From 80a73c346c436a43715d95cc30b5eed5890566f0 Mon Sep 17 00:00:00 2001 From: Samuil Petkov <57594550+SamPetkov@users.noreply.github.com> Date: Sun, 26 Jul 2026 21:54:03 +0300 Subject: [PATCH 1/2] =?UTF-8?q?Add=20reader-first=20Erd=C5=91s=20625=20man?= =?UTF-8?q?uscript=20revision?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- ...ANONICAL_V2_READER_FIRST_PROSE_REVISION.md | 387 ++++++++++++++++++ 1 file changed, 387 insertions(+) create mode 100644 625/proofs/CANONICAL_V2_READER_FIRST_PROSE_REVISION.md diff --git a/625/proofs/CANONICAL_V2_READER_FIRST_PROSE_REVISION.md b/625/proofs/CANONICAL_V2_READER_FIRST_PROSE_REVISION.md new file mode 100644 index 00000000..78f80141 --- /dev/null +++ b/625/proofs/CANONICAL_V2_READER_FIRST_PROSE_REVISION.md @@ -0,0 +1,387 @@ +# Erdős 625: canonical Version 2 reader-first prose revision + +## Status + +This is a manuscript-level exposition pass stacked on the corrected Version 2 +roadmap. It does not modify the frozen canonical TeX and does not convert the +remaining Section VIII closure theorem into a proved result. The proposed text +is copy-ready only after the attained-demand/all-deficit physical-fibre theorem +and its aggregate weight identity have been checked. + +The current proof frontier must remain explicit: + +```text +attained canonical high skeleton + <-> block support + admissible deficits + local partial physical matchings +``` + +with exact preservation of the aggregate weight. Until this theorem is closed, +the normalized second moment, the final main theorem, and the stronger Version +2 constant remain conditional. + +## Principal editorial diagnosis + +The candidate manuscript contains the right global ideas but asks the reader to +learn the proof in the wrong order. The first-moment root comparison is +conceptual. The second-moment proof is a finite disintegration followed by two +small aggregate estimates. The current exposition instead introduces much of +the technical vocabulary before the reader knows which quantity each section +is bounding. + +The revision therefore imposes four rules. + +1. **Three layers, stated immediately:** location, existence, amplification. +2. **One object dictionary before Section VIII:** profile blocks, block support, + full multiplicities, deficits, physical matching fibres, endpoint tables, + and residual attachments are never conflated. +3. **One aggregate formula per technical section:** Section VIII proves the + bare-skeleton estimate; Section IX proves the conditional attachment estimate. +4. **No obsolete machinery in the main narrative:** the near/middle split, + residual walk kernel, simple-cycle decomposition, mixed matching-cycle + encoding, and polymer terminology are removed from Version 2 unless retained + in a historical appendix. + +## The proof in one paragraph + +For \(G_n\sim G(n,1/2)\), an ordinary colouring requires every class to be +independent. A cocolouring may declare each class either independent or +complete. At density \(1/2\), the two declarations have the same probability +cost, so a partition into \(k\) classes gains an exact sign multiplicity +\(2^k\). Restricting class sizes to four consecutive phase values creates a +signed first-moment root below the ordinary colouring root by order +\(n/(\log n)^3\). An exact signed-overlap identity reduces the second moment to +partial diagonals, a matching of high cells, and a residual even-subgraph +attachment. Summing all physical high-cell fibres through one global +falling-factorial denominator gives the bare-skeleton estimate; injective +restriction outside the exposed matching gives the residual estimate. This +produces a positive-probability signed seed, and a leftover-colouring plus +bounded-differences argument amplifies it to high probability. Intersecting +that event with the ordinary chromatic lower bound yields the gap. + +That paragraph should end the first page of the paper. + +## Recommended title and abstract + +The title + +> A Polynomial-Scale Gap Between the Chromatic and Cochromatic Numbers of a +> Random Graph + +is accurate and should be retained. + +The abstract should not lead with the original `/32` constant while the +corrected theorem architecture supports a different conditional Version 2 +constant. Two copy-ready variants are supplied in +`625/arxiv/READER_FIRST_FRONT_MATTER_V2.tex`: + +- an audit-safe research-status abstract for the present branch; +- a post-closure publication abstract using the phase-resolved coefficient. + +Only the second should be used in a submission, and only after the global +Section VIII theorem is complete. + +## Recommended section order + +```text +1. Introduction and theorem +2. Phase notation and one-class asymptotics +3. Ordinary-colouring location +4. Signed four-size location +5. Exact signed-overlap identity +6. Partial diagonals and canonical high support +7. Endpoint reference transport +8. Physical high-cell fibres and all deficits +9. Residual attachment by matching restriction +10. Positive-probability seed and amplification +11. Final event intersection +Appendix A. Exact phase and entropy certificates +Appendix B. Finite fibre identities and formalisation map +Appendix C. Superseded Section VIII--IX machinery +``` + +The current Sections 2--5 may remain largely intact, but their introductions +should say which root is being located and whether a statement moves the root or +only certifies a finite approximation. Sections VIII and IX should be replaced +by the corrected aggregate route. + +## Reader contract at the end of the introduction + +The introduction should state the proof obligations in the following form. + +### Layer I: location + +- locate the ordinary-colouring root \(r_+(n)\); +- locate the signed four-size root \(r_4^{\mathrm{co}}(n)\); +- prove the phase-uniform separation + + \[ + r_+(n)-r_4^{\mathrm{co}}(n) + = + \left[ + \frac{(\log2)^2}{4}A_4(\delta_n)+o(1) + \right]\frac{n}{(\log n)^3}; + \] +- place the signed seed at the midpoint, retaining one half of this separation. + +### Layer II: existence + +For the signed-profile count \(Z_n\), prove + +\[ + \frac{\mathbb E Z_n^2}{(\mathbb E Z_n)^2} + \le + \exp\!\left\{o\!\left(\frac{n}{(\log n)^4}\right)\right\}. +\] + +Every object in Sections 6--9 exists only to prove this displayed inequality. +Those sections do not alter the first-moment roots. + +### Layer III: amplification + +Use Paley--Zygmund to obtain a possibly rare seed, then use a one-Lipschitz +cocolourable-capacity variable and leftover colouring to obtain a +high-probability upper bound for \(\zeta(G_n)\). Combine it with the independent +high-probability lower bound for \(\chi(G_n)\) by a union bound. + +## Object dictionary + +This dictionary should appear before the high-skeleton section. + +| Object | Meaning | Do not confuse with | +|---|---|---| +| profile block | one actual colour or clique class | an abstract type slot | +| block atom | type/slot label used to organise blocks | a physical vertex set | +| high-demand table | actual high-cell multiplicities \(j_{ab}\) | the endpoint table | +| block support \(P\) | matching of block pairs with positive high demand | a matching of physical vertices | +| full multiplicity \(m_e\) | \(\min\{s_e,t_e\}\) for one support edge | actual multiplicity \(j_e\) | +| deficit \(h_e\) | \(m_e-j_e\) | residual attachment size | +| partial physical matching | literal size-\(j_e\) matching of stubs | a chosen full completion | +| endpoint table \(L\) | counts of support edges by endpoint types | the full high-demand table | +| bare skeleton | exposed high-cell data before residual attachment | the complete overlap table | +| residual attachment | conditional contribution of non-high cells | the high-cell deficit fibre | + +A partial physical matching can have many full completions. The proof compares +whole finite fibres; it never chooses a canonical completion objectwise. + +## Section-by-section prose revision + +### Introduction + +Keep the historical background concise. The introduction needs only: + +1. the definition of the two parameters; +2. the Erdős--Gimbel question; +3. the theorem or current audited target; +4. the one-paragraph mechanism; +5. the three-layer reader contract; +6. a compact relation-to-prior-work paragraph. + +The detailed chronology of phase-dependent partial results should be shortened +or moved to a background subsection. It should not separate the theorem from +its proof mechanism by several pages. + +Replace ornate phrases such as `exquisitely phase-sensitive` by direct +mathematical language such as `phase-sensitive but uniform over the full +threshold window`. + +### Sections 2--4: ordinary location + +Open the block with: + +> These sections establish a lower location for the chromatic number without +> assuming that an optimal colouring has a prescribed profile. + +At the end of each section, state explicitly what has been proved for the root +and what remains to convert it into a high-probability chromatic lower bound. + +### Section 5: signed four-size root + +Begin with the extra entropy in one sentence: + +> At \(p=1/2\), declaring a class independent or complete has the same edge +> cost; summing the two choices contributes an exact factor \(2\) per class. + +Then explain why four sizes are used: they cover the entire phase interval with +a uniform entropy advantage while keeping the transportation problem finite. +Do not introduce the full overlap machinery until the root displacement has +been stated and interpreted. + +### Section 6: exact overlap identity + +State the exact identity first and explain its terms afterward: + +\[ + \operatorname{SignedOverlapWeight}(r) + = + \left(\prod_{a,b}g(r_{ab})\right)2^{\beta(H_r)}. +\] + +Then say: + +> The local product records cell multiplicities; the cycle-space factor records +> the remaining sign compatibility. This is an exact finite identity, not an +> approximation. + +### Section 7: partial diagonals + +The section title and opening should say that common whole classes are being +removed. The central-rate estimate is an input to the normalized second moment, +not a new random-graph location theorem. + +### Section 8: physical high-cell fibres + +Replace the old near/middle narrative by the following spine. + +1. canonical high support is a block matching; +2. for fixed support \(P\) and multiplicities \(j_e\), the literal physical + fibre has aggregate weight + + \[ + w(P,j) + = + \frac{\prod_{e\in P}(s_e)_{j_e}(t_e)_{j_e}} + {(n)_J\prod_{e\in P}j_e!} + \prod_{e\in P}g(j_e); + \] +3. compare \(j_e=m_e-h_e\) with the full endpoint reference; +4. use the single global denominator ratio + + \[ + \frac{(n)_{J+H}}{(n)_J}=(n-J)_H\le n^H; + \] +5. sum every deficit through one geometric product; +6. transport the full endpoint reference by square-free AM--GM. + +The result of the section is one displayed statement: + +\[ + \operatorname{BareSkeletonSum}_n + \le + \exp\!\left\{o\!\left(\frac{n}{(\log n)^4}\right)\right\}. +\] + +Every lemma should be introduced by saying which factor in this bound it +controls. + +### Section 9: residual attachment + +Begin with the injection, not with cycle terminology: + +> Deleting the exposed matching is injective on even residual edge sets. + +Then derive + +\[ + \sum_{F\ \mathrm{even}} + \prod_{e\in F\setminus M}q_e + \le + \prod_{e\notin M}(1+q_e). +\] + +Charge the local increment and even-subgraph products to the same total-\(q\) +bound. The section ends with + +\[ + \operatorname{AttachmentSum}_n + \le + \operatorname{BareSkeletonSum}_n + \exp\!\left\{ + \eta_n\frac{n}{(\log n)^4} + \right\}, + \qquad \eta_n\to0. +\] + +No residual walk kernel, simple-cycle decomposition, traversal parameter, or +polymer surrogate should appear in the Version 2 main text. + +### Sections 10--11: amplification and intersection + +Make the logic explicit: + +- the second moment gives positive probability, not high probability; +- the amplification variable is one-Lipschitz under vertex exposure; +- leftover colouring absorbs uncovered vertices at lower order; +- the chromatic and cochromatic events need not be independent; +- a union bound is sufficient. + +## Standard transition paragraphs + +Before ordinary location: + +> We first locate the ordinary colouring threshold. This part of the argument +> contains no signed structure and will later supply the lower bound for +> \(\chi(G_n)\). + +Before the signed profile: + +> We now repeat the location calculation with signed classes. The sole new +> first-moment resource is the exact choice between independent and complete +> classes. + +Before the overlap identity: + +> The first moment identifies a lower signed root. To show that this root is +> attained, we must control the overlap of two signed partitions. + +Before Section VIII: + +> After common whole classes are removed, every canonical high cell belongs to +> a matching. We sum these high cells by their literal physical matching fibres +> and their deficits from full containment. + +Before Section IX: + +> The exposed high matching fixes the only large overlap cells. What remains is +> an even residual edge set, and restriction outside the matching is injective. + +Before amplification: + +> The normalized second moment produces a seed with positive probability. The +> final step converts that seed into a high-probability cocolouring without +> changing the leading root separation. + +## Claim-status discipline + +The manuscript should use exactly three status labels. + +- **proved in the manuscript:** ordinary mathematical argument complete; +- **exactly certified:** finite identity or sign reconstructed by a deterministic + verifier; +- **formally checked:** specific Lean theorem with stated dependency and axiom + scope. + +Do not use `verified` without specifying which of these meanings is intended. + +Until closure, the front matter must say `candidate proof` or `conditional +Version 2 theorem`. After closure, remove the status language from the theorem +statement itself but retain a concise reproducibility paragraph. + +## Material to move out of the main narrative + +- the superseded near/middle split; +- the old `E_mid`, `Xi_4`, and residual-mass dichotomy; +- simple-cycle and mixed-cycle encodings used only by the old Section IX route; +- long Lean theorem-name inventories; +- CI implementation details; +- alternative support scans and near-root research programmes; +- full endpoint table and LDL-style finite ledgers. + +These remain useful audit artifacts and should be indexed in an appendix or +repository map. + +## Editorial acceptance test + +After reading the abstract and introduction, a probabilistic combinatorialist +should be able to answer: + +1. why cocolourings gain a factor \(2^k\); +2. where the \(n/(\log n)^3\) scale comes from; +3. why the proof needs a second moment; +4. what the exact signed-overlap identity separates; +5. why high support is a matching; +6. why there is one global \((n)_J\) denominator; +7. why deleting the exposed matching controls the residual even family; +8. why amplification is needed after Paley--Zygmund. + +If the reader must first parse the old cycle-walk machinery or the full Lean DAG, +the main exposition is still not reader-first. From e012545b1b453e6bf52cfc3211c0df6134bc1d2f Mon Sep 17 00:00:00 2001 From: Samuil Petkov <57594550+SamPetkov@users.noreply.github.com> Date: Sun, 26 Jul 2026 21:54:51 +0300 Subject: [PATCH 2/2] =?UTF-8?q?Add=20copy-ready=20Erd=C5=91s=20625=20front?= =?UTF-8?q?=20matter?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- 625/arxiv/READER_FIRST_FRONT_MATTER_V2.tex | 289 +++++++++++++++++++++ 1 file changed, 289 insertions(+) create mode 100644 625/arxiv/READER_FIRST_FRONT_MATTER_V2.tex diff --git a/625/arxiv/READER_FIRST_FRONT_MATTER_V2.tex b/625/arxiv/READER_FIRST_FRONT_MATTER_V2.tex new file mode 100644 index 00000000..aeefad3c --- /dev/null +++ b/625/arxiv/READER_FIRST_FRONT_MATTER_V2.tex @@ -0,0 +1,289 @@ +% Reader-first front matter for the corrected Version 2 manuscript. +% This file is not included automatically. +% +% IMPORTANT STATUS BOUNDARY +% ------------------------- +% The post-closure abstract and theorem below must not be inserted into main.tex +% until the global Section VIII attained-demand/all-deficit physical-fibre +% theorem and exact aggregate weight identity have been checked. + +% --------------------------------------------------------------------------- +% A. AUDIT-SAFE ABSTRACT FOR THE CURRENT RESEARCH BRANCH +% --------------------------------------------------------------------------- + +\begin{abstract} +For a graph $G$, let $\chi(G)$ be its chromatic number and let $\zeta(G)$ be the +least number of parts in a vertex partition in which every part induces either +an empty graph or a complete graph. We develop a full-sequence candidate proof +that, for $G_n\sim G(n,1/2)$, the difference $\chi(G_n)-\zeta(G_n)$ is bounded +below with high probability by a positive constant times +$n/(\log n)^3$. The argument has three layers. An ordinary first-moment +calculation locates the chromatic threshold; a signed four-size first moment +places the cocolouring threshold lower by order $n/(\log n)^3$; and an exact +signed-overlap identity reduces the required second moment to partial +diagonals, a matching of high cells, and a residual even-subgraph attachment. +The corrected Section VIII route sums high cells through literal physical +matching fibres, one global falling-factorial denominator, and a geometric +all-deficit estimate. The residual attachment is then controlled by injective +restriction outside the exposed matching. A rare signed seed is amplified to +high probability by leftover colouring and bounded differences. The remaining +submission-blocking step is the global reindexing of every attained canonical +high skeleton by block support, admissible deficits, and local partial physical +matchings, with exact aggregate weight preservation. Until that finite theorem +is closed, the normalized second moment and the final random-graph theorem +remain conditional. +\end{abstract} + +% --------------------------------------------------------------------------- +% B. POST-CLOSURE PUBLICATION ABSTRACT +% Delete the audit-safe abstract above and use this version only after closure. +% --------------------------------------------------------------------------- + +\iffalse +\begin{abstract} +For a graph $G$, let $\chi(G)$ be its chromatic number and let $\zeta(G)$ be the +least number of parts in a vertex partition in which every part induces either +an empty graph or a complete graph. For $G_n\sim G(n,1/2)$ we prove, uniformly +over the full independence-threshold phase, that +\[ + \chi(G_n)-\zeta(G_n) + \ge + \left[ + \frac{(\log2)^2}{8} + \bigl(\log2-D_4(\delta_n)\bigr)-o(1) + \right] + \frac{n}{(\log n)^3} +\] +with high probability. In particular, +\[ + \chi(G_n)-\zeta(G_n) + \ge + \left[ + \frac{(\log2)^2}{8} + \log\!\left(\frac{1000}{639}\right)-o(1) + \right] + \frac{n}{(\log n)^3} +\] +with high probability. The gain comes from signed colour classes: at density +$1/2$, declaring a class independent or complete has the same edge cost and +contributes an exact factor $2$ per class. An exact overlap identity separates +local cell weights from a binary cycle-space factor. We sum the canonical high +cells through literal physical matching fibres and a single global +falling-factorial denominator, and control the residual even-subgraph family by +injective restriction outside the exposed matching. The resulting second +moment gives a positive-probability signed seed, which is amplified to high +probability by leftover colouring and bounded differences. +\end{abstract} +\fi + +% --------------------------------------------------------------------------- +% C. READER-FIRST INTRODUCTION +% The theorem box is written in post-closure form; retain the explicit status +% paragraph while this remains a research draft. +% --------------------------------------------------------------------------- + +\section*{Introduction}\label{introduction-reader-first-v2} + +Let $G_n\sim G(n,1/2)$ be the random graph on $[n]$. The chromatic number +$\chi(G)$ is the least number of independent sets in a partition of $V(G)$. +The cochromatic number $\zeta(G)$ is the least number of parts in a partition in +which every part is either independent or complete. Erd\H{o}s and Gimbel asked +whether +\[ + \chi(G_n)-\zeta(G_n)\longrightarrow\infty +\] +with high probability. The question is now catalogued as Erd\H{o}s +Problem~625. + +The mechanism behind the gap is simple. Fix a partition into $k$ classes. An +ordinary colouring requires every class to be independent. A cocolouring may +declare each class independent or complete. Because $G(n,1/2)$ is invariant +under complementation, the two declarations have the same probability cost. +Summing over the sign choices therefore contributes an exact factor $2^k$. +That additional entropy lowers the signed first-moment root by order +$n/(\log n)^3$. + +The proof has three logically distinct layers. + +\begin{enumerate} +\item \emph{Location.} We compare the ordinary-colouring root $r_+(n)$ with + the signed four-size root $r_4^{\mathrm{co}}(n)$. +\item \emph{Existence.} We prove that the lower signed root is populated by + controlling one normalized second moment. +\item \emph{Amplification.} We turn the resulting positive-probability signed + seed into a high-probability cocolouring without changing the leading + root separation. +\end{enumerate} + +Writing +\[ + A_4(\delta):=\log2-D_4(\delta), +\] +the phase-uniform first-moment comparison gives +\[ + r_+(n)-r_4^{\mathrm{co}}(n) + = + \left[ + \frac{(\log2)^2}{4}A_4(\delta_n)+o(1) + \right] + \frac{n}{(\log n)^3}. +\] +We place the signed seed at the midpoint between these roots, retaining one +half of the separation. The exact entropy certificate gives the uniform bound +\[ + A_4(\delta)>\log\!\left(\frac{1000}{639}\right). +\] + +% Use the following theorem box only after the Section VIII closure theorem. +\begin{resultbox}{Theorem 1: conditional Version 2 target} +For $G_n\sim G(n,1/2)$, +\[ + \chi(G_n)-\zeta(G_n) + \ge + \left[ + \frac{(\log2)^2}{8}A_4(\delta_n)-o(1) + \right] + \frac{n}{(\log n)^3} +\] +with high probability. Consequently, +\[ + \chi(G_n)-\zeta(G_n) + \ge + \left[ + \frac{(\log2)^2}{8} + \log\!\left(\frac{1000}{639}\right)-o(1) + \right] + \frac{n}{(\log n)^3} +\] +with high probability. +\end{resultbox} + +\displayheading{What the second moment must prove} +Let $Z_n$ count the signed four-size profiles at the selected midpoint. The +entire overlap analysis is directed toward the single estimate +\[ + \frac{\mathbb E Z_n^2}{(\mathbb E Z_n)^2} + \le + \exp\!\left\{ + o\!\left(\frac{n}{(\log n)^4}\right) + \right\}. +\] +The second-moment sections do not move either first-moment root. Their purpose +is only to show that the signed first-moment advantage survives overlap. + +For two ordered signed partitions, let $r=(r_{ab})$ be the overlap table and +let $H_r$ be the bipartite support graph of cells with multiplicity at least +two. Summing the sign declarations gives the exact identity +\[ + \operatorname{SignedOverlapWeight}(r) + = + \left(\prod_{a,b}g(r_{ab})\right)2^{\beta(H_r)}, +\] +where $\beta(H_r)$ is the binary cycle rank. The local product records the cell +multiplicities, while the cycle-space factor records the remaining sign +compatibility. + +We then reduce the overlap in three stages. Common whole classes form the +partial diagonals. The remaining canonical high cells have support on a block +matching. Everything else is a residual even-subgraph attachment after the +high matching has been exposed. + +\displayheading{The high-cell dictionary} +A \emph{profile block} is an actual class; a \emph{block atom} is its type/slot +label. The \emph{block support} $P$ is the matching of block pairs carrying +positive high demand. For $e\in P$, let $s_e,t_e$ be the endpoint sizes, +\[ + m_e:=\min\{s_e,t_e\}, + \qquad + j_e:=m_e-h_e, +\] +where $j_e$ is the actual high multiplicity and $h_e$ is its deficit from full +containment. A \emph{partial physical matching} is the literal size-$j_e$ +matching of stubs in one selected block pair. An \emph{endpoint table} records +only how many support edges have each pair of endpoint types. A partial +physical matching need not have a unique full completion; the proof compares +whole finite fibres. + +For fixed support $P$ and multiplicities $j=(j_e)$, the aggregate physical +weight is +\[ + w(P,j) + = + \frac{\displaystyle + \prod_{e\in P}(s_e)_{j_e}(t_e)_{j_e}} + {\displaystyle + (n)_J\prod_{e\in P}j_e!} + \prod_{e\in P}g(j_e), + \qquad + J:=\sum_{e\in P}j_e. +\] +There is one factorial $j_e!$ per selected cell and one global denominator +$(n)_J$. + +If $H:=\sum_e h_e$, then the full-reference denominator changes by +\[ + \frac{(n)_{J+H}}{(n)_J}=(n-J)_H\le n^H. +\] +For endpoint sizes $m,m+d$, define +\[ + R_{m,d}(h) + := + \frac{\binom mh}{(d+1)(d+2)\cdots(d+h)} + 2^{-hm+h(h+1)/2}. +\] +The global comparison is +\[ + \frac{w(P,m-h)}{w_{\mathrm{full}}(P)} + \le + \prod_{e\in P}n^{h_e}R_{m_e,d_e}(h_e). +\] +The high-cell condition implies $2h_e