Erdős–Rado Sunflower Conjecture

Source-reported subject: | All-edge protected residual is nonnegative on every bad triangle | False. Relative petal profile $(0,1,1)$ has first residual $-1$. A larger ambient fixed union does not repair the intrinsic-core sign. The old threshold near $0.008802r$ is discarded. |. | All-edge protected residual is nonnegative on every bad triangle | False. Relative petal profile $(0,1,1)$ has first residual $-1$. A larger ambient fixed union does not repair the intrinsic-core sign. The old threshold near $0.008802r$ is discarded. |

Source-reported subject: | Tensorized collision inequality $T\ge P^2$, or the proposed kernel domination | The exact weighted twenty-word construction refutes the claim. The retained executable script contains the controlling word data and checks. Copositivity must not be upgraded to a general spectral assertion. |. | Tensorized collision inequality $T\ge P^2$, or the proposed kernel domination | The exact weighted twenty-word construction refutes the claim. The retained executable script contains the controlling word data and checks. Copositivity must not be upgraded to a general spectral assertion. |

Source-reported subject: Nine-edge quiet joins, sharp-root reverse factor2, same-source direct sum. The governing source reports the following scoped result or conditional interface. Its full hypotheses and exclusions remain part of this report.

11. Quiet pairs, sharp certificates, and the exact dichotomy

A mutual pair is quiet if all four corners equal $S$. The remaining five cross labels are already fixed by the root and exact-class conditions. Thus quietness is equivalent to an exact nine-edge join:

The parents are disjoint bad triangles, each with every internal label strictly larger than $S$ by inclusion.

For a base pair $e=\{a,b\}$, define

Every such root label is nonempty and strictly contained in $\sigma(a,b)$. The equal-label sharp-root energy is

This counts unordered pairs of roots of a fixed base pair. It is not a count of bad triangles.

The quiet certificate is explicit. Orient the source pair by the fixed order, say $x<y$, let $e=P_x(y)\setminus\{y\}$, choose a deterministic $c\in P_y(x)\setminus\{x\}$, and put $f=\{x,c\}$. Both members of $f$ are exact roots of $e$ with label $S$. Given $(e,f,S)$, there are at most two choices for which member of $f$ is $x$; its canonical parent containing either endpoint of $e$ recovers $y$. Therefore

The factor two is a reverse multiplicity, not a spare safety factor. A symmetric orientation convention elsewhere must not silently count both certificates.

Combining the two source types yields the exact local direct-sum inequality

This is V26.2’s strengthening of the inherited $6|D(W)|$ version, which remains a valid weaker statement. All terms refer to the same ambient subset and the same fixed canonical matchings. The four-corner and quiet maps are disjoint branches of the source; their target sets need not be disjoint from one another as sets of words.

Whole-code source (V25.1, A25). Write $N=B^r$ and choose $W=\mathcal C$, so $n=N+1$. Without invoking layers, bottom saturation, shields, or resets,

This follows directly from $U_r<N/B$. It gives the same dichotomy and constants displayed below, now without the bottom-source hypothesis. Its parents are rooted and its corner children descend below their parent floors; they are not all bottom triangles. This is a shorter dependency chain, not a solution or a transfer of bottom ancestry to all codewords.

For the near-full bottom source in Section 8, the elementary estimates at $B\ge512$ imply

If $J(W)$ denotes the number of distinct quiet packets defined next, the useful A25 dichotomy for either of these two choices of $W$ is:

or else

For example, if the first inequality fails, at most $6B^{2r}/25$ mutual pairs are nonquiet, leaving more than $B^{2r}/4$ quiet pairs. The packet and sharp multiplicities then give the displayed weaker convenient thresholds. These inherited thresholds are retained for compatibility.

The current stronger alternative uses the smaller exact-base triangle class:

Indeed, failure of the first branch gives $E_d\le2B^{2r}/25$, leaving $E_q>0.41B^{2r}$; divide by nine and two. Both versions are dichotomies of large sources, not upper-bound theorems.

Source-reported subject: Ordered cell profiles and unordered root-label pairs → ENERGY-v lower bound v>=1, upper1<=v<=B. Prerequisites reported for the scoped interface; all additional analytic/combinatorial hypotheses remain explicit in the target quotation.

The referenced context changed. The complete target statements, qualifications and statuses before and after this update are shown below.

Historical subject: Exact telescoping exposes boundary frontier. Aligned geometric scales cancel all interior moments; lower-tail and factorial-moment inequalities constrain the agreement distribution but do not yet pay for the surviving boundary.

Source-reported subject: Current packet route. A source-preserving same-scale global estimate remains open; local counts, support overlaps and all-generation ownership must be controlled.

Source-reported subject: Exact parity clean maxima. The governing source reports the following scoped result or conditional interface. Its full hypotheses and exclusions remain part of this report.

6. Exact clean endpoint: V18.4, repaired by V24.1

Let $G(r)$ be the clean pairwise-agreeing maximum and $H(r)$ the clean arbitrary maximum. Their exact values are

The active clean ceiling is

Use $K_r$, not the superseded $\lfloor\beta^r\rfloor$, when an exact clean loss matters. The simpler bounds $G(r)\le\beta^r$ and $H(r)\le2\beta^r$ remain valid.

A24 repair. The old V18.4 proof used the positive-rank estimate $H(d)\le(2/\beta)\beta^d$ at $d=0$, where it is false. The theorem is retained because the missing terminal branch has a separate proof. In the clean hierarchy, let $s$ be the minimum label rank, $m$ the number of top classes, and $k$ the least integer with $\binom{k}{s}\ge m-1$. The induction recurrence is

For $r-k\ge1$, the parity checks for $k\le4$ and the $k\ge5$ exponential envelope are valid. For $k=r$, the correct bound is $|\mathcal C|\le m\le\binom r s+1\le G(r)$: check $r=3,4$, then use $2^r+1\le G(r)$ from $r=5,6$ onward by the two-step recurrence. The full repaired proof is at V24_CLEAN_TERMINAL_REPAIR in the foundations file.

Sharpness comes from two disjoint-alphabet copies for $H(r)$ when $r\ge1$, and a three-cluster construction giving $G(d+2)\ge3H(d)=6G(d)$ for $d\ge1$. The omitted domain in the v26 summary is repaired by V27.3: at $d=0$, $G(2)=2<3H(0)=3$, so that extrapolation is false. Thus $\sqrt6$ is the exact exponential clean base. A route requiring any smaller clean exponential base is blocked, regardless of how well one improves finite prefactors.

Source-reported subject: 88,478 recurrence checks at ranks3–80 are source-reported finite support, not substitute for all-rank induction.. The retained v24 diagnostic checks 88,478 integer recurrence cases, including the formerly missing zero-residual boundary, for ranks 3 through 80. This finite check supports the explicit all-rank induction; it is not a substitute for it.

Source-reported subject: Rooted amplification and seven-state source cover include common-root alternative. The governing source reports the following scoped result or conditional interface. Its full hypotheses and exclusions remain part of this report.

20. Rooted amplification and source-preserving reset interfaces

The rooted machinery is optional for the whole-code mutual source and secondary for the ancestry-preserving route, but remains available. Every interface below requires actual common roots or the stated exact frame; a low numerical rank without the corresponding labels is insufficient.

§54: multiplicity-to-descendants. If a bad parent $\tau$ has $c$ common roots and the relevant lower bad ranks are absent as required by the layer argument, a fixed choice of pivot and maximal matching gives at least

lower-floor children. The selected supports are disjoint away from the retained pivot. This is a count of parent-to-child certificates; the same child can be selected from other parents.

§63: two-sided rooted amplification. From one rooted parent above bottom, the two-sided construction supplies more than

lower children, with pairwise disjoint private supports outside the fixed four-word frame. V18.7 groups these by their frame traces. There are at most eleven traces of size zero, one, or two on the frame, producing a canonical book, fan, or matching of size greater than $M/44$. The exact trace roles and any singleton common-root branch must be retained.

V20.4: external-root calculus. For an incident-minimal rooted parent at root $y$, with root label $R$, $|R|=h$, a word outside $\{y\}\cup\mathcal C_R(y)$ gives either a child of floor at most $h$ on one of the six two-vertex traces of the four-word frame, or the specified common-root alternative at rank $h$. This is an alternative with two types of outputs, not a six-trace cover of all source words.

V20.5 and V20.6: terminal saturation and reset. Terminal root-floor saturation gives a canonical low-floor family of at least $B^r/13$. The rebound-free rooted reset yields, from any rooted bad parent with root $x$, a canonical family of at least $B^r/16$ bad triangles whose floors are at most

The internal root-label descent can choose new parents and roots without repeatedly thinning the original source at each step. It is an important single-invocation reset theorem. It does not prove a bounded-multiplicity global flow when invoked separately on every parent in a quadratic source.

V21.5: corrected rooted source cover. The mass-preserving cover has seven states: the common-root state plus six two-vertex traces. Its source size is at least $B^r-B^{r-h}$. The common-root state can hold half the mass and must not be discarded. Comparing unordered pairs of the seven states gives twenty-eight coarse pair types, not the twenty-one pairs of six trace states alone.

The external-root calculus, seven-state cover, terminal saturation, single-invocation reset, and the specified private-support/fixed-root amplification steps are A25/C; see V25_SOURCE_CHAIN_AUDIT. The ceiling tracked during reset is the initial $\mu(x)$, not an unproved monotonicity of every newly chosen incident minimum. The more general multiplicity-to-descendants variant §54 and derivative trace refinements retain R/C where noted in the registry. Any new global use must name the source, private support, frame, choices, and output measure; the single-invocation theorem is not a global bounded-preimage flow.

Source-reported subject: Source reports V27.1/2/3 and exact finite counts; no external reexecution, formal proof or all-generation closure.. ## 30. Audit conclusions and remaining review scope

The [v27 ledger](legacy_provenance_audit_v27_pass2.txt) records the second mathematical audit and final completeness review; the [v26 ledger](legacy_provenance_audit_v26_pass1.txt) identifies preceding checks not counted again as new work. All statuses are scoped, not formal certification.

V26.1–5 retain their exact-label, four-preimage, weighted-budget, common-root, and support-overlap conclusions. V27.1 adds an eight-word rank-twenty witness attaining four; coefficient-two sharpness is not claimed. V27.2 repairs the raw-versus-refined source ambiguity, distinguishes bad pages from rooted parents, and reconstructs fixed-pivot and all-cell extraction with unchanged constants. V27.3 restores the positive-rank domain of the clean sharpness amplifier. The two summary-level repairs do not invalidate the corresponding complete, properly scoped arguments.

The new exact suite checks the four sharpness incidences, 72 ordering/coordinate/padding variants, 1,152 labelwise checks, and 66,276 low-link nonroot configurations among 87,357 eligible four-word models through rank seven. Its 191 exact boundary checks accompany analytic proofs. This geometry test is distinct from the earlier union-budget census. Data, scripts, and logs are separate.

Do not repeat these local audits without a new reason. Neither global accounting gate is closed. Derivative book-rank, scalar-closure, and unrelated higher-moment modules retain their stated R/C statuses. Whole-code and tagged-bottom sources remain different contracts; no all-generation reverse bound or support-disjoint packet extraction is available.

Historical subject: Extract an asymptotic BTR dual certificate. Find a positive combination of finite BTR, lower-tail, factorial-moment, and exact combinatorial constraints whose coefficients telescope or have bounded total mass uniformly in rank.

Source-reported subject: Three local-to-global status corrections. The source distinguishes three corrections: occurrence-disjoint edges can share a full parent; forbidding nonnegative exact-label weights is a superseded inference; and one-generation indegree does not control merged generations globally. The remaining issues are support multiplicity, the actual normalized source scale and aggregate capacity ownership.

The referenced context changed. The complete target statements, qualifications and statuses before and after this update are shown below.

Historical subject: Prove boundary stability for BTR. Control the explicit right-boundary moments left by the aligned telescoping identity strongly enough to contradict the summed BTR inequality whenever M>Aʳ.

Source-reported subject: Open descent work with exact source conditions. global reverse map into contact collisions; single-invocation reset to global-flow gap; changing cells and repeated-localization loss; Gate D source-preserving global descent; transition contract; canonical descendant genealogy; branch and frame restrictions for new extensions

Source-reported subject: | Bounded number of sunflower-free projected layers | Steiner/vector-space constructions block the claimed uniform bound. Projection class count cannot be replaced by an absolute constant. |. | Bounded number of sunflower-free projected layers | Steiner/vector-space constructions block the claimed uniform bound. Projection class count cannot be replaced by an absolute constant. |

The referenced context changed. The complete target statements, qualifications and statuses before and after this update are shown below.

Historical subject: Prove a cross-class direct-sum gain. Find a gain across exact root classes that charges total residual information and survives all retained hierarchical, many-branch, and high-rank adversarial models.

Source-reported subject: Open packet work with exact source conditions. support-disjoint extraction and total-packet gap; long quiet-component capacity gap; occurrence-to-vertex-disjoint block conversion; Gate J aggregate binary-packet compression; genuine quiet fork and normalized cells

Source-reported subject: | An occurrence matching is a vertex-disjoint block matching | Explicitly false. V26.5's nine-word code gives two packet-distinct, occurrence-disjoint edges sharing a full parent. A support multiplicity theorem is still needed. |. | An occurrence matching is a vertex-disjoint block matching | Explicitly false. V26.5's nine-word code gives two packet-distinct, occurrence-disjoint edges sharing a full parent. A support multiplicity theorem is still needed. |

The referenced context changed. The complete target statements, qualifications and statuses before and after this update are shown below.

Historical subject: Random-rainbow reduction; Pairwise-agreeing core reduction; Binomial-trace recurrence; Boundary-stability theorem → Three-petal Erdős–Rado sunflower conjecture. If the proposed boundary-stability input closes the BTR induction for I(r), the pairwise-core binomial reduction bounds F(r), and the rainbow reduction then proves the three-petal exponential bound.

Source-reported subject: Auxiliary V25.3 requires r<=2B. The governing source reports the following scoped result or conditional interface. Its full hypotheses and exclusions remain part of this report.

Auxiliary rank-seven count (V22.9.4–5 / V25.3, A25/C). The range $r\le2B$ is indispensable. At every root there are more than $B^r/36$ vertex-disjoint selected bad parents of ambient floors in $\{2,\ldots,7\}$, rooted by labels of sizes at most four. Across all roots there are more than $MB/288$ distinct parent triangles, and one floor carries more than $MB/1728$. This is a parent count, not just a count of covered words or root occurrences.

Here is the counting interface that was missing from the compressed source. Low root labels cover at least $(8-e^2)B^r$ words. A maximal matching restricted to ambient parent floor at most seven leaves, in a root class of size-$k$ label, at most

This follows either from the layer bound on a minimum-edge remainder or from FACTORIAL-CAP. Summing the main residues costs exactly

and the additive residues cost at most $2^r\le4B^{r-2}$. Thus more than $B^r/12$ words, hence $B^r/36$ parents, are matched. A fixed floor-at-most-seven parent can occur at fewer than

counted roots, because every root label is a proper subset of one minimum edge label. This proves both deduplication denominators. The exact-class cutoff and general Poisson criterion remain valid for arbitrary $r$; the concrete rank-four and rank-seven conclusions do not. This auxiliary theorem is not needed for D-OR-J.

Source-reported subject: V27.2 uses raw disjoint supports and separates bad pages from jointly rooted parents. The governing source reports the following scoped result or conditional interface. Its full hypotheses and exclusions remain part of this report.

23. One-time fixed-pivot localization and the secondary plateau route

Use the raw disjoint selector supports from Section 8: singleton pages of a raw book, off-center pairs of a raw fan, or triples of a raw matching. There are $t\ge B^r/16$ blocks, each of size at most three. Whole book or fan triangles are not disjoint; their shared pivots are excluded from these supports. The more restrictive canonical shape refinement has smaller thresholds and is not used here. This source-contract correction preserves the constants below.

The selector hypergraph has independence number at most $K_r$. Supersaturation gives a pair of blocks of codegree exceeding $6(t-2)/K_r^3$. Among at most nine actual pivot pairs, one fixed pair $x,y$ has a bad-page set $Z$, one word per distinct third block, with $xyz$ bad and

Matching internal bad triangles in each of at most $3^r$ exact profiles leaves at most $3^rK_r$ words. The all-rank margin $192(108/B)^r<1$ for $B\ge512,r\ge4$ makes that residue at most $|Z|/4$. Thus V18.5 gives a vertex-disjoint matching of parents, each rooted by both pivots, of size

These implications are A27/C under V27.2, not an unaudited upstream assumption.

Define a selector triple of parent blocks as nonplateau when some bad selector has floor below the minimum parent floor or is not rooted by both fixed pivots. In the latter case, a nonrooting pivot has a link below that minimum floor, and V27.2's low-link lemma gives a strict lower-floor child. A matching of at least $P_0/12$ nonplateau triples yields a lower-floor fan or matching of at least $P_0/36$, by selecting one of three pivot traces. Otherwise deleting a maximal such matching leaves an all-cell plateau with

Every bad selector then has floor at least the minimum source-parent floor and lies in one exact joint cell. At this scale $p\ge4K_r$, because $(B/36)^r>1024$. There is no initial single-cell pigeonhole; the optional one-cell version costs another $K_r$.

V20.3 / V25.5 is the controlling contact-absorbing plateau interface. Its input contract requires that every bad selector across three source blocks has floor at least the minimum of the three parent floors and lies in one common exact joint cell. Neither clause follows from the word “plateau” alone. Under this same-frame contract, with $p\ge4K_r$, one gets a canonical output of size at least

in the appropriate alternative: strict source-floor descent, growth of the fixed-coordinate capacity drop $\kappa$, or arrival at bottom. Flat selectors are rooted by the fixed pivots, so variable-rank fixed-root descent handles them directly. An ordinary shield or suspended-contact terminal is no longer a separate live plateau alternative.

Processing an already valid plateau remains A25/C. The raw-support extraction, its $K_r^{-3}$ constants, the nonplateau alternative, and the full plateau input contract are now A27/C. Derivative book-rank/entropy estimates are not promoted by this audit. The contact growth proof is now active at SOURCE_L02990 and V25_PLATEAU_CONTACT_AUDIT: if $S,T\subsetneq L$, $|L|=a$, and $S\cup T\subseteq L$, then the old $\kappa\le a-1$, whereas rerooting with profile $(S,L)$ gives $\kappa=a$. At contact $L=S\cup T$, the gain is exactly one. These are source-cell facts, not a new global potential. A child below a source floor $a$ may still be at or above the fixed pivot floor

and may leave the source cell. No current theorem promotes source-floor descent automatically to pivot-floor descent. A contact projection retaining one coordinate of the union is cell-dependent; do not merge such projected codes across cells without a direct-sum argument.

The fixed-pivot theorem loses $K_r^{-3}$, which is exponentially small in $r$. Paying it once is compatible with the huge frozen base. Paying it at $\Theta(r)$ successive stages loses $\exp(-\Theta(r^2))$. The route is therefore a one-time localization tool, not a repeated engine, unless a new reset theorem proves that the localization loss is nonmultiplicative.

Source-reported subject: V26.1/2 exact source label survives, unique minimum, at most4 reverse incidences and E_d<=2|D_*|; Nine-edge quiet joins, sharp-root reverse factor2, same-source direct sum → Arbitrary nonnegative exact-label weights now valid for the one-generation maps. Prerequisites reported for the scoped interface; all additional analytic/combinatorial hypotheses remain explicit in the target quotation.

Source-reported subject: Source-preserving input contract. Specify the minimal rank-r pairwise-agreeing counterexample at the fixed base, the exact chosen whole-code or tagged-bottom source, deterministic rootwise matchings and the source edges or packets being charged. State whether a theorem covers the whole source, a component or a controlled subfamily, and quantify every restriction loss.

Source-reported subject: General incident/class cutoff separate from concrete r<=2B rank4/rank7 consequences. The governing source reports the following scoped result or conditional interface. Its full hypotheses and exclusions remain part of this report.

21. Auxiliary incident-rank and singleton reductions

The incident minimum obeys the universal inherited estimate V20.7

It comes from exact root-class capacities and the distribution of their label ranks. An additional exact class-count cutoff, V22.9, says that a normalized counterexample must satisfy

This narrows a rank regime; it does not create a finite exhaustive verification of the remaining ranks.

A useful Poisson-tail criterion is: if

then $\mu(x)\le q$. Thus $r\le2B$ implies $\mu(x)\le4$, because $e^2-7<1$. In that regime $s=1$, and the retained construction gives rank-at-most-four root-labeled mass greater than $B^r/5$ in disjoint bad parents at the relevant root. The source words, root label, and actual parent matching remain part of the output.

For a projected code of rank $d$, size $m$, and minimum agreement at least $t$, protected factorial moments give, for $1\le j<t$,

If $d\le2B$ and $t\ge T\ge2$, this implies

The deletion used to establish the moment inequality is protected by the actual residual agreement bound. Do not use $t$ from an unrelated ambient code after a projection has changed it.

Source-reported subject: All-cell, base-minimum book and suspended-contact estimates have distinct conditional domains. The governing source reports the following scoped result or conditional interface. Its full hypotheses and exclusions remain part of this report.

Retained rank-compression diagnostics are:

which implies $a\le(0.005094+o(1))r+1$ at the frozen base; and

for the V18.8 canonical base-minimum-book branch, giving the diagnostic coefficient approximately 0.002838. The latter is not proved for terminal fans or matchings. The older suspended-contact estimate

and its approximately 0.003826 coefficient are retained only as conditional capacity diagnostics for that old configuration, not as a current terminal branch. None of these asymptotic coefficients is needed in the principal mutual-source argument.

Source-reported subject: Ordered cell profiles and unordered root-label pairs. The governing source reports the following scoped result or conditional interface. Its full hypotheses and exclusions remain part of this report.

15. Full joint-cell energy: exact identity, exact scope

For a base pair $e$, retain all common-root labels, not only equal ones, and define

The second sum is over unordered distinct label pairs. Choose a fixed orientation for each pivot pair $f$. Opposite-pair counting on four distinct words gives

The profile sum on the right is ordered relative to the oriented pivots. There is no extra factor two on the left's off-diagonal term. The upper bound uses $\binom a2/L\le a/2$ for a cell of size $a\le L$, followed by the fact that the cells partition the $M-2$ other words for each pivot pair.

For a subset $W$, the same equality and bound hold with its own root counts and cell sizes and with $M$ replaced by $n$ on the right. The denominators continue to use the ambient induction capacities $B^{r-\kappa}$. One does not obtain a smaller effective rank merely by restricting to $W$.

The equal-label summand in this budget is

not the unweighted $H^{=}$. V26.2 now supplies a compatible one-generation source normalization. Returning to the unweighted $H^{=}$, however, still requires a quantitative comparison of denominators or a rank-distribution theorem. Moreover, the available upper bound is cubic in the number of words, while the mutual source is quadratic. An exact identity with a too-large upper bound does not itself produce a contradiction.

This identity, its unordered/ordered conventions, and its capacity derivation are A24. New exact tests compare both sides on finite codes using rational arithmetic. These tests do not verify a conjectured restricted-budget improvement because no such improvement has been proved.

The joint-cell form is preferred to separately capped root-label fibers. The union term $|S\cup T|-1$ retains information that disappears if one replaces every cell by independent one-pivot estimates. Conversely, discarding the off-diagonal cells and then treating their capacity as unused is not justified when another part of the argument also spends the same four-vertex budget.

Source-reported subject: | Improve the clean exponential base below $\sqrt6$ | Impossible in this model. The exact clean extremal construction attains that base. Finite-prefactor improvements cannot bypass it. |. | Improve the clean exponential base below $\sqrt6$ | Impossible in this model. The exact clean extremal construction attains that base. Finite-prefactor improvements cannot bypass it. |

Source-reported subject: Five-bin source preserves actual support, frame and assigned words. The governing source reports the following scoped result or conditional interface. Its full hypotheses and exclusions remain part of this report.

8. Bottom saturation without early pigeonholing

The propagation/shield theorem §67–§68 / V21.3 starts with one unrooted bottom triangle $Q=\{u,v,w\}$. Every external word either supplies a bad bottom triangle with two vertices of $Q$, or falls into the uniquely determined shield configuration handled by disjoint two- or three-word supports. A maximal choice of those supports leaves at most $E_s^{(B)}B^r$ unmatched words.

The source-preserving version assigns every covered word once to the five bins

The first three record propagation triangles on the indicated two pivots. The last two retain the actual two-word or three-word shield supports, their chosen parents, and their assignment to the source word. The supports, frame, and assignment are part of the data, not decorative labels.

The union $W$ of assigned source words satisfies

All its associated bad witnesses are bottom rank $s$ and unrooted. This source construction is A25 under V25_SOURCE_CHAIN_AUDIT. The support order is two-word supports first, then three-word supports. The final remainder has no bad triangle of floor at most $s$ even after the retained shield pivot is included, so the layer estimate applies. This is more than merely forbidding internal three-word witnesses in the remainder.

A largest-bin argument gives the raw terminal alternatives, for $B\ge512$: a bottom book with at least $3B^r/16$ pages, a bottom fan with at least $3B^r/32$ disjoint page-pairs, or a bottom matching of at least $B^r/16$ triangles. The separate refined canonical trichotomy V17.4 guarantees only $B^r/80$, $B^r/160$, or $B^r/240$ blocks in its respective branches. V27.2 retains the larger raw supports for localization; it does not substitute the smaller refined outputs. Those thresholds remain useful for one-time localization. They are secondary outputs. The ancestry-preserving mutual-block route keeps the entire tagged union $W$, rather than discarding four bins first. The shorter untagged route can instead take $W=\mathcal C$, as in Section 11; only the former carries the original bottom-support assignments.

A book or fan has several normal forms. In a base-minimum book the spine has rank $s$ and the two sides of each page triangle have larger rank. In a base-minimum fan each paired leaf edge has rank $s$ and the two spokes are larger. A bottom diamond instead has a larger spine and two rank-$s$ spokes with different labels. Do not transfer an inequality between these forms just because each consists of three vertices and has floor $s$.

The mass of $W$ is a count of words. A later occurrence or packet count is a different measure. Every conversion between these measures must state its multiplicity; the original source assignment is retained to make such a conversion possible rather than implicit.

Source-reported subject: Exact root-class strictness and cell drop max(|S|,|T|,|S union T|-1); Exact parity clean maxima → V22.8 requires vertex-disjoint bad parent blocks and exact equal-rank distinct profiles. Prerequisites reported for the scoped interface; all additional analytic/combinatorial hypotheses remain explicit in the target quotation.

Source-reported subject: | Apply a book rank-compression bound to fans or matchings | V18.8 is branch-conditional. The missing extension must be proved, not inferred from a common floor. |. | Apply a book rank-compression bound to fans or matchings | V18.8 is branch-conditional. The missing extension must be proved, not inferred from a common floor. |

Source-reported subject: Five-bin source preserves actual support, frame and assigned words; Exact parity clean maxima; V22.8 requires vertex-disjoint bad parent blocks and exact equal-rank distinct profiles → V27.2 uses raw disjoint supports and separates bad pages from jointly rooted parents. Prerequisites reported for the scoped interface; all additional analytic/combinatorial hypotheses remain explicit in the target quotation.

Source-reported subject: Final same-scale inequality and boundary terms. State the final inequality with constants, every normalization denominator and all boundary terms. Compare its upper bound directly with the appropriate actual source lower bound. Treat s=1 separately from positive protected scales and specify small residual-rank conventions before using an asymptotic envelope.

Source-reported subject: Layers require absence of all smaller bad ranks → Rooted amplification and seven-state source cover include common-root alternative. Prerequisites reported for the scoped interface; all additional analytic/combinatorial hypotheses remain explicit in the target quotation.

Source-reported subject: | Dense-box projection with a power-sized retained branch | Prefix/hierarchical codes obstruct the proposed universal projection; agreement-graph connectivity does not fix the loss. Independent pruning incurs the known square-root loss. |. | Dense-box projection with a power-sized retained branch | Prefix/hierarchical codes obstruct the proposed universal projection; agreement-graph connectivity does not fix the loss. Independent pruning incurs the known square-root loss. |

Source-reported subject: | Sharp uniform energy constant $(3/2)^r$ | The type-class construction defeats the proposed uniform bound. A weighted or label-restricted replacement requires a different statement. |. | Sharp uniform energy constant $(3/2)^r$ | The type-class construction defeats the proposed uniform bound. A weighted or label-restricted replacement requires a different statement. |

Source-reported subject: Exact root-class strictness and cell drop max(|S|,|T|,|S union T|-1); Exact parity clean maxima → Layers require absence of all smaller bad ranks. Prerequisites reported for the scoped interface; all additional analytic/combinatorial hypotheses remain explicit in the target quotation.

Source-reported subject: | Drop common-root states from rooted saturation | V21.5 has seven states, including common roots. That state can contain a macroscopic share of the source. |. | Drop common-root states from rooted saturation | V21.5 has seven states, including common roots. That state can contain a macroscopic share of the source. |

Source-reported subject: | A six-vertex selector must give a lower child or a new internal root | The ten-word local obstruction disproves this dichotomy in the stated frame. Its word data and exact verifier are present. |. | A six-vertex selector must give a lower child or a new internal root | The ten-word local obstruction disproves this dichotomy in the stated frame. Its word data and exact verifier are present. |

Source-reported subject: One-root and two-root bottom-diamond zero classifications. The governing source reports the following scoped result or conditional interface. Its full hypotheses and exclusions remain part of this report.

18. Zero states, diamonds, and contact allocation

The zero-defect results V20.8–V20.13 convert pages into collisions that can be charged to a protected budget. The one-root and two-root zero classifications and clique bounds are A25 under V25_ZERO_STATE_AUDIT; the displayed contact bridge is independently derived there as an allocation to the same budget. Other higher-moment refinements of these source sections retain their individual registry statuses. No arbitrary two-pivot geometry is substituted for the required bottom diamond.

For a single root $u$, take a rank-$s$ page layer with labels $A_z=\sigma(u,z)$. A pair has $\delta_u=0$ only in the stated two zero forms: equal $A$-profiles with one extra agreement coordinate, or distinct profiles whose intersection has size $s-1$ and whose mutual label has the corresponding minimum size. A zero-defect clique has at most

vertices. Consequently the total nonnegative first defect on $n$ pages is at least

The polynomial clique bound is a quantitative local input, not an upper bound on the size of the entire layer.

For a two-root bottom diamond, fix $u,v$ with $|\sigma(u,v)|>s$, and require for every page $z$ that both spoke ranks are $s$ and $uvz$ is bad. In particular the two spoke labels $A_z,C_z$ are different. Let $J=\max\{r-s+1,s+1\}$. A simultaneous zero-defect clique for the two roots has at most $2J$ vertices, and the corresponding combined defect is at least

Dropping the condition that $uvz$ is bad permits equal spoke profiles and changes the zero geometry. The theorem is not a generic two-pivot estimate.

The exact two-root allocation resolves the projection slack. Put $U=\sigma(u,v)$ and

For a protected coordinate set $J\subseteq U$, the two pivot patterns are the same and their common count is denoted $t_J$. For $J\not\subseteq U$, the patterns are distinct and their counts $a_J,c_J$ are disjoint. The identity decomposes PROTECTED into $2X_k$, the terms $t_J(L-t_J)$, the two separate terms $a_J(L-a_J)+c_J(L-c_J)$, and the remaining nonnegative fiber slack. Shared pivot patterns must be counted once, not twice.

At the contact scale $k=s-1$, assume $s\ge2$, set $r'=r-s+1$, $L=B^{r'}$, and partition the contact pages by $R$ of size $s-1$, with class sizes $n_R$. Then the inherited protected-to-singleton bridge is

Here the page set and frame satisfy the bottom-diamond hypotheses. A contact cell at some higher local floor $h>s$ is not automatically covered by the same theorem. The fixed-coordinate deletion and the ambient protected scale must be checked separately.

These bounds identify a possible destination for a globally controlled descent when $s\ge2$. They do not yet supply the required global reverse map from all descendant certificates to these particular collisions.

The referenced context changed. The complete target statements, qualifications and statuses before and after this update are shown below.

Historical subject: Cross-class direct-sum inequality. A structural gain across exact root classes is active as the equivalent combinatorial formulation of boundary stability.

Source-reported subject: Current normalized route. A source-preserving same-scale global estimate remains open; local counts, support overlaps and all-generation ownership must be controlled.

The referenced context changed. The complete target statements, qualifications and statuses before and after this update are shown below.

Historical subject: Open work and evidence boundary. Primary, secondary, and audit obligations remain explicitly separate from proved-in-project claims and from the one computation whose certificate is missing.

Source-reported subject: Open work and evidence boundary. Current global accounting, normalization and singleton obligations remain open. Historical audit and linear-model tasks remain separately scoped; the missing18-word certificate is still not proof.

The referenced context changed. The complete target statements, qualifications and statuses before and after this update are shown below.

Historical subject: Binomial-trace frontier. The live recurrence, exact telescoping, endpoint constraints, proposed boundary theorem, and conditional route to the full conjecture.

Source-reported subject: Retained BTR and scalar toolbox. Historical BTR, exact telescoping and endpoint constraints remain scoped optional tools. Their old principal frontier and conditional closing assembly are inactive; current source mass flows through the two open D/J gates.

Source-reported subject: One fixed deterministic rootwise matching on the chosen W → V26.1/2 exact source label survives, unique minimum, at most4 reverse incidences and E_d<=2|D_*|. Prerequisites reported for the scoped interface; all additional analytic/combinatorial hypotheses remain explicit in the target quotation.

Source-reported subject: Paid transitions and controlled mergers. List every local transition, including internal packet returns, genuine forks, same-label children, root changes and exact-cell changes. Prove for each a definite potential decrease, paid terminal charge or bounded loop/merger rule. A new root is not automatically incident-minimal in its new code.

Source-reported subject: | Apply the concrete rank-seven count at arbitrary rank | V25.3 requires $r\le2B$. The exact-class cutoff is general, but does not extend this auxiliary counting range. |. | Apply the concrete rank-seven count at arbitrary rank | V25.3 requires $r\le2B$. The exact-class cutoff is general, but does not extend this auxiliary counting range. |

The referenced context changed. The complete target statements, qualifications and statuses before and after this update are shown below.

Historical subject: Bound exact root classes independently, including through one-coordinate pruning or exact two-pivot trace classes.. The pruning bound iterates by square roots, exact trace-class induction recreates factorial branching, and the k=1 two-pivot trace sum diverges for each fixed base.

Source-reported subject: Bound exact root classes independently, including through one-coordinate pruning or exact two-pivot trace classes.. The pruning bound iterates by square roots, exact trace-class induction recreates factorial branching, and the k=1 two-pivot trace sum diverges for each fixed base.

Source-reported subject: Layers require absence of all smaller bad ranks. The governing source reports the following scoped result or conditional interface. Its full hypotheses and exclusions remain part of this report.

7. Layer compression and the bottom anchor

Define, for $j\ge1$,

The inherited layer theorem, §53 / V16.3, says that when no bad triangle of minimum rank below $a$ is present in the relevant code, the rank-$j$ layer around any root, for $j<a$, has size at most $E_j^{(B)}B^r$. The hypothesis concerns all smaller bad ranks, not merely bad rank $j$. At global bottom the ambient minimum removes smaller pair ranks automatically.

The following numerical bounds are retained:

In particular,

The general compression and bottom-anchor argument are A25, with their reconstructed proof at V25_LAYER_AUDIT. At a root, different rank-$j$ exact classes have constant rank-$j$ cross labels; the $m$ incident distinct labels force a fixed union of size $k$ with $m\le\binom{k}{j}$. Exact-class deletion and union-minus-one deletion give the two capacities in $E_j^{(B)}$. The singleton-union boundary uses the first, not same-rank induction. The exact low-$j$ formula and tail sum remain inherited algebraic envelope refinements; the source proof uses only $E_j\le2/B$. The retained B=32 test is diagnostic, not the authority for arbitrary $B,r$.

Choose a minimum edge $xy$, with $|\sigma(x,y)|=s$. If there were no bad triangle of minimum rank $s$, every third vertex would lie in the rank-$s$ layer of $x$ or $y$. Layer compression would imply

contradicting the normalized counterexample at the active base. Therefore a bottom bad triangle exists. It is unrooted by Section 4. This is the starting object for the near-full bottom source; it is not selected by a lossy search through all label profiles.

The minimum-rank consequence from protected fibers is

In particular, $r\le2B$ forces $s=1$. This makes the singleton branch structurally central, not a negligible exceptional base case. The huge numerical value of $B$ does not make the infinite family of singleton-rank configurations finite.

Source-reported subject: | Pair-local “excess or immediate descendant” for a two-root diamond | The four-word hinge witness has two bottom bad triangles but clean cross faces. Local nonzero rank alone does not force the claimed child. |. | Pair-local “excess or immediate descendant” for a two-root diamond | The four-word hinge witness has two bottom bad triangles but clean cross faces. Local nonzero rank alone does not force the claimed child. |

The referenced context changed. The complete target statements, qualifications and statuses before and after this update are shown below.

Historical subject: Transversal and pairwise-agreeing reductions. The source first converts uniform set families to transversal codes and then concentrates exponential growth on pairwise-agreeing codes through exact equality-label classes.

Source-reported subject: Exact label incidence/nonuniform copositivity/square-root conditional closure and scalar diagnostics remain tools, not principal-source dependencies or independent budgets.. The governing source reports the following scoped result or conditional interface. Its full hypotheses and exclusions remain part of this report.

25. Label-resolved and scalar diagnostic tools still worth retaining

Several older exact tools remain useful for checking a proposed new inequality even though none currently supplies the global contradiction. They are retained in the budgets file, not promoted to an independently audited new strategy.

For each nonempty proper label $S$, let $E_S$ count its edges, let $R_{S,T}$ count isosceles triangles with labels $S,S,T$, and let $B_S$ count rainbow-edge incidences labeled $S$. §14 gives

Also

In particular, the graph of a maximal occurring label is a matching. These are label-resolved constraints, not only moment constraints on label cardinalities.

For rank $j$, write $\mathsf E_j$ for the number of edges of that rank and $\mathsf J_j$ for the isosceles count with repeated rank $j$. Then

With $L_B(r)=\sum_{d=1}^{r-1}(B^d-1)$, the source uses the weaker bound $Q_1\le N L_B(r)$. Thus a large-base counterexample is dominated in aggregate by label-rainbow triangles; this does not imply that most of them are bad rather than clean rainbows.

For $u>0$, define

The nonuniform root-weight inequality §15 says, for nonnegative vertex weights $w_x$, $a=\sum_xw_x$,

This is relevant to weighted direct sums, but copositivity is not a spectral positive-semidefiniteness assertion. The exact tensor counterexample in the legacy evidence forbids that stronger inference.

For completeness, the associated copositive matrix is

For nonempty $U$, put $q_u(U)=\mathbf1_U^\mathsf T H_u\mathbf1_U\ge0$. At a root $x$, write $U_S=\mathcal C_S(x)$, $m_S=|U_S|$, and sum only over nonempty classes. Define

Here $S<T$ is just a fixed ordering of distinct labels. The square-root branch criterion §24 is a genuine conditional alternative: put $\beta_u=(M-2)/Z_r(u)$. If a constant $K$ bounded it by $K\beta_u(M-1)$ in every minimal counterexample, copositivity would give a class of size at least $(M-1)/(K+1)$, hence $M-1\le(K+1)B^{r-1}$. If that uniform theorem is valid at a fixed base larger than $K+1$, it closes the induction at that base; one cannot change the base after assuming an incompatible hypothesis. The required bound on $\mathcal E_x(u)$ is unproved, and independent summation of pair-root capacities does not establish it.

The retained hereditary jump envelopes, water-filling minima, coordinate-entropy/Cauchy defects, full-profile multiscale decompositions, chain-versus-sharp-conflict inequalities, and fractional-shadow constraints refine this toolbox. Their detailed formulas are load-on-demand under §§17–18, 21, 23–28, 31, 35–36, 38, 40, and 44 in the theorem index. They are not additional independent budgets and are not dependencies of D-OR-J. Their live role is to test a proposed label-resolved closure against known exact identities and scalar obstructions. No unfinished scalar relaxation is being designated the next principal work item.

Source-reported subject: One fixed deterministic rootwise matching on the chosen W → Packet returns versus genuine intersecting-parent forks. Prerequisites reported for the scoped interface; all additional analytic/combinatorial hypotheses remain explicit in the target quotation.

Source-reported subject: | Bound quiet mass by a small multiple of bad-triangle mass and declare closure | V24.4 shows the uncontrolled positive $|D|$ term remains. Both channels need an upper bound or a coupled global potential. |. | Bound quiet mass by a small multiple of bad-triangle mass and declare closure | V24.4 shows the uncontrolled positive $|D|$ term remains. Both channels need an upper bound or a coupled global potential. |

Source-reported subject: One-root and two-root bottom-diamond zero classifications → Three-root silent hypotheses include ranks, clean faces and zero defects. Prerequisites reported for the scoped interface; all additional analytic/combinatorial hypotheses remain explicit in the target quotation.

Source-reported subject: | Solve the singleton regime by a large finite check | No such exhaustive certificate is present. The rank cutoff and small clean checks do not constitute that verification. |. | Solve the singleton regime by a large finite check | No such exhaustive certificate is present. The rank cutoff and small clean checks do not constitute that verification. |

Source-reported subject: V22.8 requires vertex-disjoint bad parent blocks and exact equal-rank distinct profiles. The governing source reports the following scoped result or conditional interface. Its full hypotheses and exclusions remain part of this report.

22. Quiet-cell compression and assigned bottom output

The retained V22.8 tool is narrower than a theorem about all quiet packets. Fix two pivots and a family of vertex-disjoint bad parent blocks in their exact cells. In the quiet configuration to which the theorem applies, the two root-label ranks are the same, say $\mu$, and the two labels in a profile are distinct. Cross-block quietness forces the prescribed exact labels and fixed-coordinate unions.

Let

The hypergraph of floor children on the parent blocks has the inherited independence bound

If there are $h$ source blocks, a maximal child matching removes at most three source blocks per child, and the three canonical output shapes give a selected family of size at least

The parent blocks must satisfy the stated disjointness and exact-profile conditions. An occurrence matching from Section 12 does not automatically satisfy them, because it need not give vertex-disjoint parent triples.

Combined with the appropriate source assignment, the inherited theorem produces a bottom book, fan, or matching larger than $B^r/80$, with disjoint assigned ancestral block sets. This source assignment is stronger than merely finding that many bottom triangles. It is the data needed to try to pass capacity ownership forward.

The independence bound and assigned-source conversion within this block model are A25/C, with a complete proof at V25_QUIET_CELL_AUDIT. Feasible $m$ satisfy $k_\mu(m)\le r$, and $k_\mu(m)\ge\mu+1\ge2$. The boundary $k=2$ means $m=\mu=1$; for $k\ge3$, $m\le2^k\le B^{k-2}$. No same-rank projection is used.

A floor child may use either pivot, both, or neither; “using the two pivots” in the retained source means chosen from their union with the blocks, not containing both. If the pivot-pair rank differs from $\mu$, every block gives a both-pivot book page immediately. Otherwise both-pivot triangles are clean, leaving exactly three output shapes, which explains the factor nine. At a global minimum edge, the rootwise parent matching loses at most one pivot-containing block and two exact-class exceptions. Book, fan, and hard-cell counts then yield $[B^r/4-B^{r-1}-1]/11>B^r/80$, with disjoint assigned source-block sets. The conversion from an arbitrary occurrence matching in the actual quiet graph to these vertex-disjoint blocks is still missing.