research(proximity): consolidate the latest prize campaign#503
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…odex/deltastar-next-20260710
…NERGY ladder discovery (improves with k; per-rung input = pure-Nat divisor moments) Co-Authored-By: Claude Fable 5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_013QaNxgRNS8yvBSCn6934As
…' into codex/deltastar-next-20260710 # Conflicts: # ArkLib.lean
…' into codex/deltastar-next-20260710 # Conflicts: # ArkLib.lean
…' into codex/deltastar-next-20260710 # Conflicts: # ArkLib.lean
… a SUBGROUP law
Exact-integer probe of the G96/G101 objects (depthFiber, allPairsDepthFiber,
actualDepthAnomaly) at n in {4,8,16}, r = 2..6, two primes each, plus 44
random-set control rows. Findings: odd-depth anomaly nonpositivity holds in
all 14 subgroup cells and FAILS in 16/44 random controls — the G105 consumer
hypothesis is empirically a genuine multiplicative-structure law, and the
even depths carry the whole DC mass. Per-depth positive anomalies each fit
q*Wick throughout; no DC violations anywhere.
Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
…n=32 by G106 The odd-depth nonpositivity claim holds only at n <= 16; the concurrent G106 kernel certificate (p=21523361, r=s=5) shows a positive depth-five anomaly at order 32, and DCEnergyBound itself fails at (64, 16778497, 5). Census correction appended; subgroup-vs-random contrast and budget checks survive. Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
…odex/deltastar-next-20260710
…production lower bound (520093695 bad scalars at the 31/64-predecessor) + full channel; D=3 over-budget stack stated as named residual with refutation reduction Formalizes the SYZ1 channel (docs/kb/deltastar-466-syz1-predecessor-cap-refuted-2026-07-10.md, probe_syzygy_configuration_bad_counts.py) against the literal mcaEvent: - pencil_line_agrees / not_pairJointAgreesOn_insert / mcaEvent_pencil: per-point channel - badScalar_count_ge_of_degenerate_subset / _family: counting forms (single subset + D-subset union) - badScalar_count_ge_of_constructed_stack: fully constructive n−|S| lower bound - rsCode_eq_of_agree_on_card_le: RS interpolation uniqueness at any |S| ≥ k - firstPrime_production_singleSubset_badScalar_lower: explicit production stack with ≥ 31·2^24−1 = 520093695 mcaEvent-bad scalars (≈48% of the hcount budget, one subset) - OverBudgetDegenerateStackExists + firstPrime_predecessor_cap_refuted_of_overBudget_stack: precisely named residual and the ¬hcount reduction. No claim of ¬hcount itself. All axiom-clean (propext, Classical.choice, Quot.sound). Bracket and CORE untouched. Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
# Conflicts: # ArkLib.lean
…odex/deltastar-next-20260710
Marking one matching position in a rung-(k+1) pair bijects (removeNth/ insertNth) onto (position pair) x (free common value) x (rung-k pair): Sum_eq matchCount = (k+1)^2 * #A * E_k and the population analogue, hence the signed matching moment equals (k+1)^2 * #A * (q*E_k - #A^(2k)) >= 0 unconditionally (G95 floor) — an exact positivity immune to the new DCEnergyBound counterexamples. matchCount vanishes exactly on the fully-disjoint maximal-depth sector, pinning all non-descent content at each rung to the full-depth sector. Axiom-clean, 0 sorryAx. Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
Formalizes the signed-depth ledger obstruction: summing all depth buckets is exactly the original signed census total, full graded targets are iff ungraded targets, empty fibers contribute zero, and positive floors consume residual slack rather than creating cancellation. Co-authored-by: wakesync <shadow@shad0w.xyz>
…#505 closed → #509 orbit-class profile; certificate reshaped to signed prefix/small-difference control; chaining metric-universal collapse, GM fence, ET non-contraction, few-weight no-go, spread-excess death; new positive substrate (G93 model theorem, G91F fiber bound, HM brick set, integer-lift rigidity) Note: full compile verification of the workbench cone is blocked by pre-existing broken in-tree file _HalfPredecessorRateQuarterKFourNoEightSevenRootFiber.lean (type error at :1120 + sorryAx, committed in cd5dad8) — flagged on #506; this edit is comment-only (delimiters verified balanced). Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
…ns the depth census
General-m matching moments via the G84/G85 slot machinery: at fixed
embeddings the slice bijects onto (common core word) x (rung-(r-m) pair),
so Sum_eq matchCountM m = (r)_m^2 * #A^m * E_{r-m} and the signed m-moment
equals (r)_m^2 * #A^m * (q*E_{r-m} - #A^{2(r-m)}) >= 0 unconditionally for
every m <= r (G95 floor). An m-matching forces a common sub-multiset of
size m, so row m is blind to depths > r-m: the ladder triangularizes the
depth census against the rung hierarchy, and the fully-disjoint sector
alone carries non-descent information. Axiom-clean, 0 sorryAx.
Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
…, doctrine-v3 pointers, G-number collision naming note; sync AGENTS.md copy Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
…k extended to every coset via faithful ℤ-lift of strip products; first unconditional non-Fourier counting bound on the G80Q terminal object: smallDiffPairs(b·H,W)^8 ≤ 314880·W²·n^12 at 2W² < p Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
…pth fibers
Visibility floor: every pair at cancellation depth s admits >= (r-s)_m
ordered m-matchings, constructed from the G87 maximal-cancellation
representation (padding complements + relative permutation, composed with
all m-selections). Against the G123 identities this yields the moment LP:
Sum_s (r-s)_m * depthFiber A r s <= (r)_m^2 * #A^m * E_{r-m}(A) for every
m <= r — the first linear, simultaneous, triangular constraint system on
the depth census, driven by lower-rung energies; row m = r recovers the
exact permutation envelope. The fully-disjoint sector has weight 0 in
every row: formally the only unconstrained direction. Axiom-clean.
Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
…rnel constants
Near-diagonal moment-LP rows: (r-s)! * depthFiber A r s <=
(r)_{r-s}^2 * #A^{r-s} * E_s for every s <= r (sharp at deep s, e.g.
fiber_{r-1} <= r^2 * #A * E_{r-1}); summing isolates the fully-disjoint
sector: E_r <= depthFiber A r r + explicit lower-rung descent terms.
Through the G96 weld, DCEnergyBound at any prime reduces up to explicit
descent terms to bounding the fully-disjoint equal-sum census — 'the wall
is disjoint-support cancellation' as a theorem chain. Axiom-clean.
Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
…nsus hypothesis Routes G125 isolation through the G96 weld: q*(disjoint census + explicit descent overhead) <= q*Wick + DC mass implies DCEnergyBound G r, at any prime. The production prize hypothesis is now consumable from a single statement about fully-disjoint equal-sum pairs. Axiom-clean. Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
Farkas transfer for the G124 LP: multipliers dominating a cost vector columnwise transfer the row bounds (row m=0 included, dual complete); census_claim_refuted turns any claimed deep-sector census lower bound into a mechanical multiplier search + kernel arithmetic refutation. Axiom-clean. Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
…pth 102 The trivial-energy descent series halves per step (2*(110-k)^2 <= 2^30); geometric tail + one kernel gate (2^3 headroom) give: with only trivial energy bounds, the G126 descent overhead over depths 0..102 fits inside the DC mass n^220 for any q <= 2^160. Only the seven deepest sub-full depths (s = 103..109) need conditional lower-rung energy input; 103 of 110 descent depths are free. Production obligation is now: disjoint census + seven near-full-depth energies vs Wick + DC. Axiom-clean. Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
…ng 110 Given DC-shape bounds q*E_s <= q*Wick_s + n^(2s) at rungs 102..109 only, the FULL 110-depth descent overhead fits a quarter of the DC mass (4*q*overhead <= n^220, kernel gate with ~2^14 headroom). Split-point calibration: with trivial energies the s=102 term alone is ~0.4*n^220; with its DC value ~10^-5*n^220. Tower step: DCEnergyBound at rung 110 follows from the rung-110 disjoint census + DC at eight predecessor rungs. Axiom-clean, 0 sorryAx. Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
…r by (2r-1)‼·q/n^r
The census/moment route sits between the Wick/Gaussian main term Wick_r = (2r−1)‼·n^r
(its natural upper ceiling) and the DC/Parseval floor DCfloor_r = n^{2r}/q (the exact
char-p moment the prize must lower-bound). The adversarial critic's fresh thin-prime family
probe exhibits the exact closed-form law
Wick_r / DCfloor_r = (2r−1)‼ · q / n^r
(verified to 12 sig figs on all 30 resonant rows). The q/n^r factor is exactly the
thin-subgroup ratio; (2r−1)‼ diverges super-exponentially at prize depth r ≈ ln q. So the
Wick ceiling sits a factor (2r−1)‼·q/n^r ABOVE the floor it would certify: bounding the
census by its own Wick ceiling is off from the Parseval floor by a super-exponential factor.
Complements G63 (census pinned at/above the floor from below) by pinning the Wick ceiling
unboundedly above it — the moment route is squeezed out from both sides at the thin prime.
Not a G63/G65 wrapper: G63/G65 bound the census from below; this bounds the DISTANCE between
the two ceilings the route lives between and shows it diverges. Route no-go, not prize
closure; the surviving route remains a joint row-labelled sponsor-prime Jacobi estimate.
Payload (axiom-clean over ℕ/ℝ, only Mathlib Nat.doubleFactorial):
- wick_mul_pow_eq_doubleFactorial_mul_dcFloorNum: division-free identity Wick·n^r = (2r−1)‼·n^{2r}
- wick_dcFloor_ratio_eq: exact ratio Wick/DCfloor = (2r−1)‼·q/n^r
- wick_ge_dcFloor_mul_doubleFactorial: thin regime n^r ≤ q ⇒ Wick ≥ (2r−1)‼·DCfloor
- wick_gt_dcFloor: thin regime, r ≥ 2 ⇒ Wick > DCfloor (strict)
- doubleFactorial_ratio_unbounded_step: (2(r+1)−1)‼ = (2r+1)·(2r−1)‼
- not_wickCeiling_certifies_dcFloor: packaged calibrated no-go
Axioms [propext, Classical.choice, Quot.sound] on all six; no sorry/native_decide.
Locked build 3298 jobs green. Codex P2 (vacuous thin-regime probe check) fixed: probe now
exercises 10 explicit thin rows (all strict) and hard-fails if untested or false.
Frontier/_G261WickCeilingExceedsDCFloor.lean; probe
scripts/probes/g261_wick_ceiling_exceeds_dc_floor_probe.py; KB
docs/kb/deltastar-466-g261-wick-ceiling-exceeds-dc-floor-2026-07-13.md; DISPROOF entry
[466-G261-wick-ceiling-exceeds-dc-floor].
Co-authored-by: wakesync <shadow@shad0w.xyz>
…round Co-authored-by: wakesync <shadow@shad0w.xyz>
…free, no cross-rank coupling leverage The single surviving CORE face after G252-G262 is the DC-subtracted row-labelled covariance gate, required (G214/G225) INDEPENDENTLY at rank five and rank six: Cov_r(W) = m*sum W R_r - (sum W)(sum R_r) = <W, f_r> with centered functional f_r[x] = m R_r[x] - sum R_r (zero-sum). The open hope after G262 + the Fable referee: does the JOINT constraint (one W at both ranks) over-constrain the adversary and give a sign/positivity/norm certificate leverage a single rank lacks (G205 single-depth; G253/G258 single-rank centering)? NO. W |-> (Cov_5 W, Cov_6 W) = (<W,f5>, <W,f6>) is Z-linear, and when f5,f6 are rank-two independent the nonnegative-integer kernel cone maps ONTO all four open sign quadrants. Exact minimal cell m=5, R5=(0,1,0,1,2), R6=(1,0,2,0,1): f5=(-4,1,-4,1,6), f6=(1,-4,6,-4,1), minor=15 (rank two). Four nonnegative-integer kernels, three single-class indicators, realize every (sign Cov5, sign Cov6): e_4->(+,+), e_3->(+,-), e_2->(-,+), e_0+e_3->(-,-). Bounded-support (<=4) stress at m=16,17 still hits all four quadrants; sparsity does not restore forcing. New binding invariant (not a G253/G258/G262 wrapper): rank-two independence of the two centered per-class functionals makes the joint image the whole plane. Orthogonal to G262's total-mass no-go (super-Wick mass annihilated by centering); this fences the centered JOINT gate no prior no-go covered. Axiom-clean [propext, Classical.choice, Quot.sound]; no sorry, no native_decide. Packaged joint_rank_sign_freedom now carries the nonnegativity conjuncts so the statement matches the advertised nonnegative-cone no-go (per Codex review). Scope: route no-go on the joint gate, not a sponsor-prime estimate and not prize closure. The surviving route remains a genuinely row-labelled sponsor Jacobi/cyclotomic covariance estimate at EACH rank; combining rank five and six supplies no additional sign-forcing structure. CORE OPEN / ON-BGK. Payload: Frontier/_G263JointRankSignFreedom.lean; probe scripts/probes/g263_joint_rank_sign_freedom_probe.py; KB docs/kb/deltastar-466-g263-joint-rank-sign-freedom-2026-07-13.md. Co-authored-by: wakesync <shadow@shad0w.xyz>
…r-uniform, not a fixed-depth island
G263 certified joint two-rank centered-gate sign-freedom at one minimal cell (m=5, four
hand-picked nonnegative-integer kernels realizing all four (sign Cov_5, sign Cov_6) quadrants).
G264 upgrades this to a general theorem: for ANY cyclic length m, ANY two zero-sum integer
functionals f,g : Z/m -> Z with a nonzero 2x2 minor D = f_i g_j - f_j g_i at coords i != j, the
nonnegative-integer kernel cone realizes ALL FOUR open sign quadrants of (<W,f>, <W,g>).
Closed-form Cramer witness: for (s,t) in {+-1}^2, a = s g_j - t f_j, b = t f_i - s g_i,
W = c*1 + a e_i + b e_j (c >= 0 offset). Zero-sum f,g kill the offset; Cramer gives EXACTLY
<W,f> = s D, <W,g> = t D. No decide, no fixed cell, no rank restriction. G64 forces the two
centered sponsor rank functionals independent at prize depth r=5,6, so the mechanism is r-uniform
at the actual prize face: combining the two ranks supplies no cross-rank sign-forcing leverage.
Payload Frontier/_G264JointGateSignFreedomGeneral.lean (only Mathlib.Tactic + G263 import):
covPairing_add_const (constant-offset invariance vs any zero-sum functional), covPairing_indicator,
covPairing_twoAtom, cramer_pairings (exact (sD,tD) identity), joint_gate_sign_free_of_minor,
joint_gate_all_quadrants (both-sided strict sign control, all four open quadrants),
g263_functionals_zero_sum + g263_recovered_from_general (recover the landed G263 cell), plus DISPROOF
entry, KB note, self-contained probe (4968/4968 random cells + depth-uniform r=5/6 surrogates).
Axiom-clean [propext, Classical.choice, Quot.sound]; no sorry / native_decide. Locked build 3298
jobs, forbidden-tokens/sorry-census/imports/kb-lint/diff-check all pass. Codex 5.5 review: one P2
(under-stated sign) FIXED to both-sided strict, re-reviewed clean.
Route no-go, r-uniform strengthening of G263, not prize closure. Surviving admissible route
unchanged: row-labelled sponsor Jacobi/cyclotomic covariance at each rank. CORE OPEN / ON-BGK.
Co-authored-by: wakesync <shadow@shad0w.xyz>
…le moves Prove simultaneous quotient-unit relabeling preserves the centered covariance, add an exact characteristic-p primitive-root probe, and correct G258/G264 admissibility scope. Co-authored-by: wakesync <shadow@shad0w.xyz>
…sign quadrants
The current frontier object is the adjacent-rank correlation R_r = dp_r * dp_{r-1}
(distinct from G220's single-subset-sum row). Its physical covariance
A_r = p*sum_x W_G(x) R_r(x) - (sum W_G)(sum R_r) realises all four (sign A5, sign A6)
quadrants over genuine cells including the rare (-,+) at (8,89): A5=-256, A6=+40. This
refutes the cross-rank sign lock (A6>0 => A5>0 false) and, with a (-,-) cell (8,113) and
a (+,+) cell (8,2969), the adjacent-rank forced-sign repair at both ranks. The
thinness-positivity bias is NOT refuted (all n=8 negatives have tau<=1.8) and is left OPEN.
Axiom-free integer certificate (8 theorems, all by decide, zero axioms) + pure-int probe of
record + KB note + DISPROOF entry. CORE remains OPEN / ON-BGK.
Co-authored-by: wakesync <shadow@shad0w.xyz>
…nt-rank CORE covariance
G266 realised all four sign quadrants of the adjacent-rank CORE covariance and left the
thinness-positivity bias OPEN and corroborated, deliberately not formalized (an unconditional
sign law at production primes IS the open BGK-hard target). G267 formalizes the exact FINITE
content: on the enumerated n=8 census (90 cells, 17 <= p <= 2657, p ≡ 1 mod 8) the sign-negative
set is exactly {17, 73, 89, 113}; every sign-negative cell has p-1 <= 112 (tau <= 1.75); every
cell with p-1 >= 136 (tau >= 2.125) is (+,+); the verified thin (+,+) tail reaches p-1 = 2656
(tau = 41.5), a >23x separation gap above the last sign flip.
Formal payload Frontier/_G267ThinnessSeparationCensus.lean: structure Cell + 90-cell census,
theorems census_length, neg_cells_below_threshold, thin_tail_plusPlus, neg_cells_are_exactly,
neg_cells_count, thin_tail_reaches_2656, packaged thinness_separation_census. All by decide;
every theorem depends on ZERO axioms (not even propext), no sorryAx, no native_decide. Probe
scripts/probes/g267_thinness_separation_census.py (pure int) recomputes the full census float-free,
confirms it matches the Lean table exactly, and asserts every separation fact.
Honest scope: a finite separation certificate, NOT a bound at production primes and NOT a proof of
the thinness repair; it sharpens G266's open bias into a precise, monotone, thinness-ordered
invariant (calibrated consumer of G266's data, joint over both ranks r=5,6). Surviving object is
the direct row-labelled sponsor covariance. CORE remains OPEN / ON-BGK.
Gates: lake-locked build PASS (3297 jobs, no warnings); axiom audit zero-axioms on all 7 theorems;
forbidden_tokens clean; sorry_census 0 holes; imports canonical; kb lint PASS; diff --check clean;
Codex 5.5 review: no discrete correctness issues (independently re-verified axioms + probe).
Co-authored-by: wakesync <shadow@shad0w.xyz>
…e adjacent-rank CORE covariance sign
The physical adjacent-rank covariance A_r = p·Σ_x W_G(x)R_r(x) − (Σ W_G)(Σ R_r) admits an exact
per-coordinate centered decomposition: with P(x) := (p·W_G(x) − SW)(p·R_r(x) − SR),
Σ_x P(x) = p·A_r. Split into the DC-diagonal P(0) (x=0 self-collision) and off-DC Poff = Σ_{x≠0}P(x);
then P(0) + Poff = p·A_r exactly.
Closes the "removable DC artifact" simplification: sign A_r = sign P(0) is FALSE in both directions.
Exact float-free witnesses (n=16, r=5, W_G(0)=0): (16,97) has A₅=−6285008 but P(0)=+101818368
(A₅<0<P(0)); (16,433) has A₅=+3425440 but P(0)=−215519232 (P(0)<0<A₅). A third cell (16,257) is a
genuine three-way cancellation (A₅<0, P(0)<0, Poff>0). Census over 160 genuine cells at both orders
(n=8 r∈{3,4}, n=16 r∈{5,6}): sign A_r = sign P(0) in only 120/160, sign A_r = sign Poff in 158/160,
|P(0)|>|Poff| in only 6/160. The covariance sign lives on the off-DC arcs — exactly the open
square-root-cancellation surface — and the DC coordinate is a frequently sign-opposed, negligible term.
Orthogonal to G266/G267 (quadrant/thinness census of A_r's value) and to the antipodal-count floor
route (G268): those study whether A_r is positive; G269 localizes which coordinates carry the sign,
away from the removable diagonal. Does NOT bound A₅ at production primes. CORE OPEN / ON-BGK.
Formal payload: axiom-clean integer certificate `_G269DCCoordinateSignDecoupling.lean` (10 theorems,
all by decide, ZERO axioms), self-contained float-free probe (computation of record, asserts exact
census figures), KB note, DISPROOF entry. Codex-reviewed (two P2 fixes applied: census now genuinely
spans both orders; exact census counts pinned as hard assertions).
Co-authored-by: wakesync <shadow@shad0w.xyz>
…iplicative quotient orbits Formalizes the H_const invariant from Fable's G270 orbit-influence audit: the weighted-relation profile W_G and every field-derived adjacent-rank row R_r are invariant under the order-n 2-power subgroup G <= F_p^*, so the exact per-coordinate centered mass P(x) = (p*W_G(x) - SW)(p*R_r(x) - SR) is constant on every G-coset of F_p^*. The covariance therefore splits into m = (p-1)/n distinct orbit masses rather than p-1 free coordinates: p*A_r = P(0) + sum_j n*P(g^j). Payload (all axiom-clean, [propext, Classical.choice, Quot.sound], built on the G265/G258 unit-relabel machinery so the bookkeeping follows almost for free as G270 anticipated): - sum_centeredMass: sum_x P = N*(N*sum W*R - SW*SR) = p*A_r (exact G269 identity). - centeredMass_unitRelabel: pointwise unit-relabel transport of P. - centeredMass_invariant / centeredMass_orbit_const: orbit constancy P(u*x) = P(x) under G-invariant profiles. - sum_centeredMass_invariant / orbit_decomposition_exact: relabel-invariant total + packaged resolution. Self-contained exact-integer probe verifies orbit constancy, the sum identity, and the orbit decomposition on 30 real char-p cells with a non-degenerate 16-distinct-mass witness. KB note + DISPROOF entry added. Scope: axiom-clean structure lemma, not a sponsor estimate and not prize closure. It certifies that the coordinate description reduces exactly to the orbit description with no further sign structure gained; the G270 census already showed no coarse orbit family is even sign-correlated with A_r. Minimal surviving object is the full character-weighted quotient covariance; surviving admissible route is the direct row-labelled sponsor Jacobi/cyclotomic covariance. CORE OPEN / ON-BGK. Co-authored-by: wakesync <shadow@shad0w.xyz>
…the CORE covariance sign The G270 referee census and the G271 formalizer handoff both name the same rank-1 open question: with the centered mass constant on multiplicative quotient orbits (G271), the adjacent-rank CORE gate factors through Z_m as a Plancherel character sum. Does the SINGLE order-2 (quadratic/Legendre-on-quotient) term track sign(A_r) where the coarse even/odd families cancel? Answer: NO. The correct single order-2 term is What(chi2)*conj(Rhat(chi2)), the product of the SEPARATE transforms of the G-invariant profiles W, R on the quotient (NOT the order-2 Fourier coefficient of the pointwise product P=W*R). Since chi2 is real, wchi2=What(chi2) and rchi2=Rhat(chi2) are exact integers and the term's sign is sign(wchi2*rchi2). Over the canonical n=16,r=5 census at every even-m prime p<2600 (22 cells), sign(term)=sign(A_r) in only 7/22 cells (below chance), realizing all four sign combinations of (sign A, sign term): p=929 (+,-), p=97 (-,+), p=257 (-,-), p=641 (+,+); many cells have term=0. The coarse even-family is separately target-consuming (its complementary-threshold bound is <=> A_r>0). Formalizes, axiom-clean: - charTwo / charTwoTerm (= charTwo w * charTwo r, the correct product-of-transforms object) + charTwoTerm_neg_iff_opposite_sign; - family_split, charTwo_eq_even_sub_odd, evenFamily_consumes_gate (target-consuming equivalence, model-independent); - term_sign_decouples_pos/_neg + term_sign_agrees_neg/_pos (all four sign combinations, decide, axiom-free), not_term_certifies_gate_sign; - no_fixed_order2_sign_law_either_polarity: a genuine CELLWISE no-go (SignLawHolds b ranges only over the four recorded (A,term) pairs), no polarity survives. Self-contained exact-integer probe reproduces the census and hard-asserts the four Lean witnesses. DISPROOF entry + KB doc added. Structure/route no-go: positively confirms the minimal surviving object is the full character-weighted quotient covariance; the sign lives in multi-character inter-orbit interference. CORE OPEN / ON-BGK. #499 respected: research/proximity-prize only, main untouched. Co-authored-by: wakesync <shadow@shad0w.xyz>
Co-authored-by: wakesync <shadow@shad0w.xyz>
…ORE covariance sign On Z_m = F_p^*/G (G271 orbit constancy) the centered quotient profiles give the exact-integer nonprincipal signed CORE gate signed = m*sum w_j r_j - (sum w)(sum r) (carries sign A_r) and the nonprincipal L2 energies E_W = m*sum w_j^2 - (sum w)^2, E_R = m*sum r_j^2 - (sum r)^2, both controlled by any pointwise Weil bound. Cauchy-Schwarz gives signed^2 <= E_W*E_R, so the best any termwise input delivers is the geometric mean sqrt(E_W*E_R). The exact census shows (E_W*E_R)/signed^2 is non-monotone and its worst case ESCALATES with no sponsor-uniform ceiling: K reaches 244647 at p=1153 (L1 ceiling overshoots |signed| by kappa ~ 403). So the signed gate is <= 1/sqrt(K) of the pointwise energy: no termwise/pointwise-Weil bound with sponsor-uniform slack certifies sign(A_r) -- phase cancellation between characters is load-bearing. With G272/G273/G274/G275 this retires the pointwise-estimate/conductor-stratification route class. Axiom-clean; decide-checked witnesses. CORE OPEN / ON-BGK. Co-authored-by: wakesync <shadow@shad0w.xyz>
Split the actual adjacent-rank alignment by integer lift carry. Simultaneous negation makes the carry histogram exactly symmetric, and every lawful antipodal packet has carry zero. Exact integer probes show the full nonzero-carry block is still below the residual deficit in every recorded rank-5/rank-6 cell, including positive gates at both ranks in the same (16,433) cell. The hidden carry-zero characteristic-p residual is load-bearing, so visible wraparound carries do not localize the CORE relation count. Axiom-clean structural identities and calibrated consumers; CORE remains open. Co-authored-by: wakesync <shadow@shad0w.xyz>
…t, cone is antipode-free
-1 in G (2-power subgroup) makes both sponsor factors W_G and R_r coordinate-even,
so B = p*sum W_G R_r - (sum W_G)(sum R_r) is a REAL signed inner product folding onto
the half-line: B = p*[W0*R0 + 2*sum_{x=1}^{(p-1)/2} W_G(x)R_r(x)] - SW*SR, verified exactly
(B_folded=B_direct) and realising both signs (no magnitude positivity). Fable's antipode -R
is NOT a realizable sponsor profile (0/12 cells over ranks/dilations/antipode/shift), so the
sponsor-specific odd-certificate escape hatch (G276/G279) is confirmed genuinely live while
every even/PSD/energy/operator-norm certificate is dead by the abstract law B(W,-R)=-B(W,R).
Sharpens the surviving route to a strictly-odd real-sign alignment statement on the
antipode-free cone. CORE OPEN / ON-BGK.
Payload: _G280SponsorConeAntipodeNoGo.lean (abstract covOdd_neg_right/left +
even_lower_certificate_forces_zero proved in-Lean [propext,Classical.choice,Quot.sound];
folded-pairing identity + both-sign facts by decide, zero axioms);
scripts/probes/g280_sponsor_cone_antipode_probe.py (pure int, A1/A2/A3 hard gates);
KB note + DISPROOF entry. Gates: locked build 3297 jobs PASS zero warnings; forbidden-tokens
clean; sorry census 0 holes; KB lint + check_generated PASS; imports canonical.
Co-authored-by: wakesync <shadow@shad0w.xyz>
…RE gate
Makes Fable's G279 referee calculation a kernel-checked magnitude no-go. A perfect
Eulerian zero-carry-shape theorem J0 <= pi_{r,0}*J plus the proved lawful antipodal
floor J0 >= L (G278) forces only J >= L/pi_{r,0}, which STILL undershoots the gate
J >= B_r/p by 271x-543x at rank 5 and >2e10x at rank 6 at both sponsor primes. Carry
shape is scale-invariant, carries no absolute-mass info, and is not a positivity
mechanism.
With L = antipodal_closed(n,r) (G278), B_r = n^2*C(n,r)*C(n,r-1), and num/den a safe
rational LOWER bound on the exact Eulerian probability (393/1000 <= pi_{5,0}, 73/200 <=
pi_{6,0}), the over-estimate floor L*den/num >= L/pi_{r,0} undershoots the gate exactly
when L*p*den < B_r*num; a fortiori the true floor L/pi_{r,0} does. All four sponsor
cells (n=2^30, P1=n(2^128+192)+1, P2=n(2^129+13)+1) satisfy the exact integer inequality.
Orthogonal to neighbours: G278 proves the floor but not shape sufficiency; G280 pins the
certificate sign shape (odd), while a carry-shape bound is polarity-invariant and dies
here on size, not sign.
Payload _G281CarryShapeAmplificationNoGo.lean: amplified_floor_lt_gate (abstract engine,
[propext,Classical.choice,Quot.sound]), amplified_floor_undershoots (packaged consumer),
four decide certs cert_P1r5/P1r6/P2r5/P2r6 + g281_carry_shape_insufficient_all (zero
axioms). Probe g281_carry_shape_amplification_nogo.py (exact int/Fraction, hard-exit
gates, PASS + exact-pi cross-check + shape scale-invariance); KB note
deltastar-466-g281-carry-shape-amplification-nogo-2026-07-13.md; DISPROOF entry.
Codex review caught a real P2: the certificate must use a LOWER bound on pi (not upper),
else L/(num/den) is smaller than the true floor L/pi_exact and would not imply the exact
floor undershoots. Fixed and re-verified against exact pi at all four cells.
Validation: focused + locked build PASS (3297 jobs, zero warnings); axiom audit clean
(certs zero-axiom); forbidden tokens clean; sorry census 0 holes; import check canonical;
KB lint + generated check PASS; diff-check clean; probe PASS. origin/main untouched
(#499). Magnitude no-go, not prize closure. CORE OPEN / ON-BGK.
Co-authored-by: wakesync <shadow@shad0w.xyz>
Co-authored-by: wakesync <shadow@shad0w.xyz>
Co-authored-by: wakesync <shadow@shad0w.xyz>
…onsor cone
G284 left open whether the ACTUAL sponsor cone puts 0 in its convex hull and whether the
surviving predeclared row-labelled ODD linear normal can carry a positive margin. G286 decides
both exactly. An exact rational LP over the realizable centered profiles c^{(r,a)} = p*R_r - SR
(r=1..n, a in G; the G280 family) at eight sponsor cells (n in {8,16}) shows 0 is strictly OUTSIDE
the convex hull at every cell (six: 0 not in the affine hull, nontrivial separator with explicit
positive margin; two: 0 in the affine hull but out of the convex hull, NO nontrivial linear
separator). So 0 is NOT the barycenter, and convexity alone does not even supply a linear separator.
The decisive fact is PARITY. Every realizable profile is coordinate-even (G280 Fact 1, because
-1 in G for the 2-power subgroup: thinness-essential). Under the reflection sigma: x -> -x, any
coordinate-ODD functional phi satisfies phi(c) = phi(sigma c) = -phi(c), so phi(c) = 0 for every
realizable c: an odd normal annihilates the whole cone and carries no positive margin. Dually the
separation value depends only on the even part, so any positive-margin separator is even -- and an
even functional's pairing is exactly the G276/G280 even inner product, carrying no new binding.
Net: the predeclared ODD-linear-normal route (named in the G56/formalizer handoffs) is
self-contradictory on the actual even sponsor cone. Any surviving certificate must be strictly
NON-LINEAR (odd quadratic or higher).
Formal payload _G286EvenConeOddSeparatorNoGo.lean: abstract even-cone/odd-functional dichotomy over
any Q-module with a linear involution (even_vector_odd_functional_zero,
odd_functional_no_positive_margin, odd_functional_margin_impossible, separation_depends_on_even_part,
positive_margin_normal_not_odd; axioms [propext, Classical.choice, Quot.sound]) plus zero-axiom
decide witnesses at n=8,p=113 (c 1 = c 112 = 5911; odd pairing 0, even pairing 11822). Probe
scripts/probes/g286_hull_zero_probe.py (pure exact-rational LP): gates A-D all hard SystemExit(1),
PASS. Closes the odd-linear separator route; does not bound the covariance and does not exclude a
non-linear odd certificate. CORE OPEN / ON-BGK.
Co-authored-by: wakesync <shadow@shad0w.xyz>
Add exact row-labelled kernel-input character probes and axiom-clean Lean witnesses showing canonical order-2/order-4 normals mismatch CORE in both directions and across adjacent ranks. Integrate the twisted Jacobi fanout asymptotics and retain CORE as OPEN / ON-BGK. Co-authored-by: wakesync <shadow@shad0w.xyz>
Extend the sponsor fixed-point obstruction to arbitrary odd statistics, certify exact Farkas no-gos for the complete generator-independent linear kernel span and its homogeneous quadratic extension, and retain CORE as open on the full row-labelled covariance. Co-authored-by: wakesync <shadow@shad0w.xyz>
…a counting mirage
G286 (odd linear) and G287 (canonical quadratic) close the weighted-kernel separation route one
polynomial degree at a time via exact positive Farkas circuits on the n=16 sponsor census. G289
records the structural reason and shows those no-gos carry no arithmetic content about the CORE gate.
The census has N=84 cells; the generator-independent Ramanujan vector (T2,T4,T8,T16) has only four
base coordinates because the admissible kernel index lives in the thin 2-power tower <2> <= (Z/n)^*.
Every bounded-degree span has dimension d = binom(4+D,D), far below N/2 = 42 for all low degrees. By
Cover's function-counting theorem the separable-dichotomy fraction is astronomically small (2e-19 at
d=5, 3e-10 at d=15), so a no-go is forced by dimension counting, not by the arithmetic gate.
The sharp certificate is gate independence: the same strictly-positive weights that annihilate the
real gate-signed features also annihilate the exactly-flipped gate, so one positive circuit forbids a
strict separator for a gate and its opposite alike. An exact five-cell circuit (p in {113,337,401,433},
avoiding the degenerate p=17 cell) realizes this; Lean checks the four coordinate identities for the
true gate and, via flip_gate_relation, for the negated gate.
Verdict: bounded-degree canonical features can never certify the CORE covariance sign; a surviving
certificate needs unbounded dimension, non-polynomial structure, or new row-labelled arithmetic beyond
the 2-power Ramanujan tower. Route-level no-go, orthogonal to the degree-specific G286/G287. Does not
bound the covariance at production primes. CORE OPEN / ON-BGK.
Formal payload _G289CountingMirageNoGo.lean (axioms [propext, Classical.choice, Quot.sound], no
sorryAx); exact probe scripts/probes/g289_counting_mirage_nogo.py (Cover ceiling + gate-independent
5-cell circuit, both hard-gated); KB note deltastar-466-g289-counting-mirage-nogo-2026-07-13.md;
DISPROOF entry [466-G289-counting-mirage-nogo].
Co-authored-by: wakesync <shadow@shad0w.xyz>
…uniform positive Radon circuit
G289 exhibits a five-cell positive Farkas circuit killing every fixed linear functional of the
canonical Ramanujan features (T2,T4,T8,T16) on the CORE gate, and proves gate-independence, but never
establishes the circuit is minimal. G291 supplies the missing theorem-level deep-floor content: the
census positive circuit is a minimal positive Radon circuit, pinned at the pigeonhole floor N=d+1=5
for the canonical linear feature dimension d=log2 n=4, and it is r-uniform.
Abstract layer (general Fin n -> Fin m -> Q): DeleteOneIndependent + minimal_support_of_deleteOne_
independent + full_support_of_nonzero_dependence prove any nonzero dependence, whose every one-deleted
subfamily is independent, has full support. Concrete consumer on the exact G289 census: census_delete
One_independent proves every 4-subset of the five signed feature vectors is independent via explicit
exact integer Cramer certificates (linear_combination + norm_num); census_circuit_full_support shows
the positive Farkas circuit uses all five cells and is minimal; circuit_r_uniform witnesses the
support spans both ranks r in {5,6}; census_minimal_r_uniform_no_go bundles the three facts.
Sharpens G289: with only d=4 canonical Ramanujan coordinates on the thin 2-power tower, d+1=5
sponsor-faithful cells force a positive circuit and not one cell fewer, and the pigeonhole certificate
carries no arithmetic gate content. Does not bound the covariance at production primes and does not
exclude unbounded-dimension or non-polynomial certificates. CORE OPEN / ON-BGK.
Payload _G291MinimalRadonFloorNoGo.lean (nine theorems, axioms exactly [propext, Classical.choice,
Quot.sound], no sorryAx); exact probe scripts/probes/g291_minimal_radon_floor_nogo.py; KB note
docs/kb/deltastar-466-g291-minimal-radon-floor-nogo-2026-07-13.md; DISPROOF entry
[466-G291-minimal-radon-floor-nogo].
Co-authored-by: wakesync <shadow@shad0w.xyz>
…e A_r = A_{n+1-r}
For n even (the sponsor 2-power regime), the CORE covariance sequence is an EXACT palindrome in the
rank: A_r = A_{n+1-r} for all r in [2, n-1], with W_G(x)=#{(y,z) in G^2: 2y-z=x}, R_r=dp_r*dp_{r-1},
A_r = p*sum_x W_G(x)R_r(x) - (sum W_G)(sum R_r). At the prize ranks this reads A_5 = A_{n-4} and
A_6 = A_{n-5}: the low prize rank is pinned to a high near-complementary rank. Verified exactly on
production cells (n=16 p=1297: A_5=A_12=324135568, A_6=A_11=949261584; n=32 p=193: A_5=A_28=-29432000,
A_6=A_27=-100975808; n=16 p=97: A_5=A_12=-6285008, A_6=A_11=-14107248).
Mechanism (a genuine rank-COUPLING, not a rank-blind feature): n even => -1 in G and sigma=sum(G)=0,
so subset complementation dp_r(x)=dp_{n-r}(-x) gives R_{n+1-r}(x)=R_r(-x); W_G is even, and the
bilinear centered pairing is invariant under reflecting the row while the gate is even, hence
A_{n+1-r}=A_r. This is an exact identity between DIFFERENT ranks via the complementation involution,
orthogonal to the rank-blind no-gos G289/G291 (dimension-forced canonical features) and G293
(rank-blind ordered label list).
Information: the finite low-rank census program (G266/G267/G289/G291/G293, all at r in {5,6},
n in {8,16,32}) has HALF the degrees of freedom it appears to -- its rank-5,6 covariances are
identical to the rank n-4,n-5 covariances, so observed sign-freedom/thinness-bias at the prize rank is
the complementary-rank value. And at TRUE prize depth r ~ log p on a thin cell n ~ p^{1/5.27} one has
r > n so n+1-r < 2: the palindrome is VACUOUS, the prize rank leaves the computable window entirely.
Surviving object sharpened: the certificate must live at depth r >~ n (beyond the palindrome), against
the rank-labelled row directly (BGK/Paley wall), and cannot be extracted from any finite low-rank
census whose prize ranks are pinned to their complementary partners.
Scope: structural rank-coupling identity, NOT a Jacobi estimate and NOT a prize closure. Payload
_G295RankReflectionSymmetry.lean (neg_involutive_sum, centeredCov, centeredCov_reflect_of_even = the
mechanism, W17_even, reflectR_3_6, A17_3_eq_A17_6 = exact ZMod 17 prize-adjacent A_3=A_6=-1344);
axioms exactly [propext, Classical.choice, Quot.sound], no sorryAx/custom/native_decide. Exact probe
scripts/probes/g295_rank_reflection_symmetry.py (even-n mechanism+palindrome on 8 cells, prize-rank
pins, ZMod 17 witness; PASS). KB note docs/kb/deltastar-466-g295-rank-reflection-symmetry-2026-07-13.md;
DISPROOF entry [466-G295-rank-reflection-symmetry]. CORE OPEN / ON-BGK.
Co-authored-by: wakesync <shadow@shad0w.xyz>
…census to (n-2)/2 values
G295 gives the exact per-cell palindrome A_r = A_{n+1-r} on [2, n-1] for n even.
G296 upgrades it to a census information bound: the reflection sigma r = n+1-r is a
fixed-point-free involution of the rank window Icc 2 (n-1), partitioning its n-2 slots
into (n-2)/2 orbits, so any palindromic sequence has census image of cardinality at most
(n-2)/2. General bound proved for EVERY even n >= 4 (reps_card_eq, palindrome_census_card_le_half),
not just hard-coded cells. Saturated on production cells (n=16: 7 distinct over 14 slots at
p in {97,193}; n=8: 3 over 6 slots at p=17). r-uniform sharpening of G295: with the prize rank
r ~ log p on a thin cell n ~ p^{1/5.27} escaping the window (r > n so n+1-r < 2), no low-rank
census argument can furnish an independent prize-rank value. Certificate must live at depth r >~ n
against the rank-labelled row (BGK/Paley wall). CORE OPEN / ON-BGK.
Formal payload Frontier/_G296PalindromeCensusCollapse.lean (15 theorems, load-bearing new
lemmas sigma_no_fixed_point, window_eq_reps_union_image, palindrome_image_card_le, the general
orbit count reps_card_eq : (reps n).card = (n-2)/2 for even n >= 4, and the headline
palindrome_census_card_le_half; exact ZMod 17 consumer census17_card_eq = 3). Axioms exactly
[propext, Classical.choice, Quot.sound]; no sorryAx/native_decide/custom. Probe
scripts/probes/g296_palindrome_census_collapse.py PASS. KB
docs/kb/deltastar-466-g296-palindrome-census-collapse-2026-07-13.md.
Codex 5.5 review: general (n-2)/2 bound gap (P2) fixed by adding reps_card_eq +
palindrome_census_card_le_half.
Co-authored-by: wakesync <shadow@shad0w.xyz>
…ve-energy threshold with sign not fixed by thinness
At depth r=1 the adjacent-rank row degenerates (R_1 = dp_1 * dp_0 = 1_G), so the
centered CORE covariance has the exact closed form A_1 = p*T3(G) - n^3, where
T3(G) = #{(y,z) in G^2 : 2y-z in G} = #{(y,z,w) in G^3 : 2y = z+w} counts 3-term
APs with midpoint in the multiplicative subgroup. Hence sign A_1 = sign(T3/n^2 - n/p):
the depth-1 covariance is exactly the subgroup's additive-3AP density against the
random density, the Paley/BGK object at depth one.
No-go: sign A_1 is not a function of thinness n. Fixing n=8 AND T3=24, the sign flips
across the threshold n^3/T3 = 512/24 ~ 21.33: (p,G)=(17,<9>) gives A_1=-104<0 and
(41,<3>) gives A_1=+472>0. So no depth-1 energy functional of shape p*T3 - n^3 signs
the CORE gate across sponsor primes; the certificate must use the rank-labelled row at
genuine depth. Orthogonal to G289/G291/G293 (rank-blind) and G295/G296 (rank palindrome).
Formal payload _G298Depth1EnergyThreshold.lean (centeredCov, centeredCov_indicator,
A17_neg=-104, A41_pos=472, T3_17=T3_41=24, depth1_sign_indeterminate); axioms exactly
[propext, Classical.choice, Quot.sound], no sorryAx/custom/native_decide. Exact probe
g298_depth1_energy_threshold.py PASS. KB + DISPROOF entry [466-G298]. CORE OPEN / ON-BGK.
Co-authored-by: wakesync <shadow@shad0w.xyz>
…drome census window, refuting G296's escape prose
G296's docstring/KB/DISPROOF claimed the prize rank r≈log p on a thin cell
n≈p^{1/5.27} satisfies r>n so n+1-r<2 (escapes the window), and the certificate
must live at depth r≳n. This is false at the campaign's own production parameters
(q=2^158, n=2^30, ledger depth r*≈89): 2≤89<2^30, and asymptotically log_n q grows
far slower than n, so the prize depth and its reflection σ(2^30)89=2^30-88 both lie
deep inside the window [2,n-1].
G299 kernel-checks the correction, reusing G296's window/sigma:
prizeDepth_lt_order (89<2^30), prizeDepth_mem_window, prizeDepth_reflection_eq
(σ=2^30-88 ≥2, not <2), prizeDepth_reflection_mem_window, not_escape_window
(¬(r*>n ∧ σ n r*<2)); r-uniform log bound natLog_prize_eq_five (Nat.log(2^30)(2^158)=5),
natLog_prize_lt_order, general natLog_le_of_lt_pow (q<n^n ⇒ Nat.log n q≤n),
natLog_prize_le_order, escape_would_need_pow (n<Nat.log n q ⇒ n^n≤q, unreachable
since 2^158≪(2^30)^(2^30)). Axioms exactly [propext, Classical.choice, Quot.sound].
The G295 palindrome and G296 census-collapse theorems are untouched and remain
sound; G299 only removes the incorrect frontier inference and reclassifies them as
symmetry/compression on the low-rank window. The surviving CORE object is unchanged
and correctly located at the in-window depth r*≈89. CORE OPEN / ON-BGK.
Probe scripts/probes/g299_prize_depth_in_window.py (pure integer, exponent-only,
hard-exit; PASS). Note docs/kb/deltastar-466-g299-prize-depth-in-window-2026-07-13.md.
Independently flagged by Fable referee (03:15) and G56 (04:20).
Co-authored-by: wakesync <shadow@shad0w.xyz>
…illates, forbidding any monotone/single-band certificate at the in-window prize depth
G299 proved the prize depth r*~89 and its palindromic reflection both lie inside
the low-rank window [2,n-1] at production (q=2^158, n=2^30). That reopens a
shortcut hope: if A_r were sign-definite / rank-monotone / a single sign band on
the window, in-window placement would sign the CORE covariance at prize depth
from cheap boundary data. G300 kills it.
On the exact sponsor cell (p,n)=(113,8), G=<15><=F_113^* (order 8), the centered
covariance A_r = p*sum_x W_G(x)R_r(x) - (sum W_G)(sum R_r), R_r = dp_r * dp_{r-1},
is A=[392,128,-7240,-13128,-13128,-7240,128], signs [+,+,-,-,-,-,+]: A_2=128>0,
A_4=-13128<0, A_7=128>0, an interior sign change strictly inside [2,7]. So the
covariance is not sign-definite, not rank-monotone on the reps half, not a single
sign band. Its value at the in-window prize depth r* cannot be read off from a
boundary datum or a monotonicity assumption; a certificate must consult the row
R_{r*} at genuine depth, the BGK/Paley wall. In-window does not mean sign-readable.
Not a small-cell artifact: (257,32) has the period-4 profile with eleven
strictly-interior sign changes at r in {5,7,...,25}.
Payload _G300WindowSignOscillation.lean (centeredCov, sum_zmod_eq_range reindexing,
marginals, A2_113=128/A2_pos, A4_113=-13128/A4_neg, R7_eq_R2/A7_eq_A2, headline
window_sign_oscillates); axioms exactly [propext, Classical.choice, Quot.sound],
no sorryAx/native_decide/custom. Probe g300_window_sign_oscillation.py (PASS). KB
note + DISPROOF entry [466-G300-window-sign-oscillation]. CORE OPEN / ON-BGK.
Co-authored-by: wakesync <shadow@shad0w.xyz>
…um structure Formalize coefficient-coset constancy, pointwise conservation, and the centered zero-sum family; add exact adjacent-rank transport reversals and correct the G295/G296 production-depth prose. Co-authored-by: wakesync <shadow@shad0w.xyz>
Co-authored-by: wakesync <shadow@shad0w.xyz>
Co-authored-by: wakesync <shadow@shad0w.xyz>
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@/tmp/proximity-pr-body.md