Scoped obstruction: sharp P112 endpoint dimension and complete linear-lift cost

research/p112-linear-endpoint-lifts/PROOF.md

@@ -0,0 +1,302 @@
+# Endpoint dimension and the cost of linear lifts
+
+All vector spaces in Sections 1--3 are over the field with two elements.
+The ambient form is symmetric and nondegenerate. An endpoint subspace is
+nondegenerate for the restricted form. Alternating endpoint planes are allowed.
+No positivity assumption is used.
+
+## 1. The graph and two exact certificates
+
+Use the 112 binary ports in the selected canonical word. Join distinct
+ports a,b when their binary dot product is zero. This is the P112
+source/target interaction graph used by the retained scalar identity.
+It has 5,824 ordered edges. Each vertex has a label t_a in dimension 9,
+with nondegenerate symmetric form S=I9, such that
+
+    t_a^T S t_a = 1;    t_a^T S t_b = 0 on every graph edge.
+
+The verifier reads the selected source bytes, checks all 12,544 binary
+pairings, and checks the following two certificates. No graph isomorphism
+to a different port model is assumed.
+
+**Certificate A.** The edge tensors `t_a tensor t_b` have rank 80 in the
+81-dimensional matrix space. The 80 listed edges form a basis. S is a
+nonzero null vector, so every bilinear form K that vanishes on graph edges
+is `K = c S`.
+
+**Certificate B.** There is an eight-vertex clique with vertices a_0,...,a_7.
+There are 24 further vertices b_ij, one for each edge ij of a graph J on
+eight points. J is K8 with one perfect matching removed. Vertex b_ij is
+adjacent to every a_k except a_i,a_j. Two b vertices are adjacent exactly
+when their J edges have a common endpoint. In particular J is 6-regular.
+The certificate lists the 32 distinct vertices. The verifier checks every
+required and absent edge, rather than accepting a claimed graph name.
+
+The certificates are finite premises of the following all-size arguments.
+The verifier is not a proof-assistant formalization of those arguments.
+
+## 2. Sharp dimension bound for arbitrary two-plane labels
+
+Suppose each graph vertex is assigned a nondegenerate two-dimensional
+subspace U_a in an h-dimensional space, and graph edges give orthogonal
+subspaces. The assignment need not depend linearly on the old labels.
+Then **h >= 18**. The binary tensor construction `U_a = t_a tensor F2^2`
+in the form `S tensor I2` attains 18.
+
+The eight anchor planes in Certificate B are mutually orthogonal. Their
+orthogonal direct sum E has dimension 16. Thus h >= 16. Put
+`H = E_0 perpendicular ... perpendicular E_7 perpendicular Z`.
+For h <= 17, the dimension z of Z is zero or one. The plane U_ij lies in
+`E_i + E_j + Z` because it is orthogonal to the other six anchor planes.
+
+Choose any basis for each U_ij. Its nondegenerate 2-by-2 Gram matrix is D_ij.
+Let K_ij,i be the Gram matrix of its projection to E_i, and let R_ij be
+the Gram matrix of its projection to Z. Thus
+
+    D_ij = K_ij,i + K_ij,j + R_ij.                 (1)
+
+For a fixed i, collect the six incident planes in one matrix. Their E_i
+projection has a Gram matrix K_i of rank at most two. Their Z projection
+has a Gram matrix R_i of rank at most z. Distinct incident planes are
+orthogonal, and their other anchor blocks differ. Therefore the off-diagonal
+blocks of K_i and R_i agree. Hence
+
+    H_i = K_i + R_i
+
+is block diagonal, with blocks `H_ij,i = K_ij,i + R_ij`, and
+
+    sum_(j:ij in J) rank(H_ij,i) = rank(H_i) <= 2 + z.       (2)
+
+If z=0, (1) gives `2 <= rank(H_ij,i) + rank(H_ij,j)`.
+Summing over 24 edges contradicts (2): `48 <= 16`.
+
+If z=1, choose the unit basis of Z. Write its coordinate row on U_ij as
+u_ij, so `R_ij = u_ij^T u_ij`. Characteristic two gives
+
+    D_ij = H_ij,i + H_ij,j + u_ij^T u_ij.
+
+Thus `rank(H_ij,i) + rank(H_ij,j) >= 1`. Summing over J and using (2)
+gives equality throughout: the total is 24, every vertex contributes 3,
+and every edge contributes 1. Also every u_ij is nonzero; otherwise its
+edge would need a contribution at least 2.
+
+At each vertex i there are therefore three rank-one diagonal blocks in
+H_i and three zero blocks. Let u_i be the concatenation of its six rows
+u_ij. It has a nonzero component at every zero block. Consequently
+`u_i^T` is not in the image of H_i. A rank-one update by this vector
+cannot reduce rank: if `(H_i + u_i^T u_i)x=0`, then
+`H_i x = u_i^T (u_i x)`. Since `u_i^T` is not in the image of H_i,
+we have `u_i x=0` and `H_i x=0`. The new kernel is contained in the old
+kernel. Hence
+
+    rank(K_i) = rank(H_i + u_i^T u_i) >= rank(H_i) = 3,
+
+contrary to `rank(K_i) <= 2`. This excludes h=17 and proves the bound.
+
+This theorem fixes the least ambient dimension. It does not classify all
+dimension-18 assignments or bound the cost of every new word on them.
+
+## 3. Rigidity of all coupled linear lifts
+
+Let V have dimension r >= 1. Let `A: F2^9 tensor V -> H` be linear and
+write `A(t)v = A(t tensor v)`. Assume all spaces `A(t_a)V` have dimension
+r and a nondegenerate restricted form. Require them to be orthogonal on
+every graph edge. There is no block-diagonal or independent-copy assumption.
+
+For v,w in V, the function
+
+    K_(v,w)(x,y) = <A(x tensor v), A(y tensor w)>
+
+is a bilinear form on the old nine-dimensional space and vanishes on all
+graph edges. Certificate A gives a scalar D(v,w) such that
+
+    K_(v,w)(x,y) = (x^T S y) D(v,w).              (3)
+
+Linearity shows that D is bilinear. Symmetry of the ambient form shows
+that D is symmetric. At any vertex t_a, (3) restricts to D because
+`t_a^T S t_a=1`. Nondegeneracy of the endpoint makes D nondegenerate.
+Thus the pullback Gram matrix of A is `S tensor D`, of rank 9r.
+The map A is injective, its image is nondegenerate, and h >= 9r.
+
+For every old subspace U, its image is `A(U tensor V)` and has dimension
+`r dim(U)`. Intersections, sums and orthogonal complements inside this
+image are the old ones tensored with V. This proves rigidity for every
+fibre size, every ambient dimension, and all linear couplings.
+
+## 4. The complete accounting class
+
+The cost conclusion concerns an endpoint lift of a fixed complete supplier
+and controller diagram. This condition matters. It does not cover a new
+scalar program or a new controller that reuses stages in a different way.
+
+The reference diagram includes every source-access edge, helper edge,
+copied-centre edge, restoration edge and idle child. It has ambient width m,
+W live role streams of equal volume, and paid edge ranks t_e. Each scalar
+gate uses a common exact frame on all its incident streams. In the selected
+instance, `m=73`, `W=14336`, and the complete histogram is
+
+    {1:349604, 2:60192, 3:62585, 4:20448,
+     5:11808, 7:7, 8:1152, 48:4866}.
+
+The linear lift replaces the old core by its r-fold tensor image. In the
+retained nine-stage orthogonal-slot controller, this has width 73r. At each
+retained boundary, the lifted subspace has intersection with this core equal
+to the image of the old boundary subspace. Additional ambient coordinates
+are permitted, with arbitrary slanted subspaces, nonmonotone transitions,
+and extra gates. Their entrance quotient dimension is zero and their final
+quotient dimension is their full width on every live role. The complete
+auxiliary/centre ledger must also be retained. An auxiliary input does not
+become a known-zero input. All W physical live streams persist through the
+serialized schedule. A local role can use a different owner in a later call;
+this does not move a physical stream path to a private centre. Each scalar
+operation has the same frame before and after on each incident live path.
+Private centre paths can start and end at different padding dimensions.
+Their costs are nonnegative. They are not used in the signed sum below.
+
+Each individual retained role-stream edge may use any exact native
+factorization, with any number of rank-zero adapters. Joint children that
+encode several arrays or combine different frozen edges are outside this class. No assumption that its old child is run r times is
+made. Each native child acts on the original role-stream volume; creating
+extra role streams does not create extra normalization weight. A factorization
+that crosses a retained scalar-gate boundary changes the reference diagram
+and is outside this class.
+
+### Source-bound minimum-rank bridge
+
+The selected archive is `research/community-frontier1094-audit/selected-replay.tar.gz`
+at the base commit in SOURCES.json. The verifier reads five exact members from
+this Git object. It does not use historical files with similar names. Each
+member has a SHA256 pin in CERTIFICATE.json.
+
+The source frame lemma in `notes/general-clifford-frames.tex` gives
+
+    rank B(T_V T_U^(-1)) = dim U + dim V - 2 dim(U intersect V).
+
+This is an exact Fourier-rank identity. It applies to degenerate U and V,
+and does not require the operators T_U and T_V to commute. The verifier
+checks that this note has the same bytes as the copy in the selected source
+manifest. Reverse/complement words preserve the same transition distance.
+
+For each of the two emitted local programs, the verifier reads every `climb`
+and `center` edge and computes this distance from its actual binary frame
+bases. It checks equality with the paid rank, not only a lower bound. There
+are 3,494 edges per orientation. All eight private-centre edges go between
+the specified rank-eight centre frame and the zero frame. Each has minimum
+rank eight. Copying the centre value at its high frame does not erase that
+transition. The total local rank mass is 5,586.
+
+The selected global controller contains 144 complete calls. Its 704 local
+roles per call have distinct owners in the live range [0,14336); the private
+centre range is separate. The verifier checks this assignment, the complete
+per-replica event order, and the final live-family range. Thus changing a
+helper's owner between calls changes the local role name, not the identity
+of a physical stream. The selected owner audit supplies the inherited exact
+chart and busy-window validity. This package does not repeat that audit.
+
+For each of the 7,526 idle children, the verifier reads the actual remainder
+axes, checks the matching controller row, and checks that they are the exact
+suffix of the inherited pending axes. It checks that their Gram matrix is
+identity. This is a coordinate Fourier child on that subspace, so its minimum
+rank is the number of these axes. Its entrance and exit frames have this
+exact distance. The idle histogram is `{1:2276, 3:377, 7:7, 48:4866}`.
+Adding the 144 literal local histograms gives the full histogram above and
+rank mass 1,041,408. All 510,662 reference native children are included.
+There is no unexamined histogram entry that could conceal an overcharged
+reference transition.
+
+Exact scalar identities, phases, source wrappers, and the controller's charts
+remain inherited from the selected review. The fresh check covers ranks and
+stream accounting; it is not a cold replay of the supplier or compiler.
+
+## 5. No native factorization or ambient padding improves the moment
+
+Write `p=1-b`, with `0<p<=1`, and
+
+    M(p) = (1/W) sum_e (t_e/m)^p.
+
+First consider no additional coordinates. Section 3 multiplies each
+boundary transition distance by r. Fourier rank is subadditive under exact
+Clifford composition: the B block of a product is `A1 B2 + B1 D2`, whose
+rank is at most `rank(B1)+rank(B2)`. Thus a factorization into ranks k_j
+has `sum_j k_j >= r t_e`. Since `sum_j k_j^p >= (sum_j k_j)^p`, its
+moment charge is at least `(r t_e/(rm))^p = (t_e/m)^p`.
+One rank-r t_e child attains this lower bound whenever it is available;
+granting it for free as an interface is an optimistic relaxation otherwise.
+
+Now add z>0 ambient coordinates. Put `M'=rm+z` and `lambda=rm/M'`.
+At a boundary B let q_B be its dimension outside the tensor core.
+For an old edge U->V, let `d_e=|q_V-q_U|`. The subspace distance satisfies
+
+    distance(U',V') >= r distance(U,V) + d_e.     (4)
+
+To see this, the core part of `U' intersect V'` has dimension
+`r dim(U intersect V)`, and its quotient part has dimension at most
+`min(q_U,q_V)`. Substitute in `dim U' + dim V' - 2 dim(U' intersect V')`.
+This proof allows slanted frames and does not assume an orthogonal splitting
+at each gate. The same native-factorization argument bounds an edge's charge
+below by `((r t_e+d_e)/M')^p`.
+
+Sum signed q changes separately along each physical live stream. Scalar
+operations do not change q on that path. This includes a copy from a live
+stream to a private centre: it creates a private value but preserves the
+live path and its frame. Rank-zero adapters preserve the preimage frame.
+The native transitions on each live path therefore have net change z.
+Their absolute changes sum to at least z. Summing the W live paths gives
+`sum_e d_e >= Wz`, before any private-path cost is added. Private descents
+cannot cancel this lower bound. New edges have old rank zero. Also
+`0<=d_e/z<=1`. Concavity gives, edge by edge,
+
+    ((r t_e+d_e)/M')^p
+      >= lambda (t_e/m)^p + (1-lambda) (d_e/z)^p.
+
+Summing, and using `x^p>=x` on [0,1], gives
+
+    M_lift(p) >= lambda M(p) + (1-lambda).        (5)
+
+If the reference moment is at least one, the lifted moment is at least one.
+This includes arbitrary padding size, all extra transitions and every native
+factorization at the retained boundaries. The unpadded tensor lift attains the
+same moment in the ideal native-child model. Thus the optimal complex moment
+exponent cannot increase in this class. A second unpaid direction is offset by
+the forced growth of every geometric rank and of the ambient denominator.
+
+## 6. Consequence for the selected kappa grid
+
+The exact accounting check reconstructs the selected full histogram and
+checks
+
+    M(1-b_low) < 1 < M(1-b_high),
+    b_low  = 0.001675462676808103522925,
+    b_high = b_low + 10^-24.
+
+It also checks the exact rational inequality
+
+    b_high/(1+b_high) < 0.001672660197076919.
+
+The retained balanced outer reduction requires `kappa < b/(1+b)`.
+Equation (5) therefore excludes the next 10^-18 kappa grid point and every
+larger one for this lift class, even before new scalar, chart, source-access,
+movement, restoration, or finite-invoice costs are imposed. Those costs are
+nonnegative. The original bit supplier is retained; its restriction can only
+decrease the feasible outer saving. Its raw exponent is not substituted for
+an outer kappa. This result neither changes nor reproves the selected supplier.
+
+The all-size compiler, exact common-frame realization, whole-call hosting,
+precision/tape, analytic transfer and bit H1--H5 conditions remain inherited.
+Arbitrary dimension-18 plane assignments, other interaction graphs, new scalar
+programs, and new outer reductions remain outside the no-improvement theorem.
+
+## 7. Relation to general endpoint labels
+
+The arbitrary-plane theorem in Section 2 is stronger than the linear-lift
+hypothesis as a statement about dimension. It does not imply Section 5 for
+all plane assignments. In dimension 18, a nonlinear assignment might have
+other intersections at internal frames and might admit a new scalar word.
+No classification or cost bound for those words is proved here.
+
+PR791, at `f80382efd4cd3e37c67b2a966398b55bb56af53c`, proves a separate
+rational-idempotent path formula. Its rank-two examples show that a generic
+endpoint penalty need not double. This proof does not assume that rule.
+Section 3 first forces a tensor pullback form in the linear class. The
+actual preimage-frame distances then scale by r. Without that rigidity or
+without the retained boundaries, this rank-scaling conclusion is not used.