Snapshot date: 2026-08-24
Research snapshot time: approximately 09:17 CDT
Purpose: self-contained prompt for a new research agent working in a new filesystem
Current theorem status: the universal sub-quarter Komlós theorem has not been proved
Most important positive change: the ultralight block now has unconditional hereditary and affine discrepancy (O(1)); the remaining obstruction is obtaining the same endpoint for the ultralight and heavy/upstream constraints
Primary audited benchmark: Nikhil Bansal and Haotian Jiang, Decoupling via Affine Spectral-Independence: Beck-Fiala and Komlós Bounds Beyond Banaszczyk, whose general Komlós conclusion is (\widetilde O((\log n)^{1/4}))
You are taking over an ambitious proof search. Your job is not to summarize the project again and not to generate another decorative framework. Your job is to obtain a literal universal improvement over the Bansal--Jiang quarter-power bound, ideally
[ \operatorname{disc}(A)=O(\log\log n), ]
for every real matrix (A) whose columns have Euclidean norm at most one, or at least
[ \operatorname{disc}(A)=O((\log n)^\varepsilon) \qquad\text{for some fixed }0<\varepsilon<1/4. ]
The proof must be universal, not an experiment, a density-one statement, or a structural-class result. It must return one sign vector for the original columns and control all original rows. If you build a recursive or stochastic proof, every conditioning, stopping, support change, and release must be legal. If you use a black box, state its exact hypotheses and show that the current state satisfies them. If a route fails, give the exact finite or scalable obstruction and say precisely what it refutes and what it does not refute.
The project has already spent substantial effort closing attractive but circular routes. Do not restart one merely under new terminology. In particular, do not claim progress by constructing a heavy-good law and a light-good law separately, by postulating a favorable endpoint law whose existence is equivalent to the desired common endpoint, by replacing a cylinderwise assertion with an expectation, or by calling an arbitrary unresolved residual a release.
Work in a new directory tree. Treat every imported research artifact as read-only. Copy an artifact into your own subfolder before changing it. Maintain a theorem-status ledger with at least the labels proved, proved relative to named black boxes, conditional theorem, candidate lemma, counterexample, experiment, and open. Never let a verifier's PASS be interpreted as a proof of a theorem beyond the exact algebra or finite fixture that the verifier checks.
The rest of this document tells you exactly where the project stands, what each prior research stream established, which holes were closed, which holes merely moved downstream, what the live bottlenecks are, and how to attack them without making circles.
The initial architecture split the matrix at an extremely small threshold
[ \tau=\exp(-aD^2), \qquad A=H+U, ]
where (D=\log\log n) or (D=(\log n)^\varepsilon), (H) contains entries larger than (\tau) in magnitude, and (U) contains the remaining entries. The hope was to use a strong-slack local-lemma code for (H) and a stochastic-localization or Bansal--Jiang walk for (U), all inside one common law.
The largest new theorem changes the diagnosis. The ultralight matrix (U) itself is no longer difficult in the unrestricted cube:
If (|U_{ij}|\le\tau), every column of (U) has squared load at most one, and (\tau^{-2}\gtrsim\log^2 n), then [ \operatorname{herdisc}(U)=O(1), \qquad \operatorname{lindisc}(U)=O(1). ]
This is proved, relative to the two printed Bansal--Jiang black boxes used in their large-(k) proof, after repairing an adapted-birth gap, a spectral-independence parameter error, rank arithmetic, the two-sided augmentation, and the arbitrary-center step.
Consequently, the precise wall is now:
[ \boxed{ \text{find one endpoint }y\in{\pm1}^n \text{ that is simultaneously }U\text{-good and }H\text{-good,} } ]
while also preserving any high-heavy rows, saturated faces, fixed coordinates, or other upstream receipts already accumulated.
The most developed probabilistic splice is conditional on a pinned endpoint law (\mu). If (\mu) is supported on the ultralight-good and upstream-admissible endpoints and has event-specific conditional subgaussian tails for every coded low-heavy row after conjunctions of nonneighbor heavy-good events, then the lopsided local lemma returns a common endpoint at additional heavy cost (O(D)). The conversion theorem and all scale arithmetic are proved. What is not proved is that an actual, independently specified weighted-walk endpoint law has those conditional tails.
The most developed deterministic alternative is a mixed-support-component theorem. Pure heavy and pure light components compose by a maximum. Every mixed component is also payable at (O(D)) if, before signs are chosen, its vertically stacked heavy/light matrix has a rank-(\exp(O(D^2))) approximation with row-(\ell_1) defect (O(D)). Bounded mixed-component size and small target-resolution numerical rank are sufficient. Connected Walsh fixtures show that this simple factor condition is not automatically universal.
The exact current frontier is therefore not “prove an ultralight estimate.” It is one of the following genuinely new statements:
- derive construction-level, nonneighbor-conditional heavy tails for a pinned ultralight-good endpoint sampler, with support one on all upstream constraints;
- prove a deterministic structural decomposition of arbitrary mixed heavy/light components into independently payable pieces without assuming a universally false low-rank factor receipt;
- prove a survival-unbiased statewise service theorem strong enough to convert one-row martingale tails into a small dangerous span on every retained dyadic state;
- discover a different common-endpoint theorem that survives all of the no-go fixtures below and still literally recovers the Bansal--Jiang quarter-power branch when its new hypothesis is unavailable.
Anything weaker is at best a structural-class improvement or a local lemma, not the requested universal breakthrough.
Let (A\in\mathbb R^{m\times n}), with columns (A_{*j}) satisfying
[ |A_{*j}|_2\le1 \qquad(j\in[n]). ]
Define
[ \operatorname{disc}(A) =\min_{y\in{\pm1}^n}|Ay|_\infty, ]
[ \operatorname{herdisc}(A) =\max_{J\subseteq[n]}\operatorname{disc}(A_{*,J}), ]
and
[ \operatorname{lindisc}(A) =\sup_{x\in[-1,1]^n} \min_{y\in{\pm1}^n}|A(y-x)|_\infty. ]
The requested result is a universal upper bound on (\operatorname{disc}(A)). A hereditary proof is stronger and is useful because every live-column restriction remains a valid Komlós instance. An affine proof is stronger still and is useful for refreshing from a current fractional center.
A complete proof must specify:
- the exact quantifiers over (m,n,A), column restrictions, centers, and random histories;
- every constant and its dependence on fixed parameters such as (\varepsilon);
- the actual sign vector or a finite randomized algorithm with positive success probability and a termination proof;
- how discrepancy already spent before a terminal call is charged;
- how support changes preserve every incoming wall or how those walls are paid and retired;
- why the same signs satisfy every matrix block being combined;
- how continuous-time or infinite-precision arguments are discretized, if an algorithmic claim is made;
- why the conclusion transfers to the original matrix (A), rather than only to a shadow, quotient, factor, or stacked surrogate;
- how the method reduces to or competes with the audited quarter-power benchmark on every input;
- a hostile audit against the fixtures in Section 13.
The following substitutions are forbidden:
- finite verification in place of a universal theorem;
- a terminal-only probability estimate when the recursion needs a statement at every retained cylinder;
- an expectation over frontier nodes in place of a statewise conditional bound, unless the compiler itself is formulated and proved at absolute-root-mass frontier level;
- two separately good codes or convex hulls with no literal common atom;
- a law selected by optimizing all desired heavy constraints and then called “canonical”;
- a point mass at a desired common endpoint used as evidence that a useful favorable law exists;
- an arbitrary direct recode of an actual dual owner;
- repeated reminting of the same owner, entropy clock, or release token;
- a covariance-null direction silently treated as a physical (A)-null direction;
- a low-rank statement about an owner-visible image substituted for a factorization of the complete physical residual;
- a black-box Bansal--Jiang call after conditioning or partial signing without proving direct-transfer legality and charging earlier discrepancy.
The audited public benchmark for this project is the Bansal--Jiang general Komlós bound
[ \widetilde O((\log n)^{1/4}). ]
Do not merely cite this at the end. A proposed architecture must have a literal baseline mode. One exact calibration used in the project is the following common-SDP branch.
Set
[ P=1+5\lceil\log\log n\rceil, \qquad k_p=2^{2p}, \qquad b_p=B(k_p)\sqrt P, ]
where the large-(k) Beck--Fiala service is
[ B(k)=\widetilde O(\sqrt{k},\log^{1/4}n). ]
Give the heavy magnitude bands affine spectral-independence budgets (1/(6P)), the varying-magnitude light band budget (1/6), and retain the printed global sub-isotropy, trace, and blocked-space budgets. The total is
[ \frac13+\frac16+\frac16+\frac16+\frac{P-1}{6P} =1-\frac1{6P}<1. ]
The scale identity
[ 2^{-p+1}\sqrt{k_p}=2 ]
gives
[ \sum_{p\le P}2^{-p+1}b_p \le \widetilde O(\log^{1/4}n)P^{3/2} =\widetilde O(\log^{1/4}n). ]
This is a genuine budget and recombination check. The new architecture should contain the following root-level fallback:
- inspect whether the new common-endpoint or structural receipt holds before changing the input;
- if it holds, run the new (O(D)) branch;
- otherwise, run the published Bansal--Jiang branch on the untouched original matrix.
This preserves the benchmark while extending it on a certified structural class. It does not give a universal improvement unless the new receipt is shown to hold universally or every failure is routed to another strictly better universal branch. Saying “take the better of the new branch and BJ” is calibration, not the final theorem.
The next agent should also recheck the current literature from primary sources. The benchmark above was the audited general offline Komlós result used by the project at this snapshot; literature-absence claims can become stale.
Choose a target radius (D=D(n)), normally
[ D=\max{1,\log\log(2n)} ]
or
[ D=(\log n)^\varepsilon, \qquad 0<\varepsilon<1/4. ]
Choose a fixed constant (a>0) and split entrywise at
[ \tau=e^{-aD^2}, \qquad H_{ij}=A_{ij}\mathbf1_{{|A_{ij}|>\tau}}, \qquad U=A-H. ]
The supports of (H) and (U) are disjoint entrywise. It is often useful to form the vertical stack
[ A_{\mathrm{stack}}= \begin{pmatrix}H\U\end{pmatrix}. ]
Because the split is disjoint,
[ |(A_{\mathrm{stack}})_{*j}|2^2 =|A{*j}|_2^2\le1. ]
A common signing with stack radius (R) gives
[ |Ay|\infty =|(H+U)y|\infty \le|Hy|\infty+|Uy|\infty \le2R. ]
Do not confuse (A=H+U) with the vertical stack. A theorem about the stack does imply a theorem for the physical sum, but with the explicit factor two above.
For the coded low-heavy rows, assume a fixed row-energy cap (E_0):
[ E_i=\sum_jH_{ij}^2\le E_0. ]
Rows above this energy cap are high-heavy rows and must be treated separately. They cannot be silently included in the low-heavy local-lemma block.
Since every nonzero low-heavy entry exceeds (\tau), define
[ Q=\lceil E_0\tau^{-2}\rceil, \qquad k=\lceil\tau^{-2}\rceil. ]
Each low-heavy row has fewer than (Q) incident columns, each column is incident to fewer than (k) heavy rows, and the row-intersection dependency graph has degree
[ d\le Q(k-1)=O(E_0\tau^{-4}). ]
At (\tau=e^{-aD^2}),
[ \log(d+1)=O(D^2). ]
The weighted ultralight theorem requires (\tau^{-2}\gtrsim\log^2n). This holds eventually in both requested regimes:
[ e^{2a(\log\log n)^2}\gg\log^2n ]
and
[ e^{2a(\log n)^{2\varepsilon}}\gg\log^2n. ]
Also, (e^{O(D^2)}=n^{o(1)}) for every fixed (\varepsilon<1/2). Thus all scalable hostile fixtures with (e^{O(D^2)}) active columns can be embedded by zero padding in the requested regimes. This matters: the counterexamples are not artifacts of an impossible ambient size.
The first research stream proposed a strong heavy/ultralight split at (\tau=e^{-aD^2}). It constructed or targeted a large strong-slack heavy-good code (\mathcal C_0\subseteq{\pm1}^n), then tried to control (U) inside the same law by measure-valued stochastic localization.
At a live cylinder (v), the code fiber was
[ \mathcal C_v ={y\in\mathcal C_0:y\text{ agrees with the fixed prefix at }v}, ]
with live count (h_v) and current component deficit
[ q_v=h_v-\log_2|\mathcal C_v|. ]
The current deficit, not the root deficit, is the relevant rank charge. A majority-pair posterior fixture showed that a code with root deficit one can have rare cylinders whose residual constraint dimension is large.
This phase established several useful facts:
- covariance rank equals the affine dimension of the current code support;
- (\dim\ker\Sigma_v\le q_v) for the current posterior covariance;
- a Bansal--Jiang covariance (V_t) whose range lies in (\operatorname{range}\Sigma_t) can be realized exactly in the mean of one law supported on the heavy code;
- endpoint anti-collision can be proved for cube martingales from a covariance domination inequality;
- independent heavy and light codes cannot be intersected merely by comparing entropy deficits;
- the missing theorem appeared to be an adaptive ultralight safety-or-release statement with a statewise protected-span bound.
The original handoff correctly emphasized one common law and current-cylinder accounting. Its diagnosis of the ultralight block was later superseded by the weighted theorem below, but its same-law machinery and hostile fixtures remain relevant to building the missing pinned sampler.
An integration stream proved useful but restricted results:
- a one-walk theorem through the entry cutoff (1/\log n), giving a sublogarithmic bound in that restricted regime;
- an (O(\log\log n))-scale theorem for sufficiently ultralight matrices, with assumptions such as entries bounded by a sufficiently high negative power of (\log n);
- a common-law stopped covariance realization when the posterior kernel and protected hit span fit within the Bansal--Jiang feasibility budget.
At that stage the apparent wall was a dynamic weighted low-degree provenance service around entries of size ((\log n)^{-1/4}). These results were important because they proved the same-walk architecture was not empty, but they did not yield a universal split.
The later weighted large-load theorem strictly supersedes these ultralight restrictions in the unrestricted cube: at the exponentially small cutoff used for (D=\log\log n) or (D=(\log n)^\varepsilon), (U) has affine discrepancy (O(1)).
A second major stream built a large downstream ledger around one common hard code, purification, entropy and dual-total-correlation identities, posterior-normalized Bansal--Jiang motion, protected affine contractions, and immutable owners.
Its important positive chain was:
- a stationary coupled heat bath gave a one-generation common-sign splice without a Dobrushin hypothesis;
- soft midpoint bias could be purified into uniform symmetric subcode children without changing the hidden endpoint;
- entropy deficit and reusable correlation were separated exactly;
- normalized posterior motion and its dependence bill were computed;
- a constrained posterior-dominated SDP gave an exhaustive dichotomy: either large progress or an actual protected low-dimensional dual owner;
- monotone first-hit tangency protected every saturated physical wall on the large-progress branch;
- low-total-variance and absolute small-core terminal recodes were proved;
- a stable absolute-root-mass two-state recurrence was shown conditionally to imply an (O(D)) terminal core.
The same stream also proved that the broad owner terminal is not the missing easy lemma:
- at an exposed face, positive protected variance cannot be removed by further same-law wall-preserving localization;
- allowing an unrestricted direct recode merely because an actual dual owner was returned is already Komlós-hard, even at a product root with a few fixed affine classes;
- owner-visible covariance data can hide an arbitrary Komlós residual in a null fiber;
- repeated renewal, reminting, fair reset, generic total-correlation payment, equality splitting, nonlinear shell release, and several scalar terminal routes fail in their stated scopes.
The exact residual theorem from that stream was a strictly gated immutable owner compiler or nested service. This is still a correct conditional architecture, but it is not currently the shortest route to the target. The weighted theorem moved the most actionable frontier further downstream to the common endpoint between an (O(1)) ultralight service and an (O(D)) heavy service.
A hostile-referee audit corrected several quantifier and accounting overclaims. Preserve these corrections:
- a no-remint compiler must explicitly state the fixed-prefix identity, centered residual, bank initialization, a quantitative ownership inequality, and domination of the actual physical correction by its envelope;
- a recurrence needs nonnegative coefficients, uniform constants, a uniform positive determinant gap, and one global error per depth;
- the curvature recurrence is an absolute-root-mass global frontier statement, not a theorem at every individual node;
- a claimed “pathwise anti-circle” inequality was only an expectation/amortization statement;
- the affine terminal needs the original column norm, cutoff range, and incoming-wall hypothesis;
- a pointwise positive recurrence gap with constants deteriorating in (n,A,D) does not imply an (\exp(O(D^2))) terminal core.
The audit verified conditional algebra; it did not supply the universal compiler.
The strongest positive result then appeared: a weighted extension of the Bansal--Jiang large-(k) method gives (O(\sqrt K)) hereditary discrepancy for matrices with column squared load (K\gtrsim\log^2 n). Rescaling (U) by (1/\tau) gives (\operatorname{lindisc}(U)=O(1)).
This eliminated “prove a better ultralight discrepancy estimate” as the main task. The remaining task became endpoint compatibility.
Three exact tools were then proved:
- the all-center convex refresh radius is between hereditary discrepancy and twice hereditary discrepancy, with factor two sharp;
- a full-residual low-rank plus row-(\ell_1)-defect factorization gives a legitimate terminal rounding using only original columns;
- a spectral cutoff at resolution (D^2/h) supplies such a factor receipt whenever the target-resolution numerical rank is at most (e^{O(D^2)}).
These are genuine terminals, but separate heavy and light convex hulls still do not intersect automatically, and connected Walsh groups refute a universal low numerical-rank dichotomy.
The project formulated the exact lopsided local-lemma splice for one pinned ultralight-good law. It then killed progressively stronger but tempting choices of that law:
- full exterior product domination is unaffordable for every hard-supported law on a random-block fixture;
- the uniform ultralight-good law becomes ferromagnetic on a legal private-ancilla construction;
- exact-mean maximum entropy and product I-projection coincide with that uniform law at a symmetric center, so both fail;
- an (H)-aware convex optimizer that enforces the full lopsided system is feasible exactly when a common endpoint exists, so it is the target in convex disguise.
The surviving law question is narrower and still open: the actual heavy bad event after conjunctions only of nonneighbor heavy-good events under a construction-level sampler.
A parallel deterministic stream proved that pure components compose, while mixed components are payable under a pre-sign factor receipt. It also proved two warnings:
- even the numerical inequality (\operatorname{herdisc}([H;U])\le\operatorname{herdisc}(H)+\operatorname{lindisc}(U)) is false on a legal path fixture;
- an unconditional one-row martingale tail does not imply a statewise small dangerous span, because a rare crossing can be exactly the branch on which a full Walsh core survives.
The current probabilistic missing statement can equivalently be viewed as a survival-unbiased service at dyadic live-count stopping times.
There are universal constants (C,c>0) such that, under the finite-precision and polynomial-input-size hypotheses used by the Bansal--Jiang source, if
[ V\in[-1,1]^{m\times n}, \qquad \sum_iV_{ij}^2\le K\quad(j\in[n]), \qquad K\ge c\log^2n, ]
then
[ \operatorname{herdisc}(V)\le C\sqrt K, \qquad \operatorname{lindisc}(V)\le2C\sqrt K. ]
For (V=U/\tau), declare (K=\lceil\tau^{-2}\rceil). Then
[ |U(y-x)|_\infty \le2C\tau\sqrt{\lceil\tau^{-2}\rceil} =O(1) ]
for every center (x\in[-1,1]^n) and some sign vector (y).
The proof uses a weighted version of the published large-(k) multilevel walk.
- Append the negative rows to make the result two-sided. This doubles squared load only by a constant.
- Discard entries of magnitude at most (1/n), costing at most a constant.
- Partition retained entries into dyadic magnitude subscales.
- Replace incidence-count frontiers by squared-mass frontiers. At level (\ell), maintain the monotone invariant [ \sum_{i:,i\text{ has ever reached level }\ell}V_{ij}^2\le K_\ell. ]
- Treat entry into a level as an adapted birth. Charge each born regularizer explicitly in the exponential potential.
- Use [ \alpha_\ell =6,2^\ell\max{1,K_\ell/\mu} ] for affine spectral independence; the maximum is essential at the final levels.
- Multiply each decoupling count by the squared entry scale and sum geometrically. The birth term contracts and the error term remains within the next squared-mass budget.
- The blocked rank is [ h/5+h/1000+1<h/3 ] while the walk runs, so the printed SDP remains feasible.
- Level costs sum geometrically to (O(\sqrt K)), and the tiny terminal phase costs (O(\sqrt{\mu\log n})=O(\sqrt K)).
- Obtain hereditary discrepancy by zero padding every column restriction back to the original ambient (n).
- Obtain arbitrary-center affine discrepancy from the classical exact inequality [ \operatorname{lindisc}(V)\le2\operatorname{herdisc}(V), ] not by assuming an unprinted arbitrary-start Brownian theorem.
It gives a sign vector in the unrestricted cube. It does not say that the selected sign vector lies in a heavy-good code, respects old saturated faces, has exact mean under a useful law, or has useful heavy-row conditional tails. Conditioning the walk on its ultralight success event may radically change the endpoint distribution. This is the central remaining interface.
When citing this theorem, say “proved relative to the printed Bansal--Jiang SDP-feasibility and GeneralFreedman/decoupling black boxes, with a repaired adapted-birth proof.” Do not say it is printed verbatim in their paper.
For
[ \mathcal S_\Delta(A,m) ={y\in{\pm1}^n:|A(y-m)|_\infty\le\Delta}, ]
define
[ \operatorname{crad}(A) =\sup_{m\in[-1,1]^n} \inf{\Delta:m\in\operatorname{conv}\mathcal S_\Delta(A,m)}. ]
The exact transference theorem is
[ \boxed{ \operatorname{herdisc}(A) \le\operatorname{crad}(A) \le2\operatorname{herdisc}(A).} ]
The factor two is sharp already for the one-column matrix ([1]).
For every linear objective, orient each hereditary signing toward the favorable halfspace during dyadic rounding. This gives an objective-preserving affine rounding. Finite-dimensional separation then places the center in the convex hull of relative-good signings. Carathéodory yields an exact-mean law on at most (n+1) atoms.
The theorem also holds for row-dependent discrepancy profiles. It is hereditary under column restriction.
- A hereditary receipt for one combined stacked wall matrix gives a literal common endpoint law after enlarging the wall by twice the receipt.
- The aggregate local-lemma code for the coded low-heavy rows can be refreshed from every center at relative cost (2t_L=O(D)).
- If an incoming center already lies within a low-heavy wall (T_0), the refreshed endpoint lies within (T_0+2t_L).
Applying the theorem separately to (H) and (U) gives two convex hulls containing the same center. Their vertex sets can be disjoint. Even two centrally symmetric, near-full-entropy endpoint sets can have empty intersection. A combined bound on the stacked matrix would already solve the target; separate refresh laws do not fuse.
Also, the at-most-(n+1)-atom Carathéodory law may be too sparse to have the local randomness needed for a lopsided local-lemma splice. Exact mean is not a conditional-tail certificate.
For a coded low-heavy row (i), let
[ S_i={j:H_{ij}\ne0}, \qquad E_i=\sum_jH_{ij}^2\le E_0. ]
Join two rows when their supports meet. The dependency degree is at most
[ d\le\lceil E_0\tau^{-2}\rceil (\lceil\tau^{-2}\rceil-1). ]
The product-law local lemma gives hereditary low-heavy discrepancy
[ O(\sqrt{\log(d+1)})=O(D). ]
This heavy code is a valid marginal result. It still needs to be combined with the ultralight endpoint.
At the proposed splice center (m), let (\mathcal K_{\mathrm{up}}\subseteq{\pm1}^n) be the endpoint set allowed by every uncoded high-heavy row, saturated face, fixed coordinate, and earlier owner receipt.
A support-changing endpoint law (\mu) must satisfy
[ \mu(\mathcal K_{\mathrm{up}})=1. ]
It is not enough that the old posterior law had this support, or that (m) lies in (\operatorname{conv}\mathcal K_{\mathrm{up}}). A valid interface is one of:
- same-hidden: (\mu\ll\nu) for an old law (\nu) already supported on (\mathcal K_{\mathrm{up}});
- direct support: the new sampler itself proves support one on (\mathcal K_{\mathrm{up}});
- retired rows: the high-heavy rows are constant on the current face or have a separately certified terminal bound.
No weaker mean or covariance condition transfers an upstream endpoint wall through a support change.
Fix a specified law (\mu), chosen by an explicit construction before a common endpoint is known, and assume
[ \mu{y:|U(y-m)|_\infty\le B}=1 ]
and (\mu(\mathcal K_{\mathrm{up}})=1).
Let the incoming low-heavy center wall be
[ |Hm|_\infty\le T_0. ]
For a proposed extra margin (r), define
[ \mathcal B_i={|H_iY|>T_0+r}. ]
If, for every row (i), every set (J) of nonneighbors of (i), and every positive-mass avoidance event,
[ \mu!\left( \mathcal B_i,\middle|, \bigcap_{j\in J}\overline{\mathcal B_j} \right) \le\frac1{e(d+1)}, ]
then the lopsided local lemma gives one atom (y) of the same law satisfying
[ |U(y-m)|\infty\le B, \qquad |Hy|\infty\le T_0+r, \qquad y\in\mathcal K_{\mathrm{up}}. ]
It is enough for the pinned construction to prove, for every low-heavy row (i), every set (J) of nonneighbors, every (t\ge0), and
[ F_J(t)=\bigcap_{j\in J}{|H_jY|\le T_0+t}, ]
that
[ \boxed{ \mu{|H_i(Y-m)|>t\mid F_J(t)} \le C_{\mathrm{tail}} \exp!\left(-\frac{t^2}{2\kappa E_i}\right).} ]
Then choose
[ r=\sqrt{2\kappa E_0 \log(C_{\mathrm{tail}}e(d+1))}. ]
If
[ \tau=e^{-aD^2}, \qquad \log C_{\mathrm{tail}}=O(D^2), \qquad \kappa=O(1), ]
then (r=O(D)).
The law must be fixed by a sampler or process before the desired common atom is found. Merely asserting that some supported law satisfies the lopsided inequalities is equivalent to the desired endpoint:
- a favorable law implies a common atom by the local lemma;
- a common atom gives the favorable point-mass law;
- at a symmetric root, the antipodal pair on (y,-y) also restores exact mean zero.
Therefore the phrase “choose a canonical favorable law” is not a theorem. You must define its transition rule, randomness, failure behavior, and support without invoking the desired common endpoint, and then derive the conditional tails from its provenance.
This is the cleanest current probabilistic target.
Suppose, before signs are chosen, the complete residual matrix has a factorization
[ A=Q+E, \qquad \operatorname{rank}Q\le p, \qquad |E|{\infty,1} :=\max_i\sum_j|E{ij}|\le\Delta. ]
Equivalently, in an owner factorization one may write (Q=LH_T^{\mathsf T}). The witness must describe the complete residual, not merely the covariance-visible action (AH_T).
Kernel compression keeps (Qx) fixed and leaves at most (p) fractional original coordinates. Rounding only those original Komlós columns gives
[ |A(y-x)|_\infty\le\sqrt p+2\Delta, ]
or, using the frozen affine small-core theorem, for every (0<\sigma<1),
[ \boxed{ |A(y-x)|_\infty \le\Phi(p,\sigma)+2\Delta,} ]
where
[ \Phi(p,\sigma) =C_0\sqrt{\log(8\max{1,p\lceil\sigma^{-2}\rceil})} +C_0\sigma\sqrt{p\log(32/\sigma^2)}. ]
Taking (\sigma\asymp p^{-1/2}) gives
[ \Phi(p,\sigma)=O(\sqrt{\log(2+p)}). ]
Thus
[ p\le e^{cD^2}, \qquad \Delta=O(D) ]
is an (O(D)) terminal receipt.
No column-norm bound on the quotient factor (Q) is needed. The final small-core call is made on the original columns, which retain the Komlós norm cap.
If disjoint support components have separate factor receipts, their discrepancy costs compose by a maximum, not a sum, because each physical row lies in one component.
Posterior-visible owner data factor through (AH_T). Matrices (A) and (A+K) with (KH_T=0) are indistinguishable to trace, operator norm, nuclear norm, stable rank, leverage, and row-variance statistics of the owner-visible covariance. A rank-one clone/beacon fixture places both an easy flat matrix and an arbitrary hidden Komlós core in the same visible fiber.
Therefore a small owner covariance is not a terminal. You need a signing-independent approximation of the complete residual with small rank and small row-(\ell_1) defect, or a different restricted theorem.
Let (B) be a residual matrix on (h) live columns and put (M=B^{\mathsf T}B). For (\lambda>0), let (P_\lambda) project onto eigenvalues of (M) larger than (\lambda), and write
[ Q=BP_\lambda, \qquad E=B(I-P_\lambda). ]
Then
[ \operatorname{rank}Q\le N_\lambda(M), \qquad |E|_{\infty,1}\le\sqrt{h\lambda}. ]
At the target resolution
[ \lambda=\frac{D^2}{h}, ]
the defect is at most (D). Therefore
[ N_{D^2/h}(B^{\mathsf T}B)\le e^{cD^2} ]
is a literal (O(D)) factor terminal.
Ridge effective dimension and log determinant certify the threshold rank when small:
[ N_\lambda(M) \le2\operatorname{tr}(M(M+\lambda I)^{-1}), ]
[ N_\lambda(M) \le\frac{\log\det(I+M/\lambda)}{\log2}. ]
These are positive gates, not universal dichotomies.
- Exact rank is discontinuous under near-clones and is the wrong parameter.
- Connected multi-Walsh groups satisfy the column-load and low-heavy budgets while having target-resolution numerical rank ((1-o(1))h) and extensive log determinant.
- An arbitrary number of Walsh groups can be bridge-connected, preventing free componentwise maximum composition.
- For these groups, any approximation with (O(D)) row-(\ell_1) defect may require arbitrarily large rank.
- Computing spectral data only on an owner-visible image misses a hidden Walsh fiber.
Use the spectral receipt aggressively when it passes. Do not spend time trying to prove it passes universally from the current hypotheses; that claim is already refuted.
Let (\mu) be a full-support law on a finite code (\mathcal C\subseteq{\pm1}^h), with mean (m) and covariance (\Sigma). Then
[ \operatorname{rank}\Sigma =\dim\operatorname{aff}(\mathcal C), ]
so
[ \dim\ker\Sigma \le h-\log_2|\mathcal C|. ]
Given a predictable covariance (V_t\succeq0) with
[ \operatorname{range}V_t\subseteq\operatorname{range}\Sigma_t, ]
define
[ Q_t=\Sigma_t^\dagger V_t\Sigma_t^\dagger ]
and evolve the finite-support weights by
[ d\mu_t(y) =\mu_t(y) \langle y-m_t,Q_t^{1/2}dB_t\rangle. ]
Then
[ dm_t=\Sigma_tQ_t^{1/2}dB_t, \qquad d\langle m\rangle_t =\Sigma_tQ_t\Sigma_t,dt =V_t,dt. ]
Thus a Bansal--Jiang admissible covariance can be realized exactly in the mean of one law supported on the existing heavy code. Heavy legality is pathwise because the mean remains in the code's convex hull, and a terminal vertex is a codeword.
This remains a plausible way to construct the missing pinned law. However, a complete proof must address:
- finite-support SDE existence up to weight extinction;
- predictable measurable selection of (V_t);
- range feasibility after support shrinks;
- finite stopping and discretization;
- how the weighted multilevel birth process is represented inside this posterior law;
- support one on all high-heavy and saturated-face constraints;
- event-specific heavy tails after nonneighbor avoidance conditioning.
The old endpoint anti-collision theorem is also available: if a cube martingale has covariance density (V_t\preceq\alpha\operatorname{diag}V_t), then for a terminal pattern on (J), conditionally at time (s),
[ \Pr[X_{\tau,J}=\sigma\mid\mathcal F_s] \le \prod_{j\in J} \left(\frac{1+\sigma_jX_j(s)}2\right)^{1/\alpha}. ]
This prevents certain endpoint collisions and yields min-entropy, but it does not by itself prove the nonneighbor-conditional heavy tails or subtract posterior component constraints. Do not promote it to a common-endpoint theorem without that missing work.
Form the bipartite support graph of the vertical stack ([H;U]), with row vertices and column vertices. A connected component is heavy-pure, light-pure, or mixed.
Let
[ D_H=\operatorname{herdisc}(H), \qquad L_*(U)=\sup_{J\subseteq[n]}\operatorname{lindisc}(U_J). ]
If every original mixed component (C) has, before signs are inspected, a factorization
[ A_C=Q_C+E_C, \qquad \operatorname{rank}Q_C\le p, \qquad |E_C|_{\infty,1}\le\Delta, ]
then
[ \operatorname{herdisc}([H;U]) \le \max{D_H,L_*(U),\Phi(p,\sigma)+2\Delta}. ]
The proof is hereditary because restricting columns only refines components and cannot increase rank or row-(\ell_1) defect.
Consequences:
- if every mixed component has at most (e^{cD^2}) columns, then the stack has discrepancy (O(D));
- if every mixed component passes the target-resolution spectral gate, then the stack has discrepancy (O(D));
- pure components are never a compatibility problem;
- the genuinely hard case is a large, connected, spectrally broad mixed component.
The exact deterministic question is:
[ \boxed{ \text{Can every large mixed component be decomposed into row-disjoint payable pieces,} \text{ or supplied with a broader noncircular terminal receipt?}} ]
Connected Walsh fixtures say the current rank-plus-row-(\ell_1) answer is not universally yes.
A legal scaled path-incidence heavy block plus one ultralight row gives a connected mixed component for which the heavy-optimal signs and light-optimal signs are incompatible. It refutes the numerical inequality
[ \operatorname{herdisc}([H;U]) \le \operatorname{herdisc}(H)+\operatorname{lindisc}(U). ]
The ratio in the fixture approaches two. This is not a growing lower bound, but it proves that “add the two discrepancy values” is not even a valid abstract theorem.
Let (X_t) be a continuous cube martingale with
[ d\langle X\rangle_t=\Sigma_t,dt, \qquad \Sigma_t\preceq\gamma\operatorname{diag}\Sigma_t. ]
For a row (a) with
[ \sum_ja_j^2\le E_0, \qquad \max_j|a_j|\le\rho, \qquad \frac{B\rho}{E_0}\le\frac12, ]
the regularizer
[ Y_t=\langle a,X_t\rangle +\frac{B}{2E_0}\sum_ja_j^2(1-X_t(j)^2) ]
gives
[ \Pr\left{\sup_t|\langle a,X_t\rangle|\ge B\right} \le2\exp!\left(-\frac{B^2}{9\gamma E_0}\right). ]
This closes the scalar-tail calculation under a max-entry condition (\rho\lesssim E_0/D).
The scalar tail does not imply a small dangerous span at a retained state. A legal fixture uses a common random-walk prefix and an untouched normalized Walsh core. The rare event that the prefix crosses the barrier is made exactly the branch on which the Walsh core remains live. On that branch,
[ N_T=h_T=s, ]
where every one of the (s) rows is dangerous and the live gradient rank is (s). Off the branch, both are zero. Unconditionally each row crosses with exponentially small probability, but conditionally on survival the probability is one.
Thus
[ \text{unconditional one-row tail} \centernot\Longrightarrow \text{statewise active-danger dimension}. ]
The minimal missing statement for a dyadic analysis is a survival-unbiased service: at every dyadic live-count stopping time (T_q), conditional on the retained live state,
[ \mathbb E[N_q\mid\mathcal L_{T_q}] \le pK h_{T_q}, ]
where (N_q) counts rows dangerous during that live-set service. If this held and
[ Kp\log n=o(1), ]
then Markov plus a union bound over (O(\log n)) epochs would keep the blocked fraction small. The Walsh fixture proves that this statement cannot be inferred from column incidence and unconditional scalar tails alone. It must come from the actual sampler or a rule that prevents rare-event survival selection.
Every new theorem must be tested against this table. A failure against a fixture does not refute the Komlós conjecture; it refutes the stated mechanism.
| Fixture | What it refutes | What it does not refute |
|---|---|---|
| Exact (4\times4) disjoint heavy/light codes | Adding code deficits or intersecting independently constructed large codes | A literal common-law construction |
| Pair-equality superatoms and seven-coordinate ({\pm p}) factor | Raw coordinate entropy and a false universal anti-collision constant | Component-relative entropy with exact slice completion |
| Majority-pair posterior | Charging only the root code deficit | Current-cylinder deficit (q_v=h_v-\log_2 |
| Aggregate covariance-inflation example | Confining entropy loss to a small coordinate reservoir | Subspace or component accounting |
| 487-column one-death valve | Immediate release of every forced-null direction | A predeclared delayed physical death or one immutable leakage owner |
| Ultra-diffuse paired wall | Free same-wall retirement | Root coding, a burned reservoir, or a paid structural release |
| Spectral cube-wall (I-J/h) at a near-corner | One-shot spectral-owner endpoint jumps | Infinitesimal common-law localization |
| Legal path heavy/light fixture | Choosing separate optimal signs; even naive numerical subadditivity | A simultaneous theorem on the full mixed component |
| Random (q\times q) ultralight hard wall, (q\asymp\tau^{-2}) | Full exterior Radon--Nikodym domination with (\log\Lambda=O(D^2)) for any hard-supported law | Row-tail-only lopsided conditioning |
| Private-ancilla Curie--Weiss fixture | Uniform light-good law, exact-mean maximum entropy, centered-product conditioning, product I-projection | Existence of common good endpoints or an engineered anti-ferromagnetic sampler |
| Canonical convex/Farkas alternative | Calling an optimizer that enforces all lopsided constraints a new construction | Proving those constraints from an independently specified process |
| Near-clones | Exact rank as a stable terminal invariant | Target-resolution numerical rank |
| Connected multi-Walsh groups | A universal small numerical-rank or log-determinant gate | Applying the spectral gate on instances where it actually passes |
| Rank-one beacon with hidden Walsh fiber | Owner-visible covariance scalars as full physical receipts | A pre-sign factorization of the complete residual |
| Walsh survivor-bias fixture | From unconditional one-row tails to conditional dangerous-span bounds | A genuinely survival-unbiased sampler theorem |
| Equality blocks | Nodewise pure-curvature contraction and endlessly renewable boundary service | Global frontier accounting plus permanent release |
| Exposed physical face | Further same-law positive variance in a saturated normal | A support-changing terminal with an independent certificate |
| Three-class actual-dual lift | Unrestricted direct recode of any actual owner | A genuinely restricted low-variance, small-core, or factor terminal |
| Product-root rare tube | Treating a rare (D^2)-variance terminal event as an easier premise | Using such an event after it is derived from a valid compiler |
| Nonlinear shell cage | Total-correlation or affine-hull scalars as universal physical service | A process-generated matrix--code invariant or full residual terminal |
| Fair reset and cosine renewal fixtures | Free independent renewal after purification | A correlated state with explicit accumulated receipts |
Two fixtures deserve detailed recollection.
Fix a constant light wall (B), small cutoff (\tau), take an even (L\ge B/\tau), and put
[ \eta=B/L, \qquad r=L^2, \qquad s=64L. ]
For every edge (e={u,v}) of the complete graph on (s) target signs, add (r) private ancillas and the ultralight row
[ U_e(x,z) =\frac\eta2(x_u-x_v)+\eta\sum_{a=1}^rz_{e,a}. ]
Under the uniform law on the light-good endpoints, the ancilla completion count makes the target marginal exactly
[ \mu_X(x)\propto \exp!\left(\frac{J_L}{2}\left(\sum_vx_v\right)^2\right), \qquad J_L\ge\frac1{16L}, \qquad sJ_L\ge4. ]
For the legal energy-one heavy row
[ H_0=\frac1{\sqrt s}\mathbf1_s, ]
the probability of discrepancy larger than (\sqrt s/2) tends to one. There is only one heavy row, so its dependency degree is zero and the failure is unconditioned. The light-good set is centrally symmetric; therefore uniform, exact-mean maximum entropy, and the centered-product I-projection are the same failed law.
Balanced common endpoints still exist. This is a law no-go, not a Komlós lower bound. It tells you that the sampler must be deliberately anti-ferromagnetic or otherwise subgaussian; maximum entropy is not neutral.
For heavy bad events (\mathcal B_i), nonneighbor avoidance events (F_{i,J}), and target bounds (p_i), define
[ c_{i,J}(y) =\mathbf1_{F_{i,J}}(y) (\mathbf1_{\mathcal B_i}(y)-p_i). ]
The lopsided constraints are linear in a law:
[ \mathbb E_\mu c_{i,J}(Y)\le0. ]
Finite-dimensional Farkas separation says exactly one of the following holds:
- there is an exact-mean law on the light-good set satisfying all inequalities;
- there are a vector (q) and nonnegative multipliers (\lambda_{i,J}) such that [ q\cdot(y-m)+\sum_{i,J}\lambda_{i,J}c_{i,J}(y)>0 ] for every light-good endpoint (y).
At mean zero with symmetric events, feasibility is equivalent to existence of a common endpoint. This is useful because it identifies an exact obstruction certificate, but solving the unrestricted feasibility program is not progress. A legitimate route would prove that dual certificates arising against a particular weighted-walk sampler have an additional structure that is impossible under the column-load geometry.
The cleanest noncircular target is the following. You may adjust constants and the precise sampler, but not weaken the same-endpoint or conditioning quantifiers without saying so.
Fix (D\ge1), (\tau=e^{-aD^2}), and a stopped residual on (h) live original columns. Split it as
[ A=H+U, ]
where:
- every column of the stacked split has squared norm at most one;
- (|U_{ij}|\le\tau);
- every coded low-heavy row (H_i) has energy (E_i\le E_0) and every nonzero entry exceeds (\tau);
- the current center is (m\in[-1,1]^h) with (|Hm|_\infty\le T_0=O(D));
- (\mathcal K_{\mathrm{up}}) is the endpoint set allowed by every high-heavy row, saturated face, fixed coordinate, and earlier immutable receipt.
Construct a randomized endpoint law
[ \mu=\operatorname{Law}(\mathsf S(U,H,m,\mathcal R_{\mathrm{up}};\omega)) ]
by a fully specified sampler (\mathsf S). The sampler may use the matrix and structural heavy supports, but it may not search for or condition on a common heavy/light-good endpoint, solve the full lopsided feasibility problem, or choose its law after inspecting a successful terminal atom.
Prove:
- finite termination: (\mathsf S) returns an original sign vector almost surely or has a finite discretization with explicit success and failure handling;
- ultralight hard support: for a universal (B), [ \mu{y:|U(y-m)|_\infty\le B}=1; ]
- upstream hard support: [ \mu(\mathcal K_{\mathrm{up}})=1; ]
- event-specific lopsided tails: for every coded row (i), every set (J) of nonneighbors of (i), every (t\ge0), and every positive-mass [ F_J(t)=\bigcap_{j\in J}{|H_jY|\le T_0+t}, ] one has [ \mu{|H_i(Y-m)|>t\mid F_J(t)} \le C_{\mathrm{tail}} \exp!\left(-\frac{t^2}{2\kappa E_i}\right), ] with (\kappa=O(1)) and (\log C_{\mathrm{tail}}=O(D^2)), uniformly in (m,n,A) and the stopped state;
- heredity: the same construction and constants apply after every column restriction or live-face transition used by the compiler;
- no hidden benchmark loss: the covariance budgets and error probabilities remain compatible with the Bansal--Jiang baseline branch.
Then the proved one-law local-lemma theorem gives one common atom with
[ |U(y-m)|\infty=O(1), \qquad |Hy|\infty=O(D), \qquad y\in\mathcal K_{\mathrm{up}}. ]
This would close the first-owner heavy/light splice. A complete universal proof would then still need to show that every earlier high-heavy receipt reaches this splice legally and that the process has finite stopping and one-use owner accounting, but it would remove the last new ultralight obstruction.
Full exterior product domination is false. Uniform and maximum-entropy laws are false. Bare existence of a favorable law is circular. However, none of the current no-go fixtures proves that the actual weighted-walk endpoint law fails the much narrower nonneighbor-conditional heavy-row inequality above. The Curie--Weiss fixture attacks a specific canonical law; the random-block fixture attacks full density domination; neither refutes an engineered construction-level sampler.
An honest negative breakthrough would be an actual first-owner state showing that every law generated by a broad, precisely defined class of weighted endpoint samplers violates the lopsided condition, together with a noncircular replacement theorem. A counterexample to one arbitrary convex decomposition is not enough.
The weighted theorem currently proves existence through a randomized multilevel walk and high-probability promotion bounds. Extract the precise law instead of referring to “the weighted-BJ endpoint.” Specify:
- the probability space and all random seeds;
- the SDP optimizer selection rule;
- the adapted birth and promotion stopping times;
- whether bad algorithmic outcomes are rejected, restarted, or included;
- whether conditioning on overall ultralight success is used;
- the exact terminal rounding of the last coordinates;
- the center from which the walk starts;
- whether the law has exact mean (m), approximate mean, or no mean guarantee;
- whether its hard support is the light-good set or only high-probability;
- how support one is achieved without conditioning in a way that destroys heavy tails.
This bookkeeping is not clerical. The endpoint distribution can change drastically when conditioned on global success, and the desired theorem is about that actual distribution.
Reprove the regularizer inequality inside the exact weighted process. For a heavy row (a), track
[ G_t=\sum_ja_j^2(1-X_t(j)^2), ]
and
[ Y_t^\pm =\pm\langle a,X_t-m\rangle+\beta G_t. ]
Under (\Sigma_t\preceq\gamma\operatorname{diag}\Sigma_t), the drift is negative and the quadratic variation is bounded by a constant times the energy clock. Pin all constants, including the condition (B\max_j|a_j|/E_0\le1/2), stopping near cube faces, and the effect of terminal rounding.
This scalar step is largely solved, but it must be integrated with the actual endpoint law rather than cited abstractly.
This is the real work. Explore only hypotheses supplied by the sampler. Plausible directions include:
- A commuting resampling oracle. Show that a nonneighbor heavy event can be resampled through randomness localized away from another row's support while the global ultralight wall remains hard-supported.
- A conditional Ville/Doob argument. Represent avoidance of nonneighbor alarms as a stopping event whose likelihood process has controlled covariation with the target row's exponential supermartingale.
- Construction-level anti-ferromagnetism. Modify the endpoint measure or walk potential to penalize correlated target magnetization while preserving the weighted promotion proof. The Curie--Weiss fixture says this bias must be deliberate.
- Local witness factorization of the global SDP. The weighted SDP couples all coordinates, so disjoint heavy supports are not automatically independent. Look for a decomposition of the random seed or covariance increments that makes nonneighbor avoidance conditionally harmless up to (e^{O(D^2)}).
- A restricted Farkas-dual impossibility theorem. Instead of solving the full lopsided LP, prove that any separator compatible with the weighted potential and birth invariants violates the column squared-load budget.
For every proposal, explicitly test whether it accidentally proves full exterior domination. If so, it is almost certainly too strong and already refuted.
The low-heavy local lemma does not include rows of energy above (E_0). Decide how they reach the first-owner splice:
- same-hidden common-law localization inside their existing hard code;
- monotone saturated-face tangency until those rows are constant on the live face;
- direct inclusion of their endpoint slabs in the sampler support;
- or a separately certified factor/spectral/small-core terminal.
Write the support statement before doing tail algebra. A center wall is not an endpoint wall.
If direct lopsided conditioning is intractable, formulate the exact dyadic statewise service needed by a recursive walk. The theorem must condition on the retained live state, not merely on the past before the survival decision. Either:
- design the sampler so the live-set selection is independent or negatively coupled to heavy crossings;
- randomize ownership before seeing the crossing;
- terminally pay the rare survivor core by a predeclared small-core receipt;
- or establish a complete-frontier absolute-mass inequality that is sufficient without a nodewise claim.
The Walsh survivor fixture should be run as the first unit test.
Try to prove a structural decomposition theorem weaker than global low rank:
- peel row-disjoint pure or factor-payable pieces;
- cut sparse articulation columns and charge their original-column rounding once;
- identify a bounded-treewidth, bounded-cycle-rank, or separator parameter compatible with hereditary restrictions;
- allow each connected Walsh group to be a small core, but prevent an arbitrary number of groups from being bridge-connected into one unpaid component;
- search for an (O(D))-defect decomposition whose ranks compose by a maximum across genuinely row-disjoint pieces rather than a sum.
Any structural theorem must survive the bridge-connected Walsh firewall. If it fails, isolate the exact obstruction and determine whether that obstruction itself carries an exploitable signing.
The broad owner route has been audited deeply. Reopen it only if you can state a hypothesis that excludes the rank-one hidden-fiber fixture and the three-class actual-dual hardness lift. Legitimate gates include:
- total remaining marginal variance at most (D^2);
- absolute live core size at most (e^{cD^2});
- a pre-sign full-residual factor receipt;
- a target-resolution spectral receipt that actually passes;
- a nested immutable service with one-use physical envelopes.
Do not claim that low owner dimension, low owner trace, or actual-dual provenance is enough.
Once a new lemma is proved, do not stop. Compile it into:
- a legal root-to-endpoint algorithm or existence proof;
- hereditary column restrictions;
- finite stopping;
- accumulated discrepancy accounting;
- support preservation for all matrix blocks;
- the exact (D=\log\log n) and (D=(\log n)^\varepsilon) arithmetic;
- a root-level Bansal--Jiang fallback;
- a final theorem with explicit constants and no hidden dependence on (A,m,n).
- Re-derive the Bansal--Jiang quarter-power budget and scale cancellation in Section 3.
- Re-derive the weighted theorem's adapted-birth lemma and affine parameter correction.
- Re-derive the one-law local-lemma implication and the (r=O(D)) arithmetic.
- Do not read all historical owner modules before starting; use the no-go table to avoid known circles.
Deliverable: a two-page internal sheet containing only assumptions, laws, filtrations, and conclusions.
- Extract the weighted multilevel process into a precise finite or stopped continuous algorithm.
- Write its endpoint law before conditioning on success.
- Identify exactly which event makes the ultralight wall hard-supported.
- Determine whether rejection sampling or restarts preserve any usable conditional tail.
- Mark every dependence on (H) and every upstream support constraint.
Deliverable: pseudocode plus a theorem saying exactly what law is produced, with no tail claim yet.
- Prove the scalar exponential supermartingale under the exact covariance rules.
- Track adapted births, promotions, and terminal rounding.
- Test the proof on the Curie--Weiss target row and explain why the engineered law does not become uniform on its hard wall.
Deliverable: an unconditional row-tail theorem for the actual law, or a concrete counterexample to it.
- Start with one target heavy row and one nonneighbor avoidance event.
- Compute the likelihood-ratio or resampling identity exactly.
- Bound its effect on the target exponential supermartingale.
- Generalize only after the two-event calculation works.
Deliverable: either a two-event conditional-tail lemma with constants, or an exact obstruction explaining which global covariance coupling breaks locality.
- If the conditional calculation works, build the induction over a conjunction of nonneighbors and check (\log C_{\mathrm{tail}}=O(D^2)).
- If it fails on the actual process, freeze the counterexample and pivot to a modified anti-ferromagnetic sampler or deterministic component decomposition.
- Do not rename the failed law and repeat the same entropy optimization.
Deliverable: a proved candidate lemma, or a no-go theorem that genuinely narrows the remaining sampler class and names the next replacement.
Before writing “breakthrough,” answer all of the following.
- Are the heavy and light inequalities satisfied by the same sign vector?
- If two laws or convex hulls appear, where is their literal common atom proved?
- If a support change occurs, which incoming endpoint constraints have probability one afterward?
- Is the endpoint law fixed by a stated construction before a common atom is known?
- Did you condition on global success, and if so, did you recompute every tail under the conditioned law?
- Does the law secretly solve the full lopsided feasibility problem?
- Could a point mass on the desired endpoint make the hypothesis true? If yes, the existential statement is likely circular.
- Is the claim unconditional, history-conditional, cylinderwise, or averaged over a frontier?
- If the recursion retains rare branches, have you ruled out Walsh survivor bias?
- Are nonneighbor events genuinely local under the sampler, or only disjoint in matrix support while coupled by a global SDP?
- Is every owner assigned before its favorable release event?
- Is it immutable and charged once?
- Does release delete or terminally certify a named physical component?
- Is the physical correction bounded row by row by a stated envelope?
- Are you restarting an entropy or dependence clock after conditioning?
- Does the terminal use original columns with norm at most one?
- Is a full residual factorization supplied before signs are chosen?
- Is incoming discrepancy added explicitly?
- If a quotient or shadow is rounded, where is the residual defect paid?
- Can the algorithm run the Bansal--Jiang branch on the untouched root input?
- If the new lemma fails, does the claimed universal guarantee actually improve anything?
- Are hidden polylogarithmic factors tracked honestly?
If any answer is missing, label the result conditional and do not continue building downstream on it as if it were proved.
For every theorem, create a proof object with this header:
Claim:
Status:
Universal quantifiers:
Fixed parameters and constant dependence:
Input cylinder / center / filtration:
Output endpoint or release:
Same-endpoint statement:
Support-preservation statement:
Entropy or discrepancy cost:
Termination:
External black boxes and exact hypotheses:
Known hostile fixtures:
What the claim does not prove:
Then perform three audits.
- Quantifier audit: swap the order of every existential and universal quantifier and check that the printed order is the one proved.
- Conditioning audit: write every conditional law and denominator. Treat zero-probability conditioning through a closed convex convention or exclude it explicitly.
- Transfer audit: trace the final sign vector back to the original columns and rows, including every factor-two, incoming wall, shadow, and terminal increment.
Use exact rational or symbolic checks whenever possible. Randomized floating-point experiments are diagnostics, not proofs. For every frozen module:
- record SHA-256 hashes for the theorem note, verifier, manifest, and pinned sources;
- run the checksum check before the verifier;
- disable bytecode output to avoid modifying imported folders;
- report the number and type of checks;
- state what the verifier does not verify;
- run exhaustive small cases for parity, conditional probabilities, support components, and LP/Farkas fixtures;
- separately audit asymptotic inequalities and threshold existence;
- pin all external paper source files used as black boxes.
Typical non-mutating verification invocation:
sha256sum -c SHA256SUMS.txt
PYTHONDONTWRITEBYTECODE=1 python3 verify_module.pyOn a platform without GNU sha256sum, recompute SHA-256 with an equivalent tool and compare the literal digest list.
A verifier can establish that:
- a finite fixture has the claimed exact discrepancy;
- a displayed algebraic identity holds over enumerated cases;
- a recurrence has the claimed eigenvalues;
- constants satisfy inequalities over a tested parameter range;
- imported files match frozen hashes.
It cannot by itself establish:
- a stochastic process exists with all stated measurable selections;
- a source black box applies outside its printed hypotheses;
- a finite pattern proves a universal asymptotic theorem;
- the open common-endpoint sampler exists;
- the universal Komlós theorem has been proved.
This prompt is self-contained, so the new filesystem need not preserve these paths. If the old artifacts are copied, retain their relative folder names and verify hashes before using them. The old project root was logically
komlos_breakthrough_agent_2026-08-23/
The most useful artifacts were:
| Relative artifact | Role | Status at snapshot |
|---|---|---|
integration/komlos_breakthrough_ledger_2026-08-23.md |
Older one-walk and restricted ultralight integration ledger | Historical; later ultralight frontier superseded |
universal_release_attack_2026-08-23/final_breakthrough_ledger_2026-08-23/komlos_downstream_breakthrough_ledger_2026-08-23.md |
Full owner/release synthesis and no-go ledger | Frozen |
universal_release_attack_2026-08-23/synthesis_hostile_audit/hostile_referee_audit_2026-08-24.md |
Quantifier and recurrence corrections | Audit note |
universal_release_attack_2026-08-23/published_splice_audit_2026-08-24/published_large_k_ultralight_splice_boundary_2026-08-24.md |
Initial weighted theorem and exact heavy-shadow firewall | Frozen, but use repaired hostile audit for the theorem proof |
universal_release_attack_2026-08-23/weighted_theorem_hostile_audit_2026-08-24/weighted_large_load_hostile_audit_2026-08-24.md |
Repaired weighted theorem | Frozen and central |
universal_release_attack_2026-08-23/integer_hull_release_2026-08-24/hereditary_integer_hull_transference_2026-08-24.md |
Sharp all-center convex refresh | Frozen and central |
universal_release_attack_2026-08-23/physical_owner_sdp_2026-08-24/physical_owner_sdp_fiber_boundary_2026-08-24.md |
Full-residual factor terminal and owner-fiber no-go | Frozen and central |
universal_release_attack_2026-08-23/spectral_danger_span_2026-08-24/spectral_danger_span_boundary_2026-08-24.md |
Target-resolution spectral gate and Walsh firewall | Frozen and central |
universal_release_attack_2026-08-23/heavy_compatible_exit_2026-08-24/heavy_compatible_ultralight_exit_gate_2026-08-24.md |
Exact one-law lopsided splice | Frozen and central |
universal_release_attack_2026-08-23/local_domination_law_2026-08-24/local_exterior_domination_no_go_2026-08-24.md |
Full exterior-density no-go | Frozen |
universal_release_attack_2026-08-23/canonical_light_law_tail_2026-08-24/canonical_uniform_light_law_ferromagnetic_no_go_2026-08-24.md |
Curie--Weiss no-go | Frozen |
universal_release_attack_2026-08-23/canonical_light_law_2026-08-24/canonical_convex_endpoint_law_boundary_2026-08-24.md |
Max-entropy/I-projection identification and exact Farkas boundary | Frozen |
universal_release_attack_2026-08-23/wall_safe_refresh_2026-08-24/wall_safe_refresh_boundary_2026-08-24.md |
Wall-safe exact refresh interface | Frozen |
universal_release_attack_2026-08-23/simultaneous_heavy_light_terminal_2026-08-24/simultaneous_heavy_light_terminal_boundary_2026-08-24.md |
Mixed-component theorem, path fixture, one-row tail, survivor bias | Live draft snapshot; verifier passed, no frozen manifest at snapshot |
universal_release_attack_2026-08-23/owner_downstream_synthesis_2026-08-24/no_circle_owner_downstream_ledger_2026-08-24.md |
Concise latest synthesis and BJ calibration | Live draft snapshot; verifier passed, no frozen manifest at snapshot |
| File | SHA-256 |
|---|---|
weighted_large_load_hostile_audit_2026-08-24.md |
3477ee9e11d21a956545cd9e67e6a673abc792b22a15bc2298acfbcbb3dd450c |
verify_weighted_large_load_hostile_audit.py |
08276a4e4f65cc5de0a0f28a0f2228da3e4015fabfc08a36d01d3d5c97fb246a |
hereditary_integer_hull_transference_2026-08-24.md |
c5d3824a30f98b717164660767aab02c615e517b3921d4438b5a84f017f8ec82 |
verify_hereditary_integer_hull_transference.py |
b13cef36707b6b05207fdbddb54b946969db86eb87509791ac3069711ce2ab34 |
physical_owner_sdp_fiber_boundary_2026-08-24.md |
d9a570d0f777beadf44e82cc7e6685e3b8a9cc849bf04a6ab8732631e068e90f |
verify_physical_owner_sdp_fiber_boundary.py |
e2411622246950c059b42d53ccfe6a909330efac9213da66ffe01b36b26ebed3 |
spectral_danger_span_boundary_2026-08-24.md |
ef22184300547b7139ff77d6bf5f2f1d54ccd51743b5d2b0b3d9aa2739be57a3 |
verify_spectral_danger_span.py |
dd9c61811a113f246efea94e14e01f3d17aa64044c986143dec167e4772cfd8c |
heavy_compatible_ultralight_exit_gate_2026-08-24.md |
5fead16e81f50d64d6b0efe4784e48027f8a0e220189c826870e9826263f1b01 |
verify_heavy_compatible_ultralight_exit_gate.py |
d233a107905ac8f8c6029b1dc6ef7639e201fb5bacb677d8a043f23955684734 |
local_exterior_domination_no_go_2026-08-24.md |
85e32b1deafd9a53b2f4ffa435794f2c009d5672c0249afcfd30a09736a9006a |
verify_local_exterior_domination_no_go.py |
d96a0e7ae633e2fe10c4ea7b4f0a8a67f5153e1a36d94dc4a745460c0bddf347 |
canonical_uniform_light_law_ferromagnetic_no_go_2026-08-24.md |
e33c86e3c77a6e82fb31f14a127104d1330e97dbeb1d2279a475d2a0312635ad |
verify_ferromagnetic_light_law_fixture.py |
bf1eb2cf832f66af1a68d80f00e424ef3e0f36cf0c8b6b8b754bbceba9d92179 |
canonical_convex_endpoint_law_boundary_2026-08-24.md |
da1ab551accb46e05e95093cda2323bdc647e9f3ea2b1902827f30e0c4c61fc1 |
verify_canonical_convex_endpoint_law_boundary.py |
2a4f3d92a1cc5ad008e81971b55913f27e5a1cfc1554d11b0e3e758e39494ae7 |
wall_safe_refresh_boundary_2026-08-24.md |
b44eebbad3687693e5af22b3c42944f4a941b3d4679699e5393b3579e7addc26 |
verify_wall_safe_refresh.py |
ae1ec7442a34d8ee8776c53cb62fd5e35f42df3a3fcddd9a34d4139e14a04688 |
komlos_downstream_breakthrough_ledger_2026-08-23.md |
52b0955f6c7dd0c66450db77e4e7497ca3e56feeeaf9182cd041dc4b3156891d |
verify_final_breakthrough_ledger.py |
8d7589ccc4223ba94d64422a7613c65d6f0cfb07fbe92e624038403c0dd9ebf1 |
published_large_k_ultralight_splice_boundary_2026-08-24.md |
c9d8cfe6e9321352637a34df49b460612da7aec62ad9065c2adb3591ca0f5c10 |
verify_published_large_k_ultralight_splice.py |
2c0a6be785ad0a97d0906dcb6b52b3270e7a3e0ea45a5d3122b4000f6c63f48b |
These two files were still active drafts and had no frozen manifest at the snapshot time. Treat the hashes as identifiers of the inspected version, not canonical frozen receipts.
| File | Snapshot SHA-256 | Snapshot verifier result |
|---|---|---|
no_circle_owner_downstream_ledger_2026-08-24.md |
1d1ebbba50f93727d6af67fd26d60937d24a91f924f4bab6fc5b12b01e2c5985 |
PASS no-circle owner/downstream synthesis: 68119 checks |
verify_no_circle_owner_downstream.py |
b68832b4a3621ab780370ab6a295f36c2585a419832df656f7c5a4fe87935eeb |
same run |
simultaneous_heavy_light_terminal_boundary_2026-08-24.md |
fdb1790477ed352c05f9c2386340c05f16f4e1bc1e76c5d1dfbf6523b02ac64d |
PASS: 12455 simultaneous/component/factor/source checks |
verify_simultaneous_heavy_light_terminal.py |
be41cdb3bae2ad8fe5c689ee3c732ccd6ef4d7afdc859bbb6bd111a0c9955c60 |
same run |
The following non-mutating runs were observed with bytecode disabled:
| Module | Result |
|---|---|
| Weighted large-load hostile audit | PASS: 117659 checks |
| Hereditary integer-hull transference | PASS: 2171708 checks |
| Physical owner SDP fiber boundary | PASS: 67742 checks |
| Spectral danger-span boundary | PASS: 5510 checks |
| Heavy-compatible ultralight exit gate | PASS: 638708 checks |
| Full exterior-domination no-go | PASS: 1093562 checks |
| Ferromagnetic light-law fixture | PASS total: 285 checks |
| Canonical convex endpoint-law boundary | PASS: 1773 checks |
| Wall-safe refresh boundary | PASS: 170103 checks |
| Frozen downstream synthesis integrity | PASS: 51 frozen files; 86 text guards; 62520 algebra checks in the non-deep run |
| Frozen downstream synthesis integrity, recorded source deep run | 25 source verifiers in addition to the preceding integrity checks |
| Concise owner/downstream synthesis draft | PASS: 68119 checks |
| Simultaneous heavy/light draft | PASS: 12455 checks |
Again: these passes validate pinned finite claims, algebra, source hashes, and guardrails. They do not prove the open endpoint law or the universal theorem.
At minimum, obtain and hash the source of:
- N. Bansal and H. Jiang, Decoupling via Affine Spectral-Independence: Beck-Fiala and Komlós Bounds Beyond Banaszczyk.
- N. Bansal, D. Dadush, S. Garg, and S. Lovett, The Gram--Schmidt Walk: A Cure for the Banaszczyk Blues.
- V. Reis, Optimal Vector Balancing for Zonotopes, if using the low-dimensional zonotope/small-core bridge from the downstream ledger.
The previous weighted audit pinned the following Bansal--Jiang source-file hashes:
| Source file | SHA-256 |
|---|---|
Large_k_Beck_Fiala_Conjecture.tex |
f7004ce9d089f8fb679e4688eaf2fea8467ccd5706ddaf9dda582ebe5ec90075 |
Weak_Beck-Fiala.tex |
3133e9daea1b95dd972179d79d23175e4423e4e230d25efa3f6063dffde1bb5d |
overview.tex |
aa034732d90cf5d3bb9861bdc56118269086a50c704fb5af1ef5629e2bc0de4b |
appendix.tex |
d2ec04dc3fe6b87811de28ffc5fe149a36d8c15e66223dfd8859530070e03bf7 |
Komlos_Proof.tex |
c75b2626b5739cada07189e2925add50a67656fe12d7c7ee18c0c4a1e2b83e38 |
If the current arXiv source differs, do not silently assume equivalence. Record the new version and audit the exact lemmas again.
Any one of the following would materially advance the project:
- a pinned weighted endpoint sampler with the exact nonneighbor-conditional tail and affordable constants;
- a proof that the actual weighted endpoint law has a survival-unbiased dangerous-span service at every dyadic live state;
- a deterministic decomposition theorem that handles arbitrary large mixed components and survives bridge-connected Walsh groups;
- a sharply scoped no-go theorem for a broad natural class of pinned samplers, together with a replacement theorem that is strictly narrower than the full Komlós endpoint problem.
A real sub-quarter result must prove, for every Komlós matrix, one of:
[ |Ay|\infty \le C\varepsilon(\log n)^\varepsilon \qquad\text{for a fixed }\varepsilon<1/4, ]
or
[ |Ay|_\infty\le C\log\log n. ]
It must include every upstream block, finite stopping, hereditary restrictions if used, and a benchmark recovery branch.
A complete handoff back should contain:
- a formally stated universal theorem;
- a proof with all quantifiers and constants;
- a finite algorithmic version or a rigorous finite approximation if stochastic localization is used;
- a source-by-source black-box audit;
- a hostile test suite and exact verifiers;
- a clear explanation of why no law-selection, support, survivor, owner, or terminal circle remains;
- a comparison with the Bansal--Jiang quarter-power proof at the exact budget level.
- another proof that ultralight (U) has a good signing in the unrestricted cube;
- another exact-mean convex decomposition with no heavy conditional tails;
- another entropy scalar or covariance scalar with no physical endpoint receipt;
- an expectation-only dangerous-row count;
- a new name for maximum entropy, product I-projection, or the full lopsided LP;
- an owner of small rank without a complete-residual factorization;
- a structural-class (O(D)) theorem presented as universal;
- a verifier pass for a finite fixture presented as a proof.
The project has already closed the obvious upstream holes. The ultralight block is affinely (O(1)). The low-heavy local-lemma arithmetic is (O(D)). Exact convex refresh, low-rank/row-(\ell_1) terminals, target-resolution spectral gates, low-variance terminals, small-core terminals, and the quarter-power fallback are available. The obstruction is the logic of one endpoint under conditioning and support changes.
Your first question should therefore be:
What concrete randomized mechanism produces an ultralight-good endpoint while retaining construction-level subgaussianity for a low-heavy row after every conjunction of nonneighbor heavy-good events, and while hard-supporting all earlier heavy receipts?
If you can prove that theorem, compile it immediately. If the actual weighted process refutes it, freeze the counterexample and change the mechanism, not the vocabulary. If every plausible sampler route fails, attack the large mixed-component decomposition and use the failure geometry itself as structure. In all cases, keep the Bansal--Jiang branch executable at the root and never let a new proof hole appear downstream of the one you have just closed.
The universal theorem is still open in this handoff. The opportunity is that the remaining wall is now unusually sharp:
[ \boxed{ \text{construction-level common-endpoint compatibility,} \quad\text{not ultralight discrepancy.}} ]
That is the point at which the next agent should begin.