Prove coupled E8 coordinate floors and the 28-neighbor partial-quotient case

research/e8-coupled-coordinate-floor/EXTENSIONS.md

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+# Normalization-stable partial quotients and high right rank
+
+Use PROOF.md's fixed full-E8 scalar code C, binary labels, COPY tariff and
+singleton left space C+Kδ_v. All results here are in characteristic zero.
+No claim is made that these are all possible right spaces. Actual spaces
+and ranks always discard zero-factor terms as specified in PROOF.md.
+
+## 1. Statements
+
+**Theorem Q (partial coordinate quotients).** Every finite raw exact
+factorization PQ^t=N0 satisfying
+
+```
+col(P)⊂C+Kδ_v,       col(Q)⊂C+D_A
+```
+
+has COPY≥72 in either of the following cases:
+
+1. |A|≤13 and u_v∉span_F2{u_a:a∈A}; or
+2. A⊂H_v and |A|≤28, with no further binary-rank assumption.
+
+The second statement uses the new relative nonroot 27-deletion theorem,
+the root weight-seven threshold 29, and a low-budget support argument,
+proved and certified in RELATIVE.md. The first
+allows coordinates scalar-orthogonal to v and uses only the inherited
+global root-only threshold 14. Both statements handle arbitrary proper
+subspaces of the coordinate quotient, including those created by merging
+proportional encoder columns. They are stronger than requiring the actual
+right space to be a full coordinate extension.
+
+**Theorem H (high actual rank).** If the exceptional left P rays each
+occur at most once, any scalar-active exact PQ^t=N0 with col(P)⊂C+Kδ_v and actual
+rank Q≥24 has COPY≥72. No coordinate hypothesis is imposed on Q.
+In particular this holds after left proportional merging if that actual
+rank is still at least 24. A lower bound on an unnormalized raw rank is
+not enough.
+
+Theorem H follows immediately from the strict basis floor in FINITE.md
+and PROOF.md's strict-floor corollary. Its one-sided counting premise is
+an additional-slack consequence of the PR2962 relaxation. The remainder
+of this file proves Theorem Q, whose equality-basis argument is different.
+
+## 2. Reduction to the actual coefficient space
+
+Apply left proportional merging as in PROOF.md §8. The actual reader
+space S still contains C by exactness and satisfies S⊂C+D_A. Since
+|A|≤28<57 and the code distance is 57, C∩D_A=0. Thus
+
+```
+S=C⊕D',       D'=S∩D_A,       e=dim_K D'.
+```
+
+Indeed the C component of every vector belongs to S, so subtracting it
+leaves its D_A component in S. This is the ACTUAL D' after merging;
+we do not replace it by D_A or by its coordinate support.
+
+In either case of Theorem Q, u_v is outside U_A=span_F2{u_a:a∈A}.
+For neighbors, U_A⊂u_v^⊥ whereas β(u_v,u_v)=1. Consequently every pure
+D' vector has w≤8, v∉A, and no pure vector can have u_v in its binary
+support span.
+
+If e≤1, both actual factor ranks are at most nine and PR3755's existing
+all-odd theorem gives COPY≥72. Suppose e≥2 from now on.
+
+## 3. Mixed low vectors and the feasible normal-rank budget
+
+In case 1, every mixed q=E(h)+d with w(q)≤8 has exactly weight eight
+and admits
+
+```
+h=c r_a,   a∈A, c≠0,    U(q)=u_a^⊥,
+d(a)=−2c,  d(x)=0 at every x≠a orthogonal to r_a.       (1)
+```
+
+Choose a nonzero binary annihilator z. Then Off(h)∩O_z⊂A. A nonroot
+has at most 42 zero root lines and |O_z|≥56, so it requires at least 14
+cancellations. Every nonmatching root character requires at least 24.
+Under |A|≤13 only the root's own character can occur, proving (1), with
+no mixed vector of weight at most seven. The same actual d∈D' witnesses
+that root's eligibility in PR3166 Theorem A.
+
+In case 2, **no mixed vector has weight at most seven** through |A|≤28.
+A nonroot E(h) has support at least 78, so after 28 corrections it retains
+at least 50 sites, exceeding the binary-weight-seven capacity 36. For a
+root normal, RELATIVE.md §8's exact two-character argument says at least
+29 neighbor corrections are required for binary weight at most seven.
+This assertion is weaker than an eligible-root-only classification of
+weight-eight words; that stronger classification already fails at 24.
+Nonroot weight-eight words can occur at 27 and are allowed here.
+
+Let R be the span of the scalar normals of all mixed weight-eight words
+in S, and let r=dim R, γ=c(D'). We claim in BOTH cases
+
+```
+r≤γ,           r≤8.                                  (2)
+```
+
+For γ≥8 this is immediate. For case 1 with γ≤7, all feasible words satisfy
+(1), and PR3166's weighted root-swap Theorem A gives r≤γ.
+For case 2 with γ≤7, recall e≥2. Take a minimum basis of D', with
+positive weights summing to γ. The union B of its coordinate supports
+has at most 21 sites: among partitions of an integer at most seven
+into at least two positive parts, the largest sum of support capacities
+is g(6)+g(1)=20+1=21. The verifier checks all 37 such partitions.
+Every actual correction d∈D' is supported on this SAME B⊂H_v.
+The relative nonroot threshold 27 and root-mismatch threshold 24 now
+force (1) for every feasible mixed weight-eight word, since |B|≤21.
+The weighted root-swap theorem again gives r≤γ.
+
+This minimum basis is used only to bound the support of the entire
+coefficient space. The equality-basis proof below still treats the given
+actual reader basis, with no substitution. The inherited weighted
+root-swap theorem is essential; capacity estimates alone do not prove
+this partial-quotient invoice.
+
+## 4. Every actual equality basis has the required low span
+
+Let F be ANY actual basis of S. Let ℓ be its number of vectors of weight
+at most seven, let κ be their weight sum, and put d=e−ℓ. The low members
+are pure by the no-mixed-weight-at-most-seven conclusion of §3, hence ℓ≤e. All vectors of weight at most eight lie in
+D'+E(R), whose dimension is e+r. If W is the total basis bill, counting
+low, weight-eight and weight-nine members gives
+
+```
+W ≥ κ+8(e+r−ℓ)+9(8−r)
+  = 72+8d+κ−r.                                       (3)
+```
+
+If fewer than e+r members have weight at most eight, this only increases
+the lower bound. Complete the ℓ actual low vectors to a basis of D'.
+Every added pure vector has weight at most eight, so
+
+```
+γ≤κ+8d.
+```
+
+Together with (2), (3) proves W≥72. Moreover, if d=1, e≥2 gives
+ℓ=e−1≥1 and κ≥1; using r≤8 in (3) gives W≥73. If d≥2, the same bound
+gives W≥80. Thus every ACTUAL bill-72 basis has d=0: its actual low
+members form a basis of D', and every high basis coordinate functional
+annihilates D'. This is not a substitution of a preferred minimum basis.
+It also covers binary rank eight; pure weight-eight directions do not
+create an omitted equality case.
+
+## 5. The actual free-pivot cut and the marked contradiction
+
+Take q=E(h)+d∈S supported on H_v. For case 2, d itself is supported
+on H_v. Thus E(h) vanishes on O_{u_v}. Those root rows span all eight
+scalar dimensions by the full-code rank input, so h=0. This proves
+that the cut is pure for ANY neighbor support size, independently of
+the weight-eight classification.
+
+For case 1, if q were mixed, (1) would give U(q)=u_a^⊥⊂u_v^⊥. These
+eight-dimensional hyperplanes are equal; nondegeneracy and distinct labels
+force a=v, contradicting v∉A. Hence again the cut is pure. In both cases
+
+```
+S∩K^{H_v}=D'∩K^{H_v}⊂D'.                              (4)
+```
+
+For neighbor supports this equals all D'. For case 1 the intersection
+may be proper. It need not be spanned by those actual basis members
+individually lying in the cut; functional annihilation is sufficient.
+
+In any actual equality basis, every pure low member has binary support
+span inside U_A and cannot carry u_v. Thus every rebated basis member
+j with u_v∈U(q_j) is high. Its actual coordinate functional annihilates
+D', and hence the cut (4). This is exactly PROOF.md's functional cut
+hypothesis. The generic coupling lemma proves COPY≥72 for the normalized
+factors, and nonincrease under merging proves the original raw bound.
+
+Equivalently, the only unpaid dangerous state has a free δ_v pivot and
+one rebated J_v column, with all other readers an actual bill-72 basis.
+After normalizing those encoders to δ_v and J_v, quotient exactness gives
+λ_J(q_pivot)=−1. Equations (3)–(4) instead give λ_J(q_pivot)=0. These
+statements concern the same actual factorization and the same basis.
+
+## 6. Boundary and genuine tests
+
+The sharp partial-quotient fixture in SHARPNESS.md has dim D'=2, minimum
+bill seven, no coordinate unit rays, seven feasible independent root
+normals, and actual ranks (9,10) with COPY=72. It is normalized on both
+sides. A second pivot variant includes coordinates orthogonal to v and
+has a proper cut; it is left-normalized only and its right merge lowers
+left rank to eight. The verifier records that distinction.
+
+A separate minimum-basis countercontrol shows why one cannot assume a
+cut is spanned by its individually vanishing actual basis columns for
+arbitrary coefficient spaces. RELATIVE.md supplies the 24-neighbor
+failure of the eligible-root-only classification and the 29-neighbor
+failure of the assertion that all mixed low vectors are pure. Neither control is a
+counterexample to Theorem Q, whose hypotheses and supporting boundaries
+remain explicit.
+
+Open: arbitrary partial right quotients of dimensions 10–23 without these
+support conditions; whether a different actual-basis argument crosses the 29-neighbor
+mixed-low boundary; full formal E8 and invoice assembly. Nothing
+here changes the selected frontier or proves the general two-sided C′.