Prove a sharp 1/4 error floor for contractive P104 mode deletion (no κ)

research/p104-contractive-gram-deflation/MISSING.json

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+{
+  "source_pr": 6166,
+  "source_author": "Matarener",
+  "source_head": "096938513138868f0ed27639061f1a5054900f96",
+  "source_path": "research/p104-carrier-code-uniqueness/e8/ports-d8.md",
+  "meaning": "Omitted root lines in the cited orthonormal D8 coordinates; doubled coordinates are these signs. Coordinate identification is a cited premise, not independently replayed.",
+  "signs": [
+    "+++++---",
+    "++++-+++",
+    "++-----+",
+    "--++---+",
+    "++-++-+-",
+    "-++---+-",
+    "+--+--+-",
+    "-+-++-++",
+    "-+----++",
+    "+-+-+-++",
+    "+++-++++",
+    "+---+--+",
+    "++---+--",
+    "-+-+++-+",
+    "----+++-",
+    "-++-+---"
+  ]
+}

research/p104-contractive-gram-deflation/PROOF.md

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+# A sharp error floor for contractive P104 Gram deflation
+
+## Contract and theorem
+
+Let P consist of the 56 unoriented D8 root lines e_i +/- e_j in R^8,
+and any proper subset of the 64 half-spin root lines of one fixed parity.
+Choose one root r_i of squared length 2 on every retained line. Let R
+have these roots as rows and let K=RR^T/2.
+
+A contractive rank-seven approximation means
+
+    K_D = R D R^T/2,
+    D real symmetric, 0 <= D <= I in positive-semidefinite order,
+    rank D <= 7.
+
+Then the exact minimum of max_{i,j}|K_ij-(K_D)_ij| is 1/4. Any omitted
+half-spin root q gives an attaining matrix D=I-qq^T/2. It is an orthogonal
+rank-seven projector. If the half-spin subset is empty the same witness
+works; only one omitted half-spin root is required.
+
+This is a scoped proof about coefficient error. It does not give a lower
+bound for unrestricted rank-seven matrices, nonsymmetric approximants,
+noncontractive approximants, nonlinear processing, or masked exactness.
+
+## The D8 direction bound
+
+For every unit vector u in R^8,
+
+    max_{r in D8} |r dot u|^2 >= 1/2.
+
+To prove it, arrange the absolute coordinates as x_1>=x_2>=...>=x_8>=0
+and put s=x_1+x_2. The largest absolute D8 inner product is s. Since
+the remaining coordinates are at most x_2,
+
+    1 = sum x_i^2 <= (s-x_2)^2 + 7*x_2^2 <= 2*s^2.
+
+The last bound follows because this convex quadratic on 0<=x_2<=s/2
+is bounded by the larger endpoint value, max{s^2,2*s^2}. This proves
+the direction bound. It is an elementary root-geometry inequality, not a
+new general rank-approximation theorem.
+
+## Lower bound for every admissible D
+
+The rank cap gives a unit vector u in ker D. Symmetry and 0<=D<=I imply
+
+    I-D >= u u^T.
+
+For example, use an orthonormal eigenbasis with u as a zero-eigenvalue
+vector of D. In that basis I-D-u*u^T has only nonnegative eigenvalues.
+Consequently the coefficient error E=K-K_D is positive semidefinite and
+
+    E_ii = r_i^T (I-D) r_i /2 >= (r_i dot u)^2/2.
+
+The retained D8 subsystem supplies an i with E_ii>=1/4. A diagonal entry
+already gives the required lower bound on maximum absolute entry. This
+argument covers all real admissible D, not just rational or sampled D.
+
+## Exact attaining deflation
+
+Choose a half-spin root q whose line is omitted. It has squared norm 2,
+so D=I-qq^T/2 is symmetric, idempotent, positive semidefinite and rank 7.
+It lies between zero and I. Its error is
+
+    E_ij = (r_i dot q)(r_j dot q)/4.
+
+For a D8 root the inner product with q is in {-1,0,1}. For another
+half-spin root of the same parity, the sign patterns differ in an even
+number of coordinates. Unless their lines agree, the inner product is
+also in {-1,0,1}. The line of q is absent, so every retained root has
+inner product at most one in absolute value. Thus every error entry is
+at most 1/4. Some D8 root has inner product of absolute value one, so
+the maximum is exactly 1/4. This proves sharpness constructively.
+
+## P104 binding and meaning for precision
+
+Matarener's PR6166 at 096938513138868f0ed27639061f1a5054900f96 identifies
+the selected P104 ports with all 56 D8 lines and 48 half-spin lines in
+orthonormal coordinates. Its e8/ports-d8.md lists the sixteen omitted
+lines. That coordinate identification is a cited predecessor premise,
+not a result independently checked here. MISSING.json retains just those
+factual signs so the new deflation witnesses replay without the pending PR.
+No predecessor certificate, code uniqueness, single-word weight, generalized
+support weight, native operator or supplier replay is run.
+
+The coefficient convention K=RR^T/2 is the retained P104 Gram-channel
+convention from Rohan Arun's PR1972 and Chafik Boukhalfa's PR1991 native
+channel, with its credited root/frame lineage. This theorem introduces
+contractive approximation of that channel as a changed mechanism.
+
+For every admissible D there is an input x=B e_i such that the i-th output
+bias is at least |B|/4. Thus the approximation error is not o(|B|), uniformly
+over arbitrary inputs. Increasing the arithmetic precision cannot remove
+this mathematical bias. The attaining matrices even have dyadic rational
+entries. Any supplier using this deletion must explicitly correct the
+residual or justify a different consumer/error contract. No correction
+cost is declared free and no complete signed-moment gain is claimed.
+
+The full scalar word may have other routes and cancellations. The theorem
+only concerns replacing this Gram block by K_D while retaining the other
+terms and demanding entrywise accuracy. It does not exclude changing the
+side correction, targets, source encoding, masking or complete circuit.
+
+## Evidence
+
+The standard-library verifier constructs this root set from the retained
+sign list, checks all sixteen rank-seven projectors by integer arithmetic,
+and enumerates every coefficient of each new error matrix. It checks the
+D8 inequality on the 6,560 nonzero vectors in {-1,0,1}^8. Those are finite
+corroborations of the written proof, not extrapolations to arbitrary D.
+The verifier also rejects wrong parity, duplicate missing lines and a
+retained-root deflation falsely claimed to attain error 1/4. Normal and
+optimized runs retain explicit exception checks. Source hashes bind the
+generated receipt. Independent human and formal proof-kernel review remain
+unclaimed.

research/p104-contractive-gram-deflation/README.md

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+# Sharp error floor for contractive P104 Gram deflation
+
+Discarding one real carrier mode through a positive contractive approximation
+of the P104 Gram channel has an exact maximum coefficient-error floor of
+**1/4**. Projecting away any omitted half-spin root direction attains it.
+This is a scoped obstruction with a written proof and exact finite witnesses.
+No new kappa or physical supplier is claimed.
+
+Write R for the retained roots of squared norm 2 and K=RR^T/2. The theorem
+covers K_D=RDR^T/2 with D real symmetric, 0<=D<=I, and rank D<=7. Its lower
+bound uses only the retained D8 subsystem and applies to every admissible
+real D. Its upper bound is the explicit dyadic projector D=I-qq^T/2 for an
+omitted half-spin root q. [PROOF.md](PROOF.md) gives both arguments.
+
+For some input B e_i every such approximation has output bias at least
+|B|/4. Increasing arithmetic precision cannot remove that fixed linear
+bias. This blocks treating a contractive one-mode deletion as vanishing
+roundoff while keeping the rest of the Gram decomposition fixed. A changed
+side correction or consumer must account for the residual.
+
+## Reproduce
+
+From the repository root, using CPython 3.12 and the standard library:
+
+```sh
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/p104-contractive-gram-deflation/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/p104-contractive-gram-deflation/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/p104-contractive-gram-deflation
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/p104-contractive-gram-deflation --check-source-history
+```
+
+The direct marker is `PASS_P104_CONTRACTIVE_GRAM_DEFLATION`. The registered
+replay prints `PASS_REGISTERED_RESEARCH_PACKAGE`. A clean CPython 3.12 can
+replace the absolute interpreter path. Replay reads without writing files.
+The optional `--output PATH` generates a new receipt outside the checkout;
+RESULT.json is generated and bound to every other package file's hashes.
+
+The verifier enumerates 173,056 entries of sixteen new error matrices,
+checks the sixteen exact rank-seven projectors and 6,560 integer-direction
+fixtures, and retains three rejection controls. These finite checks
+corroborate the written all-real-D proof. They do not recheck predecessors.
+
+## Scope, sources and overlap
+
+The original coverage-law-change question is changed to contractive approximate
+decoding. The connection to the current complex bottleneck is a possible
+carrier-mode saving which exact decoding excludes. This result computes its
+unavoidable residual before any native or caller expansion. It supplies no
+complete signed invoice or card-level supplier admission.
+
+Base main is `3f0152a32eeebac0df11946dbd50cabe5a7f7839`, priority revision 8.
+The selected frontier is unchanged. Historical provenance pins are in
+SOURCES.json. The mathematical statement itself covers all D8 root lines
+plus any proper subset of one half-spin shell.
+
+For the P104 application, Matarener's [PR6166](https://github.com/CrocSwap/integer-mult-bounds/pull/6166)
+at `096938513138868f0ed27639061f1a5054900f96` supplies its orthonormal D8
+coordinate identification and sixteen omitted lines. [MISSING.json](MISSING.json)
+retains those factual coordinates. This identification is a cited premise,
+not independently verified here. Its ternary uniqueness, minimum weight and
+code-support questions are separate. The new result concerns real contractive
+approximation and introduces no question from that source's open-question list.
+
+Rohan Arun's PR1972 and Chafik Boukhalfa's PR1991 supply the retained Gram
+channel convention. Their native/operator results are cited, not replayed.
+Ali Romman's PR5526 concerns Clifford-history error versus paid returns;
+it is a different approximation contract. The present result neither uses
+nor checks its call-cap or native tensor arguments.
+
+## Limits and engineering notes
+
+Unrestricted rank-seven matrices, nonsymmetric or noncontractive filters,
+masked-entry accuracy, changed source encodings and nonlinear consumers
+remain outside the theorem. The bias might be corrected elsewhere in a new
+word. No restoration, ordinary-cost, native, physical caller, precision/tape,
+compiler, BIT, outer, Lean or full multiplication-theorem replay is supplied.
+Independent human and formal proof-kernel review are unclaimed.
+
+The verifier has bounded loops, no recursion, functions below sixty lines
+and explicit failure checks. Standard-library collection allocation is a
+managed-runtime deviation from static allocation;
+root tables have at most 120 rows of eight integers, and one error histogram
+has at most 10,816 entries counted into integer bins. No shared code or
+repository-wide diagnostics configuration is changed.
+
+## Open next questions
+
+- Can a noncontractive or nonsymmetric rank-seven factor beat 1/4 in maximum
+  entry error? It would escape the positive-error argument. Start with an
+  explicit rational factor and certify every coefficient; its physical
+  realization and stability would still need a separate bill.
+- Can a changed side correction absorb the explicit rank-one residual with
+  a favorable full invoice? Start from one omitted-root projector and track
+  its residual through an actual scalar word. No such caller is supplied.
+
+## Credit
+
+This proof and verifier were produced by Nick Petalas. Matarener retains the
+P104 D8 coordinates and omitted-root list. Rohan Arun, Chafik Boukhalfa and
+their retained native/root/frame predecessors retain the Gram-channel
+construction. Ali Romman, DreamingOfClouds, Cinna Davis, Pierre-Loic Doulcet,
+icekylinx, zxv, Jacob Alan Sussman and the retained P104/helper/compiler/BIT
+contributors retain their source results. Root geometry, orthogonal
+projection and positive-semidefinite order are classical. No general
+literature-priority claim is made. Apache-2.0.