not needed, and its unavailable precise inequality remains unverified.

The regular-color observation in Section 7.3 is superseded by Symmetry obstructions, which allow permuted colors and also exclude general one-point transitivity for $r\ge100$. The compatibility warnings remain correct: completing edge-disjoint witnesses preserves independence bounds, but need not preserve their components. Partial-net incidences, proper-subset marginal consistency, or counts of near-full clusters do not force the required full-clique packing. The index's separate budgeted and bounded-exception cores remain separate under their full hypotheses.

Unresolved. No checked claim from this input excludes the surviving cores for all parameters or constructs a counterexample.

Fourth independent input: checked comparison

The fourth isolated input was read as unverified, then compared with the root index and all its linked topical notes. Its supporting notes/ files are unavailable. The checks used its displayed arguments and the established proofs, not its claims of prior auditing. The archived body is preserved unchanged.

A useful boundary example, with stronger verification

Section 7's graph $H=\overline{C_5(r-2,r-2,1,1,1)}$ has the stated order $2r-1$, independence two, clique number $r-1$, and edge count $D_r(2r-1)-1$. It actually satisfies all hereditary caps for every $r\ge3$. Its complement has maximum degree $r-1$ and $(r-1)^2+1$ edges; deleting vertices gives

$$e(\overline H[W])\ge(r-1)(|W|-r)+1 \quad(r\le|W|\le2r-1).$$

The complete proof is in Two-row structure. This makes explicit the sharpness of the $K_r$-free order bound and the failure of a naive boundary induction. It is not a new obstruction to the actual surviving cores: the family is among the existing odd-core structures, and Unit-deficit propagation already has locally colored, cap-saturating examples at this order for $r\ge4$. The separate suppression counterexample in Local-cap obstructions has the same order but different private-row behavior.

Valid results already matched or surpassed

Local independence. Sections 1--3 check, including the exact $2r$ rigidity proof, the degree-$r$ step for $P_3$ when $r\ge6$, and the harmonic greedy clique construction in Theorem 4. In that construction the accumulated complement deficit is at most $2k^2H_k$; the exceptional partition, strict random-transversal bound, and final $3r/8+r/8$ estimate are valid. These give alternative proofs, not better conclusions than the established two-row and three-row results or Hereditary span. The latter replaces

$$r\ge256k^6(1+\log k)^3 \quad\text{by}\quad r\ge64k^2$$

under the hereditary hypotheses available in a coloring, and also proves equality rigidity at order $kr$. Thus the input's statement that no general boundary theorem was proved describes only its isolated investigation. The overloaded-component consequence likewise improves to independence greater than $\sqrt r/8$; this still leaves an unbounded range, not a contradiction.

Full-clique packings. Theorem 5 is valid, including its matching-transversal classification and maximum-degree equality fact. Its direct grid argument is an alternative to the full-resolution obstruction already in Packing tools. Section 7's conditional edge charge and polynomial $D(r,h)$ also check. They omit the additional residual contribution $p_r(hr+1)$ retained in equation (P) of that note. More decisively, Global clique packing already proves

$$p\le r-\max\{3,\lfloor\sqrt r/8\rfloor\}-1\qquad(r\ge8)$$

in every color, stronger than the input's minority-only $p\le r-3$. Neither upper bound forces a large packing to exist.

Double-neighbor exceptions. The private-fiber identities and inequalities (16)--(17) are valid. One omitted case should be explicit: a double cannot have two full supporting fibers. Its $r-1$ nonneighbors in each would have to be completely joined, contradicting the external degree-one bound into a full clique. This also excludes the $d=1$, no-deficient-fiber case before applying the $sr+1$ edge budget. With one full support the double is complete to its deficient support; with two deficient supports the square-root assignment works as stated.

The resulting fourth-root lower bound is weaker than the existing cross-color excess-deficit bound, which applies to every foreign color without assuming $d\le r$. Writing $s$ for the number of deficient fibers and $E=\sum_{t_a>0}(t_a-1)$ gives

$$d=E+s+1\ge \left\lceil\frac{\sqrt{8r-7}-5}{2}\right\rceil+2 \qquad(r\ge7).$$

No upper bound on exceptions, clique-safe completion, or least-color budget follows from the independent lower bound.

Single-color obstructions. The near-equal pentagon construction for every $r\ge11$, its edge budget, and its small-set optimization check. They already appear with additional properties in Local-cap obstructions. The exceptional $2r+1$-vertex core violates the full hereditary cap and is excluded by $P_2$. These examples refute a proof from the minority budget and local caps alone, not one using every inherited cap or the actual coloring. The input's unsupported links to further scratch constructions add no independently checked claim beyond the displayed arguments.

Resolved auxiliary uncertainty and failures that remain useful

Section 7 leaves the averaged inequality

$$\alpha(G)\ge\frac{2|G|}{\overline d(G)+\omega(G)+1}$$

unproved. It is already established in Packing tools: the localized bound of Abiad--Kumar--Pragada, Theorem 3.1, followed by Jensen and $c(v)\le\omega(G)$, gives exactly this inequality. Theorem 2.2's inequality proof and Theorem 3.1 were rechecked in the primary source for this audit. The input's minimum-degree reduction is therefore not a remaining research gap. Its warning against replacing the mean degree on a maximum independent set by the graph's mean is nevertheless correct; the stronger degree-balancing counterexamples show why that substitution cannot justify the theorem. The averaged inequality forces a least-color $K_r$, not a near-complete resolution or a solution of the light-set problem.

The exact-component versus spanning-template warning also remains correct. Adding leftover edges preserves independence upper bounds but may destroy components. In the particular $(r-2)K_r$ plus independence-two-core template, $P_2$ excludes the spanning core itself, so that obstruction survives completion. It does not justify arbitrary component incidence arguments after independently pruning colors.

Research status unchanged. No checked claim combines the separate budgeted and bounded-exception normalizations, excludes the surviving core range, or supplies a counterexample coloring.