Strengthen nonlinear Eisenstein bounds using port polar ranks (no κ)

research/eisenstein-port-quadratic-floor/PROOF.md

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+# A polar-rank obstruction for quadratic messages
+
+This contribution was produced by Nick Petalas. Apache-2.0.
+
+## Contract and credited premise
+
+Use Michael Fischler / shiftybitshiftr's PR814 fixed-support nonlinear
+Eisenstein model. Port t has an independent vector X_t in F2^(b_t), with
+b_t >= 0 and N = sum_t b_t. Every fixed message bit has fixed source and
+reader port sets. Its additive charge is the rational pairing-rank price
+in that model. Receiver t recovers X_t on every input from its received
+messages and the inputs at ports orthogonal to t. Decoders can be nonlinear.
+There are no free input-dependent headers or correlated entrance promises.
+
+We use Jose Manuel / jmbarrancoml's PR2017 fixed-support block-linear
+corollary as a cited theorem: affine messages with these supports and
+independent demands cost at least the total demand. Known constants can be
+subtracted. That source result, its geometry and its numerical evidence
+are not independently proved or replayed here.
+
+## Product-affine restriction lemma
+
+For each port choose an injective affine map
+
+    X_t = a_t + M_t U_t,    U_t in F2^(d_t).
+
+Suppose all message bits become affine functions of the full independent
+product U. Then the unchanged original message charge C satisfies
+
+    C >= sum_t d_t.                                           (1)
+
+Indeed, compose the original exact encoder and decoder with these maps.
+Each receiver recovers X_t and hence U_t using a fixed left inverse of M_t.
+Orthogonal-port side information is still available: U_s determines X_s.
+All U_t vary independently. A message can retain its original source and
+reader supports and width, so its charge is unchanged. Subtract the known
+affine constants and apply the credited block-linear theorem. The maps
+are a restriction used in a lower-bound proof, not free physical gates
+or an input-dependent observation channel. A subspace coupling distinct
+ports would invalidate this argument and is excluded.
+
+## Quadratic class and theorem
+
+For every active port t, fix a scalar Boolean polynomial q_t of degree at
+most two. Write every transmitted bit in the form
+
+    Y_j(X) = ell_j(X) + sum_t lambda_jt q_t(X_t),
+
+where ell_j is affine and lambda_jt is in F2. The index j runs over all
+message bits, not just messages. The displayed function must obey its
+declared source support. Call t active when some lambda_jt is one; forms
+at inactive ports are ignored. Multiple independent quadratic forms at
+one port and cross-port quadratic monomials are outside this theorem.
+Linear and constant parts of q_t are allowed and immaterial.
+
+Define its alternating polar form
+
+    B_t(v,w) = q_t(v+w) + q_t(v) + q_t(w) + q_t(0).
+
+Let rank(B_t) = 2 r_t at active ports and r_t = 0 otherwise. Then
+
+    C >= N - sum_t r_t.                                       (2)
+
+Moreover, sum_t r_t is exactly the least total codimension of a product
+of affine subspaces on which every transmitted bit is affine. Thus (2)
+is the best bound from lemma (1) for this declared quadratic class.
+It is not asserted to be the optimal message charge.
+
+## Isotropic dimensions, including degenerate forms
+
+For a quadratic q with polar form B on V = F2^b, its restriction to
+an affine subspace a+L is affine exactly when B vanishes on L times L.
+To see this, the translated function q(a+v)+q(a) has polarization B
+on L. It is linear precisely when it is additive. Thus the translation
+a and every original affine term have no effect on the criterion.
+
+An alternating form has a radical R = ker B and a basis of r symplectic
+pairs plus b-2r radical vectors. This follows constructively: if B is
+nonzero, choose e,f with B(e,f)=1. For each remaining vector v replace it
+by v+B(v,f)e+B(v,e)f, which removes both pairings. Repeat on the remaining
+space. Each iteration removes two dimensions, so the process terminates.
+Taking one member of every pair and all radical vectors gives an
+isotropic subspace of dimension b-r.
+
+For the upper bound, the image of any isotropic L in the nondegenerate
+quotient V/R is isotropic. In a nondegenerate 2r-dimensional alternating
+space, W is contained in its perpendicular and
+
+    dim(W) + dim(W-perpendicular) = 2r,
+
+so dim(W) <= r. Consequently
+
+    dim(L) <= dim(R) + r = b-r.                               (3)
+
+This proves both the even-rank assertion and the exact maximum dimension.
+These are classical alternating-form facts; no historical priority is
+claimed for symplectic elimination or the isotropic dimension formula.
+
+## Apply the dimension calculation without coupling ports
+
+Choose a maximum isotropic L_t at every active port and the entire input
+space at every inactive port. All q_t become affine on the product, hence
+all messages do. Its dimension is N-sum_t r_t, proving (2) by (1).
+
+Conversely, suppose every message is affine on a product of affine spaces
+with direction spaces L_t. Fix a port t and vary only its coordinates.
+The polarization of bit j is lambda_jt B_t restricted to L_t. Since t
+is active, one bit has lambda_jt=1. Thus B_t vanishes on L_t. No quadratic
+term at another port can cancel this polarization because those inputs
+are held fixed. Formula (3) bounds every active d_t by b_t-r_t; inactive
+ones are at most b_t. Summing proves the exact restriction codimension.
+
+## Strict improvement over coordinate faces
+
+For any integer s >= 1, take one port of width 2s+1, with coordinates
+u_1,...,u_s,v_1,...,v_s,z, and put
+
+    q(X) = (u_1+...+u_s)(v_1+...+v_s).
+
+Its polar form has rank two, so its exact affine restriction loss is one.
+The hyperplane sum_i u_i=0 kills q while preserving independent parameters
+within that port. In contrast, fixing individual coordinates to arbitrary
+constants makes q affine only after all u coordinates or all v coordinates
+have been fixed. If one unfixed u and one unfixed v remain, their mixed
+quadratic coefficient is still one, regardless of the constants. Therefore
+the exact coordinate-face loss is s. The difference s-1 is unbounded.
+This comparison sharpens Nick Petalas / PR4895's coordinate-face obstruction
+within its own fixed-support model; it does not contradict that theorem.
+
+The first strict case is s=2, b=5. Give every one of the forty ports five
+input bits. At port zero send its four u/v bits directly to itself and
+send z+q instead of z. Every other port sends its five bits directly to
+itself. This is an exact fixed-support code: the first four messages
+recover q and hence z. The credited singleton pairing price is one, so
+its charge is 200. The new lower bound is 199, while the exact coordinate
+face bound is 198. The example distinguishes the two obstructions; it
+is not an attaining charge-199 code or a nonlinear saving construction.
+
+## Scope and next decision
+
+The result is a scoped written proof with finite exact corroboration,
+conditional on the cited fixed-support linear pricing theorem. It narrows
+the genuinely nonlinear payload direction left open by ywz65 / PR2607.
+A proposed saving d in this class needs sum_t r_t >= d. Dense monomials
+alone do not imply enough nonlinear freedom to support that saving.
+
+No physical encoder, native delivery, restoration schedule, complete signed
+supplier invoice, selected framework assumption, new kappa or multiplication
+theorem is supplied. Arbitrary quadratic vector maps, cross-port products,
+adaptive queries, errors, private headers and correlated demands are excluded.
+The general nonlinear optimum remains open. Independent human and formal
+proof-kernel review are unclaimed.
+
+Credit Michael Fischler / PR814 for the nonlinear model; Jose Manuel /
+PR2017 for the linear floor; Jonathan Lam / PR798 and Joel Pulikkan /
+GamingPuzzled for geometry and pricing; ywz65 / PR2607 for the open nonlinear
+payload direction; Nick Petalas / PR4895 for the coordinate restriction
+comparison; and the retained Eisenstein, decoder and construction lineage.

research/eisenstein-port-quadratic-floor/README.md

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+# A polar-rank obstruction for nonlinear Eisenstein messages
+
+This contribution was produced by Nick Petalas. Apache-2.0.
+
+For a fixed-support exact Eisenstein message code with one shared scalar
+quadratic form q_t at each input port, let B_t be its binary polar form.
+Every transmitted bit may combine these forms with arbitrary binary
+coefficients and an arbitrary affine function. With N independent input
+bits and the retained additive pairing price, the new obstruction is
+
+    C >= N - sum_t rank(B_t)/2.
+
+Only forms actually used by the encoder enter the sum. The proof also
+shows that this is the strongest lower bound obtainable by restricting
+each port to an affine subspace and applying the cited linear-copy floor.
+The input restriction preserves independent demands at distinct ports.
+
+This is a scoped written proof with finite exact corroboration. It builds
+on Jose Manuel / jmbarrancoml's PR2017 linear theorem and responds to ywz65 /
+PR2607's open genuinely nonlinear payload direction. Those results are
+cited premises, not independently audited or replayed here. The alternating
+form decomposition is classical; the contribution is its exact use in this
+fixed-support restriction contract and its stricter nonlinear obstruction.
+
+## First strict improvement
+
+At one port put q=(u0+u1)(v0+v1), with an additional independent coordinate z.
+Fixing individual coordinates requires two fixings to make q affine.
+The single linear equation u0=u1 suffices, and its polar rank is two.
+For forty five-bit ports, the new lower bound is **199**, compared with
+**198** from the exact coordinate-face criterion of Nick Petalas / PR4895.
+The demonstrated exact code costs **200**. No cheaper code is claimed.
+
+More generally q=(sum u_i)(sum v_i), with s coordinates in each sum,
+has exact coordinate restriction loss s and exact product-affine loss one.
+The improvement in this restriction parameter is therefore unbounded.
+The universal statement is the written argument in [PROOF.md](PROOF.md).
+The first strict s=2 case is the literal finite decoder fixture.
+
+## Reproduce
+
+From the repository root, CPython 3.12 and the standard library suffice:
+
+```sh
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/eisenstein-port-quadratic-floor/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/eisenstein-port-quadratic-floor/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/eisenstein-port-quadratic-floor
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/eisenstein-port-quadratic-floor --check-source-history
+```
+
+Other hosts may substitute a CPython 3.12 executable. Each registered command
+has a 120-second limit and generates its receipt on stdout. No generated
+receipt or replay output is retained in the checkout. The marker is
+`PASS_EISENSTEIN_PORT_QUADRATIC_FLOOR` followed by exact enumerated evidence
+and SHA256 hashes of package files and all four live source inputs.
+
+The verifier checks all **1,100** homogeneous quadratic forms through five
+variables against an independent complete subspace/truth-table oracle.
+It tests **387,405** subspace restrictions, including degenerate forms,
+and **69,677** translated restrictions through four variables. Constructed
+maximum isotropic spaces are checked directly. Another **64** three-bit
+vector maps on two independent two-bit ports receive **1,600** complete
+product-subspace checks, including equal nonzero quadratic coefficient
+columns and inactive ports. The decoder fixture checks **1,280** local
+input/receiver cases. Six malformed-form rejections and four mathematical
+boundary controls delimit the implementation and model.
+
+Local CPython 3.12.15 on macOS 27.0.1 arm64 passed both registered modes
+and both direct commands. Direct normal/optimized elapsed times were
+0.217/0.217 seconds, each below the 120-second registered limit. This is
+a replay-budget observation, not a speedup claim. Warning-strict source
+compilation and AST inspection pass; the longest function is 35 lines.
+Ruff is unavailable in the local environment. Hosted CI is pending.
+
+Finite checks corroborate the written theorem. They do not prove its
+all-width quantifiers, re-establish the cited linear theorem, or constitute
+independent human or formal proof-kernel review.
+
+## Scope and provenance
+
+[Task metadata](TASK.json) records the changed scope under
+`complex-rank-shape-decoder`. This is a nonlinear decoder side result,
+not a completed supplier or changed paid/productive histogram. The source
+question is ywz65 / PR2607; the earlier coordinate-face result supplies a
+comparison, not the assignment. The [advance claim](https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19291977)
+records the question, boundaries and overlap checks.
+
+The package starts at canonical main
+`2485805a6d47f36ea83ddde6327d453ac82d44dc`, priority revision 8. The selected
+binding-only orientation passed and resolved selected PR6030. Historical
+priority prose still mentions PR5599; the selected record governs. The
+selected frontier is unchanged. [SOURCES.json](SOURCES.json) pins four
+unchanged predecessor documents and historical catalogue/selected metadata.
+No predecessor verifier is imported or executed.
+
+The proof assumes fixed supports/widths, independent full binary demands,
+zero-error decoding, orthogonal side information and additive pairing prices.
+It permits one shared polar form per port. Cross-port quadratic products,
+multiple independent polar forms at one port, adaptive queries, private
+headers, errors and correlated inputs require a different argument.
+The new bound need not be sharp for message cost. Native operations,
+physical delivery/restoration, precision/tape, stock, compiler, analytic
+recovery and a complete signed supplier invoice remain open. There is no
+new kappa or complete multiplication theorem. Selected cold replay, Lean,
+Docker, independent human review and formal-kernel review were not run.
+
+Credit Michael Fischler / shiftybitshiftr / PR814 for the nonlinear model,
+Jose Manuel / jmbarrancoml / PR2017 for the fixed-support block-linear floor,
+Jonathan Lam / PR798 and Joel Pulikkan / GamingPuzzled for geometry and
+pricing, ywz65 / PR2607 for the nonlinear payload question, Nick Petalas /
+PR4895 for coordinate restrictions, and the retained Eisenstein, decoder
+and construction lineage. Symplectic elimination, Boolean polarization,
+and the isotropic dimension bound retain their classical status.
+
+Power of Ten inspection is scoped to the new verifier. It has no recursion,
+subprocesses, dynamic imports or writes. Its largest exhaustive dimension
+is five and its subspace list has at most 374 entries. Functions remain
+below sixty lines. Source inputs have explicit byte and inventory limits.
+CPython integer/container allocation is a managed-runtime deviation from
+static allocation; fixed mathematical domains bound live data. Warning-strict
+normal and optimized replays provide the available runtime diagnostics.
+
+## Open next questions
+
+- Determine the exact product-affine restriction dimension for two independent
+  quadratic forms at one port. Start with literal small alternating matrices
+  and a complete common-isotropic-space oracle. The single-form rank formula
+  does not determine their simultaneous restriction.
+- Construct a genuinely nonlinear exact Eisenstein code with charge below
+  total demand, if one exists. Specify source/reader sets and every additive
+  charge, then calculate the restriction certificate. No lower-charge code
+  or favorable complete physical invoice is supplied here.
+
+Structured first steps and blockers are in TASK.json.

research/eisenstein-port-quadratic-floor/SOURCES.json

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