Exact dual witnesses for the attainable-quotient floor T*(y, N)

Session 2026-09-07 (continuation). Input: main 880ec07. Scope: the attainable-quotient problem T* of hunts/quotient_certificate/RESULTS.md (PR #203), kept separate from the all-cell problem V*. Grade of the two values in Section 2: enclosure-carrying (exact rational identities plus interval arithmetic on finitely many log p), and confirmed by an independent route (Codex's rational primal certificates are upper bounds that agree to every printed digit). Everything in Sections 3 and 4 is a lower bound only where it says so, and a negative result where it says so. Nothing here is a theorem about growth in N.

Notation. Q_N = {floor(N/d) : 2 <= d <= N}; A[q, j] = floor(q/j) for q in Q_N, j <= y; L_j = log floor(N/j)!; m_q = sum_{d : floor(N/d) = q} Lambda(d) is the prime-power mass landing in cell q. The dual of T* is

maximize sum_q nu_q s.t. sum_q nu_q floor(q/j) = L_j (j <= y), nu >= 0,

m is feasible with value psi(N), and any feasible nu certifies T*(y, N) - psi(N) >= sum_q nu_q - psi(N) (BARRIER.md Lemma 1 on Q_N).

1. Basic witnesses are bases, and their gains are the tight certificate's elevations

Scope correction (2026-09-07, coordinator). The first version of this section said "every witness is a basis". That is too strong. A feasible dual measure need not be a vertex: mixtures of feasible measures are feasible, and the general feasible set is the polytope {nu >= 0 : A^T nu = L} whose vertices are the basic solutions below. What follows describes the basic (vertex) witnesses; Section 6 gives a non-basic construction.

Take y cells S with A_S (the y x y matrix (floor(s/j))_{s in S, j <= y}) invertible. The mass of any cell q is redistributed onto S by the exact rational rates

r^{(q)} = (A_S^T)^{-1} A[q, :], i.e. sum_{s in S} r^{(q)}_s floor(s/j) = floor(q/j) for all j <= y,

which preserve every moment, and its gain per unit mass is

1^T r^{(q)} - 1 = W_c(q) - 1, c = A_S^{-1} 1,

because c^T A[q, :] = 1^T A_S^{-1} A[q, :] = 1^T r^{(q)}: the primal certificate that is tight on S. So the dual witness for a basis S is nu = sum_q m_q r^{(q)}, its gain is sum_q m_q (W_c(q) - 1), and it is a valid lower bound exactly when nu >= 0 on S; if in addition W_c >= 1 on Q_N the same basis gives the upper bound and T* - psi(N) = sum_q m_q (W_c(q) - 1) exactly. This is complementary slackness written out; its use here is that it turns "validate a witness" into two finite checks with exact arithmetic.

Exactness. L_j = sum_p log(p) l^{(p)}_j with integers l^{(p)}j = sum{k >= 1} floor(N/(p^k j)), so nu^{(p)} = (A_S^T)^{-1} l^{(p)} is a rational vector for each prime p, nu_s = sum_p log(p) nu^{(p)}_s, and the gain is G = sum_p log(p) s_p with rational s_p = 1^T nu^{(p)} - #{k : p^k <= N}. The moment identities A_S^T nu^{(p)} = l^{(p)} are checked as exact rational equalities for every p; the signs of nu_s and the value of G are decided by interval arithmetic (mpmath.iv at 60 digits) on the finitely many log p. "Logarithmic right-hand sides" are never rationalized.

dual_witness.py does all of this from the basis of a floating dual simplex vertex, and refuses to report unless the identities hold exactly.

2. The two witnesses

Both are exact optima: the dual witness (lower bound) and the tight primal certificate (upper bound) come from the same basis and both verify. Codex's rational primal certificates (PR #203) gave 41.28216944 and 226.83268961 as upper bounds; they agree with these values to every digit they print.

Files: results/dual_witness_N1000_y31.json, results/dual_witness_N10000_y100.json (basis, exact primal c, per-cell exchange rates and gains as fractions, the nonzero s_p, the gain interval). Primal c at (1000, 31): c = mu on 1..6, then c_7 = -22/37, c_8 = -15/37, c_10 = 1, c_11 = -11/74, ... , c_31 = 9/74; every gain per unit has denominator 74 (the basis determinant), at (10^4, 100) denominator 8.

3. Which cells exchange mass, and why the moments do not move

(1000, 31): 18 cells are emptied, {7, 11, 13, 14, 18, 20, 23, 31} below y and {32, 34, 52, 62, 111, 125, 142, 200, 250, 333} above, at gains W_c(q) - 1 = 15/37, 63/74, 79/74, 21/74, 35/74, 67/37, 63/74, 83/74 (below) and 83/74, 22/37, 11/74, 39/37, 39/74, 39/74, 57/74, 141/74, 17/37, 1/37 (above). The largest contributions are cell 7 (the primes 127..139, 8.2 of the 41.3), then cells 13, 32, 11, 20, 23, 200 (3.1 to 4.6 each). The basis has 22 cells carrying real mass and 9 carrying none: {26, 28, 30, 35, 38, 66, 71, 100, 166}, all of the form floor(N/d) with every d in the interval composite. The fake prime measure puts mass there.

(10^4, 100): 40 cells emptied (23 below y, 17 above; three of them, 20, 23 and 120, at gain exactly 0, so the vertex is degenerate but the basis is still the support), 41 of the 100 basis cells carry no real mass (16 below y: 40, 46, 48, 56, 58, 60, 65, 70, 77, 84, 92, 94, 95, 96, 98, 100). The emptied cells above y are exactly the cells of d = 97, 89, 83, 71, 67, 49, 43, 31, 27, 23, 19, 13, 9, 8, 5, 3, 2.

The pattern, in words: the optimal fake measure removes the mass of prime powers d from their cells floor(N/d) and places it on attainable cells whose d are composite, at rates fixed by the basis; the moments are unchanged because each rate vector solves sum_s r_s floor(s/j) = floor(q/j) for all j <= y, which is a statement about integer division only. The rates are dense: a unit at cell 333 is spread over all 31 basis cells with coefficients k/37 of both signs. There is no local three- or five-cell relation behind the optimum; the gains, not the rates, are the local quantity.

4. The explicit family that fails: the Mobius prefix

The one basis with a closed form is S0 = {1, ..., y} (all attainable for y <= sqrt(N) - 1). A_{S0} is unitriangular, c = mu_{<= y}, and for q > y

R_q(t) = sum_{k <= y/t} mu(k) floor(q/(t k)), r^{(q)}_s = R_q(s) - R_q(s+1), R_q(y+1) = 0,

R_q(t) being the number of multiples of t up to q whose cofactor has no divisor in [2, y/t]. prefix_witness.py checks these closed-form rates against the exact basis solve for every cell at both sizes: equal. Moving all mass gives the explicit measure nu_s = N(s) - N(s+1) with N(t) = sum_{k <= y/t} mu(k) log floor(N/(t k))!, so the full prefix transfer is feasible exactly when t -> N(t) is nonincreasing on [1, y], and its gain is B_mu(N) - psi(N) = sum_{q > y} m_q (W_mu(q) - 1). Fractional transfer theta_q in [0, 1] is a small LP; its rationalized optimum is re-verified exactly (moments) and by enclosure (signs).

Result: this family certifies almost nothing.

At (1000, 31) M1(31) < 0, so W_mu(q) < 1 on every mass-carrying cell above y and no move gains; at (10^4, 100) M1(100) > 0, every cell gains under mu and the unconstrained transfer would exceed the true optimum five times over, but nonnegativity on S0 (the drains at small s, where R_q(s) - R_q(s+1) is large and negative) admits one cell. So "the Mobius prefix carries the barrier" fails at these two instances. Scope correction (2026-09-07, coordinator): this is a finite statement about (1000, 31) and (10^4, 100); it is not an asymptotic exclusion of prefix-type witnesses, and it says nothing about other supports or cutoffs. Section 6 shows what this family missed: it only withdrew mass from cells above y, and the sign it did not use is the productive one.

The three-cell relation R_q - R_{q-1} = R_1 for a y-rough q (the rough spike in T* form: remove theta <= m_q from q, add it to q-1 and 1, gain theta) needs q y-rough with real mass and q-1 attainable. Correction (coordinator, 2026-09-07): an earlier version said this vanishes at y = sqrt(N) because it looked only below sqrt(N). Consecutive attainable cells exist above sqrt(N): at (10^4, 100) both 102 = floor(N/98) and 103 = floor(N/97) are attainable, 103 is prime, R_103 - R_102 = e_1 = R_1, and log(97) (e_1 + e_102 - e_103) is feasible with gain log 97 = 4.5747. The actual restriction is the attainability of q-1, which above sqrt(N) holds only where consecutive quotients floor(N/d), floor(N/(d-1)) differ by one; Section 8 evaluates the family so defined.

5. What is established, what is refuted, what remains

Established (enclosure-carrying): the two exact values in Section 2, with every moment identity exact and every sign enclosed; they equal Codex's primal upper bounds, so T*(31, 1000) and T*(100, 10^4) are known exactly.

Established (elementary): Section 1, the basis characterization; Section 4, the closed-form rates and the monotonicity criterion for the prefix family.

Refuted: that the Mobius-prefix basis (equivalently, moving mass onto the cells where c = mu is tight) certifies a positive proportion of T* - psi at these sizes; it certifies 0 and 3.6%.

Not established: any witness family valid for all N with gain of order N/sqrt(y), or any lower bound on T*(y, N) - psi(N) beyond the finite values above. The exponent 3/4 measured in RESULTS.md 4.6 remains a conjecture, and nothing here extends it to supports other than y = sqrt(N). The obstacle is now precise: a witness is a basis; the optimal bases at these sizes contain 9 of 31 and 41 of 100 mass-free composite cells, chosen by the simplex, and no rule producing such bases for general (N, y) is in hand. A proof of the barrier for T* needs that rule (or a dual object not tied to a basis, such as a measure-valued construction whose moment identities hold by an arithmetic identity rather than by solving a linear system). No such object was found in this session.

6. A tractable non-basic construction: exchanges through the prefix

The rule, as a perturbation. Rows R_1, ..., R_y are unitriangular, so every attainable q > y has the unique expansion R_q = sum_{s <= y} r^{(q)}_s R_s with the integer rates of Section 4, r^{(q)}_s = R_q(s) - R_q(s+1). Put

z_q = e_q - sum_{s<=y} r^{(q)}_s e_s (a measure on Q_N, q > y).

Lemma 6. A^T z_q = 0 and 1^T z_q = 1 - W_mu(q). The zero-moment measures on Q_N are exactly the linear span of {z_q : q in Q_N, q > y}.

Proof. A^T z_q = R_q - sum_s r^{(q)}_s R_s = 0 by the expansion, and 1^T z_q = 1 - sum_s r^{(q)}_s = 1 - R_q(1) = 1 - W_mu(q). If delta has zero moments, delta' = delta - sum_{q>y} delta_q z_q is supported on {1..y} with zero moments, hence zero by unitriangularity. QED.

(The two-sided LP over the z_q, theta_q >= -m_q, reproduces 41.282169 and 226.832690 at the two sizes, fake_mass_witness.py: the numerical check of the lemma.)

Construction A (additions). delta = sum_{q in Q+} theta_q z_q with theta_q >= 0 and Q+ = {q > y : W_mu(q) < 1}. Support Q_N, A^T delta = 0, gain sum_q theta_q (1 - W_mu(q)) (exact rational), and

m + delta >= 0 iff sum_q theta_q r^{(q)}_s <= m_s for every s <= y.

It creates mass at q, which may carry no real prime power, by withdrawing theta_q r^{(q)}_s from the prefix cells. Its exact failure: the withdrawals land on the cells s in (y/2, y], where r^{(q)}_s = floor(q/s) - floor(q/(s+1)) is 0 or 1 and the real mass m_s is the Lambda-mass of one interval of length N/s^2 in [1, 4], i.e. a single log p or nothing. At (1000, 31) the best single cell is q = 40 (W_mu = -3), bound by s = 31 with m_31 = log 2 (the prime power 32), gain 4 log 2 = 2.7726; the additions LP over all 29 cells of Q+ certifies exactly that; the constant-parameter version theta_q = t on Q+ ∩ (y, Y] has t_max = 0 for every Y because prefix cell 17 (d in {56, 57, 58}, all composite) has m_17 = 0 and a positive total drain. At (10^4, 100) Q+ is empty (M1(100) > 0 makes W_mu(q) >= 1 on every cell above y), so A certifies 0. Conversely Section 4's withdrawal family is empty at (1000, 31) and certifies 8.16 at (10^4, 100). The sign of M1(y) decides which one-sided family exists, and both are capped by the top of the prefix.

Construction B (exchange a -> prefix -> b). For a cell a > y with real mass and any cell b > y, b != a,

delta = theta_b z_b - theta_a z_a, 0 <= theta_a <= m_a, theta_b >= 0.

Support Q_N and A^T delta = 0 by Lemma 6;

m + delta >= 0 iff m_s + theta_a r^{(a)}_s - theta_b r^{(b)}_s >= 0 for every s <= y (and theta_a <= m_a, which is the condition at a); gain = theta_b (1 - W_mu(b)) - theta_a (1 - W_mu(a)).

For the equal-mass exchange theta_a = theta_b = theta this reads

delta = theta (e_b - e_a) + theta sum_s (r^{(a)}_s - r^{(b)}_s) e_s, gain = theta (W_mu(a) - W_mu(b)), feasible iff theta <= m_a and m_s >= theta (r^{(b)}_s - r^{(a)}_s) for all s <= y.

At the top of the prefix, s in (y/2, y], the rate r^{(q)}_s = floor(q/s)

• floor(q/(s+1)) = #{k >= 1 : floor(q/k) = s} is the multiplicity of s

among the quotients of q. Correction (coordinator, 2026-09-07): an earlier version of this paragraph called these rates 0/1 step indicators. They are counts: they are 0 or 1 only for q < s(s+1), and at (10^4, 100), q = 5000, s = 51 the rate is 98 - 96 = 2. The implementation always used the integer rates, so no saved witness is affected; only the explanation was wrong. The condition at the top is therefore: for every s in (y/2, y], m_s >= theta (r^{(b)}_s - r^{(a)}_s) with the full multiplicities. Below y/2 the masses are of order N/s^2 and the rates are the Mobius-weighted counts of Section 4; there the condition is a finite inequality that holds for theta up to a multiple of m_a in every case computed here.

Gains established (exact rationals, every touched cell checked by enclosure, moments checked as exact identities). Pairs are chosen greedily, each step's amounts rationalized to 10^-6 and shrunk until the accumulated delta is feasible; the sum of the steps is a single witness.

Files: results/fake_mass_N1000_y31.json, results/fake_mass_N10000_y100.json (all pairs with positive gain, the greedy sequence, the exact amounts).

What it explains about the saved optima. Mass flows from cells where the Mobius certificate W_mu is high to cells where it is low, and the prefix pays the rate difference. At (10^4, 100) the seventeen emptied cells above y have W_mu = 4, 6, 5, 6, 2, 9, 8, 10, 10, 12, 20, 26, 34, 39, 64, 106, 153 and the twenty-five fake-only basis cells above y have W_mu = 1 .. 15; the first greedy exchange, 232 -> 102, moves log 43 from W_mu = 8 to W_mu = 3. At (1000, 31) every fake-only cell above y has W_mu <= -2 (35, 38: -2; 66, 71: -3; 100: -6; 166: -9) and the prime cells above y are the only sources, so the optimum empties them although W_mu < 1 there too: the removal is the refill of the top prefix cells (r^{(a)}_s >= 0 for s > y/2) that funds the additions. The mediating cells are the prefix's top, whose real masses are single logarithms or zero, and those walls are what the exact optimum balances with its dense rates; the greedy exchange respects the same walls one pair at a time and reaches three quarters and three fifths of the optimum with ten and twenty-five pairs.

Limits, stated. These are lower bounds at two instances. The greedy choice is not claimed optimal; the fraction reached is measured, not predicted; the number of steps was capped at 25; nothing extends to other N, other supports, or a growth rate. What is proved is Lemma 6 and the feasibility criteria displayed, which make any chosen family of exchanges a checkable witness.

8. One prescribed family: halving folds, and exactly where it fails

The formula. For a real-mass attainable cell a > y let q = floor(a/2) = floor(N/(2d)), attainable. Since floor(a/j) - 2 floor(q/j) = floor(a/j) mod 2, define the integer vectors

kappa_i(a) = T_i(a) - T_{i+1}(a), T_i(a) = sum_{m <= y/i} mu(m) (floor(a/(im)) mod 2), T_{y+1} = 0, g(a) = 1 + sum_{j <= y} mu(j) (floor(a/j) mod 2) = 1 + W_mu(a) - 2 W_mu(q),

and the measure on Q_N

Fold(a) = -e_a + 2 e_q + sum_{i <= y} kappa_i(a) e_i.

Lemma 7. A^T Fold(a) = 0 and 1^T Fold(a) = g(a).

Proof. The moment vector of -e_a + 2 e_q is -(floor(a/j) mod 2)_j; the prefix measure sum_j v_j rho_j with rho_j = sum_{i | j} mu(j/i)(e_i - e_{i-1}) has moment vector v (Section 6), and collecting its coefficient on cell i gives sum_{j : i | j} v_j mu(j/i) - sum_{j : (i+1) | j} v_j mu(j/(i+1)) = T_i - T_{i+1} = kappa_i for v_j = floor(a/j) mod 2. The mass of sum_j v_j rho_j is sum_j v_j mu(j), so 1^T Fold(a) = -1 + 2 + sum_j mu(j) v_j = g(a). QED. (Checked as an exact integer identity for every source at the three sizes below.)

The rule. With the prescribed source set A = {a > y : m_a > 0, g(a) > 0} and one parameter t in [0, 1],

D(t) = t sum_{a in A} m_a Fold(a).

Support: the cells a, floor(a/2) and 1..y, all attainable (the prefix must be attainable, which holds at the three sizes). A^T D(t) = 0 by Lemma 7. Net capacity on a cell c, with every multiplicity retained:

m_c + t Delta_c >= 0, Delta_c = -m_c [c in A] + 2 sum_{a in A : floor(a/2) = c} m_a + [c <= y] sum_{a in A} m_a kappa_c(a).

Hence D(t) is feasible iff t <= t* := min over cells with Delta_c < 0 of m_c / (-Delta_c) (and t <= 1), and its gain is

G(t) = t Gamma, Gamma = sum_{a in A} m_a g(a) = sum_{a in A} m_a (1 + W_mu(a) - 2 W_mu(floor(a/2))).

The drift a M1(y) cancels in g(a); what remains is a Mobius sum over the j with odd quotient floor(a/j), of size comparable to sqrt(y) at each source, so Gamma is of the order psi(N/y) sqrt(y) = N/sqrt(y) when t* is of order one.

Conditional lemma. If every cell c with Delta_c < 0 has m_c > 0, then t* > 0 and T*(y, N) - psi(N) >= t* Gamma. The hypothesis is the wall condition: no cell of Z = {c : m_c = 0} is drained by D. It is not a statement about prime density; it is an arithmetic condition on the parity profiles (floor(a/i) mod 2)_{i <= y} of the sources against the set of zero-mass prefix cells, which is where Codex's capacity analysis enters.

Exact failure at the tested sizes. The wall condition fails, and it fails at every source separately:

At the top of the prefix kappa_i(a) is the parity difference (floor(a/i) mod 2) - (floor(a/(i+1)) mod 2) in {-1, 0, 1}, one third of the top cells are walls (36 of 100 at N = 10^4), and a source's parity profile meets a wall with a -1 almost surely; the restrictions F3 (sources above 2y+1) and F4 (sources in (y, 2y+1], the withdrawals of Section 4) have t* = 0 for the same reason. Gamma at (10^4, 100) exceeds the exact optimum 226.83, which is consistent: it is the gain the family would have if the walls did not exist. The F5 witnesses (15.9387 and 5.1761) are exact (moments as integer identities, capacities by enclosure at t = 1), and they are the whole positive content of the family at these sizes.

A drain-free family (Family R). For a > y y-rough with real mass and a-1 attainable, R_a - R_{a-1} = e_1 = R_1 (a has no divisor in [2, y]; if y+1 is prime and divides a the cell y is also touched, which does not occur here), so Shift(a) = e_1 + e_{a-1} - e_a has zero moments and mass 1, and

D_R(t) = t sum_{a in Rough} m_a Shift(a), Rough = {a > y : m_a > 0, a y-rough, a-1 in Q_N},

is feasible for every t in [0, 1] with no condition at all (the only cell that loses is the source, by t m_a <= m_a), with gain t sum_{a in Rough} m_a. This is proved, unconditionally. Its values: 0 at (1000, 31) (no such cell), log 97 = 4.5747 at (10^4, 100) (a = 103), 9.9937 at (3600, 60) (a = 61, 67, 73). Small, but it is the only family here whose gain formula needs no assumption.

What this settles and what it hands over. The prescribed fold rule is correct (Lemma 7), explicit, and fails at the tested sizes for one exact reason, the walls. So the question a capacity analysis has to answer is not the size of the prime mass in the prefix cells but its zeros: the density and placement of the wall set Z in (y/2, y] and whether a source set with a prescribed arithmetic shape can have nonnegative parity drains on all of Z. Nothing here proves or refutes the barrier; the exact optima (Section 2) show that feasible witnesses of the right size exist, and the greedy exchanges (Section 6) show how they route around the walls one pair at a time.

9. Compensated folds: the coordinator's bundle, the band lemma, and a prescribed repair rule

The bundle, verified. At (1000, 31), D = Fold(76) + 2 Fold(200) + Fold(333) (compensated_fold.py --bundle 76:1,200:2,333:1): support in Q_N, A^T D = 0 as integers, gains per unit 2, 4, 0, total 10 units. Withdrawn cells with multiplicities: {2: 3, 3: 1, 5: 4, 7: 7, 9: 3, 11: 3, 12: 3, 14: 3, 16: 3, 20: 3, 23: 1, 27: 1, 76: 1, 200: 2, 333: 1}; every wall has D >= 0 (at 19: -1 + 0 + 1, at 25: 1 - 2 + 1). With M_c the product of the prime bases of the prime powers in cell c (so m_c = log M_c), the largest scale is eps* = log(M_20)/3 = log(7)/3, and feasibility at every withdrawn cell is the integer inequality M_c^3 >= 7^{d_c}, all of which hold (equality at c = 20). Gain (10/3) log 7 = 6.486367163517711017... . Fold(333) has zero gain and is the repair: it refills 17, 19, 22, 25, 30 and drains no wall.

Lemma 8 (band structure of a fold at the top of the prefix). For a real-mass cell a > y and a prefix cell w with y/2 < w < y, w != floor(a/2),

Fold(a)w = sum{k >= 1 : floor(a/k) = w} (-1)^{k+1},

and Fold(a)_y = floor(a/y) mod 2 (the last prefix rate is floor(q/y), not a difference); the destination floor(a/2), if it lies in the range, carries an additional +2. Proof. For w > y/2 the prefix rate of any q is r^{(q)}_w = floor(q/w) - floor(q/(w+1)) = #{k : floor(q/k) = w}, and with q = floor(a/2), floor(q/k) = floor(a/(2k)), so Fold(a)_w = r^{(a)}_w - 2 r^{(q)}_w = #{k : floor(a/k) = w} - 2 #{k even : floor(a/k) = w}. QED. Checked on every source and every such w at the three sizes (255, 1715, 725 cases, 0 violations). So for w > sqrt(a) a fold drains the wall w exactly when the unique k with floor(a/k) = w is even, and refills it when k is odd: the cells a in [kw, k(w+1)) with k odd are the refills of w. Below y/2 the action is the full Mobius-weighted kappa_w(a) of Lemma 7.

Rule C (band-matched repair), prescribed. Start from the profitable folds, coefficient 1 each (P = {a > y : m_a > 0, g(a) > 0}). Walk the walls w in increasing order; at the first wall with D_w < 0, add the fold of the real-mass cell r > y with Fold(r)_w > 0 chosen by the fixed key (drains no wall, then largest g(r), then smallest r), with the integer multiplicity ceil(-D_w / Fold(r)_w); repeat until no wall is negative or the chosen wall has no untried refill, which is reported as the failure. The scale is eps* = min over withdrawn cells of m_c / (-D_c) and the gain is eps* (sum_P g(a) + sum_R c_r g(r)), the second sum being the repairs' gain or sacrifice. Variant C0 uses the key "smallest r".

Results.

At (1000, 31) Rule C reproduces the mechanism of the coordinator's bundle with a different repair set (all five profitable sources plus 2 Fold(333)) and a slightly smaller gain, because its scale is set by the repair's own mass (2 units withdrawn from cell 333, m = log 3) rather than by cell 20.

Conditional Lemma 9. Assume (W_top): every wall lies in (y/2, y]. Assume (R_clean): every wall w has a real-mass refill r_w > y with Fold(r_w)_w >= 1 that drains no wall. Then Rule C closes every wall in at most |Z| repair steps, its bundle is feasible with eps* = min over withdrawn cells of m_c/(-D_c) > 0, and its gain is eps* (sum_P g(a) + sum_w c_w g(r_w)) with c_w = ceil(-D_w / Fold(r_w)w) computed at the moment the wall is treated. Proof. A clean repair increases no wall's deficit, so each wall is treated once and no new deficit appears; feasibility on the mass cells holds for eps below the stated minimum, and the gain is the total mass of the bundle times eps* (Lemma 7). QED. The net withdrawal of the bundle at a mass cell c is sum{a in P} kappa_c(a) + sum_w c_w kappa_c(r_w) (plus -1 or -c_w at the sources themselves), so eps* is the smallest of m_c over these totals; at (1000, 31) that minimum is attained at the repair source.

Exact failure, and the first unproved step. (W_top) is false at (10^4, 100) and at (3600, 60): the walls 33 and 30 lie at or below y/2, where a fold's action is the Mobius-weighted parity sum kappa_w = T_w - T_{w+1}, T_w = sum_{m <= y/w} mu(m) (floor(a/(wm)) mod 2) (three terms at w = 33: parities at 33, 66, 99), which is negative for most sources and for most repairs of other walls; the wall becomes a sink, its deficit grows as other walls are repaired (11, 12, 25, 22 at N = 10^4), and the finite supply of cells with kappa_33 > 0 is exhausted. A lower-half wall is a prime-power-free interval (N/(w+1), N/w] of length about N/w^2 >= 4, i.e. a prime gap of that size near N/w; such gaps exist for every N, so (W_top) cannot be assumed in general, and the walls it fails at sit in a band just below y/2. The first unproved step is therefore: a supply bound for lower-half walls, an arithmetic statement that for each wall w <= y/2 the real-mass cells r with kappa_w(r) > 0 outweigh, in the units of Rule C, the drains that the profitable sources and the other repairs put on w. That is a capacity question in Codex's lane; nothing here proves it, and the failure above is the exact instance where it bites. Not promoted: the labels 76, 200, 333 of the coordinator's bundle, and 333 as Rule C's repair, are facts at one input.

10. The compensated bundle at (10^4, 100): why Rule C misses it, and the credit identity

Verified. With U = sum of the 28 profitable folds (each once, sum U = 91),

D = 2U + 57 F_163 + 7 F_232 + 5 F_270 + 74 F_434 + 40 F_625

• 22 F_1250 + 12 F_2500 + 4 F_5000

checks in my implementation (compensated_fold.py --N 10000 --y 100): support in Q_N, all 100 moments zero as integers, every wall D_w >= 0, total gain sum D = 615 units. With M_c the product of prime bases in cell c, the scale is eps* = log(M_43)/146 = log(229)/146, binding UNIQUELY at cell 43, and every negative coordinate satisfies M_c^{146} >= 229^{d_c} (equality only at 43). Gain E >= (615/146) log 229 = 22.888623508122310848... . The binding cell 43 is a MASS cell (m_43 = log 229), not a wall: the ceiling is drift on the mass cells, not wall closure.

Why Rule C fails, precisely. Cell 33 is a lower-half wall (33 < y/2 = 50) with 2U_33 = -22. It has exactly 6 fold refills (kappa_33 > 0: cells 232, 434, 1250, 1428, 2000, 3333), each contributing only +1 per unit, and NONE is clean: every one drains another wall. Five restrictions in Rule C each block the bundle, and each must be relaxed:

1. Seed multiplicity fixed at 1. The feasible repair set lives at the doubled deficit; with a single seed the credits at cells 54, 62 are too small (see the identity below).
2. One repair source per wall-visit. Cell 33 needs +22 assembled from several of its six +1 refills at once; no single source supplies it while staying feasible elsewhere.
3. The (r, w) "tried" set. It forbids ever raising a source's coefficient or reusing it; the bundle puts 57 and 74 on single cells.
4. The clean-first key. It steers toward refills that drain no wall, but cell 33 has none, so Rule C can never even begin to fill it cleanly.
5. Integer ceil per single wall instead of a joint solve. A one-source fill of cell 33 with 22 units would drain 54 and 62 by 22; only a balanced multi-source block cancels those drains down to 2 and 4.

"No untried refill" is thus an algorithmic stopping condition, not an impossibility: the bundle exists, Rule C's search shape cannot reach it.

The grouped-cancellation identity. The repair block R = 57 F_163 + ... + 4 F_5000 is negative at exactly two walls:

R_54 = -2 = -(2U)_54, R_62 = -4 = -(2U)_62, R >= 0 at every other wall.

So at 54 and 62 the repair spends the entire doubled-seed credit and the final value is 0; these are the credits Codex was told to retain, and this is why they cannot be discarded. Everywhere else R only adds to the walls. It follows that the seed multiplier k = 2 is the UNIQUE feasible one for this R:

k = 1: (U + R)_54 = -1 < 0 and (U + R)_62 = -2 < 0 (under-credited); k = 2: (2U + R)_w >= 0 for all 36 walls (feasible); k >= 3: (3U + R)_33 = -33 + 22 = -11 < 0 (re-opens cell 33).

The repair block itself has zero moments and mass equal to the sum of its gains, 433, so total gain 2(91) + 433 = 615, and the drift it deposits on the mass cells is what cell 43 caps.

Lemma 10 (seed-credit repair, sufficient). Let U be the profitable-fold seed, k a positive integer, and R a nonnegative integer combination of folds of real-mass cells such that (i) (kU + R)_w >= 0 at every wall w, and (ii) k is minimal for (i). Then D = kU + R is a feasible dual witness for T*(y, N) with scale eps* = min over mass cells c with D_c < 0 of m_c/(-D_c) and gain eps* (k sum_a g(a) [over profitable a] + mass R). Proof. Support in Q_N and A^T D = 0 by Lemma 7 (each fold contributes both); at walls D >= 0 by (i); at mass cells D + (1/eps*) m >= 0 by the definition of eps*; the gain is eps* times the mass of D, which is k times the seed mass plus the mass of R by Lemma 7. QED. This is a sufficient condition, applied to the actual bundle: (i) holds with k = 2, (ii) is the k-uniqueness above, and the two facts together certify E >= 22.8886... . It is not necessary (a witness may keep no wall tight), and it does not by itself produce R.

Parameter range proved, and the first remaining gap. Proved at this input: the bundle is feasible with the stated eps* and gain; k = 2 is minimal for this R; the ceiling is a mass cell. The source labels 163, 232, 270, 434, 625, 1250, 2500, 5000 are NOT promoted to any other N. The gap Lemma 10 does not fill is the existence of a wall-closing repair block R whose only wall negatives are covered by kU: this is a nonnegative-integer transportation feasibility whose supply side is the lower-half refills (the six kappa_33 > 0 cells and their analogues), and whether they can be combined to fill every lower-half wall while their side-drains cancel to within the seed credit is the capacity question in Codex's lane. Nothing here shows that supply is sufficient or insufficient in general; at this finite input it IS sufficient, by the exhibited R.

11. The 132.729535 witness in the signed fold basis: the excluded move is the un-fold

The expansion. The folds {Fold(a) : a in Q_N, a > y} are a basis of the 98-dimensional zero-moment space at (10^4, 100), because delta_a = -x_a + 2 sum_{b : floor(b/2) = a} x_b makes the coefficient map triangular. So the prefix-exchange witness of Section 6 has a unique signed fold expansion, solved by the descending recurrence the coordinator gave,

x_q = 2 x_{2q} + 2 x_{2q+1} - delta_q, q in Q_N, q > y (missing indices 0).

signed_fold.py solves it from the witness's saved exact steps and checks reconstruction at EVERY cell including the prefix: exact, and sum_a x_a g(a) equals the witness gain 26545907/200000 = 132.729535. There are 41 nonzero coefficients, 24 positive and 17 negative, with |x|_1 = 138.00.

Corrections (coordinator, 2026-09-08). An earlier version of this section over-read the negatives. Four fixes, each material:

• A positive fold DOES deposit above y: +Fold(a) puts +2 at floor(a/2), which exceeds y once a > 2y (e.g. Fold(434) contributes +2 at 217). The nonnegative family is not barred from cells above y; it is barred from the specific SIGNED combinations below.
• A negative coefficient x_a < 0 is a fold-basis coordinate, NOT a net deposit at cell a. In this witness delta_113 = delta_128 = delta_151 = 0 even though x_113, x_128, x_151 < 0: the recurrence delta_a = -x_a + 2 x_{2a} + 2 x_{2a+1} makes the net at a independent of the sign of x_a.
• 66.338409... is Codex's proved UPPER BOUND on the nonnegative-fold family, not an established optimum; the 17 negatives have not each been proved indispensable.
• "prime cell" below means a cell with positive prime-power mass, m_q > 0, which is a statement about the interval (N/(q+1), N/q] containing a prime power, not about the label q. Cell 103 has m = log 97 because that interval contains the prime 97.

The 17 negative coefficients. With those cautions, the negative coordinates are the fold directions the nonnegative family cannot use (its optimum is bounded by 66.34); realizing 132.73 needs them, and Section 12 accounts for exactly how much each one is worth. Their cells:

(y, 2y]: 102, 113, 123, 125, 126, 128, 129, 133, 142, 144, 151, 156, 163, 172 (14, destination floor(a/2) in the prefix top-half) (2y, .]: 227, 256, 303 (3, with floor = 113, 128, 151, themselves negative cells)

so the negatives form a two-level halving tree: three tail deposits feed the band deposits, which feed the prefix top-half. This is what "push a deposit up into the fake cells" costs in fold coordinates.

One parameterized signed exchange with proved cancellation. Pair a source with its half-destination twin: for k in (y/2, y] with both 2k and 2k+1 attainable,

X_k = Fold(2k+1) - Fold(2k) = -e_{2k+1} + e_{2k} + (kappa(2k+1) - kappa(2k)),

and the half-destination cancels exactly, because floor((2k+1)/2) = floor(2k/2) = k, so the two +2 e_k terms subtract to zero (proved, not fitted; checked on all eight attainable twins, destination gone in every case). A_top X_k = 0 by Lemma 7, and the gain is g(2k+1) - g(2k). This is the atomic prime-to-fake lift underlying the negative coefficients: it deposits at the fake cell 2k using the un-fold e_{2k}, funded by the down-fold at 2k+1.

Concrete capacity argument. X_k's negative coordinates are the source 2k+1 and the cells where kappa(2k+1) - kappa(2k) < 0; the half-destination k is cancelled. It is feasible from m iff every negative cell has m_c > 0, and then its scale is eps* = min over those cells of m_c/(-X_c) -- absence of a zero-mass withdrawal guarantees SOME positive scale, not the full source mass (that coincides only when the binding coefficient is 1). At (10^4, 100) exactly one twin qualifies: X_51 = Fold(103) - Fold(102), whose only negative cell is 103 (m_103 = log 97 > 0 because the interval (N/104, N/103] holds the prime 97; coefficient -1; the prefix correction is nonnegative), so eps* = log 97 and gain (g(103)-g(102)) eps* = log 97 = 4.5747 -- the rough shift of Section 8 in fold coordinates.

Proved parameter domain, and the gap. Proved: for every k in (y/2, y] with 2k, 2k+1 attainable, X_k has zero moments and a cancelled destination, and it is a feasible witness with scale eps* = min_{X_c<0} m_c/(-X_c) and gain eps*(g(2k+1)-g(2k)) whenever every negative cell has positive prime-power mass; at (10^4, 100) X_51 is the only such instance, equal to the rough shift. Not proved: the other 16 un-folds have a negative cell at a zero-mass position, so a single X_k is infeasible there and they enter only inside a coordinated bundle whose zero-mass drains cancel -- the lower-half supply question of the capacity lane. Source labels are facts at this one input, not a formula in N.

12. The h-decomposition of the witness, and the confluence triple

Codex's upper certificate is beta_q = B_q / 1387 on the 52 cells B (listed in grouped_signed.py). With g_a = sum_q Fold(a)_q and

h_a = -g_a - (F^T beta)_a = -g_a - sum_q Fold(a)_q beta_q,

the reduced-cost identity holds for EVERY signed x with nu = m + Fx >= 0:

g^T x = beta^T m + sum_a h_a (-x_a)_+ - sum_a h_a (x_a)_+ - beta^T nu,

by (F^T beta)_a = -g_a - h_a. grouped_signed.py checks h >= 0 on all 98 folds and the identity on the saved witness:

132.729535 = 66.338409 (beta^T m, Codex's upper-bound base)

• 152.056717 (sum over negatives of h_a |x_a|)
• 58.708394 (sum over positives of h_a x_a)
• 26.957196 (beta^T nu, the slack against the certificate).

Two consequences. First, an exact upper handle on the WHOLE signed family: since h >= 0, x_a >= 0 terms and beta^T nu >= 0 only subtract, so

g^T x <= beta^T m + sum_{x_a < 0} h_a |x_a|

for every feasible x. The excess of any signed witness over the upper-bound base 66.338 is at most the h-weighted negative mass. Second, the h-weights say which negatives matter: h_a = 0 at the three cells 113, 163, 303, so those un-folds carry no gain leverage (pure feasibility bookkeeping); the six largest weighted contributions h_a |x_a| are at 102 (24.12), 123 (19.32), 126 (19.18), 133 (16.45), 142 (13.79), 144 (13.28). These weights are diagnostic, not the gain of any single move (correction: a negative coordinate is a fold-basis coefficient, and delta_113 = delta_128 = delta_151 = 0 despite x < 0 there).

One grouped signed construction: the confluence triple. For an even cell b above y with m_b = 0, let k = b/2 and take

C(b) = Fold(2b) + Fold(2k+1) - Fold(b),

the double 2b and the odd twin 2k+1 as sources and the un-fold at b. It has zero moments by Lemma 7 (each fold contributes), and it is more than a twin: adding the double Fold(2b) is what lifts it past the rough shift. Its measure withdraws at 2b, at 2k+1, and at the negative entries of the combined prefix correction kappa(2b) + kappa(2k+1) - kappa(b); the destination deposits +3 at b and +2 - 2 at the prefix cell k (the +2 from Fold(2k+1) and -2 from -Fold(b) offset, though the kappa residue at k does NOT fully vanish, so feasibility is "no zero-mass cell drained", verified, not exact cancellation at k). Scale eps* = min over withdrawn cells of m_c / (-C(b)_c); gain eps* (g(2b) + g(2k+1)

• g(b)).

At (10^4, 100) the only even zero-mass b whose double and twin both carry mass, with positive gain units, is b = 102: 2b = 204 = floor(N/49) (m = log 7, 49 = 7^2), twin = 103 = floor(N/97) (m = log 97), gain units g(204) + g(103) - g(102) = 4 + 3 - 2 = 5, binding cell 204 so eps* = log 7, total gain 5 log 7 = 9.729551. This beats the rough shift log 97 = 4.5747 by adding the double source, and it is a genuine three-fold group, not a twin, not the whole witness, and not the rough shift.

Proved, and the remaining gap. Proved at this input: the identity and the excess bound above (h >= 0 verified on all 98 folds), and C(102) as a feasible signed witness of gain 5 log 7 with every withdrawal on a mass cell. The confluence rule is prescribed arithmetically (sources 2b and 2k+1, destination b, weights +1, +1, -1), and its capacity condition is explicit: m_{2b} > 0, m_{2k+1} > 0, and the prefix residue kappa(2b) + kappa(2k+1) - kappa(b) nonnegative off the mass cells. What is NOT proved, and the reason 132.73 needs the full coupled bundle: at (10^4, 100) b = 102 is the ONLY cell meeting the capacity condition, so the confluence does not by itself compose to a large witness; its supply (double and twin both massed, residue on mass cells) is the same lower-half availability question the capacity lane owns, now sharpened to a condition on the pair (2b, 2k+1). Source labels are facts at this one input, not a formula in N.

13. The halving escalator and the source-exhaustion bound

The control J = Fold(232) + 2 Fold(116) - Fold(102) decomposes the archived exchange 232 -> 102 (its gain 18.806001 is greedy step 1 exactly). Fold(232) deposits +2 at the empty intermediate 116 = floor(232/2); 2 Fold(116) consumes exactly those +2 (net 0 at 116) and carries the mass on. That is one step of the halving escalator.

Lemma 11 (escalator telescoping). For a > y let a = a_0 > a_1 > ... be the halving chain a_{j+1} = floor(a_j/2), and J the first index with a_J <= y. Then

S(a) := sum_{j=0}^{J-1} 2^j Fold(a_j) = -e_a + 2^J e_{a_J} + sum_{j=0}^{J-1} 2^j kappa(a_j),

every intermediate cell a_1, ..., a_{J-1} cancelling. Proof. 2^j Fold(a_j) puts -2^j at a_j and +2^{j+1} at a_{j+1}; consecutive terms cancel at each interior a_j, leaving -e_a (j=0), +2^J e_{a_J} (the last destination), and the accumulated prefix corrections. QED. So a whole chain of empty intermediate cells is self-funded, and S(a) moves one unit from a to the single prefix cell a_J = floor(a/2^J). Verified for a = 232 (chain 232, 116, 58, ...; the intermediate 116 is absent from S(232), G_S = 7, deposit 2^2 = 4 at 58).

The collection, and how the un-fold pays for itself. S(232) alone is infeasible: its prefix corrections drain the zero-mass wall 33. The un-fold -Fold(102) that completes J refills cell 33 (kappa_33(102) = -1, so -Fold(102) contributes +1 there) while depositing at the destination 102. So each archived exchange J_b = S(a) - Fold(b) is one escalator plus one un-fold that simultaneously deposits at b and repairs the wall the escalator opened; J_102, J_123, J_126 are all feasible with gain 18.806 and binding cell 232.

Lemma 12 (source exhaustion -- the capacity estimate; domain and formula corrected after review, 2026-09-08). Assume {1, ..., y} is attainable, a is attainable and above y, and D is a finite set of attainable cells above y EXCLUDING a. For weights theta_b >= 0 write T = sum_b theta_b, with absent theta entries zero. In the collection

C = sum_{b in D} theta_b (S(a) - Fold(b))

the exact coordinate at a is

C_a = -T - 2 (theta_{2a} + theta_{2a+1}),

since S(a) carries -1 at a (Lemma 11) and the two parent labels 2a, 2a+1, if present in D, contribute -2 theta_b at a = floor(b/2) through -theta_b Fold(b), a withdrawal at a. Disjointness from the DESCENDING chain does not exclude those parents: at a = 102, D = {204} the coordinate is -3, not -1. Feasibility m + C >= 0 gives T + 2(theta_{2a} + theta_{2a+1}) <= m_a, hence T <= m_a, and the always-valid bound is

gain(C) = sum_b theta_b (G_S(a) - g(b)) <= m_a * max({0} union {G_S(a) - g(b) : b in D}), G_S(a) = sum_{j<J} 2^j g(a_j),

where the 0 covers an empty D and a negative maximum (a = 103, D = {204}: G_S = 3, g(204) = 4; the all-zero weights are feasible with gain 0 <= 0, whereas a bound without the 0 would read -log 97).

This is independent of |D|. A single source cell delivers at most m_a * max({0} union {G_S(a) - g(b)}) whether its mass is sent to one destination or split across many: the destinations choose only WHERE the deposit lands, not how much total gain is available. At a = 232 (m = log 43): G_S = 7, g(102) = g(123) = g(126) = 2, so the bound is 5 log 43 = 18.806, and the collection over all three destinations indeed totals 18.806 -- the SAME as one exchange (coordinate -3 at 232 by the formula, scale log 43 / 3).

Direction of the bound, and what it does and does not say. Lemma 12 is an UPPER capacity bound for gains obtainable from the restricted source collection sum_b theta_b (S(a) - Fold(b)). It is not a lower bound on T*(y, N) - psi(N). The lower bound at this input comes from the feasible finite example itself: each J_b = S(232) - Fold(b), b in {102, 123, 126}, is a feasible witness, so E >= 5 log 43 = 18.806. The full-capacity 232 exchange empties that source AT THAT STEP; it does not prohibit a later exchange from refilling cell 232 (a superposition can carry a positive contribution at a source from another collection's -Fold(a)), so feasibility of a sum does not by itself grant each collection a separate original-mass budget.

Source count, restricted. A count of sources follows only under explicitly justified shared budgets without refills: pool every term with source cell a and impose T_a <= m_a after pooling. A concrete sufficient no-refill restriction is that every destination set excludes every counted source cell. With H_a = max({0} union {G_S(a) - g(b) : b in D_a}) and a positive uniform bound M on m_a H_a over the counted sources, a family of gain G then needs k >= ceil(G / M) distinct source CELLS (not necessarily distinct prime bases); if M = 0 such a family cannot have positive gain. No assertion is made that the archived 132.729535 witness satisfies these restrictions: that was not checked, and the earlier explanatory claim about why it draws on many primes is withdrawn.

Canonicity of the signed escalator expansion (replaces the earlier "gap"). S(a) and -z_a (Section 6: z_a = e_a - sum_s r^{(a)}_s e_s) have identical coordinates above y, namely -1 at a and 0 elsewhere, and both have zero moments. Their difference is therefore prefix-supported with zero moments, and the prefix matrix (floor(s/j))_{s, j <= y} is unitriangular, hence invertible, so the difference is zero: S(a) = -z_a. Consequently every zero-moment delta supported on Q_N has the unique signed expansion

delta = -sum_{a > y} delta_a S(a),

by matching coordinates above y and invoking the same prefix invertibility. This is standard prefix linear algebra (checked by back-substitution on the seven controls a = 102, 103, 116, 123, 126, 204, 232, and by escalator.py::z_identity_check). Transport pairings, which exchange goes with which destination, can be non-unique; the signed escalator expansion is canonical, and its uniqueness is not an open problem. The remaining issue for general superpositions is the control of refills and shared capacities described above, not uniqueness; its resolution is not a prerequisite for the finite result here.

Evidence label (corrected). escalator.py evaluates scales and gains with point-valued mpmath at 40 digits and serializes floats; it does not carry interval enclosures. Moments, coordinates and the telescoping are exact integer/rational checks. The reviewer's independent exact capacity inequalities (M_c >= 43^{-J_b(c)} for each exchange, M_c^3 >= 43^{-C_c} for the sum, equality only at 232) and Arb enclosure of the gain, [18.806000578467812, 18.806000578467813], support the finite statement; no enclosure is claimed by this file. Source labels are facts at this one input, not a formula in N.

7. Reproduce

OPENBLAS_NUM_THREADS=1 OMP_NUM_THREADS=1 MKL_NUM_THREADS=1 VECLIB_MAXIMUM_THREADS=1 \ .venv/bin/python dual_witness.py --N 1000 --y 31 --output out.json # 12 s ... --N 10000 --y 100 ... # 37 s .venv/bin/python prefix_witness.py --N 1000 --y 31 --output out.json # 0.2 s ... --N 10000 --y 100 ... # 14 s .venv/bin/python fake_mass_witness.py --N 1000 --y 31 --output out.json # 30 s ... --N 10000 --y 100 ... # 100 s .venv/bin/python fold_family.py --N 1000 --y 31 --output out.json # 0.1 s ... --N 10000 --y 100 ...; ... --N 3600 --y 60 ... # 0.1 s each

tests/test_certificate_lp_dual_witness.py pins the (1000, 31) witness (exact identities, sign enclosure, the gain interval, the prefix family's zero) and Construction B's established gain at (1000, 31).