[Lane T] Force the rate-half pair floor and isolate its biform gates

[AllenGrahamHart]

Workboard scope

workboard_item: T
row: symbolic rate-half half-distance profile; official N=2^41, k=2^40, t=2^39
object: LINE
architecture: DIRECT
atom_or_cell: symmetric-Hankel core-one quadratic support profile
status: PROVED
impact: ROUTE_CUT
claimed_bound: pair union at least 3rho/2-1; floor and first strict boundaries reduce to explicit split biforms with a sparse scalar weld and connected-rank dichotomy

Exact result

This extends the half-distance Hankel/LineRay program at one precisely typed
residual profile: fixed core one, primitive parameter degree e, scalar
residual degree two, and minimum quadratic gap u=4.

The pair argument couples both oriented rank-two locator pencils, excludes
the rho+3 cell, and then uses one concave incidence obstruction to prove

|S_alpha union S_beta| >= 3rho/2-1.

Every assigned-center codeword line contains at most three supported slopes,
and every pair has at least rho+1 full-locator-expanding third slopes.

The two first pair boundaries are reduced further:

floor equality:        bidegree (e-2, rho/2-3)
first strict boundary: bidegree (e-1, rho/2-2).

The floor profile has at least 3rho/2-3 split domain rows and 2e
full-degree split zero-excess parameter fibers. The first strict profile has
at least rho split domain rows and rho/2+2 full-degree split zero-excess
parameter fibers. Positive padding is retained: every such fiber is a scalar
multiple of its actual inside-support locator times its padded-heavy factor.

For both biforms, the root data must pass exact full-support coefficient-MDS
kernel tests in both directions. Along fixed-domain rows,

G(t,x)=lambda_x product_(delta in A_x)(t-delta),

and every scaled coefficient vector lies in one punctured RS evaluation code
on the domain set. Along zero-excess parameter fibers,

G(delta,X)=zeta_delta product_(x in B_delta)(X-x),

and every scaled coefficient vector lies in one punctured RS evaluation code
on the slope set. Equivalently, each printed coefficient-barycentric matrix
must have a kernel vector with every coordinate nonzero. The two systems
constrain one common biform; passing either separately is not a realizability
certificate.

At the official row, the fixed-domain matrix sizes are

extremal: 100743818300669342078294 x 824633720829
strict:    50371909150884426853035 x 549755813888,

and the parameter-direction matrix sizes are

extremal: 50371909150609548946088 x 366503875926
strict:   25185954575671278348969 x 274877906946.

The two coefficient systems are now welded exactly. On every nonincidence,
lambda_x P_x(delta)=zeta_delta F_delta(x); eliminating zeta_delta
produces a sparse matrix W with two nonzero entries per row. Common-biform
realizability is equivalent to [Krow; W]lambda=0 with full support, and the
parameter-direction coefficient gate then follows automatically.

The complement incidence graph is connected in both profiles, so

rank W in {|X|-1, |X|}.

Full rank excludes the boundary. At rank |X|-1, the kernel is unique up to
scalar and automatically full-support, leaving one projective vector to test
against Krow and the retained source/Hankel equations. At the official row,
the weld has at least 201487636602438195784362 extremal rows and
75557863726738957139970 strict rows. These counts are not rank proofs.

As finite calibration, the exact e=7,d_A=1 smooth cyclic ledger and 500
degree-preserving switched ledgers were full rank over F_337 and F_421
for the fixed-domain gate. The weld itself was full rank in the cyclic ledger
and 200 switched ledgers. This is evidence, not the official all-profile
theorem.

Why it belongs in Lane T

This is a rigorous profile-level route cut with exact live-integer
specialization. The full source proof is pinned to
AllenGrahamHart/rs-mca-prize-dag@cd318d6155d9f96ff986926dfec5a0b58f54a408
through the two coefficient gates and
AllenGrahamHart/rs-mca-prize-dag@77a26cfa85c9b1954345f9dd3027a75c59fa8943
through the scalar weld and connected-rank dichotomy. All thirty-one
statement/proof/verifier/probe SHA-256 values are printed in the note.

The biforms and coefficient matrices are phrased in the repository's
base-field split-pencil language, but they are residual reductions, not BC
payments. Existing ray-collapse results identify the deduplicated target with
LineRay; they do not bound it, and this PR does not claim otherwise.

Validation

Passed:

python3 experimental/scripts/verify_rate_half_core_one_quadratic_pair_floor_v1.py --check
python3 -O experimental/scripts/verify_rate_half_core_one_quadratic_pair_floor_v1.py --check
python3 experimental/scripts/verify_rate_half_core_one_quadratic_pair_floor_v1.py --tamper-selftest
python3 -O experimental/scripts/verify_rate_half_core_one_quadratic_pair_floor_v1.py --tamper-selftest
python3 -m py_compile experimental/scripts/verify_rate_half_core_one_quadratic_pair_floor_v1.py
git diff --check

The tamper replay rejects 5/5 formula-contract mutations. The proof is
analytic; the verifier checks exact identities, endpoint ranges, both biform
dimensions, both coefficient-matrix directions, split counts, weld counts,
connectivity margins, a finite-field rank/tamper control, and official
integers.

Nonclaims and next adapter

This does not change a v4 atom, pay a finite adjacent row, prove the active
deduplicated LineRay census, prove primitive shift-pair control, or move a
leaderboard score. It also does not assert that every active first-match cell
already satisfies the profile interface. The next integration step is to
identify the exact first-match owner and then prove which weld-rank branch
occurs. In the nullity-one branch, it remains to exclude the unique
projective scalar vector using Krow and the retained source/Hankel
identities.

[AllenGrahamHart]

Lane-T extension added at 07ea0f2.

The new pinned packet extends the extremal biform theorem to every off-line supported slope:

Q(delta,X)=chi A_delta B_delta R_delta,
G(delta,X)=zeta A_delta H_delta R_delta,
gcd_X(Q_delta,G_delta)=A_delta R_delta.

All actual-support intersections are transverse. After one copy of every actual-support and padding point is removed, Bezout leaves an exact projective residual cycle of degree four.

The contracted second-kind numerator gives:

QB-Lambda G=L_U0 P_F,
Res_X(Q,P_F)=c a^(2d+1)D_1,
Res_X(Q,G)=c E_4 product_(delta off line) ell_delta^(n-a_delta).

Thus the four-core is exactly the regular Kronecker correction quartic: E_4 ~ S_B^2 in the double-root arm and E_4 ~ S_1S_2 in the two-simple arm.

The packet also records the route fence that marked orders eight and seven occur in abstract symmetric affine Kronecker pencils. A closure therefore needs the Hankel/source/split-fiber structure, not determinant multiplicity alone. This remains a proved profile-level route cut, not a LineRay payment or endpoint movement.

Replay:

python3 experimental/scripts/verify_rate_half_core_one_quadratic_pade_quartic_v1.py --check
python3 -O experimental/scripts/verify_rate_half_core_one_quadratic_pade_quartic_v1.py --check
python3 experimental/scripts/verify_rate_half_core_one_quadratic_pade_quartic_v1.py --tamper-selftest

[AllenGrahamHart]

Lane-T follow-up added at 3f9267d (source pin edf8a35cc).

The existing Pade/quartic packet now includes three further proved route cuts:

correction-free supported slope:
  ord_gamma(D_1)=c_gamma,
  rank dotPhi(Q_min^2 A B)=c_gamma,
  radical=<R_gamma>;

coefficient plane:
  dim((W_q cap ker M_gamma)/<Q_gamma>) <= floor(c_gamma/2),
  rank E_gamma=e at ordinary rank-one loss;

separated double-root correction:
  P_F(t,x_*)=D_1 C_0,
  M(t)U(t)=D_1 C(t),  deg_t C<=3,
  Smith type [2] at each correction root.

The last statement assumes S_B squarefree and gcd(g_*,S_B)=1. Nonreduced/shared corrections and the two-simple correction remain open. No LineRay payment or endpoint movement is claimed.

Replay:

python3 experimental/scripts/verify_rate_half_core_one_quadratic_pade_quartic_v1.py --check --source-root /path/to/rs-mca-prize-dag
python3 -O experimental/scripts/verify_rate_half_core_one_quadratic_pade_quartic_v1.py --check --source-root /path/to/rs-mca-prize-dag
python3 experimental/scripts/verify_rate_half_core_one_quadratic_pade_quartic_v1.py --tamper-selftest

Current replay: 123 checks, 16 source hashes, 7/7 formula mutations rejected.

[AllenGrahamHart]

Lane-T extension added at a0f6ef6 (source pin 62dc0233c).

The existing quadratic packet now adds the exact heavy-row adapter on the separated double-root locus:

J=gcd(Lambda,g_*S_B^2),  j=deg J<=3,
G(t,x_*)=(g_*S_B^2/J)T_j,  deg T_j<=j.

Thus all but at most three heavy-row roots are prescribed, and the added coefficient-RS coordinate introduces only j+1<=4 scalar unknowns. This is an augmented rank gate, not a full-rank claim.

The same packet records a separate exact Layer-A route fence. At m=2,rho=7,T=9,a=13, Q(Z,X)=Z^2-X^4 on thirteen points of mu_16 gives 26 pointwise-saturated incidence rows on 24 coefficients, but

ker E={A(X)(Z^2-X^4): deg A<=3},
rank E=20,  nullity E=4.

This retires only the generic implication “row surplus plus saturation implies rank.” It does not realize the canonical pair-union blocks or the endpoint Hankel/source constraints, so the structured Lane-T theorem remains live.

Replay now gives 200 checks, 20 pinned source hashes in normal and optimized Python, and 12/12 formula mutations rejected.

[AllenGrahamHart]

Follow-up pushed in 6684929.

This extends the same Lane-T packet with two exact atoms:

• the augmented heavy-row coefficient-MDS condition is equivalent to the single barycentric remainder test H | R_lambda, or B_H lambda = 0;
• on the squarefree, support-disjoint, center-disjoint double-root locus, passage forces R_lambda = c g_* S_B^2 with c != 0.

The source pin is now AllenGrahamHart/rs-mca-prize-dag@a97f137e71b761d76ad2c5a657a7d897add2480d with 24 exact statement/proof hashes. Normal replay, python -O, source-root replay, and all 15 mutation gates pass (220 exact checks). The center-overlap, nonreduced/shared, and two-simple loci remain explicit nonclaims.

[AllenGrahamHart]

Further sharpening pushed in 1b2f340.

Re-auditing the fully cancelled resultant removes the center-disjoint restriction: on the complete squarefree, supported-disjoint double-root locus, G(t,x_*) != 0 for every center-overlap degree j=0,1,2,3. A center correction root is not off-line supported, so the exact factorization still gives resultant order two, while a zero heavy row would force order at least three. Off-line supported roots remain excluded by the exact actual/padding dichotomy.

The source pin is now 39c001bfb6196de1739cc68118dc78dc730e25fd with 26 exact hashes. All 220 checks, optimized replay, source replay, and 15 mutation gates pass. The remaining separated wall is now precisely whether a nonzero R_lambda can satisfy H | R_lambda; zero collapse is no longer a survivor.

[AllenGrahamHart]

Two further exact Lane-T atoms are included in b6e67a0.

1. At each separated correction root tau, the heavy row has exact order 2 - ord_tau Lambda, equal to the order already supplied by H. Hence gcd(T_j,S_B)=1; the free overlap form contributes no hidden correction contact.
2. The endpoint/source ledger gives deg gcd(S_B,Lambda)<=1 and deg gcd(g_*,Lambda)<=1. Separatedness makes these disjoint, so j<=2: the former four-scalar case is impossible and the heavy row has at most three scalar coefficients.

The pin is now c0bcb637256420ce8407619385a58afd39f7fde0 with 30 exact hashes. Normal, optimized, and source-root replay pass (205 checks), as do all 15 mutation gates. The remaining separated wall is a nonzero, correction-coprime remainder divisibility problem.

[AllenGrahamHart]

Latest packet sharpening is in a725ba3.

• The full three-center source identity shows gcd(S_B,Lambda)=1, not merely an endpoint-center cap. Since at most one center lies in g_*, the separated overlap is now j<=1; the exact heavy row is controlled by a nonzero constant or linear T_j (at most two scalar coefficients).
• A squarefree root shared by g_* and S_B is reduced to one local scalar kappa_tau=(F_0/z^2) mod z. The recurrence gives (F_i/z^2) mod z=x_*^i kappa_tau, and full D_1 divisibility holds iff kappa_tau=0; then the cubic quotient extends with Smith type [3]. Otherwise the canonical image has exact order two.

Source pin: 688bbbb511a795b3dbc80fbd2737d9fc44b9f58e, 34 exact hashes. Normal, optimized, and source-root replay pass (190 checks), with 16/16 mutation gates rejected.

[AllenGrahamHart]

Squarefree shared-root closure and unification are pushed in 0e58b85.

The Cycle-166 scalar is forced to vanish: at a shared root the regular symmetric Hankel block has corank one and determinant order three. A Schur-complement argument makes u^T M u vanish to order at least three, while its order-two coefficient is kappa_tau U_tau(x_*); simplicity of the padded root gives U_tau(x_*) != 0, hence kappa_tau=0 and local Smith type [3].

As a result, every squarefree double-root packet, shared or separated, obeys the same exact gate:

j<=1, G(t,x_*)=(g_*S_B^2/J)T_j, with nonzero constant/linear T_j, gcd(T_j,S_B)=1, and passage iff H | R_lambda.

Source pin: 3da4ed86f6d93dcf88574ac03b2362a0512c7cae, 38 hashes. All 190 checks, optimized/source replay, and 17 mutation gates pass. Nonreduced S_B, the global nonzero remainder decision, and the two-simple arm remain explicit nonclaims.

[AllenGrahamHart]

Lane-T packet extended at ec45a26 (source pin AllenGrahamHart/rs-mca-prize-dag@f68d685e78da099e0a4dff362cc90ed2601a341f).

New exact results:

• Squarefree overlap is no longer a free case split: J=gcd(Lambda,g_*S_B^2)=gcd(Lambda,g_*) and deg J=d_A in {0,1}. Thus d_A=0 has a nonzero scalar remainder quotient and d_A=1 a nonzero quotient of degree at most one.
• An unshared nonreduced correction is reduced to exactly two Hasse jets kappa_2,kappa_3; both vanish iff the cubic quotient extends, with local Smith type [4]. The packet does not assert their vanishing.
• Factoring the paired biform over F(X) forces every irreducible factor Q_j to satisfy 3e n_j >= (3p-3+d_A)m_j, split completely on every classified row, and split over U_0 on every clean parameter fiber. One factor has m_j >= ceil(e/3)=61083979321 for d_A=0, or m_j >= ceil(3e/7)=78536544842 for d_A=1. This rules out bounded-degree common-factor mechanisms, including the mechanism behind the Layer-A route-fence fixture, but is not an exclusion.

Replay passed under normal and optimized Python: 230 exact checks, 44/44 source hashes, and 22/22 hostile formula mutations rejected in both modes. git diff --check passes.

No LineRay payment, adjacent endpoint movement, v4 atom, or leaderboard claim is added. The live wall is now a macroscopic two-directionally split factor plus the printed nonzero constant/affine remainder gate; shared nonreduced and two-simple profiles remain separate.

[AllenGrahamHart]

Lane-T packet extended at f605c93 (source pin AllenGrahamHart/rs-mca-prize-dag@2ab0a04e5b84c09fc1027764253d8ba4cc3d0122).

Two further exact route cuts are now included:

• On the unshared nonreduced locus, if the regular specialized symmetric block has corank one, the minimal-locator identity gives U_tau(x_*) != 0 and the order-four Schur complement forces both obstruction jets kappa_2,kappa_3 to vanish. Thus a nonzero-jet survivor must have regular corank at least two. That higher-corank locus, shared nonreduced roots, and the two-simple arm remain open.
• The extremal paired biform has constant content. Consequently every irreducible factor lies exactly on n_j=ceil((3p-3+d_A)m_j/(3e)), with no domain-degree slack, and the complete factor multiset has exactly one of three profiles: one large odd factor; two large odd plus one small odd; or one huge even plus one small odd. All remaining factors are ordinary even. This classifies but does not exclude the three profiles.

Normal, optimized, and source-root replay pass: 25,826 checks, 48/48 pinned source files, exhaustive classification of 25,504 small integer partitions (776 feasible), and 31/31 hostile formula mutations rejected in both normal and optimized modes. git diff --check passes.

No LineRay payment, adjacent endpoint movement, v4 atom, or leaderboard claim is added.

[AllenGrahamHart]

Added Section 49 in commit 8f83b95: all-rank domain-map birationality.

For every minimal tensor presentation, the domain coefficient map is basepoint-free and its degree d_X onto the image divides n. Every U_0 point is unramified in its normalization fiber. Each unit of scheme-theoretic fiber degree outside U_0 is rooted by all m incident parameter hyperplanes, so it consumes at least m from the exact residual norm divisor. That divisor has degree 3en-Rm=2e-7=2m-3, allowing at most one outside fiber-degree unit globally. Hence d_X divides either R or R+1; the official gcds with n are both one, so d_X=1.

Together with Section 48, both coefficient maps are birational in every tensor rank. Repeated row types can occur only as distinct normalization branches over singular image points. This does not bound those singularities, exclude Shape A, pay an atom, or move an endpoint.

The compact checker passes in normal and optimized Python with 630 calibration cases, the exact rank-27 fence, both birationality ledgers, and 32/32 source hashes; 15/15 hostile mutations are rejected in both modes. py_compile and git diff --check pass.

[AllenGrahamHart]

Section 50 adds a proved rank-three weighted incidence/genus router (commit da8a6f1). After the two coefficient-map birationality results, equal-image classes on the rational plane curves give a weighted C4-free bipartite graph with exact row degrees m, column degrees n-w_P, total weighted deficit 2e-7, and delta-genus collision budgets on both sides. Consequences include at least 10 image vertices on each side and at least e+7 parameter-branch weight on at least two zero-deficit vertices. This is a structural reduction only: it does not exclude Shape A or tensor rank >=4. Compact replay: 630 calibration cases, rank fence and geometry ledgers, 37/37 source pins, 18/18 hostile mutations.

[AllenGrahamHart]

Section 51 adds a proved macroscopic separation-rank floor (commit 3abd608). The first-degree primitive locator has e+1 independent coefficient vectors. On Shape A, d_A=1 partitions U_0 into exactly three source-root classes; the dual-MDS identity puts each evaluated locator row in one of three fixed quadratic multiples of the evaluated G-row span. Injective evaluation gives e+1 <= 3 sr(G), hence sr(G) >= 61,083,979,322 officially. This excludes all rank-two/rank-three and every other low-rank Shape-A model, but does not exclude the remaining macroscopic-rank interval. Compact replay: 630 calibration cases, 42/42 source pins, 21/21 hostile mutations.

[AllenGrahamHart]

Section 52 adds the proved three-class restricted Koszul/source-Gram router (commit 0f673f6). Each Shape-A source class alone spans V; the three quadratic source shifts generate all degree-e forms, so dim ker Phi=3 sr(G)-(e+1). The two endpoint isotropy equations make the three classwise weighted locator Gram matrices proportional, with common image in the inverse prolongation J and rank between max(0,2r-(n+2)) and dim J. At the minimum live rank the Koszul kernel has dimension 2; from rank 137,438,953,472 the common Gram matrix is forced nonzero. No Shape-A exclusion is claimed. Compact replay: 630 calibration cases, 47/47 source pins, 25/25 hostile mutations.

[AllenGrahamHart]

Section 53 adds the proved coefficient syzygy-bundle classification (commit 7bccbc2). Quadratic generation gives H^1(E(2))=0, so every geometric splitting degree of the basepoint-free coefficient series is 1, 2, or 3. The counts satisfy c1+c2+c3=r-1, c1+2c2+3c3=e-2, 2c1+c2=3r-(e+1), with dim J=c1. At minimum rank exactly two profiles remain: (1,0,61083979320) and (0,2,61083979319); nonzero common source Gram selects the first. No Shape-A exclusion is claimed. Compact replay: 630 calibration cases, 52/52 source pins, 27/27 hostile mutations.

[AllenGrahamHart]

Sections 54-55 materially raise and classify the Lane-T Shape-A frontier (commit e63028d). Locator X-interpolation maps an (n+1)-dimensional family into the restricted Koszul kernel; either n+2 source-class projection has rank at least r-1, forcing r >= (e+1)/2 = 91,625,968,982 and excluding the former one-third boundary. Equality in the projection squeeze then determines every live syzygy profile: c1=2r-e, c2=e-r-1, c3=0, so S_1 V=S_(e-1). Each small class has exact interpolation rank r-1 and yields a unique projective domain coefficient defect with the explicit barycentric values in (279). No Shape-A exclusion is claimed. Compact replay: 630 calibration cases, 62/62 source pins, 30/30 hostile mutations.

[AllenGrahamHart]

Lane-T update at 97f3287 adds Section 56 and a separate compact replay.

New proved reduction:

• every restricted-Koszul coordinate image is V_gamma={f in V:f(gamma)=0};
• each small-class defect from Section 55 is canonically G(gamma,X);
• the large-class interpolation rank is exactly r-1 or r-2;
• in the r-2 branch, G(gamma_0,X)=(X-x_*)B_0(X) with B_0 in W_X, deg B_0<=n-1, and the cubic heavy-row residual factors as T_3=ell_(gamma_0)T_2, deg T_2=2.

This does not exclude Shape A. It splits the next attack into (i) compatibility of the three actual center fibers in the common coefficient space, and (ii) exclusion of the multiplication chain B_0,(X-x_*)B_0 in W_X.

Replay:

• new source pins: 11/11;
• new hostile mutations: 13/13;
• finite-field dual-RS toy ranks: small/large = 3/2;
• prior packet remains 630 cases, 62/62 pins, 30/30 hostile mutations.

[AllenGrahamHart]

Lane-T follow-up at 8f82c4a adds Section 57 and closes the lower branch of Section 56.

If rank T_(gamma_0)=r-2, then G(gamma_0,X)=(X-x_*)B_0(X). Together with the padded center locator, this creates a residual common point (gamma_0,x_*) of Qbar and G. It is not among the mandatory intersections over off-line supported slopes, so it survives in Z_4. But the exact resultant theorem gives pi_*Z_4=div(E_4)=2div(S_B), supported only at the off-center collision tau. Contradiction.

Thus all three class maps have exact rank r-1, image V_gamma, and sole defect G(gamma,X).

Updated replay: 18/18 source pins and 16/16 hostile mutations; prior 630-case packet remains green.

[AllenGrahamHart]

Lane-T update at 35466b5 adds Section 58.

For each assigned center, the complete center locator is coprime to the corresponding split-biform fiber:
gcd_X(Qbar(gamma,X),G(gamma,X))=1.
The only non-classified case is the large padded root x_*, excluded by the Section 57 residual-support argument. Hence G(gamma_0,x_*) != 0 and the exact cubic heavy residual is nonzero at gamma_0.

Specializing the Padé identity gives exact nonzero quotients:
B_src(gamma)=L_Mgamma C_gamma,
P_F(gamma)=chi_gamma R_gamma C_gamma,
where R_gamma=1 for the two small classes and R_(gamma_0)=X-x_*.

Updated compact replay: 25/25 source pins and 20/20 hostile mutations.

[AllenGrahamHart]

Lane-T update at b0fd91e adds Section 59.

At the large center, Qbar and P_F share X-x_*. The formal Padé resultant c a^(2d+1)D_1 has exact order one there: a is a unit and D_1=g_*S_B^2 has a simple center root. Therefore (gamma_0,x_*) is their unique common point in that fiber, with local intersection length one.

Consequences:

• C_0(x_*) != 0;
• B_src(gamma_0,x_*) != 0;
• the source-numerator pencil does not send x_* to the large center.

Updated compact replay: 30/30 source pins and 23/23 hostile mutations.

[AllenGrahamHart]

Lane-T update at 930a7c9 adds Section 60.

After removing the homogeneous fixed gcd H of the two parameter coefficients of B_src, the primitive numerator defines a morphism phi:P^1_X -> P^1_t of degree D=R-1-h.

Its three center fibers are L_Mgamma Cbar_gamma; on U_0 they contain exactly the three source classes. The residuals are pairwise coprime with exact degrees d-1-h,d-1-h,d-2-h, and their locator products satisfy the unique three-term relation with all coefficients nonzero. Also H has no root on U_0 or at x_*, and phi(x_*) != gamma_0.

Updated compact replay: 35/35 source pins and 28/28 hostile mutations.

[AllenGrahamHart]

Added Section 61 at f58bbb8: center residue-pairing common-kernel router.

For each of the three Shape-A source classes, the locator-interpolation map is now the restriction of an explicit symmetric residue form

beta_gamma(f,h)=sum_(x in M_gamma)
 R_gamma(x)f(x)h(x)/(G(gamma,x)L_Mgamma'(x)).

The form has exact rank n and sole radical span{G(gamma,X)}; the large-class proof retains the two-dimensional dual-RS parity space and uses the already-proved G(gamma_0,x_*) != 0 to cancel X-x_*. If kappa is the intersection dimension of the three restricted orthogonal complements, then

rank T = n+1-kappa,
r >= ceil((5e-3-2kappa)/6).

At the current lower boundary r=(e+1)/2, survival requires kappa >= e-3 = 183251937960; kappa <= e-4 would raise the rank floor by one. No such bound, Shape-A exclusion, or endpoint movement is claimed. Projective distinctness of the center fibers alone is explicitly not treated as a common-kernel proof.

Replay: 40/40 source pins at local 1bf81a36f; compact finite-field ranks (full, restricted, combined)=(4,2,5); 34/34 hostile mutations rejected.

[AllenGrahamHart]

Added a further PROVED Lane-T router in commit 5800ddf: Section 62 rewrites the three center residue kernels as one owner-sensitive Pade intersection

K_cap = S_n intersect J(W_X + varphi W_X + varphi^2 W_X)^perp,
varphi = -K/J,    dim(W_X + varphi W_X + varphi^2 W_X) = 3r.

At the lower boundary the multiplier space has dimension 274877906946, its residue orthogonal has dimension 549755813884, and a survivor requires intersection dimension at least 183251937960. This is a normal form, not an exclusion; Section 62 explicitly records that ambient dimension alone is insufficient.

Replay result: 45/45 pinned source files, toy ranks (9,9,9), official dimensions (274877906946,549755813884,183251937960), and 40/40 hostile formula mutations pass.

python3 experimental/scripts/verify_rate_half_shape_a_center_fiber_defect_v1.py --check --tamper-selftest --source-root /path/to/rs-mca-prize-dag

[AllenGrahamHart]

Important correction/route fence added in 02d65eb (Section 63). The Section-62 common-kernel floor is automatic Pade/RS parity, not an ambient alignment that can be bounded away.

Modulo L_U0,

J Qbar = q_varphi G,
J U_Q subset E_3,
S_n^perp = S_d,  d=R-n-2.

Writing

xi = dim(E_3 intersect J S_d) - (e+1) >= 0,

the exact formulas are

rank T = dim ker(Phi) - xi,
kappa = n+e+2-3r+xi.

At the lower boundary, kappa=e-3+xi, so the complete e-3=183251937960 floor is mandatory. A dimension-only continuation of Section 62 is therefore fenced; exclusion needs an independent collision/Hankel, fixed-factor, or split-row incompatibility. Neither value of xi alone excludes Shape A.

Replay: 50/50 pinned source files, parity toy ranks 3/2, official dimensions (549755813887,183251937964,91625968982,183251937960), and 46/46 hostile formula mutations pass.

[AllenGrahamHart]

Lane-T Shape-A program banked on the public prize DAG (head AllenGrahamHart/rs-mca-prize-dag@8c3a30f9a; node cluster prefix ...quadratic_gap_four_double_root_nonreduced_unshared_collision_shape_a_*, 31 PROVED nodes, every one with primary + independent verifiers).

The two structural kills, for the extremal paired-biform profile A (one large odd factor):

• Tensor rank two is excluded for every official survivor (..._tensor_rank_two_biform_exclusion): distinct projective row types of a primitive rank-two pencil have disjoint m=e-2 root sets among the 3e slopes, and 4(e-2)>3e for e>=9 caps the types at three; each type owns at most n domain rows, but Shape A requires R=(9e-7)/2=3n+7. Sharp onset e=9; the e=7 fence is retained.
• The parameter map is birational in every surviving rank (..._all_excess_parameter_map_birationality): active slopes occur in complete reduced fibers, so the map degree divides m=e-2 and one of 3e, 3e-1; both official gcds are 1.

Supporting stack (same cluster): global-subgroup Euler/genus floor; Z_4=2B with h^0(O_C(2B))=1; natural residual-section and characteristic-free route fences; omitted-recurrence/norm and bordered-Hankel flag presentations; scalar-weld residual-MDS flag; all-excess parameter-MDS gate; degree-ledger rank route fence; rank-three projective-frame and birational-singularity routers.

Net: any Shape-A kernel surviving the Lane-T pair floor is a tensor-rank->=3, birationally-parametrized object, and the pre-registered next attack is an image-geometry or source/Hankel obstruction against exactly that class. No LineRay payment, endpoint movement, or leaderboard claim is added.