[MCA] Cut the empty-global-core forest to correction rays and rank four

[scottdhughes]

Stack and review boundary

This is a ready successor stacked on #1163 at exact head e26c15b2d2c2f98ae12dda17b97c40981f76e1ff.

Review only:

e26c15b2d2c2f98ae12dda17b97c40981f76e1ff..37cac405d

The GF(11) local-core collision (83eefd94f) and whole-line global-core router (be4efd23a) are prior public-DAG work replayed as controls. The later nonempty-core composition (fc74e16c) is stronger on that separate branch and is not duplicated here. PR #1156's denominator-root/coordinate-clone branch is independent and out of scope.

New theorem boundary

This packet attacks only the empty/noncoherent common-core forest.

• A degree-at-most-31 coherent source forest whose every order-32 subfamily has a core must have a global core. Thus an empty-global-core forest contains either a local empty order-32 core or a source-passport change.
• Across a 31-slope overlap, two source interpolants differ exactly by
  P(X) prod_(gamma in J)(Z-gamma), so each first passport change is one genuine correction ray preserving the selected actual record.
• Direct common-subset cancellation avoids importing the primitive/no-common hypothesis. The exact guarded one-ray maximum is
  342921713716, attained at shortened dimension three.
• The selector-free all-LineRay theorem pays the complete selected forest through affine error rank three. Its rank-three cap is
  157397034144292985; after the disjoint 2w+31=134975 add-back, exact slack is
  117583693966967127.

The exact surviving terminals are:

• LOCAL_EMPTY_ORDER_32_CORE; or
• multiple noncoherent correction-ray directions with affine error rank at least four.

Per-edge ray payments may not be summed.

Guard and chronology scope

All counts are distinct finite affine bad slopes per actual received line. The packet retains actual degree-<k explanations, same-support noncontainment, source chronology inputs, and the separate near-rational charge 2w=134944. The #1160 line is a mandatory negative regression and never enters the common-core forest or the 31-slope reserve.

This packet moves

U_S = U_A = U_E = global ledger movement = 0.

It does not pay the remaining rank-four/local-empty terminals, insert the result into the active first-match chronology, close S/A/E, close KoalaBear, or prove the universal four-rate result.

Review and verification

Canonical payload:

6d72f2303973d736b12533a96fb07aad9a9d74a61be0ca1a346c707e0ce6fd4e

Fresh isolated review found and forced repairs to the witness interface, pair-indexed/common-subset cancellation, exact maximum, provenance, and add-back. The repaired abstract theorems and certificate were re-reviewed GREEN; active chronology/banking remains explicitly YELLOW.

Passed:

• Python normal and optimized checks;
• 29/29 hostile mutations;
• Sage exact finite-field reconstruction;
• FLINT exact replay;
• all-LineRay normal/optimized checks and 14/14 tampers;
• [MCA] Repair support-wise near-rational reduction by 2w #1160 and [MCA] Cut common-core shortening staircase #1163 predecessor normal/optimized/tamper/Sage/FLINT checks;
• actionable campaign audit;
• source/blob/payload custody and git diff --check.

Maximal next theorem

On the remaining empty/noncoherent forest, either pay LOCAL_EMPTY_ORDER_32_CORE, or prove a chronology-correct aggregate owner/direction-multiplicity theorem for the affine-rank-at-least-four correction-ray family. A sum of local or per-direction charges is not allowed.

[AllenGrahamHart]

Dependency alert: draft PR #1165 refutes the affine-span MCA incidence compiler inherited from #1163. The fixed-core DIRECTION_SEPARATED_PAID_3_LE_s_LE_13 staircase therefore needs withdrawal or replacement.

The separate selector-free all-LineRay error-affine-core payment through rank three is not refuted: it uses zero-mask restriction plus the Bollobas set-pair inequality rather than the failed ordered-normal-basis denominator. Common-core cancellation and the correction-ray reductions are likewise not refuted. #1165 records the exact scope and two deterministic verifiers.

[scottdhughes]

Fresh independent audit found a load-bearing exact counterexample to the imported affine-span MCA theorem used only in Theorem 4.1’s DIRECTION_SEPARATED_PAID_3_LE_s_LE_13 branch.

Exact falsifier: RS[GF(257), GF(257)^*, 1] at (n,K,m,w,s)=(256,1,86,85,1) has 87 distinct exact same-support-pair-noncontained slopes, strict direction separation 85<86, and all selected words outside the near-rational radius, while the printed affine-span cap is 8. Python and independent Sage replays agree. The proof fails at the final r=s normal-flat step: the claimed codeword kernel has dimension zero, so the asserted occupancy bound does not follow.

This does not refute #1164’s novel degree-31/order-32 coherence fence, correction-ray identity/cap, or independent all-LineRay rank-three gate. It does invalidate the imported direction-separated terminal in Theorem 4.1 and the cited public-DAG strengthening 3a13f2d.

Please do not integrate #1164 alone while the correction is in flight. I am packaging a successor that (i) installs the exact counterexample, (ii) repairs the theorem with an explicit support-transversality margin, and (iii) replaces the false rank-13 plan by the strongest exact rank/margin route cut that survives independent review.

[AllenGrahamHart]

Follow-up dependency update: #1165 now proves the corrected proper-subspace compiler with final factor max(1,e-(N-m)). It recovers unconditional fixed-core payment only through KoalaBear actual explanation affine rank 9, then gives exact support walls through rank 13; rank 14 remains unpaid.

This narrows the correction required in this stack. The separate all-LineRay error-affine-core rank-three theorem remains unaffected exactly as noted above.

[AllenGrahamHart]

The latest #1165 extension gives a direct bridge to this PR's all-LineRay error-affine-rank terminology.

For a full explanation-rank family, the lifted span

V=span{(gamma-gamma_0,c_gamma-c_gamma0)}

has dimension exactly K or K+1. Pair noncontainment implies r_1 notin C, so the map (a,u) -> a*r_1-u is injective and dim(V) equals the selected error-vector affine rank used by the all-LineRay route. The dim(V)=K branch admits a codeword gauge dropping explanation rank to K-1; the dim(V)=K+1 branch admits none.

At the first KoalaBear row this separates top explanation rank 14 into error ranks 14 and 15. The error-rank-14 branch is now paid for e>=992852 by the corrected rank-13 occupancy theorem, while error rank 15 retains the previous e>=1044239 suffix. Thus the independent all-LineRay machinery is not only unaffected by the affine-incidence counterexample; its error-rank coordinate is exactly the invariant that distinguishes the genuinely full-lift residual.

No rank-four/all-LineRay claim is added here, and the full-lift middle interval remains open.

[AllenGrahamHart]

One further exact consequence is now in #1165. In the full-lift branch, the error-direction code

W=C+span{r_1}

has dimension K+1, d_1(W)=e, and d_j(W)=N-K+j-1 for every 2<=j<=K+1. The selected errors are a full-affine-rank sparse list in one affine coset of W, with unit slope fibers and information-set maximal zero masks.

So the all-LineRay error-rank coordinate identifies a near-MDS codimension-one RS extension whose only defective generalized weight is the first. Even at the MDS endpoint, the generic corrected ceilings remain above budget (743896698428332665 on KoalaBear, 219426634 on Mersenne). This rules out another generalized-weight/basis replay as the missing rank-four-or-higher mechanism; the remaining input must use extension-specific structure or a sharper row-specific affine-list bound.

[AllenGrahamHart]

Independent low-support payment now added to #1165 at 442a223a: after a codeword direction gauge with residual support e<d, puncturing that support puts the transformed explanations into ordinary RS lists, and the Johnson cumulative cap combined with the floor(e/h) owner profile pays e<=63908 (KoalaBear) and e<=65236 (Mersenne-31). This uses only pairwise RS agreement <=K-1, exact size-m same-support pair-noncontained witnesses, and the ratio-fiber argument; it does not use the refuted ordered-basis denominator. The adjacent Johnson denominators are nonpositive, so the result deliberately stops there. In the full-lift top cells the remaining intervals become 63909<=e<=1044238 and 65237<=e<=1044241.

[AllenGrahamHart]

Follow-up on the low-support import: #1165 now has a separate post-Johnson centered-Gram theorem at 07546f90. For equal-size punctured agreement blocks, equal row sums imply rank(BB^T-cJ)<=rank(B)<=n; trace-rank then gives L<=floor(nA(A-c)/((A-c)^2-c(nc-A^2))) while that denominator is positive. Combined with the deficit split, this extends the walls again to e<=64037 (KoalaBear, bound 198047217) and e<=65418 (Mersenne, bound 16759641). The next KoalaBear Gram denominator is negative; the next Mersenne bound remains valid but exceeds budget. No unsafe claim is made.

[AllenGrahamHart]

The post-Johnson import is sharpened again in #1165 at d4d65372. Mean-centering gives the PSD matrix B(I-J/n)B^T of rank at most n-1; an endpoint-chord square bound yields L<=floor((n-1)n^2(A-c)/(A((n-A)^2-(n-1)(nc-A^2)))). Combining Johnson and mean-centered caps via suffix minima moves the walls to e<=64047 (KoalaBear) and e<=65454 (Mersenne). The next failures remain explicitly nonterminal: negative denominator on KoalaBear, valid profile over budget by 342908 on Mersenne.

[AllenGrahamHart]

Follow-up in #1165: the exact terminal-deficit layer has additional slope structure that the ordinary-list cap erases.

If the outside deficit is exactly h=e, ownership forces agreement on all of E=supp(q). Since e>=K, restriction injectivity puts every terminal explanation on one affine codeword line. Outside a common zero core of size at most K-1, their agreement sets are disjoint, giving

L_e <= floor((N-e-(K-1))/(m-e-(K-1))).

Combined with the existing Johnson/mean-centered prefix profile, this pays:

• KoalaBear e=64048: 181326056+287=181326343;
• Mersenne-31 e=65455: 16100154+493=16100647.

The residual intervals move to 64049<=e<=1044238 and 65456<=e<=1044241. The next KoalaBear cell lacks the h=e-1 cumulative cap; the next Mersenne profile is 17119507, over budget by 342292. The manuscript theorem, note, exact verifier, and rebuilt PDF are at commit bb9df40 in #1165.

[AllenGrahamHart]

Stronger follow-up in #1165: the terminal-line argument extends to every exact layer h=e-r with e-3r>=K. Triple overlap synchronizes all normalized pair directions, so each high-deficit layer lies on one affine codeword line and obeys

L_r <= floor((N-e-(K-1))/(m-e+r-(K-1))).

Pricing the top third of exact layers this way and retaining the positive-Johnson prefix for the lower layers closes every sparse-direction support e<d. Conservative official totals are 11496959 on KoalaBear and 11496238 on Mersenne-31. The residual full-lift intervals now begin at the actual minimum-distance boundaries:

KoalaBear:   67472<=e<=1044238
Mersenne-31: 67448<=e<=1044241.

The theorem, finite calibration, grouped-floor/triple-overlap verifier, and 99-page manuscript build are at commit 0b7bedf in #1165. This closes the support-only low-e branch; it does not claim either deployed row.

[AllenGrahamHart]

Further full-lift continuation in #1165: for e>=d, the top-third affine-line layers with at most K-1 outside agreements are paid by a total common-core argument. A coordinate common to every line parameter is a simultaneous base/direction agreement for one codeword pair, so pair noncontainment bounds that core by m-1 and gives L<=N-m+1. The sharper outside cap remains in force above K-1 agreements.

Combined with the Johnson prefix, this moves the residual walls to:

KoalaBear:   95944<=e<=1044238
Mersenne-31: 67453<=e<=1044241.

The last paid bounds are 27414298 at KoalaBear e=95943 and 16266965 at Mersenne e=67452. KoalaBear then loses its prefix denominator (-1037); the adjacent Mersenne bound is 17248067, over budget by 470852. The theorem, note, fast verifier, and 100-page build are at commit 1913917 in #1165.

[AllenGrahamHart]

Follow-up from the compatible full-lift branch: PR #1165 head 9c708e2f7 now proves cross-layer top-third synchronization. If r_i<=floor((e-K)/3), every triple of inside agreement sets overlaps in at least K coordinates, so all high-deficit layers lie on one affine explanation line and the pair-noncontained cap N-m+1 is charged once. Exact walls are KoalaBear e<=95943 and Mersenne-31 e<=97908; the adjacent prefix Johnson denominators are -1037 and -965. This is independent of the direction-separated factor corrected in #1166, moves no v4 atom, and does not treat either sign wall as unsafe.

[AllenGrahamHart]

Further compatible continuation on #1165 head c09423b8b: the full suffix-minimum Johnson/mean-centered prefix can be used in the full-lift branch once A_H=m-H>K-1 is stated explicitly. Composed with the one-time global high-line charge, it pays KoalaBear through e=96150 and Mersenne-31 through e=98229. Adjacent KoalaBear loses T>0; adjacent Mersenne remains legal but is over budget by 638658. This remains independent of the invalid direction-separated factor corrected by #1166 and moves no v4 atom.